extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 2 (2020), 197 – 204 doi:10.17398/2605-5686.35.2.197 available online october 20, 2020 on angular localization of spectra of perturbed operators m.i. gil’ department of mathematics, ben gurion university of the negev p.o. box 653, beer-sheva 84105, israel gilmi@bezeqint.net received june 29, 2020 presented by manuel gonzález accepted september 17, 2020 abstract: let a and ã be bounded operators in a hilbert space. we consider the following problem: let the spectrum of a lie in some angular sector. in what sector the spectrum of ã lies if a and ã are “close”? applications of the obtained results to integral operators are also discussed. key words: operators, spectrum, angular location, perturbations, integral operator. ams subject class. (2010): 47a10, 47a55, 47b10. 1. introduction and preliminaries let h be a complex separable hilbert space with a scalar product (. , .), the norm ‖.‖ = √ (. , .) and unit operator i. by b(h) we denote the set of bounded operators in h. for an a ∈ b(h), a∗ is the adjoint operator, ‖a‖ is the operator norm and σ(a) is the spectrum. we consider the following problem: let a and ã be “close” operators and σ(a) lie in some angular sector. in what sector σ(ã) lies? not too much works are devoted to the angular localizations of spectra. the papers [5, 6, 7, 8] should be mentioned. in particular, in the papers by e.i. jury, n.k. bose and b.d.o. anderson [5, 6] it is shown that the test to determine whether all eigenvalues of a complex matrix of order n lie in a certain sector can be replaced by an equivalent test to find whether all eigenvalues of a real matrix of order 4n lie in the left half plane. the results from [5] have been applied by g.h. hostetter [4] to obtain an improved test for the zeros of a polynomial in a sector. in [7] m.g. krein announces two theorems concerning the angular localization of the spectrum of a multiplicative operator integral. in the paper [8] g.v. rozenblyum studies the asymptotic behavior of the distribution functions of eigenvalues that appear in a fixed angular region of the complex plane for operators that are close to normal. as applications, he calculates the asymptotic behavior of the spectrum of two classes of operissn: 0213-8743 (print), 2605-5686 (online) c© the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.35.2.197 mailto:gilmi@bezeqint.net https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 198 m.i. gil’ ators: elliptic pseudo-differential operators acting on the sections of a vector bundle over a manifold with a boundary, and operators of elliptic boundary value problems for pseudo-differential operators. it should be noted that in the just pointed papers the perturbations of an operator whose spectrum lie in a given sector are not considered. below we give bounds for the spectral sector of a perturbed operator. without loss of the generality it is assumed that β(a) := inf re σ(a) > 0. (1.1) if this condition does not hold, instead of a we can consider perturbations of the operator a1 = a + ic with a constant c > |β(a)|. for a y ∈b(h) we write y > 0 if y is positive definite, i.e., infx∈h,‖x‖=1 (y x,x) > 0. let y > 0. define the angular y -characteristic τ(a,y ) of a by cos τ(a,y ) := inf x∈h,‖x‖=1 re(y ax,x) |(y ax,x)| . the set s(a,y ) := {z ∈ c : |arg z| ≤ τ(a,y )} will be called the y -spectral sector of a. lemma 1.1. for an a ∈b(h), let condition (1.1) hold and y be a positive definite operator, such that (y a)∗+y a > 0. then σ(a) lies in the y -spectral sector of a. proof. take a ray z = reit (0 < r < ∞) intersecting σ(a), and take the point z0 = r0e it on it with the maximum modulus. by the theorem on the boundary point of the spectrum [1, section i.4.3, p. 28] there exists a normed sequence {xn}, such that axn −z0xn → 0 , (n →∞). hence, re(y axn,xn) |(y axn,xn)| = re r0e it(y xn,xn) r0|(y xn,xn)| + �n = cos t + �n with �n → 0 as n →∞ . so z0 is in s(a,y ). this proves the lemma. example 1.2. let a = a∗ > 0. then condition (1.1) holds. for any y > 0 commuting with a (for example y = i) we have (y a)∗ + y a = 2y a and re(y ax,x) = |(y ax,x)|. thus cos τ(a,y ) = 1 and s(a,y ) = {z ∈ c : arg z = 0}. on angular localization of spectra 199 so lemma 1.1 is sharp. remark 1.3. suppose a has a bounded inverse. recall that the quantity dev(a) defined by cos dev(a) := inf x∈h,x 6=0 re(ax,x) ‖ax‖‖x‖ is called the angular deviation of a, cf. [1, chapter 1, exercise 32]. for example, for a positive definite operator a one has cos dev(a) = 2 √ λmλm λm + λm , where λm and λm are the minimum and maximum of the spectrum of a, respectively (see [1, chapter 1, exercise 33]). besides, in exercise 32 it is pointed that the spectrum of a lies in the sector |arg z| ≤ dev(a). since |(ax,x)| ≤ ‖ax‖‖x‖, lemma 1.1 refines the just pointed assertion. 2. the main result let a be a bounded linear operator in h, whose spectrum lies in the open right half-plane. then by the lyapunov theorem, cf. [1, theorem i.5.1], there exists a positive definite operator x ∈b(h) solving the lyapunov equation 2 re(ax) = xa + a∗x = 2i. (2.1) so re(xax,x) = ((xa + a∗x)x,x)/2 = (x,x) (x ∈h) and cos τ(a,x) = inf x∈h,‖x‖=1 (x,x) |(xax,x)| = 1 supx∈h,‖x‖=1 |(xax,x)| ≥ 1 ‖ax‖ . put j(a) = 2 ∫ ∞ 0 ‖e−at‖2dt. now we are in a position to formulate our main result. theorem 2.1. let a,ã ∈b(h), condition (1.1) hold and x be a solution of (2.1). then with the notation q = ‖a− ã‖ one has cos τ(ã,x) ≥ cos τ(a,x) (1 −qj(a)) (1 + qj(a)) , provided qj(a) < 1. 200 m.i. gil’ the proof of this theorem is based on the following lemma. lemma 2.2. let a,ã ∈ b(h), condition (1.1) hold and x be a solution of (2.1). if, in addition, q‖x‖ < 1, (2.2) then cos τ(ã,x) ≥ cos τ(a,x) (1 −‖x‖q) (1 + ‖x‖q) . proof. put e = ã − a. then q = ‖e‖ and due to (2.1), with x ∈ h, ‖x‖ = 1, we obtain re(x(a + e)x,x) ≥ re(xax,x) −|(xex,x)| = (x,x) −|(xex,x)| ≥ (x,x) −‖x‖‖e‖‖x‖2 = 1 −‖x‖q. (2.3) in addition, |(x(a + e)x,x)| ≤ |(xax,x)| + ‖x‖‖e‖‖x‖2 = |(xax,x)| ( 1 + ‖x‖q |(xax,x)| ) (‖x‖ = 1). but |(xax,x)| ≥ |re(xax,x)| = re(xax,x) = (x,x) = 1. hence |(x(a + e)x,x)| ≤ |(xax,x)| ( 1 + ‖x‖q re(xax,x) ) ≤ |(xax,x)|(1 + ‖x‖q). now (2.3) yields. re(xãx,x) |(xãx,x)| ≥ (1 −‖x‖q) |(xax,x)|(1 + ‖x‖q) (‖x‖ = 1), provided (2.2) holds. since cos τ(ã,x) = inf x∈b,‖x‖=1 re(xãx,x) |(xãx,x)| , we arrive at the required result. on angular localization of spectra 201 proof of theorem 2.1 note that x is representable as x = 2 ∫ ∞ 0 e−a ∗te−atdt [1, section 1.5]. hence, we easily have ‖x‖ ≤ j(a). now the latter lemma proves the theorem. 3. operators with hilbert-schmidt hermitian components in this section we obtain an estimate for j(a) (a ∈b(h)) assuming that a ∈b(h) and ai := (a−a∗)/i is a hilbert-schmidt operator, (3.1) i.e., n2(ai) := (trace(a 2 i)) 1/2 < ∞. numerous integral operators satisfy this condition. introduce the quantity (the departure from normality) gi(a) := [ 2n22 (ai) − 2 ∞∑ k=1 |im λk(a)|2 ]1/2 ≤ √ 2n2(ai), where λk(a) (k = 1, 2, . . .) are the eigenvalues of a taken with their multiplicities and ordered as |im λk+1(a)| ≤ |im λk(a)|. if a is normal, then gi(a) = 0, cf. [2, lemma 9.3]. lemma 3.1. let conditions (1.1) and (3.1) hold. then j(a) ≤ ĵ(a), where ĵ(a) := ∞∑ j,k=0 g j+k i (a)(k + j)! 2j+kβj+k+1(a)(j! k!)3/2 . proof. by [2, theorem 10.1] we have ‖e−at‖≤ exp [ −β(a)t ] ∞∑ k=0 gki (a)t k (k!)3/2 (t ≥ 0). 202 m.i. gil’ then j(a) ≤ 2 ∫ ∞ 0 exp[−2β(a)t] ( ∞∑ k=0 gki (a)t k (k!)3/2 )2 dt = 2 ∫ ∞ 0 exp[−2β(a)t]   ∞∑ j,k=0 g k+j i (a)t k+j (j!k!)3/2  dt = ∞∑ j,k=0 2(k + j)!g j+k i (a) (2β(a))j+k+1(j! k!)3/2 , as claimed. if a is normal, then gi(a) = 0 and with 0 0 = 1 we have ĵ(a) = 1 β(a) . the latter lemma and theorem 2.1 imply corollary 3.2. let a,ã ∈b(h) and let the conditions (1.1), (3.1) and qĵ(a) < 1 hold. then cos τ(ã,x) ≥ (1 −qĵ(a)) (1 + qĵ(a)) cos τ(a,x). 4. integral operators as usually l2 = l2(0, 1) is the space of scalar-valued functions h defined on [0, 1] and equipped with the norm ‖h‖ = [∫ 1 0 |h(x)|2dx ]1/2 . consider in l2(0, 1) the operator ã defined by (ãh)(x) = a(x)h(x) + ∫ 1 0 k(x,s)h(s)ds (h ∈ l2,x ∈ [0, 1]), (4.1) where a(x) is a real bounded measurable function with a0 := inf a(x) > 0, (4.2) and k(x,s) is a scalar kernel defined on 0 ≤ x,s ≤ 1, and∫ 1 0 ∫ 1 0 |k(x,s)|2ds dx < ∞. (4.3) on angular localization of spectra 203 so the volterra operator v defined by (v h)(x) = ∫ 1 x k(x,s)h(s)ds (h ∈ l2,x ∈ [0, 1]), is a hilbert-schmidt one. define operator a by (ah)(x) = a(x)h(x) + ∫ 1 x k(x,s)h(s)ds (h ∈ l2,x ∈ [0, 1]). then a = d + v, where d is defined by (dh)(x) = a(x)h(x). due to lemma 7.1 and corollary 3.5 from [3] we have σ(a) = σ(d). so σ(a) is real and β(a) = a0. moreover, n2(ai) = n2(vi) ≤ n2(v ) = [∫ 1 0 ∫ 1 x |k(x,s)|2ds dx ]1/2 . here vi = (v −v ∗)/2i. thus, gi(a) ≤ gv := √ 2n2(v ) and ‖a− ã‖≤ q0 := [∫ 1 0 ∫ x 0 |k(x,s)|2ds dx ]1/2 . simple calculations show that under consideration ĵ(a) ≤ ĵ0 := ∞∑ j,k=0 g j+k v (k + j)! 2j+ka j+k+1 0 (j! k!) 3/2 . making use of corollary 3.2 and taking into account that in the considered case cos τ(a,x) = 1, we arrive at the following result. corollary 4.1. let ã be defined by (4.1) and the conditions (4.2) and (4.3) hold. if, in addition, q0ĵ0 < 1, then σ(ã) lies in the angular sector{ z ∈ c : |arg z| ≤ arccos (1 −q0ĵ0) (1 + q0ĵ0) } . example 4.2. to estimate the sharpness of our results consider in l2(0,1) the operators (ah)(x) = 2h(x) and (ãh)(x) = (2 + i)h(x) (h ∈ l2,x ∈ [0, 1]). 204 m.i. gil’ σ(a) consists of the unique point λ = 2 and so cos(a,x) = cos arg λ = 1. we have j(a) = 2 ∫ ∞ 0 e−4tdt = 1/2 and q = 1. by corollary 3.2 cos τ(ã,x) ≥ 1 − 1/2 1 + 1/2 = 1/3. compare this inequality with the sharp result: σ(ã) consists of the unique point λ̃ = 2 + i. so tan(arg λ̃) = 1/2, and therefore cos(arg λ̃) = 2/( √ 5). acknowledgements i am very grateful to the referee of this paper for his (her) deep and helpful remarks. references [1] yu.l. daleckii, m.g. krein, “stability of solutions of differential equations in banach space”, vol. 43, american mathematical society, providence, r. i., 1974. [2] m.i. gil’, “operator functions and operator equations”, world scientific publishing co. pte. ltd., hackensack, new jersey, 2018. [3] m.i. gil’, norm estimates for resolvents of linear operators in a banach space and spectral variations, adv. oper. theory 4 (1) (2019), 113 – 139. [4] g.h. hostetter, an improved test for the zeros of a polynomial in a sector, ieee trans. automatic control ac-20 (3) (1975), 433 – 434. [5] e.i. jury, n.k. bose, b.d.o. anderson, a simple test for zeros of a complex polynomial in a sector, ieee trans. automatic control ac-19 (1974), 437 – 438. [6] e.i. jury, n.k. bose, b.d.o. anderson, on eigenvalues of complex matrices in a sector, ieee trans. automatic control ac-20 (1975), 433 – 434. [7] m.g. krein, the angular localization of the spectrum of a multiplicative integral in hilbert space (in russian) funkcional. anal. i prilozhen 3 (1) (1969), 89 – 90. [8] g.v. rozenblyum, angular asymptotics of the spectrum of operators that are close to normal, j. soviet math. 45 (3) (1989), 1250 – 1261. introduction and preliminaries the main result operators with hilbert-schmidt hermitian components integral operators � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 1 (2020), 35 – 42 doi:10.17398/2605-5686.35.1.35 available online january 7, 2020 on h3(1) hankel determinant for certain subclass of analytic functions d. vamshee krishna 1,@, d. shalini 2 1 department of mathematics, gis, gitam university visakhapatnam530 045, a.p., india 2 department of mathematics, dr. b. r. ambedkar university srikakulam532 410, a.p., india vamsheekrishna1972@gmail.com , shaliniraj1005@gmail.com received february 21, 2019 presented by manuel maestre accepted september 3, 2019 abstract: the objective of this paper is to obtain an upper bound to hankel determinant of third order for any function f, when it belongs to certain subclass of analytic functions, defined on the open unit disc in the complex plane. key words: analytic function, upper bound, third hankel determinant, positive real function. ams subject class. (2010): 30c45, 30c50. 1. introduction let a denotes the class of analytic functions f of the form f(z) = z + ∞∑ n=2 anz n (1.1) in the open unit disc e = {z : |z| < 1}. let s be the subclass of a consisting of univalent functions. in 1985, louis de branges de bourcia proved the bieberbach conjecture also called as coefficient conjecture, which states that for a univalent function its nthtaylor’s coefficient is bounded by n (see [4]). the bounds for the coefficients of these functions give information about their geometric properties. for example, the nth-coefficient gives information about the area where as the second coefficient of functions in the family s yields the growth and distortion properties of the function. a typical problem in geometric function theory is to study a functional made up of combinations of the coefficients of the original function. the hankel determinant of f for @ corresponding author issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.35.1.35 mailto:vamsheekrishna1972@gmail.com mailto:shaliniraj1005@gmail.com mailto:vamsheekrishna1972@gmail.com https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 36 d. vamshee krishna, d. shalini q ≥ 1 and n ≥ 1 was defined by pommerenke [20], which has been investigated by many authors, as follows. hq(n) = an an+1 · · · an+q−1 an+1 an+2 · · · an+q ... ... ... ... an+q−1 an+q · · · an+2q−2 . (1.2) it is worth of citing some of them. ehrenborg [7] studied the hankel determinant of exponential polynomials. noor [18] determined the rate of growth of hq(n) as n → ∞ for the functions in s with bounded boundary rotation. the hankel transform of an integer sequence and some of its properties were discussed by layman (see [13]). it is observed that h2(1), the feketeszegö functional is the classical problem settled by fekete-szegö [8] is to find for each λ ∈ [0, 1], the maximum value of the coefficient functional, defined by φλ(f) := |a3 − λa22| over the class s and was proved by using loewner method. ali [1] found sharp bounds on the first four coefficients and sharp estimate for the fekete-szegö functional |γ3 − tγ22|, where t is real, for the inverse function of f defined as f−1(w) = w + ∑∞ n=2 γnw n when f−1 ∈ s̃t(α), the class of strongly starlike functions of order α (0 < α ≤ 1). in recent years, the research on hankel determinants has focused on the estimation of |h2(2)|, where h2(2) = a2 a3 a3 a4 = a2a4 −a23, known as the second hankel determinant obtained for q = 2 and n = 2 in (1.2). many authors obtained an upper bound to the functional |a2a4−a23| for various subclasses of univalent and multivalent analytic functions. it is worth citing a few of them. the exact (sharp) estimates of |h2(2)| for the subclasses of s namely, bounded turning, starlike and convex functions denoted by r, s∗ and k respectively in the open unit disc e, that is, functions satisfying the conditions ref ′(z) > 0, re { zf′(z) f(z) } > 0 and re { 1 + zf′′(z) f′(z) } > 0 were proved by janteng et al. [11, 10] and determined the bounds as 4/9, 1 and 1/8 respectively. for the class s∗(ψ) of ma-minda starlike functions, the exact bound of the second hankel determinant was obtained by lee et al. [15]. choosing q = 2 and n = p + 1 in (1.2), we obtain the second hankel determinant for the p-valent function (see [24]), as follows. h2(p + 1) = ap+1 ap+2 ap+2 ap+3 = ap+1ap+3 −a2p+2, on h3(1) hankel determinant 37 the case q = 3 appears to be much more difficult than the case q = 2. very few papers have been devoted to the third order hankel determinant denoted by h3(1), obtained for q = 3 and n = 1 in (1.2), also called as hankel determinant of third kind, namely h3(1) = a1 a2 a3 a2 a3 a4 a3 a4 a5 (a1 = 1). expanding the determinant, we have h3(1) = a1(a3a5 −a24) + a2(a3a4 −a2a5) + a3(a2a4 −a 2 3), (1.3) equivalently h3(1) = h2(3) + a2j2 + a3h2(2), where j2 = (a3a4 −a2a5) and h2(3) = (a3a5 −a24). babalola [2] is the first one, who tried to estimate an upper bound for |h3(1)| for the classes r, s∗ and k. as a result of this paper, raza and malik [22] obtained an upper bound to the third hankel determinant for a class of analytic functions related with lemniscate of bernoulli. sudharsan et al. [23] derived an upper bound to the third kind hankel determinant for a subclass of analytic functions. bansal et al. [3] improved the upper bound for |h3(1)| for some of the classes estimated by babalola [2] to some extent. recently, zaprawa [25] improved all the results obtained by babalola [2]. further, orhan and zaprawa [19] obtained an upper bound to the third kind hankel determinant for the classes s∗ and k functions of order alpha. very recently, kowalczyk et al. [12] calculated sharp upper bound to |h3(1)| for the class of convex functions k and showed as |h3(1)| ≤ 4135, which is far better than the bound obtained by zaprawa [25]. lecko et al. [14] determined sharp bound to the third order hankel determinant for starlike functions of order 1/2. motivated by the results obtained by different authors mentioned above and who are working in this direction (see [5]), in this paper, we are making an attempt to obtain an upper bound to the functional |h3(1)| for the function f belonging to the class, defined as follows. definition 1.1. a function f(z) ∈ a is said to be in the class q(α,β,γ) with α, β > 0 and 0 ≤ γ < α + β ≤ 1, if it satisfies the condition that re { α f(z) z + βf ′(z) } ≥ γ, z ∈ e. (1.4) 38 d. vamshee krishna, d. shalini this class was considered and studied by zhigang wang et al. [26]. in obtaining our results, we require a few sharp estimates in the form of lemmas valid for functions with positive real part. let p denotes the class of functions consisting of g, such that g(z) = 1 + c1z + c2z 2 + c3z 3 + · · · = 1 + ∞∑ n=1 cnz n, (1.5) which are analytic in e and reg(z) > 0 for z ∈ e. here g is called the caratheodòry function [6]. lemma 1.2. ([9]) if g ∈ p, then the sharp estimate |ck −µckcn−k| ≤ 2, holds for n,k ∈ n = {1, 2, 3, . . .}, with n > k and µ ∈ [0, 1]. lemma 1.3. ([17]) if g ∈ p, then the sharp estimate |ck − ckcn−k| ≤ 2, holds for n,k ∈ n, with n > k. lemma 1.4. ([21]) if g ∈ p then |ck| ≤ 2, for each k ≥ 1 and the inequality is sharp for the function g(z) = 1+z 1−z , z ∈ e. in order to obtain our result, we refer to the classical method devised by libera and zlotkiewicz [16], used by several authors. 2. main result theorem 2.1. if f(z) = z + ∑∞ n=2 anz n ∈ q(α,β,γ), (α, β > 0 and 0 ≤ γ < α + β ≤ 1) then |h3(1)| ≤ 4t21 [ k1α 6 + k2α 5 + k3α 4β + k4α 3β2 + k5α 2β3 + k6αβ 4 + k7β 5 (α + 2β)2(α + 3β)3(α + 4β)2(α + 5β) ] , where k1 = 2, k2 = 2(18β + 1), k3 = 2(132β + 15), k4 = 2(511β + 87), k5 = (2179β+490), k6 = 12(203β+56), k7 = 12(93β+30) and t1 = (α+β−γ). proof. let f(z) = z+ ∑∞ n=2 anz n ∈ q(α,β,γ). by virtue of definition 1.1, there exists an analytic function g ∈ p in the open unit disc e with g(0) = 1 and re{g(z)} > 0 such that 1 α + β −γ { α f(z) z + βf ′(z) −γ } = g(z) (2.1) on h3(1) hankel determinant 39 using the series representation for f and g in (2.1), upon simplification, we obtain ∞∑ n=2 (α + nβ)anz n−2 = (α + β −γ) ∞∑ n=1 cnz n−1. (2.2) the coefficient of zt−2, where t is an integer with t ≥ 2 in (2.2) is given by at = (α + β −γ)ct−1 (α + tβ) , with t ≥ 2. (2.3) substituting the values of a2, a3, a4 and a5 from (2.3) in the functional given in (1.3), it simplifies to h3(1) = (α + β −γ)2 [ c2c4 (α + 3β)(α + 5β) − (α + β −γ)c32 (α + 3β)3 − c23 (α + 4β)2 − (α + β −γ)c21c4 (α + 2β)2(α + 5β) + 2(α + β −γ)c1c2c3 (α + 2β)(α + 3β)(α + 4β) ] . (2.4) on grouping the terms in the expression (2.4), in order to apply the lemmas, we have h3(1) = t 2 1 [ c4(c2 − t1c21) (α + 2β)2(α + 5β) − c3 (α + 4β)2 { c3 − t1(α + 4β)c1c2 (α + 2β)(α + 3β) } (2.5) + c2(c4 − t1c22) (α + 3β)3 − c2 (α + 3β)(α + 4β)2 { c4 − t1(α + 4β)c1c3 (α + 2β)(α + 4β) } + (d1α 6 + d2α 5 + d3α 4β + d4α 3β2 + d5α 2β3 + d6αβ 4 + d7β 5)c2c4 (α + 2β)2(α + 3β)3(α + 4β)2(α + 5β) ] , with d1 = 1, d2 = (18β − 1), d3 = (133β − 19), d4 = 4(129β − 35), d5 = 2(554β − 249), d6 = 8(156β − 107), d7 = 4(144β − 143) and t1 = (α + β −γ). on applying the triangle inequality in (2.5), we have∣∣∣h3(1)∣∣∣ ≤t21[ |c4||(c2 − t1c21)|(α + 2β)2(α + 5β) + |c3|(α + 4β)2 ∣∣∣∣c3 − t1(α + 4β)c1c2(α + 2β)(α + 3β) ∣∣∣∣ + |c2||(c4 − t1c22)| (α + 3β)3 + |c2| (α + 3β)(α + 4β)2 ∣∣∣∣c4 − t1(α + 4β)c1c3(α + 2β)(α + 4β) ∣∣∣∣ (2.6) + |d1α6 + d2α5 + d3α4β + d4α3β2 + d5α2β3 + d6αβ4 + d7β5||c2||c4| (α + 2β)2(α + 3β)3(α + 4β)2(α + 5β) ] . 40 d. vamshee krishna, d. shalini upon using the lemmas given in (1.2), (1.3) and (1.4) in the inequality (2.6), it simplifies to |h3(1)| ≤ 4t21 [ k1α 6 + k2α 5 + k3α 4β + k4α 3β2 + k5α 2β3 + k6αβ 4 + k7β 5 (α + 2β)2(α + 3β)3(α + 4β)2(α + 5β) ] , (2.7) with k1 = 2, k2 = 2(18β + 1), k3 = 2(132β + 15), k4 = 2(511β + 87), k5 = (2179β + 490), k6 = 12(203β + 56), k7 = 12(93β + 30) and t1 = (α + β −γ). this completes the proof of the theorem. remark 2.2. for the values α = 1−σ, β = σ, γ = 0, so that (α+β−γ) = 1 in (2.7), we obtain |h3(1)| ≤ 4 [ 63σ6 + 312σ5 + 411σ4 + 414σ3 + 188σ2 + 44σ + 4 (1 + σ)2(1 + 2σ)3(1 + 3σ)2(1 + 4σ) ] . (2.8) remark 2.3. choosing σ = 1 in the expression (2.8), it coincides with the result obtained by zaprawa [25]. acknowledgements the authors are extremely grateful to the esteemed reviewers for a careful reading of the manuscript and making valuable suggestions leading to a better presentation of the paper. references [1] r.m. ali, coefficients of the inverse of strongly starlike functions, bull. malays. math. sci. soc., (2) 26 (1) (2003), 63 – 71. [2] k.o. babalola, on h3(1) hankel determinant for some classes of univalent functions, in “inequality theory and applications 6” (ed. cho, kim and dragomir), nova science publishers, new york, 2010, 1 – 7. [3] d. bansal, s. maharana, j.k. prajapat, third order hankel determinant for certain univalent functions, j. korean math. soc. 52 (6) (2015), 1139 – 1148. [4] l. de branges, a proof of bieberbach conjecture, acta math. 154 (1-2) (1985), 137 – 152. 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[26] zhi-gang wang, chun-yi gao, shao-mou yuan, on the univalency of certain analytic functions, j. inequal. pure appl. math. 7 (1) (2006), article 9, 1 – 4. introduction main result � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 2 (2020), 127 – 135 doi:10.17398/2605-5686.35.2.127 available online may 7, 2020 extreme and exposed points of l(nl2∞) and ls( nl2∞) sung guen kim department of mathematics, kyungpook national university daegu 702-701, south korea sgk317@knu.ac.kr received november 26, 2019 presented by jesús m.f. castillo accepted april 10, 2020 abstract: for every n ≥ 2 this paper is devoted to the description of the sets of extreme and exposed points of the closed unit balls of l(nl2∞) and ls(nl2∞), where l(nl2∞) is the space of n-linear forms on r2 with the supremum norm, and ls(nl2∞) is the subspace of l(nl2∞) consisting of symmetric n-linear forms. first we classify the extreme points of the closed unit balls of l(nl2∞) and ls(nl2∞), correspondingly. as corollaries we obtain |ext bl(nl2∞)| = 2 (2n) and |ext bls(nl2∞)| = 2 n+1. we also show that exp bl(nl2∞) = ext bl(nl2∞) and exp bls(nl2∞) = ext bls(nl2∞) . key words: n-linear forms, symmetric n-linear forms, extreme points, exposed points. ams subject class. (2010): 46a22. 1. introduction let n ∈ n,n ≥ 2. we write be for the unit ball of a real banach space e and the dual space of e is denoted by e∗. an element x ∈ be is called an extreme point of be if y,z ∈ be with x = 12 (y + z) implies x = y = z. we denote by ext be the set of all the extreme points of be. an element x ∈ be is called an exposed point of be if there is a f ∈ e∗ so that f(x) = 1 = ‖f‖ and f(y) < 1 for every y ∈ be\{x}. it is easy to see that every exposed point of be is an extreme point. we denote by exp be the set of exposed points of be. we denote by l(ne) the banach space of all continuous n-linear forms on e endowed with the norm ‖t‖ = sup‖xk‖=1 |t(x1, · · · ,xn)|. ls( ne) denote the closed subspace of all continuous symmetric n-linear forms on e. let us say about the history of the classifications of extreme and exposed points of the unit ball of continuous (symmetric) multilinear forms on a banach space. kim [1] initiated and classified ext bls(2l2∞) and exp bls(2l2∞), where ln∞ = r n with the supremum norm. it was shown that ext bls(2l2∞) = exp bls(2l2∞). kim [2, 3, 4, 5] classified ext bls(2d∗(1,w)2), ext bl(2d∗(1,w)2), exp bls(2d∗(1,w)2), and exp bl(2d∗(1,w)2), where d∗(1,w) 2 = r2 with the octagonal norm ‖(x,y)‖w = max { |x|, |y|, |x|+|y| 1+w } . kim [6, 7] classified ext bls(2r2h(w)) issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.35.2.127 mailto:sgk317@knu.ac.kr https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 128 s. g. kim and ext bl(2r2 h(w) ), where where r 2 h(w) = r2 with the hexagonal norm ‖(x,y)‖h(w) = max{|y|, |x|+ (1−w)|y|}. kim [8, 9, 10] classified ext bls(2l3∞), ext bls(3l2∞) and ext bl(3l2∞). it was shown that every extreme point is exposed in each space. kim [11] characterized ext bl(2ln∞) and ext bls(2ln∞). recently, kim [12] classified ext bl(2l3∞) and showed exp bl(2l3∞) = ext bl(2l3∞). 2. the extreme and exposed points of the unit ball of l(nl2∞) let l2∞ = {(x,y) ∈ r2 : ‖(x,y)‖∞ = max(|x|, |y|)}. for n ≥ 2, we denote wn := {[(1,w1), . . . , (1,wn)] : wj = ±1 for j = 1, . . . ,n}. note that wn has 2n elements in sl2∞ ×···×sl2∞. recall that the krein-milman theorem [13] say that every nonempty compact convex subset of a housdorff locally convex space is the closed convex hull of its set of extreme points. hence, the unit ball of l2∞ is the closed convex hull of {(1, 1), (−1, 1), (1,−1), (−1,−1)}. theorem 2.1. let n ≥ 2 and t ∈l(nl2∞). then, ‖t‖ = sup w∈wn |t(w)|. proof. it follows that from the krein-milman theorem and multilinearity of t. let z1, . . . ,z2n be an ordering of the monomials xl1 · · ·xljyk1 · · ·ykn−j with {l1, · · · , lj,k1, · · · ,kn−j} = {1, · · · ,n}. note that {z1, . . . ,z2n} is a basis for l(nl2∞). hence, dim(l(nl2∞)) = 2n. if t ∈l(nl2∞), then, t = 2n∑ k=1 akzk for some a1, . . . ,a2n ∈ r. by simplicity, we denote t = (a1, · · · ,a2n )t. let w1, . . . ,w2n be an ordering of the elements of wn. let m(z1, . . . ,z2n ; w1, . . . ,w2n ) = [zi(wj)] be the 2n × 2n matrix. note that, for every t ∈l(nl2∞), m(z1, . . . ,z2n ; w1, . . . ,w2n )t = (t(w1), . . . ,t(w2n )) t. here, (�1, . . . ,�2n ) t denote the transpose of (�1, . . . ,�2n ). extreme and exposed points of l(nl2∞) and ls(nl2∞) 129 theorem 2.2. let n ≥ 2. then, ext bl(nl2∞) = { m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(�1, . . . ,�2n ) t : �j = ±1, j = 1, . . . , 2n } . proof. claim 1: m(z1, . . . ,z2n ; w1, . . . ,w2n ) is invertible. consider the equation m(z1, . . . ,z2n ; w1, . . . ,w2n )(t1, . . . , t2n ) t = (0, . . . , 0)t. (*) let a1, · · · ,a2n be a solution of (*) and let t = ∑2n k=1 akzk ∈l( nl2∞). then, t(wj) = 0 j = 1, . . . , 2 n. by theorem 2.1, ‖t‖ = 0, hence t = 0. since z1, . . . ,z2n are linearly independent in l(nl2∞), we have aj = 0 for all j = 1, . . . , 2n. hence, the equation (*) has only zero solution. therefore, m(z1, . . . ,z2n ; w1, . . . ,w2n ) is invertible. claim 2: m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(�1, . . . ,�2n ) t is an extreme point for �j = ±1, (j = 1, . . . , 2n). let t := m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(�1, . . . ,�2n ) t. since m(z1, . . . ,z2n ; w1, . . . ,w2n )t = (�1, . . . ,�2n ) t, t(wj) = �j for j = 1, . . . , 2 n. by theorem 2.1, ‖t‖ = sup 1≤j≤2n |t(wj)| = sup 1≤j≤2n |�j| = 1. suppose that t = 1 2 (t1 + t2) for some tk ∈ bl(nl2∞) (k = 1, 2). we may write t1 = m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(�1, . . . ,�2n ) t + (δ1, . . . ,δ2n ) t and t2 = m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(�1, . . . ,�2n ) t − (δ1, . . . ,δ2n )t for some δj ∈ r (j = 1, . . . , 2n). note that (tk(w1), . . . ,tk(w2n )) t = m(z1, . . . ,z2n ; w1, . . . ,w2n )tk for k = 1, 2. 130 s. g. kim therefore, (t1(w1), . . . ,t1(w2n )) t = (�1, . . . ,�2n ) t + m(z1, . . . ,z2n ; w1, . . . ,w2n )(δ1, . . . ,δ2n ) t and (t2(w1), . . . ,t2(w2n )) t = (�1, . . . ,�2n ) t −m(z1, . . . ,z2n ; w1, . . . ,w2n )(δ1, . . . ,δ2n )t. hence, for j = 1, . . . , 2n, t1(wj) = �j + (z1(wj), . . . ,z2n (wj))(δ1, . . . ,δ2n ) t, and t2(wj) = �j − (z1(wj), . . . ,z2n (wj))(δ1, . . . ,δ2n )t. it follows that, for j = 1, . . . , 2n, 1 ≥ max{|t1(wj)|, |t2(wj)|} = |�j| + |(z1(wj), . . . ,z2n (wj))(δ1, . . . ,δ2n )t| = 1 + |(z1(wj), . . . ,z2n (wj))(δ1, . . . ,δ2n )t|, which shows that (z1(wj), . . . ,z2n (wj))(δ1, . . . ,δ2n ) t = 0 for j = 1, . . . , 2n. hence, m(z1, . . . ,z2n ; w1, . . . ,w2n )(δ1, . . . ,δ2n ) t = 0 . therefore, (δ1, . . . ,δ2n ) t = m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(0, . . . , 0)t = (0, . . . , 0)t. hence, tk = t for k = 1, 2. therefore, t is extreme. suppose that t ∈ ext bl(nl2∞). note that (t(w1), . . . ,t(w2n )) t = m(z1, . . . ,z2n ; w1, . . . ,w2n )t. extreme and exposed points of l(nl2∞) and ls(nl2∞) 131 claim 3: |t(wj)| = 1 for all j = 1, . . . , 2n. if not. there exists 1 ≤ j0 ≤ 2n such that |t(wj0 )| < 1. let δ0 > 0 such that |t(wj0 )| + δ0 < 1. let t1 = m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1 × (t(w1), . . . ,t(wj0−1),t(wj0 ) + δ0,t(wj0+1), . . . ,t(w2n )) t and t2 = m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1 × (t(w1), . . . ,t(wj0−1),t(wj0 ) −δ0,t(wj0+1), . . . ,t(w2n )) t. hence, t1(wj0 ) = t(wj0 ) + δ0,t2(wj0 ) = t(wj0 ) −δ0,t1(wj) = t2(wj) = t(wj) (j 6= j0). obviously, t 6= tk for k = 1, 2. by theorem 2.1, ‖tk‖ = 1 for k = 1, 2 and t = 1 2 (t1 + t2), which is a contradiction. therefore, t = m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(t(w1), . . . ,t(w2n )) t with |t(wj)| = 1 for all j = 1, . . . , 2n. kim [10] characterized ext bl(3l2∞). notice that using wolfram mathematica 8 and theorem 2.2, we can exclusively describe ext bl(nl2∞) for a given n ≥ 2. for every t ∈l(nl2∞), we let norm(t) := { [(1,w1), . . . , (1,wn)] ∈wn : |t((1,w1), . . . , (1,wn))| = ‖t‖ } . we call norm(t) the set of the norming points of t. corollary 2.3. (a) let n ≥ 2. ext bl(nl2∞) has exactly 2 (2n) elements. (b) let n ≥ 2 and t ∈ l(nl2∞) with ‖t‖ = 1. then t ∈ ext bl(nl2∞) if and only if norm(t) = wn. theorem 2.4. ([4]) let e be a real banach space such that ext be is finite. suppose that x ∈ ext be satisfies that there exists an f ∈ e∗ with f(x) = 1 = ‖f‖ and |f(y)| < 1 for every y ∈ ext be\{±x}. then x ∈ exp be. 132 s. g. kim theorem 2.5. let n ≥ 2. then, exp bl(nl2∞) = ext bl(nl2∞). proof. let t ∈ ext bl(nl2∞) and let f := 1 2n ∑ 1≤j≤2n sign(t(wj))δwj ∈l( nl2∞) ∗. note that 1 = ‖f‖ = f(t). let s ∈ ext bl(nl2∞) be such that |f(s)| = 1. we will show that s = t or s = −t. it follows that 1 = |f(s)| = | 1 2n ∑ 1≤j≤2n sign(t(wj))s(wj)| ≤ 1 2n ∑ 1≤j≤2n |s(wj)| ≤ 1, which shows that s(wj) = sign(t(wj)) (1 ≤ j ≤ 2n) or s(wj) = −sign(t(wj)) (1 ≤ j ≤ 2n). suppose that s(wj) = −sign(t(wj)) (1 ≤ j ≤ 2n). it follows that s = m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(s(w1), . . . ,s(w2n )) t = m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(−sign(t(w1)), . . . ,−sign(t(w2n )))t = m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(−t(w1), . . . ,−t(w2n ))t = −t. note that if s(wj) = sign(t(wj)) (1 ≤ j ≤ 2n), then s = t. by theorem 2.4, t is exposed. extreme and exposed points of l(nl2∞) and ls(nl2∞) 133 3. the extreme and exposed points of the unit ball of ls(nl2∞) let n ≥ 2 and un := { [(1, 1), (1, 1), . . . , (1, 1)], [(1,−1), (1, 1), . . . , (1, 1)], [(1,−1), (1,−1), (1, 1), . . . , (1, 1)], [(1,−1), (1,−1), (1 − 1), (1, 1), . . . , (1, 1)], . . . , [(1,−1), (1,−1), . . . , (1,−1), (1, 1)], [(1,−1), (1,−1), . . . , (1,−1), (1,−1)] } . note that un has n + 1 elements in sl2∞ ×···×sl2∞. theorem 3.1. let n ≥ 2 and t ∈ls(nl2∞). then, ‖t‖ = sup u∈un |t(u)| . proof. it follows that from theorem 2.1 and symmetry of t. for j = 0, . . . ,n, we let fj = ∑ {l1,··· ,lj,k1,··· ,kn−j}={1,··· ,n} xl1 · · ·xljyk1 · · ·ykn−j. then, {f0, . . . ,fn} is a basis for ls(nl2∞). hence, dim(ls(nl2∞)) = n + 1. if t ∈ls(nl2∞), then, t = n∑ j=0 bjfj for some b0, . . . ,bn ∈ r. by simplicity, we denote t = (b0, · · · ,bn)t. for j = 0, . . . ,n, we let uj = [(1,u1), . . . , (1,un)] ∈un, where uk = −1 for 1 ≤ k ≤ j and uk = 1 for j + 1 ≤ k ≤ n. let m(f0, . . . ,fn; u0, . . . ,un) = [fi(uj)] be the (n + 1) × (n + 1) matrix. note that, for every t ∈ls(nl2∞), m(f0, . . . ,fn; u0, . . . ,un)t = (t(u0), . . . ,t(un)) t. by analogous arguments in the claim 1 of theorem 2.2, m(f0, . . . ,fn; u0, . . . ,un) is invertible. 134 s. g. kim theorem 3.2. let n ≥ 2. then, ext bls(nl2∞) = { m(f0, . . . ,fn; u0, . . . ,un) −1(�0, . . . ,�n) t : �j = ±1,j = 0, . . . ,n } . proof. it follows by theorem 3.1 and analogous arguments in the claims 2 and 3 of theorem 2.2. notice that using wolfram mathematica 8 and theorem 3.2, we can exclusively describe ext bls(nl2∞) for a given n ≥ 2. for every t ∈ls(nl2∞), we let norm(t) := { [(1,u1), . . . , (1,un)] ∈un : |t((1,u1), . . . , (1,un))| = ‖t‖ } . we call norm(t) the set of the norming points of t. corollary 3.3. (a) let n ≥ 2. ext bls(nl2∞) has exactly 2 n+1 elements. (b) let n ≥ 2 and t ∈ ls(nl2∞) with ‖t‖ = 1. then t ∈ ext bls(nl2∞) if and only if norm(t) = un. theorem 3.4. let n ≥ 2. then, exp bls(nl2∞) = ext bls(nl2∞). proof. let t ∈ ext bls(nl2∞) and let f := 1 n + 1 ∑ 0≤j≤n sign(t(uj))δuj ∈ls( nl2∞) ∗. note that 1 = ‖f‖ = f(t). by analogous arguments in the proof of theorem 2.5, f exposes t. therefore, t is exposed. questions. (a) let n ≥ 2 and �1, . . . ,�2n be fixed with �j = ±1, (j = 1, . . . , 2n). is it true that ext bl(nl2∞) = { m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(�1, . . . ,�2n ) t : z1, . . . ,z2n,w1, . . . ,w2n are any ordering } ? (b) by theorem 2.2, m(z1, . . . ,z2n ; w1, . . . ,w2n ) −1(�1, . . . ,�2n ) t is extreme if z1, . . . ,z2n , w1, . . . ,w2n are any ordering. similarly, we may ask the following: let n ≥ 2 and δ0, . . . ,δn be fixed with δk = ±1, (k = 0, . . . ,n). is it true that ext bls(nl2∞) = { m(f0, . . . ,fn; u0, . . . ,un) −1(δ0, . . . ,δn) t : f0, . . . ,fn,u0, . . . ,un are any ordering } ? extreme and exposed points of l(nl2∞) and ls(nl2∞) 135 acknowledgements the author is thankful to the referee for the careful reading and considered suggestions leading to a better presented paper. references [1] s.g. kim, the unit ball of ls(2l2∞), extracta math. 24 (2009), 17 – 29. [2] s.g. kim, the unit ball of ls(2d∗(1,w)2), kyungpook math. j. 53 (2013), 295 – 306. [3] s.g. kim, extreme bilinear forms of l(2d∗(1,w)2), kyungpook math. j. 53 (2013), 625 – 638. [4] s.g. kim, exposed symmetric bilinear forms of ls(2d∗(1,w)2), kyungpook math. j. 54 (2014), 341 – 347. [5] s.g. kim, exposed bilinear forms of l(2d∗(1,w)2), kyungpook math. j. 55 (2015), 119 – 126. [6] s.g. kim, the unit ball of l(2r2 h(w) ), bull. korean math. soc. 54 (2017), 417 – 428. [7] s.g. kim, extremal problems for ls(2r2h(w)), kyungpook math. j. 57 (2017), 223 – 232. [8] s.g. kim, the unit ball of ls(2l3∞), comment. math. 57 (2017), 1 – 7. [9] s.g. kim, the geometry of ls(3l2∞), commun. korean math. soc. 32 (2017), 991 – 997. [10] s.g. kim, the geometry of l(3l2∞) and optimal constants in the bohnenblusthill inequality for multilinear forms and polynomials, extracta math. 33 (1) (2018), 51 – 66. [11] s.g. kim, extreme bilinear forms on rn with the supremum norm, period. math. hungar. 77 (2018), 274 – 290. [12] s.g. kim, the unit ball of the space of bilinear forms on r3 with the supremum norm, commun. korean math. soc. 34 (2) (2019), 487 – 494. [13] m.g. krein, d.p. milman, on extreme points of regular convex sets, studia math. 9 (1940), 133 – 137. introduction the extreme and exposed points of the unit ball of l(nl2) the extreme and exposed points of the unit ball of ls(nl2) � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae article in press available online december 14, 2022 a note on isomorphisms of quantum systems martin weigt ∗ department of mathematics and applied mathematics, summerstrand campus (south) nelson mandela university, port elizabeth (gqeberha), south africa martin.weigt@mandela.ac.za , weigt.martin@gmail.com received july 4, 2022 presented by martin mathieu accepted november 9, 2022 abstract: we consider the question as to whether a quantum system is uniquely determined by all values of all its observables. for this, we consider linearly nuclear gb∗-algebras over w∗-algebras as models of quantum systems. key words: quantum system, observables, gb∗-algebra, jordan homomorphism. msc (2020): 46h05, 46h15, 46h35, 47l10, 47l30, 81p05, 81p16. 1. introduction the main objective of this paper is to determine whether all values of all observables in a quantum system are sufficent to determine the quantum system uniquely. to answer this question, we first have to find a suitable mathematical framework in which to reformulate the question. in the well known formalism of haag and kastler, a quantum system takes on the following form: the observables of the system are self-adjoint elements of a ∗-algebra a with identity element 1, and the states of the system are positive linear functionals φ of a for which φ(1) = 1. this is well in agreement with the hilbert space formalism, where the observables are linear operators on a hilbert space h, and all states are unit vectors in h. since observables are generally unbounded linear operators on a hilbert space (such as position and momentum operators, which are unbounded linear operators on the hilbert space l2(r)), one requires the ∗-algebra a above to at least partly consist of unbounded linear operators on some hilbert space. the question is then what ∗-algebra of unbounded linear operators one must take to house the observables of the quantum system under consideration. a candidate can be found among the elements in the class of gb∗-algebras, which are locally ∗ this work is based on the research supported wholly by the national research foundation of south africa (grant number 132194). issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) mailto:martin.weigt@mandela.ac.za mailto:weigt.martin@gmail.com https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 2 m. weigt convex ∗-algebras serving as generalizations of c∗-algebras, and were first studied by g.r. allan in [2], and later by p.g. dixon in [6] to include non locally convex ∗-algebras (see section 2 for the definition of a gb∗-algebra). every gb∗-algebra a[τ] contains a c∗-algebra a[b0] which is dense in a (see section 2). in [13], the author motivated why one can model a quantum system as a gb∗-algebra a[τ] which is nuclear as a locally convex space (referred to as a linearly nuclear gb∗-algebra for here on). in addition to this, it would be useful to have that a[b0] is a w ∗-algebra (i.e., a von neumann algebra): since a[τ] is also assumed to be locally convex, a can be faithfully represented as a ∗-algebra b of closed densely defined linear operators on a hilbert space (see [6, theorem 7.11] or [10, theorem 6.3.5]). if we denote this ∗-isomorphism by π : a → b, then π(a[b0]) = bb, where bb is the ∗-algebra of all bounded linear operators in b, and is a von neumann algebra (this follows from [6, theorem 7.11], or [10, theorem 6.3.5]). let x ∈ a be self-adjoint. then π(x) is a self-adjoint element of b and π(x) = ∫ σ(π(x)) λd pλ. now (1 + y∗y)−1 ∈ bb for all y ∈ b. by [7, proposition 2.4], it follows that all y ∈ b are affiliated with bb. therefore, pλ ∈ bb for all λ ∈ σ(π(x)). the spectral projections, pλ, λ ∈ σ(π(x)), are important for determining the probability of a particle in a certain set (see [8, postulate 4, p. 13]). so far, the observables of a quantum mechanical system are self-adjoint elements of a locally convex ∗-algebra a[τ] (more specifically, in our case, a linearly nuclear gb∗-algebra with a[b0] a w ∗-algebra). in fact, one can sharpen this by noting that if x,y ∈ a are self-adjoint (i.e., observables), then x ◦ y = 1 2 (xy + yx) is again self-adjoint, i.e., an observable. in 1932, j. von neumann and collaborators proposed that a jordan algebra be used to house the observables of a quantum system (see [5, introduction]). a linear mapping φ : a → a with φ(x ◦ y) = φ(x) ◦ φ(y) for all x,y ∈ as, where as denotes the set of self-adjoint elements of a, is called a jordan homomorphism. we note here that as is a jordan algebra with respect to the operation ◦ above. if, in addition, φ is a bijection, then φ is called a jordan isomorphism, i.e., an isomorphism of jordan algebras. a jordan isomorphism is therefore an isomorphism of quantum systems (see [5, introduction]). observe that a linear map φ is a jordan homomorphism if and only if φ(x2) = φ(x)2 for all x ∈ a. we already know that all possible values of an observable, when considered as a self-adjoint unbounded linear operator on a hilbert space, are in the spectrum of the observable. an interesting question is therefore if a quantum system is uniquely determined by the values/measurements of its observables. isomorphisms of quantum systems 3 to answer this question in our setting of a linearly nuclear gb∗-algebra a[τ] with a[b0] a w ∗-algebra (an abstract algebra of unbounded linear operators), one requires a notion of spectrum of an element which is an analogue of the notion of spectrum of a self-adjoint unbounded linear operator on a hilbert space. the required notion is the allan spectrum of an element of a locally convex algebra. the values/measurements of a self-adjoint element x ∈ a (i.e., an observable) are therefore in σa(x), the allan spectrum of x, as defined in definition 2.3 below in section 2. if no confusion arises, we write σ(x) instead of σa(x). the above question can be reformulated as follows: let a[τ] be a linearly nuclear gb∗-algebra with a[b0] a w ∗-algebra. let φ : a → a be a bijective self-adjoint linear map such that σ(φ(x)) = σ(x) for all x ∈ as, where as is the set of all self-adjoint elements of a. is φ a jordan isomorphism? below, in corollary 3.6, we answer this question affirmatively for the case where a[τ] has the additional property of being a fréchet algebra, i.e., a complete and metrizable algebra. we do not require the gb∗-algebra to be linearly nuclear in this result. the above result is similar to results which are partial answers to a special case of an unanswered question of i. kaplansky: if a and b are banach algebras with identity, and φ : a → b is a bijective linear map such that spb(φ(x)) = spa(x) for all x ∈ a, is it true that φ is a jordan isomorphism? here, spa(x) refers to the spectrum of x, which is the set {λ ∈ c : λ1 − x is not invertible in a}. the answer to this question remains unresolved for c∗-algebras, but it has been shown, by b. aupetit, to have an affirmative answer if a and b are von neumann algebras (see [3, theorem 1.3]). for the physical problem under consideration, we have to replace the spectrum of x in kaplansky’s question with the allan spectrum of x, as explained above. we refer the reader to [4] for an excellent introduction to kaplansky’s problem. section 2 of this paper contains all the background material required to understand the discussion in section 3, where the main result is presented. 2. preliminaries in this section, we give all background material on generalized gb∗algebras (gb∗-algebras, for short) which is required to understand the main results of this paper. gb∗-algebras were introduced in the late sixties by g.r. allan in [2], and taken further, in the early seventies, by p.g. dixon in 4 m. weigt [6, 7]. recently, the author, along with m. fragoulopoulou, a. inoue and i. zarakas, published a monograph on gb∗-algebras [10] containing much of the developed theory on this topic. almost all concepts and results in this section are due allan and dixon, and can be found in [1, 2, 6]. we will, however, use [10] as a reference. a topological algebra is an algebra which is a topological vector space and in which multiplication is separately continuous. if a topological algebra is equipped with a continuous involution, then it is called a topological ∗-algebra. a locally convex ∗-algebra is a topological ∗-algebra which is locally convex as a topological vector space. we say that a topological algebra is a fréchet algebra if it is complete and metrizable. definition 2.1. ([10, definition 3.3.1]) let a[τ] be a unital topological ∗-algebra and let b∗ denote a collection of subsets b of a with the following properties: (i) b is absolutely convex, closed and bounded; (ii) 1 ∈ b, b2 ⊂ b and b∗ = b. for every b ∈ b∗, denote by a[b] the linear span of b, which is a normed algebra under the gauge function ‖ · ‖b of b. if a[b] is complete for every b ∈b∗, then a[τ] is called pseudo-complete. an element x ∈ a is called bounded, if for some nonzero complex number λ, the set {(λx)n : n = 1, 2, 3, . . .} is bounded in a. we denote by a0 the set of all bounded elements in a. a unital topological ∗-algebra a[τ] is called symmetric if, for every x ∈ a, the element (1 + x∗x)−1 exists and belongs to a0. definition 2.2. ([10, definition 3.3.2]) a symmetric pseudo-complete locally convex ∗-algebra a[τ], such that the collection b∗ has a greatest member, denoted by b0, is called a gb ∗-algebra over b0. every c∗-algebra is a gb∗-algebra. an example of a gb∗-algebra, which generally need not be a c∗-algebra, is a pro-c∗-algebra. by a pro-c∗-algebra, we mean a complete topological ∗-algebra a[τ] for which the topology τ is defined by a directed family of c∗-seminorms. another example of a gb∗-algebra which is not a pro-c∗-algebra is the locally convex ∗-algebra lω([0, 1]) = ∩p≥1lp([0, 1]) defined by the family of seminorms {‖ · ‖p : p ≥ 1}, where ‖ · ‖p is the lp-norm on lp([0, 1]) for all p ≥ 1. isomorphisms of quantum systems 5 if a is commutative, then a0 = a[b0] [10, lemma 3.3.7(ii)]. in general, a0 is not a ∗-subalgebra of a, and a[b0] contains all normal elements of a0, i.e., all x ∈ a such that xx∗ = x∗x [10, lemma 3.3.7(i)]. definition 2.3. ([10, definition 2.3.1]) let a[τ] be topological algebra with identity element 1 and x ∈ a. the set σa(x) is the subset of c∗, the one-point compactification of c, defined as follows: (i) if λ 6= ∞, then λ ∈ σa(x) if λ1 −x has no bounded inverse in a; (ii) ∞∈ σa(x) if and only if x /∈ a0. we define ρa(x) to be c∗ \σa(x). if there is no risk of confusion, then we write σ(x) to denote σa(x). proposition 2.4. ([10, theorem 3.3.9, theorem 4.2.11]) if a[τ] is a gb∗-algebra, then the banach ∗-algebra a[b0] is a c∗-algebra, which is sequentially dense in a. moreover, (1 + x∗x)−1 ∈ a[b0] for every x ∈ a and b0 is the unit ball of a[b0]. the next proposition has to do with extensions of characters of the commutative c∗-algebra a[b0] to the gb ∗-algebra a, which could be infinite valued. proposition 2.5. ([10, proposition 2.5.4]) let a[τ] be a commutative pseudocomplete locally convex ∗-algebra with identity. then, for any character φ on a0, there exists a c∗-valued function φ′ on a having the following properties: (i) φ′ is an extension of φ; (ii) φ′(λx) = λφ′(x) for all λ ∈ c (with the convention that 0.∞ = 0); (iii) φ′(x + y) = φ′(x) + φ′(y) for all x,y ∈ a for which φ′(x) and φ′(y) are not both ∞; (iv) φ′(xy) = φ′(x)φ′(y) for all x,y ∈ a for which φ′(x) and φ′(y) are not both 0,∞ in some order; (v) φ′(x∗) = φ′(x) for all x ∈ a (with the convention that ∞ = ∞). 6 m. weigt 3. the main result the following example is an example of a linearly nuclear gb∗-algebra over a w∗-algebra, which is not a c∗-algebra. example 3.1. consider a family {hα : α ∈ λ} of finite dimensional hilbert spaces. then, for every α ∈ λ, we have that b(hα) is a finite dimensional c∗-algebra, and hence a linearly nuclear space, with respect to the operator norm ‖ · ‖α. let a = παb(hα). then a is a pro-c∗-algebra in the product topology τ, when all b(hα) are equipped with their operators norms ‖·‖α [9, chapter 2]. furthermore, a[τ] is linearly nuclear since it is a product of linearly nuclear spaces. observe that xξ = (xα(ξα))α for all ξ = (ξα)α ∈ h, where h is the direct sum of the hilbert spaces hα. note that h is itself a hilbert space. now a[b0] = { x = (xα)α ∈ a : supα‖xα‖α < ∞ } = ⊕αb(hα), and this is a von neumann algebra with respect to the norm supα‖xα‖α. lemma 3.2. if x is a self-adjoint element of a gb∗-algebra a[τ], then x is a projection if and only if σ(x) ⊆{0, 1}. proof. let x ∈ a be a projection and let b be a maximal commutative ∗-subalgebra of a containing x. then σb(x) = σa(x) (see [10, proposition 2.3.2]) and b is a gb∗-algebra over the c∗-algebra bb = a[b0] ∩b (see [6]). let m0 denote the character space of the commutative c ∗-algebra bb. then, by proposition 2.5 and [10, corollary 3.4.10], it follows that σb(x) = { x̂(φ) = φ′(x) : φ ∈ m0 } = { φ(x) : φ ∈ m0 } ⊆{0, 1}. the second equality above follows from the fact that x ∈ a[b0], due to the fact that x is a projection, and therefore x ∈ bb. therefore σa(x) ⊆{0, 1}. now assume that σa(x) ⊆ {0, 1}. let b be a maximal commutative ∗subalgebra of a containing x. then σb(x) = σa(x). like above, we have that { x̂(φ) = φ′(x) : φ ∈ m0 } = σb(x) = σa(x) ⊆{0, 1} isomorphisms of quantum systems 7 for all characters φ on a[b0]. therefore x̂ is an idempotent function. since x 7→ x̂ is an algebra ∗-isomorphism [10, theorem 3.4.9], we get that x is an idempotent element of a. therefore x is a projection because x is self-adjoint. if a and b are ∗-algebras and φ : a → b a linear map such that φ(x2) = φ(x)2 for all self-adjoint elements x in a, then φ is a jordan homomorphism [3, page 922]. we require this in the proof of proposition 3.3 below. proposition 3.3. let a[τ] be a gb∗-algebra with a[b0] a w ∗-algebra, and let b be a topological ∗-algebra. suppose further that the multiplications on a and b are jointly continuous. if φ : a → b is a continuous linear mapping which maps projections to projections, then φ is a jordan homomorphism. proof. let s be a self-adjoint element in a[b0]. by the spectral theorem, and the fact that a[b0] is a w ∗-algebra, there is a sequence (sn) of finite linear combinations of orthogonal projections in a[b0] such that sn → s in norm [11, theorem 5.2.2], and hence also with respect to the topology τ on a, since the restriction of the topology τ to a[b0] is weaker than the norm topology of a[b0]. therefore φ(s 2 n) = φ(sn) 2 for every n. hence, since φ is continuous, and since the multiplications on a and b are jointly continuous, it follows that φ ( s2 ) = φ ( lim n→∞ s2n ) = φ ( lim n→∞ sn )2 = φ(s)2. this holds for any self-adjoint element s ∈ a[b0]. by the paragraph following lemma 3.2, φ|a[b0] is a jordan homomorphism. let x ∈ a. then there is a sequence (xn) in a[b0] such that xn → x. since φ is continuous, a[b0] is dense in a, and the multiplications on a and b are jointly continuous, it follows that φ(x2) = φ(x)2. this holds for every x ∈ a, and therefore φ is a jordan homomorphism. we say that an element x in a gb∗-algebra a[τ] is positive if there exists y ∈ a such that x = y∗y. the following proposition is required to prove theorem 3.5 below. proposition 3.4. ([12, proposition 7]) let a[τ1] and b[τ2] be fréchet gb∗-algebras. if φ : a → b is a linear mapping which maps positive elements of a to positive elements of b, then φ is continuous. 8 m. weigt theorem 3.5. let a[τ] be a fréchet gb∗-algebra with a[b0] a w ∗algebra, and let φ : a → a be a self-adjoint linear map such that σ(φ(x)) ⊆ σ(x) for all x ∈ as, where as is the set of all self-adjoint elements of a. then φ is a jordan isomorphism. proof. by hypothesis and [10, proposition 6.2.1], it follows that if x ∈ a is a positive element, then σ(φ(x)) ⊂ σ(x) ⊆ [0,∞], and therefore φ(x) is a positive element in a. therefore φ maps positive elements of a to positive elements of a. by proposition 3.4 and the fact that a is a fréchet gb∗algebra, it follows that φ is continuous. we now show that if p ∈ a is a projection, then φ(p) is also a projection in a. if p ∈ a is a projection, then p and φ(p) are self-adjoint elements in a. therefore, by lemma 3.2, σ(p) ⊆ {0, 1}. since σ(φ(p)) ⊆ σ(p), we get that σ(φ(p)) ⊆{0, 1}. by lemma 3.2 again, φ(p) is a projection. since a[b0] is a w ∗-algebra and the multiplication on a is jointly continuous (because a is a fréchet algebra), it follows from proposition 3.3 that φ is a jordan homomorphism. the following corollary is the desired result of this section, and affirms that all quantum mechanical isomorphisms, in the context of fréchet gb∗-algebras, are jordan isomorphisms. corollary 3.6. let a[τ] be a fréchet gb∗-algebra with a[b0] a w ∗algebra, and let φ : a → a be a bijective self-adjoint linear map such that σ(φ(x)) = σ(x) for all x ∈ as, where as is the set of all self-adjoint elements of a. then φ is a jordan isomorphism. in [3], b. aupetit proved that any bijective linear map φ : a → b between von neumann algebras a and b, satisfying spb(φ(x)) = spa(x) for all x ∈ a, is a jordan homomorphism. observe that φ need not be self-adjoint. the proof of aupetit’s result in [3] is complicated and relies on a deep spectral characterization of idempotents in a semi-simple banach algebra (see [3, theorem 1.1]). if we additionally assume that φ is self-adjoint, then one has a much simpler proof of his result, namely, the proof of corollary 3.6 for the case where a[τ] is a von neumann algebra. acknowledgements the author wishes to thank the referee for his/her meticulous reading of the manuscript, which significantly enhanced its quality. isomorphisms of quantum systems 9 references [1] g.r. allan, a spectral theory for locally convex algebras, proc. london math. soc. (3) 15 (1965), 399 – 421. [2] g.r. allan, on a class of locally convex algebras, proc. london math. soc. (3) 17 (1967), 91 – 114. [3] b. aupetit, spectrum-preserving linear mappings between banach algebras or jordan-banach algebras, j. london math. soc. (2) 62 (2000), 917 – 924. [4] m. bresar, p. semrl, spectral characterization of idempotents and invertibility preserving linear maps, exposition. math. 17 (1999), 185 – 192. [5] o. bratteli, d.w. robinson, “ operator algebras and quantum statistical mechanics ”, vol. 1, texts and monographs in physics, springer-verlag, new york-heidelberg, 1979. [6] p.g. dixon, generalized b∗-algebras, proc. london math. soc. (3) 21 (1970), 693 – 715. [7] p.g. dixon, unbounded operator algebras, proc. london math. soc. 23 (1971), 53 – 69. [8] z. ennadifi, “ an introduction to the mathematical formalism of quantum mechanics ”, master of science thesis, mohammed v university, rabat, morocco, 2018. [9] m. fragoulopoulou, “ topological algebras with involution ”, northholland mathematics studies 200, elsevier science b.v., amsterdam, 2005. [10] m. fragoulopoulou, a. inoue, m. weigt, i. zarakas, “ generalized b∗algebras and applications ”, lecture notes in mathematics, 2298, springer, cham, 2022. [11] r.v. kadison, j.r. ringrose, “ fundamentals of the theory of operator algebras ”, academic press, 1996. [12] m. weigt, on nuclear generalized b∗-algebras, in “ proceedings of the international conference on topological algebras and their applications–ictaa 2018 ” (edited by mart abel), math. stud. (tartu), 7, est. math. soc., tartu, 2018, 137 – 164. [13] m. weigt, applications of generalized b∗-algebras to quantum mechanics, in “ positivity and its applications ”, trends math., bikhauser/springer, cham, 2021, 283 – 318. introduction preliminaries the main result � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 2 (2022), 185 – 194 doi:10.17398/2605-5686.37.2.185 available online april 25, 2022 on a class of power associative lcc-loops o.o. george 1,@, j.o. olaleru 1 j.o. adéńıran 2, t.g. jaiyéo. lá 3 1 department of mathematics, university of lagos, akoka, nigeria 2 department of mathematics, federal university of agriculture abeokuta 110101, nigeria 3 department of mathematics, obafemi awolowo university ile ife 220005, nigeria oogeorge@unilag.edu.ng , jolaleru@unilag.edu.ng , ekenedilichineke@yahoo.com adeniranoj@unaab.edu.ng , jaiyeolatemitope@yahoo.com , tjayeola@oauife.edu.ng received august 20, 2021 presented by a. turull accepted april 7, 2022 abstract: let lwpc denote the identity (xy · x) · xz = x((yx · x)z), and rwpc the mirror identity. phillips proved that a loop satisfies lwpc and rwpc if and only if it is a wip pacc loop. here, it is proved that a loop q fulfils lwpc if and only if it is a left conjugacy closed (lcc) loop that fulfils the identity (xy · x)x = x(yx · x). similarly, rwpc is equivalent to rcc and x(x · yx) = (x · xy)x. if a loop satisfies lwpc or rwpc, then it is power associative (pa). the smallest nonassociative lwpc-loop was found to be unique and of order 6 while there are exactly 6 nonassociative lwpcloops of order 8 up to isomorphism. methods of construction of nonassociative lwpc-loops were developed. key words: left (right) conjugacy closed loop, power associativity, lwpc-loop, rwpc-loop. msc (2020): 20n02, 20n05. 1. introduction a quasigroup (q, ·) consists of a non-empty set q with a binary operation (·) on q such that given x,y ∈ q, the equations a ·x = b and y ·a = b have unique solutions x,y ∈ q respectively. we shall sometimes refer to (q, ·) as simply q. it is usual to set x = a\b and y = b/a. a loop is a quasigroup with a two-sided neutral element 1. we write xy for x ·y and stipulate that · have lower priority than juxtaposition among factors to be multiplied-for instance, x ·yz stands for x(yz). for an overview on loop theory, see [1, 7, 11]. if a is an element of a loop q, then la : x 7→ ax permutes q and is called the left translation of a. similarly, ra : x 7→ xa is called the right translation of a. the loop q is said to be a left conjugacy closed (lcc) if the @ corresponding author issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.37.2.185 mailto:oogeorge@unilag.edu.ng mailto:jolaleru@unilag.edu.ng mailto:ekenedilichineke@yahoo.com mailto:adeniranoj@unaab.edu.ng mailto:jaiyeolatemitope@yahoo.com mailto:tjayeola@oauife.edu.ng mailto:oogeorge@unilag.edu.ng https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 186 j.o. adéńıran, o.o. george, t.g. jaiyéo. lá, j.o. olaleru left translations are closed under conjugation (i.e., for all x,y ∈ q, there exists z ∈ q such that lxlyl−1x = lz). similarly, right conjugacy closed (rcc) loops are those in which the right translations are closed under conjugation. a loop is said to be conjugacy closed (cc) if it is both lcc and rcc. kinyon and kunen [8] thoroughly analyzed power associative cc-loops (pacc-loops). a loop is power associative (pa) if each of its elements generates a (cyclic) subgroup. by [8], the structure of pacc-loops heavily depends upon the structure of wip elements. that motivated phillips [12] to find a short equational basis for wip pacc loops. now, wip stands for weak inverse property. a loop q is said to be a wip-loop if it satisfies the equivalent identities x(yx)ρ = yρ and (xy)λx = yλ, where x ·xρ = 1 = xλ ·x for all x ∈ q. by [12], a loop q is a wip pacc-loop if and only if it fulfils the laws (xy ·x) ·xz = x((yx ·x)z) , (lwpc) zx · (x ·yx) = (z(x ·xy))x. (rwpc) the purpose of this paper is to initiate the study of loops that fulfil only one of the latter identities. our main result is theorem 2.2 that proves that an lwpc loop is power associative, and that lwpc loops are exactly the lcc loops in which (xy · x)x = x(yx · x). a mirror result holds for rwpc loops. during our preliminary search for lwpc-loops, we found two lcc-loops (that are lwpc-loops) of orders 6 and 8 with the property that their right nuclei are abelian groups which are of index two. a general construction of such loops can be found in drápal [4]. these loops were constructed by means of an arbitrary abelian group and two permutations that satisfy some constraints (cf. proposition 5.1). we shall be adopting this construction to show that an infinite series of lwpc-loop is feasible. if q is a loop and α,β and γ permute q, then (α,β,γ) is said to be an autotopism of q if α(y)β(z) = γ(yz) for all y,z ∈ q. autotopisms of q can be composed through componentwise multiplication, and thus they form a group called the autotopism group denoted by atp(q). the fact that left translations are closed under conjugation can be expressed equationally by the law x ·y(x\z) = xy/x · z (cf. [6]). this law may also be written as x ·yz = (xy/x) ·xz. hence, q is lcc if and only if( r−1x lx, lx,lx ) ∈ atp(q) for all x ∈ q. (1) on a class of power associative lcc-loops 187 writing lwpc as (x((y/x)/x) ·x) ·xz = x ·yz implies that this law holds if and only if ( rxlxr −2 x ,lx,lx ) ∈ atp(q) for all x ∈ q. (2) characterizations of lcc and lwpc by autotopisms are of crucial importance for the proof of the main result. 2. lwpc-loop, rwpc-loop and their properties the following criterion for power associativity will be useful to establish the main result in theorem 2.2. lemma 2.1. let x be an element of a loop q. suppose that xλ = xρ, and denote the latter element by x−1. suppose that for each i ≥ 1 any bracketing of i occurrences of x yields the same element, and denote this element by xi. similarly, let any bracketing of i occurrences of x−1 yield an element x−i. set x0 = 1 = (x−1)0 and (x−i)−1 = xi. finally, suppose that yjy−i = yj−i = y−iyj whenever j ≥ i ≥ 1 and y ∈ {x, x−1}. then x generates a subgroup of q, and the element xi attains the usual meaning of the ith power, for every integer i. proof. first note that xjx−i = xj−i = x−ixj holds for any positive integers i and j since if j < i, then y−jyi = yi−j = yiy−j may be used, with y = x−1. we have to show that xi ·xjxk = xixj ·xk for any i,j and k. if any of them is zero, the corresponding power is equal to 1 and the equality holds. if all of i,j and k are positive, then the equality follows from the assumption on bracketing. if two or more exponents are negative, replace x with y = x−1. thus only the case with exactly one of exponents negative needs to be solved. this means to verify that each of the ensuing triples associates, under the assumption that all of i,j and k are positive integers,( xi,xj,x−k ) , ( xi,x−j,xk ) and ( x−i,xj,xk ) . the leftmost and the rightmost triples are mirror symmetric. hence, only the first two triples will be considered. now, xi ·xjx−k = xixj−k = xi+j−k = xi+jx−k = xixj ·x−k and xi ·x−jxk = xix−j+k = xi−j+k = xi−jxk = xix−j ·xk. 188 j.o. adéńıran, o.o. george, t.g. jaiyéo. lá, j.o. olaleru theorem 2.2. let q be a loop. then (i) q fulfills lwpc ⇔ q is lcc and (xy ·x)x = x(yx ·x)︸︷︷︸ pλ(x,y) for all x,y ∈ q. (ii) q fulfills rwpc ⇔ q is rcc and x(x ·yx) = (x ·xy)x︸︷︷︸ pρ(x,y) for all x,y ∈ q. if q fulfills lwpc or rwpc, then q is power associative. proof. setting z = 1 in the lwpc law yields (xy · x)x = x(yx · x). to prove (i) thus means to show lwpc ⇔ lcc under the assumption of r2xlx = lxr 2 x, for all x ∈ q. the equivalence of lcc and lwpc follows immediately from the expression of the identities via autotopisms as in (1) and (2), since r2xlx = lxr 2 x implies that rxlxr −2 x = rxr −2 x lx = r −1 x lx . nothing else is needed to get (i). point (ii) follows by a mirror argument. let us assume that q fulfills lwpc. to prove that q is power associative, let us start by showing that for k ≥ 1 any bracketing of i occurrences of x yields the same element. proceed by induction. cases k = 1 and k = 2 are clear. the case k = 3 follows from lwpc by setting y = z = 1. assume k ≥ 4. we need to verify that xixj = xxk−1 whenever i ≥ 2 and i + j = k. if i ≥ 3, express xi as xxi−2 ·x and set z = xj−1. lwpc yields xixj = (xxi−2 ·x) ·xxj−1 = x((xi−2x ·x) ·xj−1) = xxk−1. assume i = 2 and set y = 1 in lwpc. then, x2 · xxj−1 = x(x2xj−1) = xxk−1. lwpc with z = 1 implies xx = (xxρ · x)x = x(xρx · x). hence x = xρx ·x and 1 = xρx. therefore, xρ = xλ = x−1. by lemma 2.1 it remains to verify that xjx−i = xj−i = x−ixj whenever j ≥ i ≥ 1. proceed by outer induction on j ≥ 2 and inner induction on i ≥ 1. to get the case i = 1, start from xxj−1 = xj = (xxj−2 ·x) ·xx−1 = x · (xj−2x ·x)x−1 = x ·xjx−1. by cancelling, xjx−1 = xj−1. therefore (x−1xj ·x−1)x−1 = x−1(xjx−1 ·x−1) = xj−3 = xj−2x−1, and that yields x−1xj ·x−1 = xj−2 = xj−1x−1. thus, x−1xj = xj−1. assume i ≥ 2. note that xx−i = x−(i−1) follows from the induction assumption since on a class of power associative lcc-loops 189 this is the same as y−1yi = yi−1, where y = x−1. hence x ·xj−i = xj−i+1 = xjx−(i−1) = (xxj−2x) ·xx−i = x · (xj−2x ·x)x−i = x ·xjx−i. by cancelling, xj−i = xjx−i. to finish up , first observe that x−i ·xj−ix−i = xj−3i = xj−ix−i ·x−i. indeed, if j − i ≤ 0, this follows from the earlier part of the proof. if j − i > 0, then xj−ix−i = xj−2i = x−ixj−i by the induction assumption (where a switch to y = x−1 is needed if j−i < i). since j−2i may be treated similarly as j−i, the expression of xj−3i follows. now, x−ixj = xj−i may be obtained by cancellation from (x−ixj ·x−i)x−i = x−i(xjx−i ·x−i) = x−i ·xj−ix−i = xj−ix−i ·x−i. corollary 2.3. a cc-loop is a power associative wip-loop if and only if it fulfills the laws (xy ·x)x = x(yx ·x) and x(x ·yx) = (x ·xy)x. proof. this follows from theorem 2.2. corollary 2.4. a cc-loop is a power associative wip-loop if and only if it is a wip lwpc-loop (alternatively, a wip rwpc-loop). proof. this follows from theorem 2.2 and the fact that lcc and rcc are equivalent in a wip-loop. corollary 2.5. (i) a loop is a lwpc-loop if and only if the conjugates of its left translations are left translations and any left translation commutes with the square of its corresponding right translation. (ii) a loop is a rwpc-loop if and only if the conjugates of its right translations are right translations and any right translation commutes with the square of its corresponding left translation. (iii) a loop is a wip pacc-loop if and only if the conjugates of both its left translations and right translations are left translations and right translations respectively and left and right translations commute with the squares of their corresponding right and left translations respectively. proof. this follows from theorem 2.2. 190 j.o. adéńıran, o.o. george, t.g. jaiyéo. lá, j.o. olaleru 3. construction of lwpc-loops suppose that g and r are abelian groups and that f : g × g → r is a mapping. call it zero preserving if f(x, 0) = 0 = f(0,x) for all x ∈ g. say that f is additive on the right if f(x,y + z) = f(x,y) + f(x,z) for all x,y,z ∈ g. say that f is additive if it is both right and left additive. say that f is quadratically triadditive on the left if g : g×g×g −→ r (x,y,z) 7−→ f(x + y,z) −f(x,z) −f(y,z) is a triadditive symmetric mapping (symmetric means that permuting x,y and z has no influence upon the value g(x,y,z)). define the radical rad(f) as the set of all x ∈ g such that f(x,y) = 0 = f(y,x) for all y ∈ g. theorem 3.1. let r be a subgroup of an abelian group g, and let f : g×g → r be such that rad(f) ≤ r, f is zero preserving and right additive. then x ·y = x + y + f(x,y) defines upon g an lcc loop. this loop is associative if and only if f is biadditive, and conjugacy closed if and only if f is quadratically triadditive on the left. the lwpc law is fulfilled if and only if f(2x + y,x) = 2f(x,x) + f(y,x) for all x,y ∈ g. (3) if g is an elementary abelian 2-group, then (3) always holds, while for g of odd order, (3) is equivalent to f(x + y,x) = f(x,x) + f(y,x) for all x,y ∈ g. proof. by [3, theorem 5.3] and [5, corollary 2.2], only the part relating to (3) needs to be considered. since f is additive on the right, (xy ·x)x = 3x + y + f(x,y) + f(x + y,x) + f(2x + y,x) , x(yx ·x) = 3x + y + f(y,x) + f(x + y,x) + f(x, 2x + y) . hence (xy · x)x = x(yx · x) if and only if (3) holds. by theorem 2.2, this means that (3) characterizes the lwpc loops. let it be satisfied. if 2z = 0 for all z ∈ g, then (3) is trivially true. furthermore, setting y = 0 implies that 2f(x,x) = f(2x,x). on a class of power associative lcc-loops 191 by adding with itself both the left and the right hand sides and using the additivity on the right, we obtain f(2x + y, 2x) = f(2x, 2x) + f(y, 2x) for all x,y ∈ g. if g is of odd order, then 2x may be replaced by x. theorem 3.1 provides a tool how to construct lwpc loops that are not conjugacy closed. by [5, theorem 4.6], the construction of theorem 3.1 covers all lcc loops q for which there exists a prime p and a central subloop z such that |z| = p and q/z is an elementary abelian pgroup. classification of such loops up to isomorphism was considered in [5, section 5], while section 8 of the same paper proves that all (left) bol loops of order 8 are lcc, and that all of them may be obtained by the construction of theorem 3.1. by [8], a nonassociative wip pacc loop is of order divisible by 16. by the library of loops package [9] of gap [10] there are, up to isomorphism, 19 left conjugacy closed loops of order 8 that are not right conjugacy closed. six of them are bol loops discovered by burn [2] and described in [5, section 8]. we have verified that of the remaining 13 loops none fulfills the lwpc law, and exactly one fulfills the law x(x ·yx) = (x ·xy)x. note that if q is a loop from theorem 3.1, then the law x(x·yx) = (x·xy)x. holds if and only if 3f(x,x) + f(x,y) + f(y,x) = f(x,x) + f(x,y) + f(2x + y,x). in characteristic 2, this is always true. the following thus holds: proposition 3.2. a loop of order 8 is lwpc if and only if it is left bol. each such loop fulfills the law x(x · yx) = (x · xy)x. there are exactly six isomorphism classes of nonassociative lwpc loops of order 8. none of them is conjugacy closed or moufang, and in each of them the left nucleus nλ is of order 4. consider now a loop that possesses a right nucleus of index two, and suppose that the nucleus is isomorphic to an abelian group (g, +). it is easy to see (cf. [4, proposition 4.2]) that such a loop is isomorphic to a loop g[f,g] that is defined upon {0, 1}×g by (0,x)(0,y) = (0,x + y) , (0,x)(1,y) = (1,g(x) + y) , (1,x)(0,y) = (1,x + y) , (1,x)(1,y) = (0,f(x) + y), for all x,y ∈ g, where f and g are permutations of g such that g(0) = 0. by 192 j.o. adéńıran, o.o. george, t.g. jaiyéo. lá, j.o. olaleru [4, proposition 5.1], such a loop g[f,g] is lcc if and only if g2 = idg and both y + f(y + z) = f ( z + g(y) ) + g(y) , x + f(y + z) = f−1 ( z + f(y) ) + f(y) (4) are true for all x,y,z ∈ g. proposition 3.3. let f and g be permutations of g, where g is an abelian group. suppose that g(0) = 0, g2 = idg and that (4) holds. the loop g[f,g] fulfills the lwpc law if and only if g ( f(x + y) + x ) = x + f ( g(y) + x ) , f ( g ( f(x) + y ) + x ) = f(x) + g ( f(y) + x ) (5) for all x,y ∈ g. proof. by theorem 2.2, the only step to do is to verify that the two equalities hold if and only if (ab ·a)a = a(ba ·a) whenever a = (ε,x) and b = (η,y) where x,y ∈ g and ε,η ∈{0, 1}. the case ε = η = 0 is clear. assume ε = 0 and η = 1. then ab ·a = (1,x + g(x) + y) and (ab ·a)a = (1, 2x + g(x) + y), while ba ·a = (1, 2x + y) and a(ba ·a) = (1, 2x + y + g(x)) = (ab ·a)a. thus, a = (1,x) may be assumed. suppose first that η = 0. then ab ·a = (0,f(x + y) + x) and (ab · a)a = (1,g(f(x + y) + x) + x), while ba · a = (0,f(x + g(y)) + x) and a(ba ·a) = (1,f(x + g(y)) + 2x). thus, the equality holds if and only if g(f(x + y) + x) = f(x + g(y)) + x. suppose now that η = 1. then, ab ·a = (1,g(f(x) + y) + x) and (ab ·a)a = (0,f(g(f(x) + y) + x) + x), while ba ·a = ( 1,g ( f(y) + x ) + x ) , a(ba ·a) = ( 0,f(x) + g ( f(y) + x ) + x ) , yielding thus the second equality of (5). assume now that f(x) = −x. then (5) holds if and only if g(−x) = −g(x) for every x ∈ g. proposition 3.3 together with [4, proposition 5.7] immediately yield the following statement: on a class of power associative lcc-loops 193 theorem 3.4. let g be a permutation of an abelian group g, g(0) = 0, such that g2(x) = x and g(−x) = −g(x) for every x ∈ g. suppose also that g(x) 6= −x for at least one x ∈ g. define an operation · upon {0, 1}×g by (0,x)(η,y) = ( η,gη(x) + y ) and (1,x)(η,y) = ( η, (−1)ηx + y ) for all x,y ∈ g and η ∈{0, 1}. the operation · describes a nonassociative lwpc loop in which the right nucleus is equal to {(0,x); x ∈ g} and the left nucleus is equal to { (0,x) : g(x + y) = g(x) −y for every y ∈ g } . surprisingly, the case g = idg fulfills the assumptions of theorem 3.4. for each n ≥ 3 the operation (ε,x)(η,y) = ( ε + η, (−1)εηx + y ) thus yields an lwpc loop upon z2 × zn. such a loop is never conjugacy closed since the left nucleus is trivial, while the right nucleus coincides with {0}×zn. the least nonassociative lwpc loop is of order 6. up to isomorphism this is the only nonassociative lwpc loop of order 6. the latter fact has been verified by using the loops package [9] of gap [10]. questions 3.5. (a) can wip lwpc loops be described (as a loop variety) by equations that do not use division and/or inverses? (b) what is the least odd order for which there exists a nonassociative lwpc loop? acknowledgements we acknowledge the valuable suggestions and recommendations of the anonymous referee which have improved the presentation and structural arrangements of this work. references [1] r.h. bruck, “ a survey of binary systems ”, springer-verlag, berlingottingen-heidelberg, 1958. [2] r.p. burn, finite bol loops, math. proc. cambridge philos. soc. 84 (3) (1978), 377 – 385. [3] p. csörgo, a. drápal, left conjugacy closed loops of nilpotency class two, results math. 47 (2005), 242 – 265. 194 j.o. adéńıran, o.o. george, t.g. jaiyéo. lá, j.o. olaleru [4] a. drápal, on left conjugacy closed loops with a nucleus of index two, abh. math. sem. univ. hamburg 74 (2004), 205 – 221. [5] a. drápal, on extraspecial left conjugacy closed loops, j. algebra 302 (2006), 771 – 792. [6] e.g. goodaire, d.a. robinson, a class of loops which are isomorphic to all loop isotopes, canadian j. math. 34 (1982), 662 – 672. [7] t.g. jaiyéo. lá, “ a study of new concepts in smarandache quasigroups and loops ”, infolearn (ilq), ann arbor, mi, 2009. [8] m.k. kinyon, k. kunen, power-associative, conjugacy closed loops, j. algebra 304 (2006), 671 – 711. [9] g.p. nagy, p. vojtěchovský, the loops package, computing with quasigroups and loops in gap 3.4.1. https://www.gap-system.org/manuals/pkg/loops/doc/manual.pdf [10] the gap group, gap groups, algorithms, programming, version 4.11.0. http://www.gap-system.org [11] h.o. pflugfelder, “ quasigroups and loops: introduction ”, sigma series in pure mathematics, 7, heldermann verlag, berlin, 1990. [12] j.d. phillips, a short basis for the variety of wip pacc loops, quasigroups related systems 14 (2006), 73 – 80. https://www.gap-system.org/manuals/pkg/loops/doc/manual.pdf http://www.gap-system.org introduction lwpc-loop, rwpc-loop and their properties construction of lwpc-loops � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 2 (2022), 211 – 221 doi:10.17398/2605-5686.37.2.211 available online november 2, 2022 topological hausdorff dimension and poincaré inequality c.a. dimarco 1000 e. henrietta rd., mathematics department, monroe community college rochester, ny 14623, usa cdimarco2@monroecc.edu received february 26, 2022 presented by g. plebanek accepted october 2, 2022 abstract: a relationship between poincaré inequalities and the topological hausdorff dimension is exposed—a lower bound on the dimension of ahlfors regular spaces satisfying a weak (1,p)-poincaré inequality is given. key words: poincaré inequality, metric space, cantor sets, topological dimension, hausdorff dimension, bi-lipschitz map, ahlfors regular. msc (2020): primary 28a80, 28a75; secondary 28a78, 54f45. 1. introduction let (x,d) be a separable metric space. the subscript of dim indicates the type of dimension, and we set dim ∅ = −1 for every dimension. poincaré inequalities are the forms of the fundamental theorem of calculus that work in general metric spaces. indeed, a one-dimensional poincaré inequality is a direct consequence of the fundamental theorem of calculus: remark 1.1. let f : [a,b] → r be differentiable. the intermediate value theorem gives a point c ∈ [a,b] with f(c) = − ∫ b a f, the average of f on [a,b]. the fundamental theorem of calculus then yields − ∫ b a ∣∣∣∣f(x) −− ∫ b a f ∣∣∣∣ dx ≤ (b−a)− ∫ b a |f ′| , which is inequality (1.1) found below, with p = λ = k = 1. there is an inherent connection between poincaré inequalities and topological hausdorff dimension because both concepts take connectivity into account. in order to discuss poincaré inequalities, we include the following definition, which can be found in [4, p. 55]. issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.37.2.211 mailto:cdimarco2@monroecc.edu https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 212 c.a. dimarco definition 1.2. given a real valued function u in a metric space x, a borel function ρ : x → [0,∞] is an upper gradient of u if |u(x) −u(y)| ≤ ∫ γ ρ ds for each rectifiable curve γ joining x and y in x. to prove the main result, we will use the upper pointwise dilation as a suitable upper gradient (see [2, p. 342]). fact 1.3. if f : x → r is a locally lipschitz function, the upper pointwise dilation lip f(x) = lim sup r→0 sup y∈b(x,r) |f(x) −f(y)| r is an upper gradient of f. the following definition of a weak poincaré inequality is from [4, p. 68], and a broader definition can be found in [2, p. 84]. definition 1.4. let (x,µ) be a metric measure space and let 1 ≤ p < ∞. say that x admits a weak (1,p)-poincaré inequality if there are constants 0 < λ ≤ 1 and k ≥ 1 so that − ∫ λb |u−uλb| dµ ≤ k(diam b) ( − ∫ b ρp dµ )1/p (1.1) for all balls b ⊂ x, for all bounded continuous functions u on b, and for all upper gradients ρ of u, where uλb is the average value of u on the set λb. also assume µ(b(x,r)) > 0 whenever r > 0. it is not difficult to show that if a space supports a weak poincaré inequality, then it is connected, and ∂b(x,r) 6= ∅ whenever r < 1 2 diam x [5, proposition 8.1.6]. such spaces are also quasiconvex, i.e., any two points can be connected by a curve of controlled length [5, theorem 8.2.3]. like the hausdorff dimension, poincaré inequalities are preserved by bi-lipschitz maps, but the constants λ and k may change after application of a lipschitz map. for a precise statement, see [2, proposition 4.16]. recently, results have surfaced that explain the relationship between poincaré inequalities and some particular fractals. mackay, tyson, and wildrick investigated the potential presence of poincaré inequalities on various carpets —metric measure spaces that are homemorphic to the standard sierpinsḱı topological hausdorff dimension 213 carpet. in short, a carpet of this kind is constructed in the same manner as the sierpinsḱı carpet, except at each step the scaling factor need not be 1/3. requiring that the sequence of scaling factors a = (a1,a2, . . . ) contain only reciprocals of odd integers that decrease to zero, one obtains a carpet (sa, |·|,µ) with euclidean metric | · | and measure µ, where µ arises as the weak limit of normalized lebesgue measure on the precarpets. for the construction, see [8]. they provided a complete characterization of these carpets in terms of (1,p)-poincaré inequalities as follows. theorem 1.5. (mackay, tyson, wildrick [8]) (i) the carpet (sa, | · |,µ) supports a (1, 1)-poincaré inequality if and only if a ∈ `1. (ii) the following are equivalent: (a) (sa, | · |,µ) supports a (1,p)-poincaré inequality for each p > 1. (b) (sa, | · |,µ) supports a (1,p)-poincaré inequality for some p > 1. (c) a ∈ `2. to see how topological hausdorff dimension is related to connectivity, one need only consider theorem 3.6 in [1]. that theorem gives an equivalent definition of topological hausdorff dimension for separable metric spaces: dimth x = min { d : ∃a ⊂ x such that dimh a ≤ d− 1 and dimt(x \a) ≤ 0 } . a significant advantage of imposing a poincaré inequality like (1.1) is the flexibility that exists in choosing the function u and one of its upper gradients ρ. to apply (1.1) to the topological hausdorff dimension of a given space x, one can apply the inequality to the boundary of an arbitrary open set u of x to determine a lower bound on dimh ∂u. if a non-trivial lower bound on dimh ∂u is achieved, then so is a lower bound on dimth x. in the next section we apply this technique and exploit the poincaré inequality to accomplish exactly that goal. a closely related concept was recently investigated by lotfi in [7], which generalized the topological hausdorff dimension by combining the definitions of topological dimension and µ-hausdorff dimension. they presented upper and lower bounds for the so-called µ-topological hausdorff dimension of the sierpinsḱı carpet, and gave a large class of measures µ, where the associated µtopological hausdorff dimension of the sierpinsḱı carpet coincides with these lower and upper bounds. 214 c.a. dimarco the main result requires that a space x satisfies a weak (1,p)-poincaré inequality, and that it is ahlfors regular. the following definition can be found in [4, p. 62]. definition 1.6. if x is a metric space admitting a borel regular measure µ such that c−1rb ≤ µ(br) ≤ crb for some constant c ≥ 1, for some exponent b > 0, and for all closed balls br of radius 0 < r < diam x, then x is called ahlfors b-regular. an ahlfors b-regular space has hausdorff dimension b [4, p. 62], and is doubling : definition 1.7. a metric measure space (x,d,µ) is doubling if there is c > 0 such that 0 < µ(b(x, 2r)) ≤ cµ(b(x,r)) for all x ∈ x and for all r > 0. there is much interplay between ahlfors regularity and weak(1,p)-poincaré inequalities in metric spaces. for example, in [6], lohvansuu and rajala recently studied the duality of moduli in this context, where the ahlfors regularity constant is assumed to be greater than one. they proved that there is something of a dual relationship, with exponents p and p∗ = p p−1 , between the path modulus and the modulus of separating surfaces. it can be challenging to obtain nontrivial lower bounds on the topological hausdorff dimension. in the presence of ahlfors regularity, however, this problem becomes more tractable. we now state the main result, which provides a lower bound in terms of the regularity and poincaré constants. theorem. let (x,µ,d) be a complete, ahlfors b-regular, (1,p)-poincaré metric measure space. then dimth x ≥ b−p + 1. due to ahlfors regularity, equality is achieved if p = 1 because dimth x ≤ dimh x = b. on the other hand, it is not clear whether a space exists that yields equality for any p > 1. 2. preliminaries the symbol b(x,ε) denotes the open ball centered at x of radius ε. for x ∈ rn, the euclidean modulus of x is denoted |x|. unless otherwise stated, distance in the metric space y is denoted dy or simply d. we use the notation topological hausdorff dimension 215 fe = − ∫ e f dµ = 1 µ(e) ∫ f dµ for the average value of an integrable function f on e ⊂ x, where (x,d,µ) is a metric measure space. for any a ⊂ (x,d), the set aδ is the δ-neighborhood of a in x. the symbol χu represents the characteristic function of any u ⊂ x. in order to define topological hausdorff dimension, we include the definition of hausdorff dimension: definition 2.1. the p-dimensional hausdorff measure of x is hp(x) = lim δ→0 inf { ∞∑ j=1 (diam ej) p : x ⊂ ∞⋃ j=1 ej and diam ej ≤ δ for all j } ; the hausdorff dimension of x is dimh x = inf{p : hp(x) = 0}. an interesting combination of the hausdorff and topological dimensions, called topological hausdorff dimension, was introduced in [1]: dimth x = inf{d : x has a basis u such that dimh ∂u ≤ d− 1 ∀u ∈u}. by theorem 4.4 in [1], the topological hausdorff dimension always falls between the topological dimension (dimt x) and the hausdorff dimension (dimh x): theorem 2.2. (balka, buczolich, elekes[1])for any metric space x, dimt x ≤ dimth x ≤ dimh x. (2.1) in certain favorable circumstances, the hausdorff and topological hausdorff dimensions are additive under products. for any product space x ×y , we use the metric d((x1,y1), (x2,y2)) = max(dx(x1,x2),dy (y1,y2)). for sake of completeness, we include theorem 4.21 from [1] and several product formulas for hausdorff dimension (see e.g. [3, chapter 7]). fact 2.3. if e ⊂ rn, f ⊂ rm are borel sets, then dimh(e ×f) ≥ dimh e + dimh f. 216 c.a. dimarco let dimhx be the upper box-counting dimension of x (see e.g. [3]). fact 2.4. for any sets e ⊂ rn and f ⊂ rm dimh(e ×f) ≤ dimh e + dimbf. we call a cantor set in [0, 1] uniform if it is constructed in the same way as the usual middle-thirds example, allowing for any scaling factor 0 < r < 1/2. since uniform cantor sets have equal hausdorff and upper box dimensions, facts 2.3 and 2.4 yield the following formula. fact 2.5. if f ⊂ r is a uniform cantor set, then for any e ⊂ rn dimh(e ×f) = dimh e + dimh f. (2.2) in light of facts 2.3 and 2.4, we observe the following convenient additivity property. fact 2.6. if x ⊂ rn and y ⊂ rm are borel sets with dimh x = dimbx, dimh(x ×y ) = dimh x + dimh y. (2.3) the condition dimh x = dimbx holds for a wide variety of spaces. theorem 2.7. if x is a nonempty separable metric space, then dimth(x × [0, 1]) = dimh(x × [0, 1]) = dimh x + 1 . (2.4) in particular, for any value c > 2, r = x × [0, 1] can be chosen such that dimth r = c. the first equality in (2.4) is due to balka, buczolich, and elekes [1]. because dimh[0, 1] = dimb[0, 1] = 1, the second equality in (2.7) is readily obtained considering fact 2.6. recall that the hausdorff dimension is invariant under bi-lipschitz maps. definition 2.8. an embedding f is l-bi-lipschitz if both f and f−1 are l-lipschitz, and we say f is bi-lipschitz if it is l-bi-lipschitz for some l. topological hausdorff dimension 217 3. a lower bound on topological hausdorff dimension for poincaré ahlfors regular spaces to provide a nontrivial lower bound on dimth x, it suffices to consider an arbitrary bounded basis element u for the topology on x, and show that dimh ∂u ≥ b−p, where b and p are the regularity and poincaré constants of x, respectively. theorem 3.1. let (x,µ,d) be a complete, ahlfors b-regular, (1,p)poincaré metric measure space. then dimth x ≥ b−p + 1. proof. let u be basis for the topology on x, and consider a bounded element u ∈u, u 6= x. choose δ > 0 small enough that δ < 1 2 diam(u), and both u \ (∂u)δ and uδ c are nonempty. let 0 < λ ≤ 1 and k ≥ 1 be as in definition 1.4, and choose z0 ∈ u \ (∂u)δ. choose r > 0 large enough that b(z0,r) ⊃ uδ and b(z0,r) \ uδ 6= ∅, and put b = b (z0,r/λ). then r is large enough that uδ ⊂ λb = b(z0,r). fix an arbitrary finite covering d of ∂u by open balls as follows: d = {di = b(xi, 2ri) : xi ∈ ∂u}, 2ri ≤ δ for all i . (3.1) we will show that there is a constant c > 0 such that ∑ i(diam di) b−p ≥ c. note that x is doubling because it is ahlfors regular, and x is proper because it is complete and doubling [5, lemma 4.1.14]. therefore ∂u is compact because it is closed and bounded. given a finite covering d of ∂u satisfying (3.1), define the functions ui(x) = min { d(x,dci ) ri , 1 } and u = max ( max i ui,χu ) . notice that ui is 1 ri -lipschitz, u is bounded, and u is continuous because d is a finite covering. considering that 0 ≤ u ≤ 1, we have 0 ≤ uλb ≤ 1, and hence − ∫ λb |u−uλb| dµ ≥ 1 µ(λb) (∫ {x∈λb:u(x)=1} |u−uλb| dµ ) + 1 µ(λb) (∫ {x∈λb:u(x)=0} |u−uλb| dµ ) = 1 µ(λb) [ (1 −uλb)µ ( {u(x) = 1} ) + uλbµ ( {u(x) = 0} )] 218 c.a. dimarco ≥ 1 µ(λb) min { µ ( {u(x) = 1} ) ,µ ( {u(x) = 0} )} ≥ 1 µ(λb) min { µ(λb ∩u),µ ( λb ∩ (uδ)c )} (3.2) ≥ 1 µ(λb) min { µ(u),µ ( λb \uδ )} . the fact that x is b-regular provides a constant m ≥ 1 with m−1rb ≤ µ(br) ≤ mrb for any ball of radius r. in particular µ(λb) ≤ mrb, and µ(u) > 0 because u is open and non-empty. also, recall that δ and r were chosen so that λb \uδ = b(z0,r) \uδ is open and nonempty. so there is a point z1 and an integer n > 0 such that b(z1, 1/n) ⊂ λb \uδ . applying regularity gives µ(λb \uδ) ≥ µ(b(z1, 1/n)) ≥ 1 mnb . (3.3) in light of (3.2) and (3.3), we see that − ∫ λb |u−uλb| dµ ≥ 1 µ(λb) min{µ(u),µ ( λb \uδ ) } ≥ 1 mrb min { µ(u), 1 mnb } = c′, (3.4) where the constant c′ > 0 is independent of the covering d. next, we show that − ∫ λb |u−uλb|dµ ≤ c′′ ∑ i r b−p i for some c ′′ > 0. to this end, recall that the upper pointwise dilation of any locally lipschitz function f is denoted lip f, and note that lim sup y→x |f(x) −f(y)| d(x,y) = lim sup r→0 sup y∈b(x,r) |f(y) −f(x)| d(y,x) ≥ lim sup r→0 sup y∈b(x,r) |f(y) −f(x)| r = lip f(x) . (3.5) topological hausdorff dimension 219 the fact that ui is 1 ri -lipschitz, along with equation (3.5), show lip ui(x) ≤ 1 ri for all x. also lip u ≤ maxi lip ui, and lip ui(x) = 0 for x /∈ di. ahlfors regularity implies µ(di) ≤ m(2ri)b for all i, and therefore∫ b |lip u|p dµ = ∫ b (lip u)p dµ ≤ ∫ b [ max i (lip ui) ]p dµ ≤ ∫ b ∑ i (lip ui) p dµ ≤ ∑ i ∫ x (lip ui) p dµ ≤ ∑ i µ(di)r −p i ≤ 2 bm ∑ i r b−p i . (3.6) finally, with the poincaré inequality (1.1), (3.4), and (3.6), the regularity lower bound µ(b) ≥ m−1 (r/λ)b gives c′ ≤− ∫ λb |u−uλb| dµ ≤ k(diam b) ( − ∫ b |lip u|p dµ )1/p ≤ k (2r/λ) µ(b)1/p (∫ b |lip u|p dµ )1/p ≤ k (2r/λ) m−1/p (r/λ) b/p ( 2bm ∑ i r b−p i )1/p ≤ k (2r/λ) m−1/p (r/λ) b/p (2bm) 1/p (∑ i r b−p i )1/p = c′′ (∑ i r b−p i )1/p . (3.7) therefore 0 < c ≤ ∑ i r b−p i , where c = ( c′/c′′) p is independent of the covering d. suppose µ(x) < ∞. we will show that for any di ∈ d, the radius ri is bounded above by a constant multiple of diam di, where the constant depends only on x. to this end, consider the ball sidi, where si = (diam di) −1. then sidi has radius ri diam di , and ahlfors regularity provides 1 m ( ri diam di )b ≤ µ(sidi) ≤ µ(x) < ∞ , ri ≤ m 1/bµ(x) 1/b diam di . (3.8) 220 c.a. dimarco in light of (3.8) it is evident that 0 < c ≤ ∑ i r b−p i ≤ ∑ i ( m 1/bµ(x) 1/b )b−p (diam di) b−p , and hence 0 < ∑ i(diam di) b−p. therefore dimh ∂u ≥ b−p for any such u, from which it follows that dimth x ≥ b−p + 1. if µ(x) = ∞, put e = b(z0,a), 0 < a < diam x, and notice that e is complete and inherits both the ahlfors b-regularity and (1,p)-poincaré properties from x (with the same constants m,b,p, and λ). by ahlfors regularity µ(e) ≤ mab < ∞, so e satisfies the assumptions of the theorem in the case that has already been proven. finally, monotonicity of th-dimension shows that dimth x ≥ dimth e ≥ b−p + 1 . if p = 1, then equality holds in theorem 3.1 because (2.1) guarantees that dimth x ≤ dimh x = b, but whether equality can be achieved for some (1,p)-poincaré space (x,µ) with p > 1 is a mystery. question 3.2. is there a number p > 1 with a space (x,µ) for which equality holds in theorem 3.1? in order to answer question 3.2, one needs a supply of spaces that support weak (1,p)-poincaré inequalities for p > 1. theorem 1.5 provides one source of potential examples. it is tempting to try to answer question 3.2 with a carpet sa = (sa, |·|,µ) that supports a weak (1,p)-poincaré inequality with p > 1. a problem arises, however, once one computes the th-dimension of this space. indeed, since sa is ahlfors 2-regular [8], dimh sa = 2, and in order to have equality in theorem 3.1, we would need dimth sa = 3 −p. let ca be the cantor set in [0, 1] obtained from the sequence of scaling factors a. since (ca × [0, 1]) ⊂ sa we see that dimth sa ≥ dimth(ca×[0, 1]) = 2 by monotonicity and additivity of th-dimension. therefore dimth sa = 2, and the equation dimth sa = 3−p is untenable because we assumed p > 1. acknowledgements this paper is based on a part of a phd thesis written by the author under the supervision of leonid kovalev at syracuse university. topological hausdorff dimension 221 references [1] r. balka, z. buczolich, m. elekes, a new fractal dimension: the topological hausdorff dimension, adv. math. 274 (2015), 881 – 927. [2] a. björn, j. björn, “ nonlinear potential theory on metric spaces ”, ems tracts in mathematics 17, european mathematical society (ems), zürich, 2011. [3] k. falconer, “ fractal geometry ”, second edition, mathematical foundations and applications, john wiley & sons, inc., hoboken, nj, 2003. [4] j. heinonen, “ lectures on analysis on metric spaces ”, universitext, springer-verlag, new york, 2001. [5] j. heinonen, p. koskela, n. shanmugalingam, j.t. tyson, “ sobolev spaces on metric measure spaces. an approach based on upper gradients ”, new mathematical monographs 27, cambridge university press, cambridge, 2015. [6] a. lohvansuu, k. rajala, duality of moduli in regular metric spaces, indiana univ. math. j. 70 (3) (2021), 1087 – 1102. [7] h. lotfi, the µ-topological hausdorff dimension, extracta math. 34 (2) (2019), 237 – 254. [8] j.m. mackay, j.t. tyson, k. wildrick, modulus and poincaré inequalities on non-self-similar sierpiński carpets, geom. funct. anal., 23 (3) (2013), 985 – 1034. introduction preliminaries a lower bound on topological hausdorff dimension for poincaré ahlfors regular spaces � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 1 (2020), 43 – 54 doi:10.17398/2605-5686.35.1.43 available online january 29, 2020 a remark on prime ideals s.c. lee 1, r. varmazyar 2,@ 1 department of mathematics education and institute of pure and applied mathematics jeonbuk national university, jeonju, jeonbuk 54896, south korea 2 department of mathematics, khoy branch, islamic azad university khoy 58168-44799, iran scl@jbnu.ac.kr , varmazyar@iaukhoy.ac.ir received september 25, 2019 presented by consuelo mart́ınez accepted january 14, 2020 abstract: if m is a torsion-free module over an integral domain, then we show that for each submodule n of m the envelope em (n) of n in m is an essential extension of n. in particular, if n is divisible then em (n) = n. the last condition says that n is a semiprime submodule of m if n is proper. let m be a module over a ring r such that for any ideals a, b of r, (a ∩ b)m = am ∩ bm. if n is an irreducible and weakly semiprime submodule of m, then we prove that (n :r m) is a prime ideal of r. as a result, we obtain that if p is an irreducible ideal of a ring r such that a2 ⊆ p (a is an ideal of r) ⇒ a ⊆ p, then p is a prime ideal. key words: prime ideal; generalized prime submodule; semiprime submodule; weakly semiprime submodule. ams subject class. (2010): 13c05, 18e40, 13b30, 16d60, 13b25. 1. introduction and preliminaries throughout this paper, all rings are commutative with identity and all modules are unitary. for definitions, examples, of prime, weakly prime, semiprime, and weakly semiprime submodules of r-modules, and relations among them we refer the reader to [8]. in this paper, we do not deal with weakly prime, but we deal with prime, semiprime, and weakly semiprime submodules of r-modules. more precisely, let’s consider the three conditions, for a submodule n of an rmodule m, given as follows: 1. for a ∈ r, m ∈ m with am ∈ n, either a ∈ (n :r m) or m ∈ n. 2. for each a ∈ r, m ∈ m with a2m ∈ n, am ∈ n. 3. for each a ∈ r with a2m ⊆ n, am ⊆ n. @ corresponding author issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.35.1.43 mailto:scl@jbnu.ac.kr mailto:varmazyar@iaukhoy.ac.ir mailto:varmazyar@iaukhoy.ac.ir https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 44 s.c. lee, r. varmazyar a proper submodule n of an r-module m is said to be prime, semiprime, and weakly semiprime if it satisfies the condition (1), the condition (2), and the condition (3), respectively. we can summarize the results as follows: prime ⇒ : semiprime ⇒ : weakly semiprime moreover, if n is a prime submodule of an r-module m then it follows from [11, proposition 2.4] that (n : m) is a prime ideal of r. by [11, proposition 2.4], every prime submodule is semiprime and by [4, theorem 11] there are some semiprime submodules that are not prime. it was shown in [8] that every semiprime submodule is weakly semiprime and the converse is not true in general. the concept of prime number in the ring of integers is generalized to the one of prime ideal of a ring. in turn, the concept of prime ideal of a ring is generalized to the one of prime submodule of a module. several authors have studied the subject for a couple of decades and many results have been given to explore the nature of prime ideals and related concepts. section 2 is devoted to primes and generalized primes. if m is a finitely generated multiplication module over a ring r, then it is shown that for any submodule n of m, radm (n) = √ (n :r m)m (theorem 2.1). let r be a ring. let m be an r-module. assume that m is a multiplication r-module, and that for any ideals a, b of r, (p :m b) ∩ (p :m a) ⊆ p . then we show that p is a generalized prime submodule of an r-module m if and only if (p :r m) is a prime ideal of r (theorem 2.2). in section 3, we concerned about semiprime submodules. let m be a torsion-free module, or a projective module over an integral domain r. then we show that for every submodule n of m, em (n) is an essential extension of n, and in particular, for every divisible submodule n of m, em (n) = n and em (n) is the injective envelope of n (theorem 3.1, corollary 3.1). in section 4, we deal with weakly semiprimes. if m is a module over a ring r such that for any ideals a, b of r, (a ∩ b)m = am ∩ bm and if n is an irreducible and weakly semiprime submodule of m, then we prove that (n :r m) is a prime ideal of r (theorem 4.1). in particular, if p is an irreducible ideal of a ring r such that a2 ⊆ p (a is an ideal of r) ⇒ a ⊆ p, then we show that p is a prime ideal of r (corollary 4.2). a remark on prime ideals 45 notations: let spec(m) be the spectrum of prime submodules of an r-module m. for a submodule n of an r-module m and an ideal a of a ring r, we write (n :r m) = {a ∈ r : am ⊆ n}, (n :m a) = {x ∈ m : rx ∈ n for all r ∈ a}, v (n) = {p ∈ spec(m) : p ⊇ n}, √ a = {a ∈ r : an ∈ a for some positive integer n}. for undefined terminologies and notations, we refer the reader to references [10, 6]. 2. primes and generalized primes this section is dedicated to the relationship between primes and generalized primes. if r is a ring and a is an ideal of r, then it is well-known that the radical√ a of a is an ideal of r (see, for example, [2, p. 33]), and √ a = ∩p∈v (a)p (see, for example, [2, corollary 2.12]). definition 2.1. let n be a submodule of an r-module m. then the radical of n in m is defined to be the intersection of prime submodules of m containing n, and is denoted by radm (n). that is, radm (n) = ∩p∈v (n)p . remark 2.1. for any r-module m, radm (m) = m because v (m) = ∅. definition 2.2. for any r-module m, the radical of m is defined to be the radical of the zero submodule in m and is denoted by rad(m). that is, rad(m) = radm (0) = ∩p∈spec(m)p. an ideal a of r is called a radical ideal if a = √ a (see [2, p. 33]). definition 2.3. a submodule n of an r-module m is called a radical submodule of m if radm (n) = n. every module is the radical submodule of itself. an r-module m is called a multiplication module if for each submodule n of m, n = (n :r m)m (see [1, 3]). 46 s.c. lee, r. varmazyar theorem 2.1. let m be a finitely generated multiplication module over a ring r. then for any submodule n of m, radm (n) = ∩p∈v (n:rm)(pm) = (∩p∈v (n:rm)p)m = √ (n :r m)m. proof. a map ϕ : spec(r/(n :r m)) → spec(m/n) defined by ϕ(p/(n :r m)) = (pm)/n is an order-preserving 1:1 correspondence (see [3]), so that the first equality follows. the second one follows from the fact that m is a multiplication module, and the third one follows from the fact in the first paragraph of this section. corollary 2.1. ([2, corollary 2.12]) if r is a ring and a is an ideal of r, then √ a = ∩p∈v (a)p. let r be a ring, then every prime ideal of r has the condition: a∩b ⊆ p (a,b are ideals of r) ⇒ a ⊆ p or b ⊆ p. however, a prime submodule p of m does not satisfy the following condition in general: a∩b ⊆ p (a,b are submodules of m) ⇒ either a ⊆ p or b ⊆ p. (2.1) an example of this is given below. example 2.1. let r be an integral domain. then r2( = r ⊕ r) is a torsion-free r-module, so that the submodule 〈(0, 0)〉 is a prime submodule. moreover, (r × 〈0〉) ∩ (〈0〉 × r) = 〈(0, 0)〉, but r × 〈0〉 6= 〈(0, 0)〉, and 〈0〉×r 6= 〈(0, 0)〉. this shows that the zero submodule of r2 does not satisfies the condition (2.1). a proper submodule p of an r-module m is called generalized prime if p satisfies condition (2.1). lemma 2.1. assume that m is a multiplication r-module. if p is a prime submodule of an r-module m, then p is a generalized prime submodule of m. a remark on prime ideals 47 proof. assume that a∩b ⊆ p . let a, b be ideals of r such that a = am and b = bm. then (ab)m ⊆ am∩bm = a∩b ⊆ p , so that ab ⊆ (p :r m). notice that (p :r m) is a prime ideal of r. then either a ⊆ (p :r m) or b ⊆ (p :r m). hence either am ⊆ p or bm ⊆ p , that is, either a ⊆ p or b ⊆ p . therefore, p is generalized prime. corollary 2.2. let r be a ring. then every prime ideal of r is generalized prime. not every generalized prime ideal of a ring r is prime. an example of this is given below. example 2.2. let p be a prime number. consider the ring zp2 = {0, 1, 2, . . . , p 2 − 1}. then its only ideals are 〈0〉, 〈p〉, and zp2 . hence it is easy to see that the zero ideal 〈0〉 is not prime, but generalized prime. lemma 2.2. assume that for any ideals a, b of r, (p :m b) ∩ (p :m a) ⊆ p. if p is a generalized prime submodule of an r-module m, then (p :r m) is a prime ideal of r. proof. let ab ⊆ (p :r m). then (ab)m ⊆ p , so that am ⊆ (p :m b) and bm ⊆ (p :m a). by our hypothesis am ∩ bm ⊆ p . since p is generalized prime, either am ⊆ p or bm ⊆ p . so, either a ⊆ (p :r m) or b ⊆ (p :r m). therefore (p :r m) is a prime ideal of r. corollary 2.3. let r be a ring. assume that for any ideals a, b of r, (p :r b) ∩ (p :r a) ⊆ p. if p is a generalized prime ideal of r, then p is a prime ideal of r. let’s summarize the results as follows: theorem 2.2. let r be a ring. let m be an r-module. assume that m is a multiplication r-module, and that for any ideals a, b of r, (p :m b) ∩ (p :m a) ⊆ p. then p is a generalized prime submodule of m if and only if (p :r m) is a prime ideal of r. 48 s.c. lee, r. varmazyar proof. (⇒) : this follows from lemma 2.2. (⇐) : this follows from [3, corollary 2.11] and lemma 2.1. corollary 2.4. let r be a ring. assume that for any ideals a, b of r, (p :r b) ∩ (p :r a) ⊆ p. then p is a prime ideal of r if and only if p is a generalized prime ideal of r. 3. semiprimes in this section, we consider what conditions can be given to an r-module m and to its submodule n, in order for a submodule n of an r-module m to be a semiprime submodule. let m be an r-module, and let n be a submodule of m. then recall that the envelope of n in m is defined by em (n) := 〈 ∪r∈r ( rm ∩ (n :m r) )〉 = 〈 gm (n) 〉 (see [8, p. 3744]). for every submodule n of an r-module m, n ⊆ em (n) ⊆ m. in particular, em (m) = m for every r-module m. lemma 3.1. for every proper submodule n of an r-module m, the following assertions are equivalent: (a) em (n) = n. (b) n is semiprime. (c) for each r ∈ r, m ∈ m with rkm ∈ n for some positive integer k, rm ∈ n. proof. (a) ⇒ (b) : let r ∈ r, m ∈ m with r2m ∈ n. then rm ∈ rm ∩ (n :r r) ⊆∪a∈r(am ∩ (n :r a)) ⊆ em (n) = n. (b) ⇒ (a) : to show that em (n) = n, it suffices to prove that ∪r∈r(rm∩ (n :m r)) ⊆ n. let x ∈ ∪r∈r(rm ∩ (n :m r)). then there exist r ∈ r, m ∈ m such that x = rm and x ∈ (n :r r). so, r2m = rx ∈ n. by assumption, x = rm ∈ n. (b) ⇒ (c) : let r ∈ r and m ∈ m with rkm ∈ n for some positive integer k. if k ≥ 3, then r2(rk−2m) = rkm ∈ n; hence by (b) rk−1m = r(rk−2m) ∈ n. by proceeding this way, we can get r2m ∈ n. hence by (b) again we get rm ∈ n. (c) ⇒ (b) is obvious. a remark on prime ideals 49 example 3.1. consider the ring z of integers. let n > 1. then nz is a semiprime ideal of z if and only if n is a product of distinct prime numbers. for example, 12z is not semiprime (for 22 · 3 ∈ 12z, but 2 · 3 /∈ 12z), but 6z is semiprime. if e is divisible, then not every submodule of e is divisible. for example q is divisible as a z-module but its submodule z is not. if m is a torsion-free module over an integral domain r, then every submodule of m is also torsion-free. it is well-known [10] that every r-module m has an injective envelope e(m), but not necessarily unique. (see [12] for examples). it is fairly well-known that every nonsingular module m over r has the unique injective envelope e(m) (see [7, lemma 2]). every torsion-free module m over an integral domain r is nonsingular, so that its injective envelope e(m) is unique. theorem 3.1. let m be a torsion-free module over an integral domain r. then the following statements are true: (1) for every submodule n of m, em (n) is an essential extension of n. (2) for every divisible submodule n of m, em (n) = n and em (n) is the injective envelope of n. proof. (1) let n be any submodule of m, and let u be a nonzero submodule of em (n). we can take a nonzero element x ∈ u. then there are a1, . . . ,an ∈ r and x1, . . . ,xn ∈ gm (n) such that x = a1x1 + · · ·+ anxn. we may assume that all xi’s are nonzero. for each i ∈ zn, there is ri ∈ r such that xi ∈ rim ∩(n :m ri). moreover, all ri’s are nonzero and rixi ∈ n. since m is torsion-free over an integral domain r, (r1 · · ·rn)x 6= 0 and (r1 · · ·rn)x = (a1r2 · · ·rn)(r1x1) + · · · + (anr1 · · ·rn−1)(rnxn) ∈ u ∩n. hence n is essential in em (n). (2) let n be any divisible submodule of m. then n is torsion-free divisible. hence it follows from [10, proposition 2.7] that n is injective. hence by (1) and [10, theorem 2.17], em (n) = n. moreover, notice that em (n) is also injective. then by (1) em (n) is the injective envelope of n. in theorem 3.1 (2), the condition “divisible” cannot be deleted. to see this let r = z, m = q, and n = z. then it is not difficult to see that em (n) = q 6= n. 50 s.c. lee, r. varmazyar by [5, exercise 9g, section 9, chapter i], every projective module over an integral domain r is torsion-free. corollary 3.1. let m be a projective module over an integral domain r. then (1) and (2) in theorem 3.1 hold. compare the following corollary 3.2 (3) with [10, theorem 2.8]. corollary 3.2. let r be an integral domain. then the following assertions are true: (1) every torsion-free divisible r-module is injective ([10, proposition 2.7]). (2) for every ideal a of r, er(a) is an essential extension of a. (3) let a be an ideal of r. then a is injective if and only if it is divisible. corollary 3.3. let m be a torsion free module over an integral domain r. then every injective submodule n of m is semiprime and em (n) is the injective envelope of n. proof. by [10, proposition 2.6], n is divisible. corollary 3.4. let r be an integral domain. then every divisible ideal of r is semiprime and injective. 4. weakly semiprimes in this section, we will consider weakly semiprimes. lemma 4.1. for each proper submodule n of an r-module m the following assertions are equivalent: (a) n is weakly semiprime. (b) for each ideal a of r with a2m ⊆ n, am ⊆ n. (c) for each ideal a of r with a2 ⊆ (n :r m), a ⊆ (n :r m). (d) (n :r m) is a semiprime ideal of r. proof. (a) ⇒ (b) : let n be any weakly semiprime submodule of m, and let a be any ideal of r with a2m ⊆ n. let x ∈ am. then there exist a1, . . . ,an ∈ a and x1, . . . ,xn ∈ m such that x = a1x1 + · · · + anxn. for a remark on prime ideals 51 each i ∈ zn, a2im ⊆ a 2m ⊆ n. since n is weakly semiprime, aim ⊆ n. in particular, aixi ∈ n. hence x = a1x1 + · · · + anxn ∈ n. this shows that am ⊆ n. the rest of the proof can be proved easily. corollary 4.1. if every proper ideal of a ring r is semiprime, then every proper submodule of a module over the ring r is weakly semiprime. proof. let p be a proper submodule of a module m over the ring r. assume that a is any ideal of r with a2m ⊆ p . then a2 ⊆ (p :r m). moreover, (p :r m) is a proper ideal of r, so by hypothesis (p :r m) is semiprime. hence a ⊆ (p :r m), so that am ⊆ p . by lemma 4.1, p is weakly semiprime. if p is a weakly semiprime submodule of an r-module m, then it is clear that (p :r m) is a proper radical ideal of r, that is, √ (p :r m) = (p :r m). it is known (see [9, corollary 2.7] ) that, over a commutative ring whose proper ideals are semiprime, every proper submodule of a nonzero module is intersection of prime submodules. let r be a dedekind domain, and let a be a nonzero proper ideal of r. it is well-known [2, corollary 11.9] that a is a finite product of (not necessarily distinct) prime ideals. hence a is semiprime if only if a is a finite product of distinct prime ideals. definition 4.1. let p be a submodule of an r-module m. then the residue class r-module m/p is called a weakly semiprime module if p is weakly semiprime. lemma 4.2. for each submodule p of an r-module m, the following assertions are equivalent: (a) the residue class r-module m/p is weakly semiprime. (b) for each ideal a of r with a2(m/p ) = 0, a(m/p ) = 0. (c) for each ideal a of r with am ⊆ (p :m a), am ⊆ p . (d) for each ideal a of r with am + (p :m a) = (p :m a), am ⊆ p . (e) for each ideal a of r with a(m/(p :m a)) = 0, m/(p :m a) = 0. an r-module m is called meet irreducible or simply, irreducible if m 6= 0 and the intersection of two nonzero submodules of m is always nonzero. a 52 s.c. lee, r. varmazyar submodule n of an r-module m is called irreducible if the quotient r-module m/n is irreducible. let m be an r-module. then for any ideals a, b of r (a∩b)m ⊆ am ∩bm. however, the converse is not true in general. if a submodule n of an r-module m is weakly semiprime, then by lemma 4.1, (n :r m) is semiprime. in order (n :r m) to be prime, we need more conditions on n as in the following results: theorem 4.1. let m be a module over a ring r such that for any ideals a, b of r, (a∩b)m = am ∩bm. if n is an irreducible and weakly semiprime submodule of m, then (n :r m) is a prime ideal of r. proof. let us write p = (n :r m). clearly, p 6= r. suppose that p is not prime. then there exist a,b ∈ r such that ab ∈ p, a /∈ p, b /∈ p. then am * n, so that there exists m ∈ m such that am /∈ n. this implies that a(m + n) 6= 0 + n in m/n, so that (p + ra)(m/n) 6= 0 and by the same argument (p + rb)(m/n) 6= 0. hence (p + ra)(m/n) ∩ (p + rb)(m/n) 6= 0. however, ((p + ra) ∩ (p + rb))2(m/n) ⊆ (p + ra)(p + rb)(m/n) ⊆ p(m/n) = 0. since n is weakly semiprime, it follows from lemma 4.2 that ((p + ra) ∩ (p + rb))(m/n) = 0, so that (p + ra)(m/p) ∩ (p + rb)(m/n) = 0. this contradiction shows that p is prime. corollary 4.2. let r be a ring and p be an ideal of r. if p is an irreducible ideal of r such that z2 ∈ p (z ∈ r) implies z ∈ p, then p is a prime ideal of r. the following lemma is the statement just prior to [3, theorem 1.6], but we include and prove here to use it in our classification and to get another result. lemma 4.3. if m is a projective r-module, then (∩λ∈λaλ)m = ∩λ∈λ(aλm) for any nonempy collection of ideals aλ (λ ∈ λ) of r. a remark on prime ideals 53 proof. let f be an r-free module and n be a submodule of f such that f = m ⊕ n. it can be easily proven that (∩λ∈λaλ)f = ∩λ∈λ(aλf) for any nonempy collection of ideals aλ ( λ ∈ λ) of r. this implies that (∩λ∈λaλ)m ⊕ (∩λ∈λaλ)n = ∩λ∈λ(aλ(m ⊕n)) = ∩λ∈λ(aλm ⊕aλn). (4.1) it follows that (∩λ∈λaλ)m = ∩λ∈λ(aλm) and (∩λ∈λaλ)n = ∩λ∈λ(aλn). for otherwise, we can take x ∈∩λ∈λ(aλm)\(∩λ∈λaλ)m and y ∈∩λ∈λ(aλn)\(∩λ∈λaλ)n. then by (4.1) x+y ∈ (∩λ∈λaλ)m⊕(∩λ∈λaλ)n = ∩λ∈λ(aλm⊕aλn), so that for all λ ∈ λ, there exist uλ ∈ aλm and vλ ∈ aλn such that x + y = uλ + vλ. hence x − uλ = vλ − y ∈ m ∩ n = 0, so that x = uλ and y = vλ. hence x ∈∩λ∈λ(aλm) and y ∈∩λ∈λ(aλn). this is a contradiction. if m is a faithful multiplication module, then we can get the same result as in lemma 4.3 (see [3, theorem 1.6]). corollary 4.3. let m be a projective module or a faithful multiplication module over a ring r. if n is an irreducible and weakly semiprime submodule of m, then (n :r m) is a prime ideal of r. as a special case of corollary 4.3, corollary 4.2 holds, as well. in other form, if p is an irreducible ideal of a ring r such that a2 ⊆ p (a is an ideal of r) ⇒ a ⊆ p, then p is a prime ideal of r. acknowledgements the authors appreciate that the referee gave us some remarks, which made us much more logical reasoning toward the draft of the paper. references [1] a. barnard, multiplication modules, j. algebra 71 (1) (1981), 174 – 178. [2] d. eisenbud, “ commutative algebra ”, graduate texts in mathematics 150, springer-verlag, new york, 1995. [3] z. el-bast, p.f. smith, multiplication modules, comm. algebra 16 (4) (1988), 755 – 779. 54 s.c. lee, r. varmazyar [4] j. jenkins, p.f. smith, on the prime radical of a module over a commutative ring, comm. algebra 20 (12) (1992), 3593 – 3602. [5] s.t. hu, “ introduction to homological algebra ”, holden-day, inc., san francisco-london-amsterdam, 1968. [6] t.y. lam, “ lectures on modules and rings ”, graduate texts in mathematics 189, springer-verlag, new york, 1999. [7] s.c. lee, d.s. lee, direct sums of indecomposable injective modules, bull. austral. math. soc. 62 (1) (2000), 57 – 66. [8] s.c. lee, r. varmazyar, semiprime submodule of a module and related concepts, j. algebra appl. 18 (8) (2019), 1950147, 11 pp. [9] r.l. mccasland, p.f. smith, on isolated submodules, comm. algebra 34 (8) (2006), 2977 – 2988. [10] d.w. sharpe, p. vámos, “ injective modules ”, cambridge tracts in mathematics and mathematical physics 62, cambridge university press, londonnew york, 1972. [11] h.a. tavallaee, r. varmazyar, semi-radicals of submodules in modules, i. j. engineering science 19 (1) (2008), 21 – 27. [12] f. wang, h. kim, two generalizations of projective modules and their applications, j. pure appl. algebra 219 (6) (2015), 2099 – 2123. introduction and preliminaries primes and generalized primes semiprimes weakly semiprimes � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 1 (2020), 55 – 67 doi:10.17398/2605-5686.35.1.55 available online october 25, 2019 on the projectivity of finitely generated flat modules a. tarizadeh department of mathematics, faculty of basic sciences, university of maragheh p.o. box 55136–553, maragheh, iran ebulfez1978@gmail.com received october 21, 2018 (revised july 18, 2019) presented by juan b. sancho accepted september 9, 2019 abstract: in this paper, the projectivity of a finitely generated flat module of a commutative ring is studied through its exterior powers and invariant factors and then various new results are obtained. specially, the related results of endo, vasconcelos, wiegand, cox-rush and puninski-rothmaler on the projectivity of finitely generated flat modules are generalized. key words: exterior power, invariant factor, projectivity, s-ring, specialization cone, generalization cone. ams subject class. (2010): 13c10, 19a13, 13c11, 13e99. 1. introduction the main purpose of the present paper is to investigate the projectivity of finitely generated flat modules of a commutative ring. this topic has been the main subject of many articles in the literature over the years and it is still of current interest, see e.g. [3, 5, 6, 7, 11, 12, 13]. note that in general there are finitely generated flat modules which are not projective, see example 2.9, also see [4, tag 00ny] as another example (note that our example is so simple than the cited one; it is also applicable for other purposes). the main motivation to investigate the projectivity of finitely generated flat modules essentially originates from the fact that every finitely generated flat module over a local ring is free, see theorem 2.2. this result together with theorem 2.8 play a major role in this paper. in this paper, the projectivity of a finitely generated flat module of a commutative ring is studied through its exterior powers and invariant factors and then we obtain various new and interesting results. one of the features of this study is that some major results in the literature on the projectivity of finitely generated flat modules are generalized. specially, theorem 3.1 generalizes [5, theorem 1], theorem 3.2 improves a little [12, theorem 2.1], issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.35.1.55 mailto:ebulfez1978@gmail.com https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 56 a. tarizadeh theorem 3.3 generalizes [13, theorem 2], [3, proposition 2.3], it also generalizes [11, proposition 5.5 and corollary 5.6] in the commutative case, and finally corollary 3.10 generalizes [13, theorem 2]. in fact, theorem 3.3 can be viewed as a generalization of all of the above mentioned results. this theorem is one of the main novel contributions of this paper and has many non-trivial consequences. theorems 3.16, 3.18 and 3.22 are another interesting results of this paper. for reading the present paper having a reasonable knowledge on the exterior powers of a module is necessary. in this paper, all rings are commutative. 2. preliminaries we need the following material in the next section. lemma 2.1. let m be a finitely generated r-module, let i = annr(m) and let s be a multiplicative subset of r. then s−1i = anns−1r(s −1m). proof. it is well known and easy. a projective r-module is also called r-projective. we also use a similar terminology for free and flat modules. unlike the kaplansky theorem [8] which states that every projective module over a local ring is free, but this is not true for flat modules. for example, the field of rationals q is zp-flat but it is not zp-free where p is a non-zero prime ideal of the ring of integers z. in spite of this, in the finite case we have the following interesting result which can be considered it as the analogue of the kaplansky theorem for flat modules. theorem 2.2. every finitely generated flat module over a local ring is free. proof. see [9, theorem 7.10] or [4, tag 00nz]. lemma 2.3. let φ : r → s be a morphism of rings and m a finitely generated flat r-module. then anns(m ⊗r s) = annr(m)s. proof. it is a local property implied by lemma 2.1 and theorem 2.2. let m be an r-module, n ≥ 2 a natural number and let jn be the rsubmodule of m⊗n generated by the collection of pure tensors of the form projectivity of finitely generated flat modules 57 x1 ⊗ . . .⊗xn with xi = xj for some i 6= j. the quotient r-module m⊗n/jn is called the n-th exterior power of m and it is denoted by λnr(m) or simply by λn(m) if there is no confusion on the base ring r. write λ0(m) = r and λ1(m) = m. the canonical r-multilinear map η : mn → λn(m) given by (x1, . . . ,xn) 7→ x1 ∧ . . . ∧ xn := x1 ⊗ . . . ⊗ xn + jn is clearly alternative. the r-module λn(m) together with the map η satisfy in the following universal property. for each alternative r-multilinear map φ : mn → n then there exists a unique morphism of r-modules φ′ : λn(m) → n such that φ = φ′ ◦η. if m is a finitely generated r-module, then λn(m) is a finitely generated r-module. if r → s is a morphism of rings and m is an r-module, then λn(m) ⊗r s as s-module is canonically isomorphic to λns(m ⊗r s). it is also well known that if m is a projective (resp. flat) r-module, then for each natural number n, λn(m) is a projective (resp. flat) r-module. if m is a r-module then the n-th invariant factor of m, denoted by in(m), is defined as the annihilator of the n-th exterior power of m, i.e., in(m) = annr ( λn(m) ) . remark 2.4. if m is a finitely generated flat r-module then theorem 2.2 leads us to a function ψ : spec r → n = {0, 1, 2, . . .} which is defined as p 7→ rankrp (mp). it is called the rank map of m. it is obvious that the rank map is continuous if and only if it is locally constant (i.e., for each prime ideal p of r then there exists an open neighborhood u ⊆ spec(r) of that point such that rankrq (mq) = rankrp (mp) for all q ∈ u). it is well known that supp ( λn(m) ) = {p ∈ spec(r) : rankrp (mp) ≥ n}. if φ : r → s is a morphism of rings then the induced map spec(s) → spec(r) given by p 7→ φ−1(p) is denoted by φ∗. the jacobson radical of a ring r is denoted by j(r). an ideal i of a ring r is called a pure ideal if the canonical ring map r → r/i is a flat ring map. pure ideals are quite interesting and play an important role in commutative and non-commutative algebra (for instance, in classifying gelfand rings and their dual rings). an ideal i of a ring r is a pure ideal if and only if ip = 0 or ip = rp, for each prime ideal p of r. hence, i is an idempotent ideal. theorem 2.5. an ideal i of a ring r is a pure ideal if and only if ann(f)+ i = r for all f ∈ i. proof. it is a local property implied by theorem 2.2. 58 a. tarizadeh corollary 2.6. let m be a finitely generated flat r-module with annihilator i. then i is a pure ideal. proof. it is a local property implied by theorem 2.2 and theorem 2.5. lemma 2.7. the annihilator of a finitely generated projective module is generated by an idempotent element. proof. it is deduced from [2, p. 132, proposition 3.1]. the following result is well known, see [1, chap. ii, §5.2, théorème 1], [4, tag 00nx] and [12, proposition 1.3]. as a contribution, we provide a new proof of this result. theorem 2.8. let m be a finitely generated flat r-module. then the following are equivalent: (i) m is r-projective. (ii) the invariant factors of m are finitely generated ideals. (iii) the rank map of m is locally constant. proof. (i) ⇒ (ii). it is well-known that λn(m) is a finitely generated projective r-module and so by lemma 2.7, in(m) is a principal ideal. (ii) ⇒ (iii). it suffices to show that the rank map of m is zariski continuous. by corollary 2.6, in(m) is an idempotent ideal. thus there exists some a ∈ in(m) such that (1 − a)in(m) = 0. clearly a = a2 and in(m) = ra. by remark 2.4, ψ −1({n}) = supp n ∩ ( spec(r) \ supp n ′ ) where n = λn(m) and n ′ = λn+1(m). but supp n = spec(r) \v (1 −a). moreover, supp n ′ = v ( in+1(m) ) since n ′ is a finitely generated r-module. therefore ψ−1({n}) is an open subset of spec r. (iii) ⇒ (i). apply theorem 2.2 and [4, tag 00nx]. in the following we give an example of a finitely generated flat module which is not projective. it should be noted that finding such examples of modules is not as easy as one may think at first. example 2.9. let r = ∏ i≥1 a be an infinite product of copies of a nonzero ring a and let i = ⊕ i≥1 a which is an ideal of r. if f = (fi) ∈ i, then there exists a finite subset d of {1, 2, 3, . . .} such that fi = 0 for all projectivity of finitely generated flat modules 59 i ∈{1, 2, 3, . . .}\d. clearly f = fg where g = (gi) ∈ i with gi = 1 for all i ∈ d and gi = 0 for all i ∈{1, 2, 3, . . .}\d. hence, i is a pure ideal of r (i.e., r/i is a finitely generated flat r-module). if r/i is r-projective then by lemma 2.7, there exists a sequence e = (ei) ∈ r such that i = re. thus there exists a finite subset e of {1, 2, 3, . . .} such that ei = 0 for all i ∈ {1, 2, 3, . . .}\e. clearly {1, 2, 3, . . .} \ e 6= ∅. pick some k ∈ {1, 2, 3, . . .}\ e. there is some r = (ri) ∈ r such that (δi,k)i≥1 = re where δi,k is the kronecker delta. in particular, 1a = rkek = rk0a = 0a. this is a contradiction. therefore r/i is not r-projective. 3. projectivity: main results throughout this section, m is a finitely generated flat r-module. the following technical result generalizes [5, theorem 1]. theorem 3.1. let r → s be an injective ring map. then m is r-projective if and only if m ⊗r s is s-projective. proof. let m ⊗r s be s-projective. without loss of generality, we may assume that r ⊆ s is an extension of rings. first we shall prove that i = annr(m) is a principal ideal. by lemma 2.3, is = l where l = anns(n) and n = m ⊗r s. hence by lemma 2.7, there is an idempotent e ∈ s such that is = se. let j = s(1 − e) ∩r. clearly ij = 0. we have i + j = r. if not, then there exists a prime ideal p of r such that i + j ⊆ p. thus, by theorem 2.2, ip = 0. therefore the extension of is under the canonical map s → s⊗r rp is zero. thus there exists an element s ∈ r\p such that se = 0 and so s = s(1 − e). hence s ∈ j. but this is a contradiction. therefore i + j = r. it follows that there is an element c ∈ i such that c = c2 and i = rc. now, let n ≥ 1. we have λn(m) is a finitely generated flat r-module. moreover, λn(m) ⊗r s is s-projective because it is canonically isomorphic to λns(m ⊗r s). thus, by what we have proved above, in(m) is a principal ideal. hence, by theorem 2.8, m is r-projective. the reverse is easy and well known. the following result is also technical and generalizes [12, theorem 2.1]. theorem 3.2. let j be an ideal of r which is contained in the jacobson radical of r. if m/jm is r/j-projective, then m is r-projective. 60 a. tarizadeh proof. first we shall prove that i = annr(m) is a principal ideal. by lemma 2.3, l = i + j where l = annr(m/jm). also, by lemma 2.7, annr/j(m/jm) = l/j is a principal ideal. this implies that i = rx + i∩j for some x ∈ r since l/j = (i + j)/j is canonically isomorphic to i/(i ∩j). i = rx: let m be a maximal ideal of r. by theorem 2.2, im is either the whole localization or the zero ideal. if im = 0 then (rx)m = 0 since rx ⊆ i. if im = rm then i is not contained in m. thus rx is also not contained in m since i ∩ j ⊆ j ⊆ m. hence (rx)m = rm. therefore i = rx. now let n ≥ 1 and let n = λn(m). then n/jn is r/j-projective, because n/jn as r/j-module is canonically isomorphic to λn r/j (m/jm) and λn r/j (m/jm) is r/j-projective. but n is a finitely generated flat r-module. therefore, by what we have proved above, in(m) = annr(n) is a principal ideal. thus the invariant factors of m are finitely generated ideals and so by theorem 2.8, m is r-projective. motivated by the grothendieck’s relative point of view, then we obtain the following result which (beside theorems 2.2 and 2.8) is one of the most powerful results on the projectivity of finitely generated flat modules. theorem 3.3. let φ : r → s be a ring map whose kernel is contained in the jacobson radical of r. then m is r-projective if and only if m ⊗r s is s-projective. proof. let m⊗r s be s-projective. clearly m/jm is a finitely generated flat r/j-module and m/jm ⊗r/j s ' m ⊗r s is s-projective where j = ker φ. moreover r/j can be viewed as a subring of s via φ. therefore, by theorem 3.1, m/jm is r/j-projective. then by applying theorem 3.2, we get that m is r-projective. the reverse is easy and well known. the above theorem has many consequences. recall that a ring r is called an s-ring (“s” referes to sakhajev) if every finitely generated flat r-module is r-projective. corollary 3.4. let φ : r → s be a ring map whose kernel is contained in the jacobson radical of r. if s is an s-ring then r is as well. proof. if m is a finitely generated flat r-module then m ⊗r s is a finitely generated flat s-module and so, by the hypothesis, it is s-projective. therefore by theorem 3.3, m is r-projective. projectivity of finitely generated flat modules 61 corollary 3.5. if there exists a prime ideal p of a ring r such that the kernel of the canonical ring map r → rp is contained in the jacobson radical of r, then r is an s-ring. proof. by theorem 2.2, every local ring is an s-ring, then apply corollary 3.4. corollary 3.6. if there exists a prime ideal p of a ring r such that the canonical ring map r → rp is injective, then r is an s-ring. proof. it is an immediate consequence of corollary 3.5. corollary 3.7. every integral domain is an s-ring. proof. it is an immediate consequence of corollary 3.6. corollary 3.8. if the jacobson radical of a ring r contains a prime ideal p of r, then r is an s-ring. proof. clearly ker π ⊆ p where π : r → rp is the canonical ring map. therefore by corollary 3.5, r is an s-ring. another proof. by corollary 3.7, r/p is an s-ring. thus by corollary 3.4, r is an s-ring. remark 3.9. let s be a subset of a ring r. the polynomial ring r[xs : s ∈ s] modulo i is denoted by s(−1)r where the ideal i is generated by elements of the form sx2s −xs and s2xs − s with s ∈ s. we call s(−1)r the pointwise localization of r with respect to s. amongst them, the pointwise localization of r with respect to itself, namely r(−1)r, has more interesting properties; for further information please consult with [10]. note that wiegand [13] utilizes the notation r̂ instead of r(−1)r. clearly η(s) = η(s)2(xs + i) and xs + i = η(s)(xs + i) 2 where η : r → s(−1)r is the canonical map and the pair (s(−1)r,η) satisfies in the following universal property: “for each such pair (a,φ), i.e. φ : r → a is a ring map and for each s ∈ s there is some c ∈ a such that φ(s) = φ(s)2c and c = φ(s)c2, then there exists a unique ring map ψ : s(−1)r → a such that φ = ψ ◦η”. now let p be a prime ideal of r and consider the canonical map π : r → κ(p) where κ(p) is the residue field of r at p. by the above universal property, there is a (unique) ring map ψ : s(−1)r → κ(p) such that π = ψ ◦ η. thus η induces a surjection between the corresponding spectra. this, in particular, implies that the kernel 62 a. tarizadeh of η is contained in the nilradical of r. using this, then the following result generalizes [13, theorem 2]. corollary 3.10. if there exists a subset s of r such that m ⊗r s(−1)r is s(−1)r-projective, then m is r-projective. proof. it is an immediate consequence of theorem 3.3. in what follows we get some new results essentially based on the referee’s excellent comments. corollary 3.11. let φ : r → s be a morphism of rings such that the induced map φ∗ has the dense image. then m is r-projective iff m ⊗r s is s-projective. proof. it is an immediate consequence of theorem 3.3, because from equality im φ∗ = spec(r) we get that the kernel of φ is contained in j(r), the jacobson radical of r. lemma 3.12. let { rk,mk } k be a family of local rings. then the kernel of the canonical ring map π : r = ∏ k rk → ∏ k rk/mk is the jacobson radical of r. proof. if the sequence x = (xk) ∈ r is a member of ker π then xk ∈ mk for all k. to prove x ∈ j(r) it suffices to show 1 + xy is invertible in r for all y = (yk) ∈ r. for each k, there exists some zk ∈ rk such that (1 + xkyk)zk = 1 because rk is a local ring. it follows that (1 + xy)z = 1 where z = (zk). conversely, let x ∈ j(r). for each k, then mk := π−1k (mk) is a maximal ideal of r because the ring map r/mk → rk/mk induced by the canonical projection πk : r → rk is an isomorphism. therefore x ∈ mk for all k. corollary 3.13. let x ⊆ spec(r) be a subset. then the following statements are equivalent: (i) m ⊗r s is s-projective where s = ∏ p∈x κ(p). (ii) m ⊗r s′ is s′-projective where s′ = ∏ p∈x r/p. (iii) m ⊗r s′′ is s′′-projective where s′′ = ∏ p∈x rp. projectivity of finitely generated flat modules 63 if moreover ⋂ p∈x p ⊆ j(r), then the above statements are equivalent with the following: (iv) m is r-projective. proof. it is an immediate consequence of theorem 3.3. the subsets min(r) and max(r) are typical examples which satisfy the hypothesis of corollary 3.13. corollary 3.14. consider the following commutative diagram of rings r // �� s′ �� s φ // t in which the kernel of φ is contained in the jacobson radical of s. if m ⊗r s′ is s′-projective, then m ⊗r s is s-projective. proof. if m ⊗r s′ is s′-projective then it is easy to see that (m ⊗r s′) ⊗s′ t ' m ⊗r t ' (m ⊗r s) ⊗s t is t-projective. but m ⊗r s is a finitely generated flat s-module. therefore by theorem 3.3, m ⊗r s is s-projective. definition 3.15. if x is a subset of spec(r) then we call ⋃ p∈x v (p) the specialization cone of x and it is denoted by xs. dually, we call ⋃ p∈x λ(p) the generalization cone of x and it is denoted by xg where λ(p) = {q ∈ spec(r) : q ⊆ p}. theorem 3.16. let x ⊆ spec(r) be a subset. put s := ∏ p∈x κ(p) and s′ := ∏ p∈xs κ(p). then m ⊗r s is s-projective iff m ⊗r s′ is s′-projective. proof. consider the canonical injective ring map t = ∏ p∈x r/p → s. then by theorem 3.1, m ⊗r s ' (m ⊗r t) ⊗t s is s-projective iff m ⊗r t is t-projective. by the axiom of choice, we obtain a function σ : xs → x such 64 a. tarizadeh that σ(p) ⊆ p for all p ∈ xs and σ(p) = p for all p ∈ x. for each p ∈ x, consider the canonical injective ring map r/p → ∏ q∈σ−1(p) r/q. then we get the canonical injective ring map t → t ′ = ∏ p∈xs r/p. again by theorem 3.1, m ⊗r t is t-projective iff m ⊗r t ′ is t ′-projective. similarly above, by applying theorem 3.1 to the canonical injective ring map t ′ → s′, we get that m ⊗r t ′ is t ′-projective iff m ⊗r s′ is s′-projective. corollary 3.17. let p be a prime ideal of a ring r and put s :=∏ q∈v (p) κ(q). then m ⊗r s is s-projective. proof. it is an immediate consequence of theorem 3.16. theorem 3.18. let x ⊆ spec(r) be a subset. put s := ∏ p∈x κ(p) and s′ := ∏ p∈xg κ(p). then m ⊗r s is s-projective iff m ⊗r s′ is s′-projective. proof. the kernel of the canonical ring map t = ∏ p∈x rp → s is the jacobson radical of t , see lemma 3.12. therefore by theorem 3.3, m⊗rs ' (m ⊗r t) ⊗t s is s-projective if and only if m ⊗r t is t-projective. by the axiom of choice, there exists a function σ : xg → x such that p ⊆ σ(p) for all p ∈ xg and σ(p) = p for all p ∈ x. for each p ∈ x, consider the canonical injective ring map rp → ∏ q∈σ−1(p) rq. then we get the canonical injective ring map t → t ′ = ∏ p∈xg rp. thus by theorem 3.1, m ⊗r t is t-projective iff m ⊗r t ′ is t ′-projective. again by lemma 3.12, the kernel of the canonical ring map t ′ → s′ is contained in the jacobson radical of t ′. hence by theorem 3.3, m ⊗r t ′ is t ′-projective if and only if m ⊗r s′ is s′-projective. corollary 3.19. let p be a prime ideal of a ring r and put s :=∏ q∈λ(p) κ(q). then m ⊗r s is s-projective. proof. it is an immediate consequence of theorem 3.18. projectivity of finitely generated flat modules 65 corollary 3.20. let x ⊆ y ⊆ spec(r) be two subsets such that either y ⊆ xs or y ⊆ xg. put s := ∏ p∈x κ(p) and s′ := ∏ p∈y κ(p). then m ⊗r s is s-projective iff m ⊗r s′ is s′-projective. proof. if y ⊆ xs then xs = ys, so apply theorem 3.16 in this case. but if y ⊆ xg then xg = yg, and so apply theorem 3.18. lemma 3.21. let φ : r → s be a morphism of rings and put s′ :=∏ p∈im φ∗ κ(p). then m ⊗r s is s-projective if and only if m ⊗r s′ is s′-projective. proof. if p ∈ im φ∗ and q ∈ (φ∗)−1(p) then we have the canonical ring map κ(p) → κ(q) which is injective since every ring map from a field into a non-zero ring is injective. then we get the canonical injective ring map κ(p) → ∏ q∈(φ∗)−1(p) κ(q). so we get the canonical injective ring map s′ → t =∏ q∈spec(s) κ(q) which fits in the following commutative diagram r φ // �� s �� s′ // t where the unnamed arrows are the canonical morphisms. it is easy to see that the kernel of the canonical morphism s → t is the nilradical of s which is contained in the jacobson radical of s. then the assertion is deduced by twice using of corollary 3.14. given a subset x ⊆ spec(r), denote x(1) := (xs)g and x(1) := (xg)s, and inductively x(n) := (x(n−1))(1) and x(n) := (x(n−1))(1). note that in general, x(n) 6= x(n). for example, if x = {2z} ⊆ spec(z) then x(1) = {0, 2z} but x(1) = spec(z). theorem 3.22. let φ : r → s be a morphism of rings and x = im φ∗. assume there exists some n ≥ 1 such that ⋂ p∈x(n) p is contained in the jacobson radical of r. if m ⊗r s is s-projective then m is r-projective. 66 a. tarizadeh proof. by lemma 3.21, m ⊗r s is s-projective iff m ⊗r s′ is s′projective where s′ := ∏ p∈x κ(p). by the successive applications of theorems 3.16 and 3.18, eventually after finite times we get that m ⊗r s′ is s′-projective iff m ⊗r t is t-projective where t = ∏ p∈x(n) κ(p). but the kernel of the canonical ring map r → t is equal to ⋂ p∈x(n) p. thus by theorem 3.3, m is r-projective. theorem 3.23. let φ : r → s be a morphism of rings and x = im φ∗. suppose there exists some n ≥ 1 such that ⋂ p∈x(n) p is contained in the jacobson radical of r. if m ⊗r s is s-projective, then m is r-projective. proof. it is proven exactly like theorem 3.22. acknowledgements the author would like to give sincere thanks to the referee for very careful reading of the paper and for his/her very valuable and excellent suggestions and comments which greatly improved the paper. i would also like to thank professors wolmer vasconcelos and jesús m.f. castillo for various scientific correspondences during the writing of this paper. references [1] n. bourbaki, “algèbre commutative”, chapitres 1 à 4, springer, berlin, 2006. [2] h. cartan, s. eilenberg, “homological algebra”, princeton university press, princeton, n. j., 1956. [3] s.h. cox jr., r.l. pendleton, rings for which certain flat modules are projective, trans. amer. math. soc. 150 (1970), 139 – 156. [4] a.j. de jong et al., stacks project. http://stacks.math.columbia.edu [5] s. endo, on flat modules over commutative rings, j. math. soc. japan 14 (3) (1962), 284 – 291. [6] a. facchini et al., finitely generated flat modules and a characterization of semiperfect rings, comm. algebra 31 (9) (2003), 4195 – 4214. [7] s. jondrup, on finitely generated flat modules, math. scand. 26 (1970), 233 – 240. [8] i. kaplansky, projective modules, ann. of math. 68 (1958), 372 – 377. [9] h. matsumura, “commutative ring theory”, cambridge university press, cambridge, 1989. http://stacks.math.columbia.edu. projectivity of finitely generated flat modules 67 [10] j.p. olivier, anneaux absolument plats universels et épimorphismes à buts réduits, in “séminaire samuel. algèbre commutative”, tomme 2 (6) (19671968), 1 – 12. [11] g. puninski, p. rothmaler, when every finitely generated flat module is projective, j. algebra 277 (2004), 542 – 558. [12] w. vasconcelos, on finitely generated flat modules, trans. amer. math. soc. 138 (1969), 505 – 512. [13] r. wiegand, globalization theorems for locally finitely generated modules, pacific j. math. 39 (1) (1971), 269 – 274. introduction preliminaries projectivity: main results � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 1 (2020), 21 – 34 doi:10.17398/2605-5686.35.1.21 available online october 16, 2019 identities in the spirit of euler a. sofo college of engineering and science, victoria university p. o. box 14428, melbourne city, victoria 8001, australia anthony.sofo@vu.edu.au received march 26, 2019 presented by jesús m.f. castillo accepted september 17, 2019 abstract: in this paper we develop new identities in the spirit of euler. we shall investigate and report on new euler identities of weight p+ 2, for p an odd integer, but with a non unitary argument of the harmonic numbers. some examples of these euler identities will be given in terms of riemann zeta values, dirichlet values and other special functions. key words: polylogarithm function, recurrence relations, euler sums, zeta functions, dirichlet functions, multiple zeta values. ams subject class. (2010): 11m06, 11m32, 33b15. 1. introduction in a previous paper, [16] we investigated families of integrals, where the integrand is the product of an inverse trigonometric or inverse hyperbolic trigonometric and the polylogarithmic function, j(a,δ,p,m) = ∫ 1 0 ym−1f(y) lip ( δy2am ) dy , for a ∈ r+, δ = ±1, p ∈ n, m ∈ r+ and where f(y) = arctan (ym) or tanh−1 (ym). it was demonstrated that integrals of products of inverse trigonometric and polylogarithmic functions can be associated with euler sums. it is well known that integrals with polylogarithmic integrands can be associated with euler sums. therefore in the spirit of euler we shall investigate integrals of the type i (a,δ,p,q) = ∫ 1 0 ln x x lip(x) liq (δx a) dx (1.1) for a ∈ r+, δ = ±1, p ∈ n, q ∈ n. some examples are highlighted, almost none of which are amenable to a computer mathematical package. we shall issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.35.1.21 mailto:anthony.sofo@vu.edu.au https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 22 a. sofo also develop new euler identities for sums of the type ∞∑ n=1 h (2) n 2 np , ∞∑ n=1 (−1)n+1 h(2)n 2 np , ∞∑ n=1 h (2) n−1 2 (2n− 1)p (1.2) and again, some examples are highlighted, almost none of which are amenable to a computer mathematical package. this work also extends the results given in [7], where the author examined integrals with positive arguments of the polylogarithm. devoto and duke [4] also list many identities of lower order polylogarithmic integrals and their relations to euler sums. some other important sources of information on polylogarithm functions are the works of [9], [10], and [16]. the famous euler identity [5], for unitary argument of the harmonic numbers, states eu (m) = ∞∑ n=1 hn nm = (m 2 + 1 ) ζ (m + 1) − 1 2 m−2∑ j=1 ζ(m− j)ζ(j + 1) . (1.3) the famous euler identity was further extended in the work of [1]. relatively recently multiple zeta values (mzvs) were studied and developed by [8], [21] and others, for example, [11]. mzvs are defined by ζ (i1, i2, . . . , ik) = ∑ n1>n2···>nk≥1 1 ni11 n i2 2 . . .n ik k for positive integers ik and i1 > 1 with weight ∑ ik and length or depth k. for arbitrary p ∈ n and q ≥ 2 the euler sum sp,q = ∞∑ n=1 h (p) n nq (1.4) is readily expressible in terms of mzvs, that is, sp,q+sq,p = ζ(p)ζ(q)+ζ(p+q). euler developed many relations, including ζ(2, 1) = ζ(3) , s2,3 = 3ζ(2)ζ(3) − 9 2 ζ(5) . integrals of polylogarithms 23 it appears that at weight eight, s2,6 cannot be reduced to zeta values and their products. we also note that we may define alternating mzvs with signs in the numerator as ζ ( i1, i2, . . . , ik ) = ∑ n1>n2>···>nk≥1 (−1)n1+n2+···+nk ni11 n i2 2 . . .n ik k . therefore an alternating mzv converges unless the first entry is an unbanned one, and we also have ζ ( 1 ) = − ln 2 and ζ (n) = η(n) = ( 1 − 21−n ) ζ(n) for n ≥ 2, so that, for example ζ ( 3, 1, 2 ) = ∑ n1>n2>n3≥1 (−1)n1+n3 n31n2n 2 3 . for arbitrary integer weight p ≥ 1, q ≥ 1 we shall define alternating euler sums as, s (p,q) := ∞∑ n =1 (−1)n+1 h(p)n nq . (1.5) there are some special cases where the linear euler sum (1.4) is reducible to zeta values. for odd weight w = (p + q) we have, bw (p,q) = ∞∑ n=1 h (p) n nq = 1 2 ( 1 + (−1)p+1 ) ζ(p)ζ(q) + (−1)p [ p2 ]∑ j=1 ( p + q − 2j − 1 p− 1 ) ζ(p + q − 2j)ζ(2j) (1.6) + (−1)p [ p2 ]∑ j=1 ( p + q − 2j − 1 q − 1 ) ζ(p + q − 2j)ζ(2j) + ζ(p + q) 2 ( 1 + (−1)p+1 ( p + q − 1 p ) + (−1)p+1 ( p + q − 1 q )) , where [z] is the integer part of z. for alternating euler sums and specified odd weights we have some particular identities. sitaramachandra rao, [12] gave the identity, for s(p,q), when p = 1 and for odd weight 1 + q as, 2s(1,q) = (1 + q)η(1 + q) − ζ(1 + q) − 2 q 2 −1∑ j=1 η(2j) ζ(1 + q − 2j) (1.7) 24 a. sofo and in another special case, gave the integral s(1, 1 + 2q) = 1 (2q)! ∫ 1 0 ln2q(x) ln(1 + x) x(1 + x) dx. in the case where p and q are both positive integers and p+q is an odd integer, flajolet and salvy [6] gave the identity: 2s(p,q) = (1 − (−1)p) ζ(p)η(q) + 2(−1)p ∑ i+2k=q ( p + i− 1 p− 1 ) ζ(p + i) η(2k) (1.8) + η(p + q) − 2 ∑ j+2k=p ( q + j − 1 q − 1 ) (−1)jη(q + j) η(2k) , where η(0) = 1 2 , η(1) = ln 2, ζ(1) = 0, and ζ(0) = −1 2 in accordance with the analytic continuation of the riemann zeta function. we define the alternating zeta function (or dirichlet eta function) η(z) as η(z) := ∞∑ n=1 (−1)n+1 nz = ( 1 − 21−z ) ζ(z) . the following euler identities for harmonic numbers at half integer values have been given in [19]. lemma 1. for δ = ±1, a ∈ r+, r ∈ n and m a positive odd integer, w(a,δ,m,r) = ∑ n≥1 δn+1h (r) an nm then w ( 1 2 , 1,m, 1 ) = eu (m) + (−1)m+1s(1,m) + m−1∑ k=2 (−1)m−kζ(k)η(m + 1 −k) , (1.9) and w ( 1 2 ,−1,m, 1 ) = ( 1 − 21−m ) eu (m) + (−1)m+1s(1,m) + m−1∑ k=2 (−1)m−kζ(k)η(m + 1 −k) . (1.10) integrals of polylogarithms 25 therefore the main aim of this paper is to develop new euler identities for the sums (1.2) and represent the solution of the integral (1.1), in terms of special functions, for various values of the parameters (a,δ,p,q). first we define some special functions that we will encounter in the body of this paper. the lerch transcendent, φ(z,t,a) = ∞∑ m=0 zm (m + a)t is defined for |z| < 1 and <(a) > 0 and satisfies the recurrence φ(z,t,a) = z φ(z,t,a + 1) + a−t. the lerch transcendent generalizes the hurwitz zeta function at z = 1, φ(1, t,a) = ∞∑ m=0 1 (m + a)t and the polylogarithm, or de-jonquière’s function, when a = 1, lit(z) := ∞∑ m=1 zm mt , t ∈ c when |z| < 1 ; <(t) > 1 when |z| = 1 . let hn = n∑ r=1 1 r = ∫ 1 0 1 − tn 1 − t dt = γ + ψ(n + 1) = ∞∑ j=1 n j(j + n) , h0 := 0 be the nth harmonic number, where γ denotes the euler-mascheroni constant, h (m) n = ∑n r=1 1 rm is the mth order harmonic number and ψ(z) is the digamma (or psi) function defined by ψ(z) := d dz {log γ(z)} = γ′(z) γ(z) and ψ(1 + z) = ψ(z) + 1 z , moreover, ψ(z) = −γ + ∞∑ n=0 ( 1 n + 1 − 1 n + z ) . the polygamma function ψ(k)(z) = dk dzk {ψ(z)} = (−1)k+1k! ∞∑ r=0 1 (r + z)k+1 26 a. sofo and has the recurrence ψ(k)(z + 1) = ψ(k)(z) + (−1)kk! zk+1 . the connection of the polygamma function with harmonic numbers is, h(α+1)z = ζ(α + 1) + (−1)α α! ψ(α)(z + 1) , z 6= {−1,−2,−3, . . .} , (1.11) and the multiplication formula is ψ(k)(pz) = δm,0 ln p + 1 pk+1 p−1∑ j=0 ψ(k) ( z + j p ) (1.12) for p a positive integer and δp,k is the kronecker delta. the work in this paper also extends the results of [7], [20]. other works including, [2], [3], [13], [14], [15], [17], and [18] cite many identities of polylogarithmic integrals and euler sums. 2. integral identities and euler sums theorem 1. for a ∈ r+, δ = {−1, 1}, p, q, positive integers, then i(a,δ,p,q) = ∫ 1 0 ln x lip(x) liq (δx a) x dx = (−1)p+1 ap ζ(2) lip+q(δ) (2.1) + (−1)pp ap+1 ∞∑ n=1 δn han np+q+1 + (−1)p ap ∞∑ n=1 δn h (2) an np+q − p∑ k=2 (−1)p−k(p + 1 −k) ap+2−k ζ(k) lip+q+2−k(δ) . where lip+q (δ) is the polylogarithm, han and h (2) an are the harmonic numbers. proof. by the definition of the polylogarithmic function we have i(a,δ,p,q) = ∞∑ n=1 δn nq ∞∑ j=1 1 jp ∫ 1 0 xj+an−1 ln x dx = − ∞∑ n=1 δn nq ∞∑ j=1 1 jp(j + an)2 integrals of polylogarithms 27 and by partial fraction decomposition, i(a,δ,p,q) = − ∞∑ n=1 δn nq ∞∑ j=1   p(−1)p+1 (an)pj(j+an) + (−1)p (an)p(j+an)2 + ∑p k=2 (−1)p−k(p+1−k) (an)p+2−kjk   . now i(a,δ,p,q) = − ∞∑ n=1 δn nq   p(−1)p+1 han (an)p+1 + (−1)p ψ′(an+1) (an)p + ∑p k=2 (−1)p−k(p+1−k) (an)p+2−k ζ(k)   = (−1)p+1 ap ζ(2) lip+q(δ) + (−1)pp ap+1 ∞∑ n=1 δn han np+q+1 + (−1)p ap ∞∑ n=1 δn h (2) an np+q − p∑ k=2 (−1)p−k(p + 1 −k) ap+2−k ζ(k) lip+q+2−k(δ), and theorem 1 is proved. in the next few corollaries we investigate various special values of the parameters (a,δ,p,q) which will yield solutions to i (a,δ,p,q) that are expressible in terms of the riemann zeta and other special functions. we shall also present new euler type identities for the sums (1.2). corollary 1. for a = 1, δ = 1, p, q, positive integers with arbitrary weight p + q, then i(1, 1,p,q) = ∫ 1 0 ln x lip(x) liq(x) x dx = (−1)p (s2,p+q + ps1,p+q+1) + (−1)p+1ζ(2)ζ(p + q) (2.2) − p∑ k=2 (−1)p−k(p + 1 −k)ζ(k)ζ(p + q + 2 −k) , where sa,b is the linear euler sum (1.4). 28 a. sofo proof. here we note that lim(1) = ζ(m) and the sums s2,p+q = ∑∞ n=1 h (2) n np+q and s1,p+q+1 = ∑∞ n=1 hn np+q+1 . remark 1. for a = 1, δ = 1, p, q, positive integers with p + q an odd integer, then i(1, 1,p,q) = ∫ 1 0 ln x lip(x) liq(x) x dx = p(−1)p eu (p + q + 1) + (−1)p bw (2,p + q) + (−1)p+1ζ(2)ζ(p + q) (2.3) − p∑ k=2 (−1)p−k(p + 1 −k)ζ(k)ζ(p + q + 2 −k) , where eu (·) is the euler identity (1.3) and bw (·, ·) is the identity (1.6). example 1. i(1, 1, 4, 2) = i(1, 1, 2, 4) = s2,6 − 2ζ(3)ζ(5) + 7 6 ζ(8), i(1, 1, 4, 4) = s2,8 − 2ζ(3)ζ(7) + 33 10 ζ(10) − 2ζ2(5), i(1, 1, 4, 5) = i(1, 1, 5, 4) = ζ(4)ζ(7) + ζ(2)ζ(9) − 3ζ(11). corollary 2. for a = 1, δ = −1, p, q, positive integers, then i(1,−1,p,q) = ∫ 1 0 ln x lip(x) liq(−x) x dx = p(−1)p+1s(1,p + q + 1) (2.4) + (−1)p+1s(2,p + q) + (−1)pζ(2)η(p + q) + p∑ k=2 (−1)p−k(p + 1 −k)ζ(k)η(p + q + 2 −k), where η(·) is the dirichlet eta function and s(·, ·) is the alternating linear euler sum. we note that when we have odd weight (p + q), s(·, ·) may be replaced with the identity (1.8). integrals of polylogarithms 29 proof. here we note that lim(−1) = −η(m) and the sum s(m,p + q) =∑∞ n=1 (−1)n+1h(m)n np+q and may be replaced with the identity (1.8) in the case when we have odd weight (p + q). example 2. i(1,−1, 2, 4) = 2ζ(2)η(6) −s(2, 6) − 2s(1, 7), i(1,−1, 4, 2) = − 359 48 ζ(8) − 2ζ(3)η(5) −s(2, 6) − 4s(1, 7), i(1,−1, 2, 3) = 43 32 ζ(2)ζ(5) − 2ζ(7), i(1,−1, 3, 4) = 7 8 ζ(4)ζ(5) + 5 2 ζ(9) − 249 128 ζ(2)ζ(7). corollary 3. for a = 2, δ = 1, p, q, positive integers, then i(2, 1,p,q) = ∫ 1 0 ln x lip(x) liq ( x2 ) x dx = (−1)p+1 2p ζ(2)ζ(p + q) (2.5) + p(−1)p2q−1 (s1,p+q+1 −s(1,p + q + 1)) + (−1)p2q−1 (s2,p+q −s(2,p + q)) − (−1)p p∑ k=2 (−1)k(p + 1 −k) 2p+2−k ζ(k)ζ(p + q + 2 −k), where s·,· and s(·, ·) are the linear euler and alternating euler sums (1.4) and (1.5) respectively. in the case when we have odd weight (p + q) then we may utilize s1,p+q+1 = eu (p + q + 1) is the euler identity (1.3), s2,p+q = bw (2,p + q) is the identity (1.6) and s(·, ·) is obtained from the identity (1.8). proof. here we note that lim(1) = ζ(m) and the sums s·,· and s(·, ·) are the linear euler and alternating euler sums (1.4) and (1.5) respectively. in the case of odd weight (p + q) the sums s1,p+q+1 = ∑∞ n=1 hn np+q+1 , s2,p+q = bw (2,p + q) = ∑∞ n=1 h (2) n np+q and s(a,b) = ∑∞ n=1 (−1)n+1h(a)n nb . 30 a. sofo example 3. i(2, 1, 3, 3) = 47 4 ζ(3)ζ(5) − 211 8 ζ(8) − 4s2,6 + 4s(2, 6) + 12s(1, 7), i(2, 1, 4, 2) = 415 24 ζ(8) − 31 4 ζ(3)ζ(5) + 2s2,6 − 2s(2, 6) − 8s(1, 7), i(2, 1, 3, 2) = 11 32 ζ(5) − 7 16 ζ(2)ζ(3), i(2, 1, 0,q) = 2q−1 (eu (q) −s(1,q)) , i(2, 1, 5, 0) = 107 256 ζ(7) + 1 4 ζ(4)ζ(3) − 37 64 ζ(5)ζ(2). the aim now, is to obtain the new euler identities for the sums (1.2), hence consider the following corollary. corollary 4. from corollary 3, let q = 0, δ = ±1, and p, a positive odd integer, then w ( 1 2 , 1,p, 2 ) = ∑ n≥1 h (2) n 2 np (2.6) = p(−1)p+122−pζ(2)ζ(p) + 2p(−1)p (eu (p + 1) −s(1,p + 1)) + 2(−1)p (bw (2,p) −s(2,p)) + (−1)p+1 p−1∑ k=3 (−1)k(p + 1 −k) 2p−k ζ(k)ζ(p + 2 −k), where eu (·), s(·, ·) and bw (·, ·) are the same as in corollary 3. proof. proceeding as in theorem 1, i(a,δ,p, 0) = ∫ 1 0 ln x lip(x) li0 (δx a) x dx = − δ a2 ∑ n≥1 1 np ∑ j≥1 δj (n + aj)2 = − δ a2 ∑ n≥1 1 np φ ( δ, 2, 1 + n a ) , where φ ( δ, 2, 1 + n a ) is the lerch transcendent. we have, i(a,δ,p, 0) =   − 1 a2 ∑ n≥1 ψ′( na +1) np for δ = 1 , 1 4a2 ∑ n≥1 ψ′( n a +1) np ( ψ′(a+n 2a ) −ψ′( 2q+n 2a ) ) for δ = −1 . (2.7) integrals of polylogarithms 31 now for a = 2 and δ = 1, we have i(2, 1,p, 0) = − 1 4 ∑ n≥1 ψ′(n 2 + 1) np = 1 4 ∑ n≥1 h (2) n 2 np − 1 4 ζ(2)ζ(p) and equating with (2.5) we obtain the desired result (2.6). also, since ∑ n≥1 h (2) n np = 21−p ∑ n≥1 h (2) n 2 np ( 1 − (−1)n+1 ) we obtain the second euler identity w ( 1 2 ,−1,p, 2 ) = ∑ n≥1 (−1)n+1h(2)n 2 np = w ( 1 2 , 1,p, 2 ) − 21−p bw (2,p) = ( 1 + (−2)p+1 ) ζ(2)ζ(p) − 2pw ( 1 2 ,−1,p + 1, 1 ) + (−2)2−pw(2, 1,p, 2) + (−2)1−p bw (2,p) + p∑ k=2 (−2)2−k(p + 1 −k)ζ(k)η(p + 2 −k) . similarly, for the third euler sum identity in (1.2) we have ∞∑ n =1 h (2) n−1 2 (2n− 1)p = 1 2 ( w ( 1 2 , 1,p, 2 ) + w ( 1 2 ,−1,p, 2 )) . example 4. ∑ n≥1 hn 2 n6 = 135 128 ζ(7) − 1 16 ζ(2)ζ(5) − 1 4 ζ(3)ζ(4), ∑ n≥1 (−1)n+1hn 2 n6 = 119 128 ζ(7) − 1 32 ζ(2)ζ(5) − 7 32 ζ(3)ζ(4), 32 a. sofo ∑ n≥1 h (2) n 2 n5 = ζ(3)ζ(4) − 21 16 ζ(2)ζ(5) + 107 64 ζ(7), ∑ n≥1 (−1)n+1h(2)n 2 n5 = 7 8 ζ(3)ζ(4) − 13 8 ζ(2)ζ(5) + 147 64 ζ(7), ∞∑ n=1 h (2) n−1 2 (2n− 1)5 = 15 16 ζ(3)ζ(4) − 47 16 ζ(2)ζ(5) + 127 64 ζ(7). corollary 5. for a = 1 2 , δ = ±1, p, q, positive integers with p + q an odd integer, then i (1 2 ,δ,p,q ) = ∫ 1 0 ln x lip(x) liq ( δx 1 2 ) x dx =   2p(−2)pw ( 1 2 , 1,p + q + 1, 1 ) − (−2)pζ(2)ζ(p + q) +(−2)pw ( 1 2 , 1,p + q, 2 ) − ∑p k=2(−1) p−k(p + 1 −k)ζ(k)ζ(p + q + 2 −k) for δ = 1 , (−2)p+1w ( 1 2 ,−1,p + q + 1, 1 ) + (−2)pζ(2)η(p + q) +(−2)pw ( 1 2 ,−1,p + q, 2 ) + ∑p k=2(−1) p−k(p + 1 −k)ζ(k)η(p + q + 2 −k) for δ = −1 , where w(·, ·, ·, ·) is evaluated from corollary (4). proof. the proof follows from (2.1). example 5. in these examples we utilize some results from example 4: i( 1 2 , 1, 5, 0) = 16ζ(3)ζ(4) + 218ζ(2)ζ(5) − 391ζ(7), integrals of polylogarithms 33 i ( 1 2 ,−1, 5, 0 ) = 371ζ(7) − 12ζ(3)ζ(4) − 210ζ(2)ζ(5), i ( 1 2 , 1, 0, 5 ) = 107 16 ζ(7) + ζ(3)ζ(4) − 37 16 ζ(2)ζ(5), i ( 1 2 ,−1, 0, 5 ) = − 147 16 ζ(7) − 7 8 ζ(3)ζ(4) + 41 16 ζ(2)ζ(5), i ( 1 2 , 1, 3, 2 ) = 75 2 ζ(2)ζ(5) − 64ζ(7), i ( 1 2 ,−1, 3, 2 ) = 63ζ(7) − 37ζ(2)ζ(5). remark 2. the integral i(a,δ,p,q) has been represented in terms of special functions. for particular values of the constants (a,δ,p,q) the integral (1.1) has been expressed in closed form in terms of riemann zeta and dirichlet eta functions. some examples are given for the solution of the integral (1.1), most of which are not amenable to a mathematical computer package. finally we have developed new identities for the euler sums (1.2) in the spirit of euler (1.3). acknowledgements the author is thankful to a referee for the careful reading and considered suggestions leading to a better presented paper. references [1] borwein, d.; borwein, j. m..; girgensohn, r. explicit evaluation of euler sums. proc. edinburgh math. soc. (2) 38 (1995), no. 2, 277–294. [2] bailey, d. h. borwein, j. m. computation and structure of character polylogarithms with applications to character mordell-tornheimwitten sums. math. comp. 85 (2016), no. 297, 295–324. [3] choi, junesang log-sine and log-cosine integrals. honam math. j. 35 (2013), no. 2, 137–146. [4] devoto, a. duke, d. w. table of integrals and formulae for feynman diagram calculations. riv. nuovo cimento (3) 7 (1984), no. 6, 1–39. [5] euler, l. meditationes circa singulare serierum genus, novi comm. acad. sci. petropol. 20 (1776), 140-186; reprinted in opera omnia, ser. i, vol. 15, b. g.teubner, berlin, 1927, pp. 217-267. 34 a. sofo [6] flajolet, p. salvy, b. euler sums and contour integral representations. experiment. math. 7 (1998), no. 1, 15–35. [7] freitas, p. integrals of polylogarithmic functions, recurrence relations, and associated euler sums. math. comp. 74 (2005), no. 251, 1425– 1440. 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[21] zagier, d. values of zeta functions and their applications, in first european congress of mathematicians, vol ii (paris, 1992), birkhauser, boston, 1994, pp. 497-512. introduction integral identities and euler sums � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 2 (2020), 185 – 195 doi:10.17398/2605-5686.35.2.185 available online september 24, 2020 mackey continuity of convex functions on dual banach spaces: a review a.j. wrobel ∗ 15082 east county road 600n, charleston, illinois, 61920-8026, united states a.wrobel@alumni.lse.ac.uk https://www.researchgate.net/profile/andrew_wrobel3/research received december 13, 2019 presented by david yost accepted september 13, 2020 abstract: a convex (or concave) real-valued function, f, on a dual banach space p∗ is continuous for the mackey topology m (p∗,p) if (and only if) it is mackey continuous on bounded subsets of p∗. equivalence of mackey continuity to sequential mackey continuity follows when p is strongly weakly compactly generated, e.g., when p = l1 (t), where t is a set that carries a sigma-finite measure σ. this result of delbaen, orihuela and owari extends their earlier work on the case that p∗ is either l∞ (t) or a dual orlicz space. an earlier result of this kind is recalled also: it derives mackey continuity from bounded mackey continuity for a nondecreasing concave function, f, that is defined and finite only on the nonnegative cone l∞+ . applied to a linear f, the delbaen-orihuelaowari result shows that the convex bounded mackey topology is identical to the mackey topology, i.e., cbm (p∗,p) = m (p∗,p); here, this is shown to follow also from grothendieck’s completeness theorem. as for the bounded mackey topology, bm (p∗,p), it is conjectured here not to be a vector topology, or equivalently to be strictly stronger than m (p∗,p), except when p is reflexive. key words: dual banach space, convex bounded mackey topology, convergence in measure, economic equilibrium. ams subject class. (2010): primary 46b99, 46e30, 52a41; secondary 46a70. 1. introduction nonmetric topologies on the norm-dual, p∗, of a real banach space (p) can become much more manageable when restricted to bounded sets. for example, given a convex subset of p∗, or a real-valued concave function on p∗, the bounded weak* topology, bw∗ := bw (p∗,p), can serve to show that the set in question is weakly* closed, or that the function is weakly* upper semicontinuous. in economic theory, such uses of the krein-smulian theorem are made in [7, proposition 1.1, theorem 4.4 and theorem 4.7], [12, proposition 1 and example 5], [14, lemma 4.1] and [15, section 6.2]. in applications of economic ∗ formerly of the london school of economics. issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.35.2.185 mailto:a.wrobel@alumni.lse.ac.uk https://www.researchgate.net/profile/andrew_wrobel3/research https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 186 a.j. wrobel equilibrium models, this can be an indispensable tool for verifying that the production sets that describe the technologies are weakly* closed, and that the profit and cost functions are weakly* semicontinuous (which is needed for equilibria to exist, and for the dual pairs of programmes to have no duality gaps): see [11, lemma 6.1] and [15, lemmas 6.2.3–6.2.5].1 when p is l1(t,σ), the space of integrable real-valued functions on a set t that carries a sigma-finite measure σ—and so p∗ is the space of essentially bounded functions l∞(t)—another useful “bounded” topology on l∞ is the bounded mackey topology, bm ( l∞,l1 ) . this is because a convex (or concave) real-valued function, f, is continuous for the “plain” mackey topology, m ( l∞,l1 ) , if (and only if) it is bm ( l∞,l1 ) -continuous, i.e., m ( l∞,l1 ) continuous on bounded sets—or, equivalently, if (and only if) f is continuous along bounded sequences (in l∞) that converge in measure (on subsets of t of finite measure). thus the reduction to bounded sets provides direct access to the methods of integral calculus, which can greatly simplify verification of mackey continuity [12, example 5]. and, in economic equilibrium analysis, mackey continuity of a concave utility or production function is essential for representing the price system by a density, as is done in [3] and [13]. in addition, the use of convergence in measure furnishes economic interpretations of mackey continuity [12, section 4 and section 5]. when the convex function f is defined and finite on the whole space l∞, the equivalence of m ( l∞,l1 ) -continuity to bm ( l∞,l1 ) -continuity is a result of delbaen and owari [7, proposition 1.2], which is quoted here as theorem 2; they also extend it to the case of a dual orlicz space instead of l∞ [7, theorem 4.5] and apply it in the mathematics of finance [7, theorem 4.8]. their ingenious argument shows first that the sublevel sets of the conjugate function, f#, are uniformly integrable and hence weakly compact (in l1); it then follows that f is mackey continuous (on l∞) by the moreau-rockafellar theorem (on the conjugacy between continuity and inf-compactness). an earlier result that derives mackey continuity from its bounded version applies to a nondecreasing concave function, f , that is defined (and finite) only on the nonnegative cone l∞+ . quoted here as theorem 1, it requires a different method—one which relies on mackey continuity of the lattice operations in l∞ as well as on the monotonicity of f [12, proposition 1 and proposition 3 with example 2 and example 4]. (this case is different because f need not have a finite concave extension to the whole space l∞, and the 1a similar use of bw∗ is made in part i of the work on energy storage whose part ii is [11]. mackey continuity of convex functions 187 mackey interior of l∞+ is empty—except when l ∞(t) is finite-dimensional, i.e., when t consists of a finite number of atoms of σ.) for a finite-valued convex (or concave) f defined on the whole space, the equivalence of mackey continuity to bounded mackey continuity extends to the case of a general dual banach space, p∗, as the domain of f: see delbaen and orihuela [6, theorem 8]. (equivalence to sequential mackey continuity follows when p is strongly weakly compactly generated [6, corollary 11].) a fortiori, those linear functionals (on p∗) that are continuous for the bounded mackey topology, bm∗ := bm (p∗,p), are actually continuous for m∗ := m (p∗,p), i.e., belong to p . it follows that the convex bounded mackey topology, cbm (p∗,p), is identical to the “plain” mackey topology 2 : cbm∗ = m∗ for every p . 3 implicit in [12, proposition 1], this result is derived here more simply from grothendieck’s completeness theorem (proof of proposition 1). it does not follow that bm∗ equals m∗ because bm∗ is not known to be a vector topology and, indeed, it is conjectured not to be one, unless p is reflexive (conjecture 1). there are, then, two different methods of “upgrading” the bounded mackey continuity to full, unqualified mackey continuity, each with its own limitations and its own area of applicability: • the method of [12] applies to a nondecreasing concave function that is finite only on the cone l∞+ (or, more generally, on the nonnegative cone p∗+ of a dual banach lattice p ∗ on which m (p∗,p) is locally solid, i.e., is a vector-lattice topology). • the method of [6] and [7] applies to an everywhere-finite convex (or concave) function on a dual banach space p∗; it is based on the fenchellegendre conjugacy (and on using convergence in measure when p∗ is l∞ or a dual orlicz space). finally, the case of a dual banach lattice different from l∞, in which the norm-bounded sets differ from the order-bounded ones, is addressed briefly in the remarks at the end of section 2; these include an outline of the delbaenowari analysis for dual orlicz spaces in [7]. 2 for p = l1 only, that cbm ( l∞,l1 ) = m ( l∞,l1 ) has been shown earlier by methods specific to this space, in [5, iii.1.6 and iii.1.9] and in [16, theorem 5]. 3 it also follows that bm∗-continuity upgrades to m∗-continuity not only for linear functionals but also for general linear maps, i.e., every bm∗-continuous linear map of p∗, into any topological vector space, is m∗-continuous (on p∗). this is because, for a linear map of a space with topologies of the forms bt and cbt , its bt -continuity implies cbt -continuity [5, i.1.7], and because cbm∗ = m∗. 188 a.j. wrobel 2. mackey continuity derived from bounded continuity quoted below are the continuity-upgrade results for finite nondecreasing concave functions on l∞+ [12] and for finite convex functions on l ∞ [7] or on any banach dual space p∗ [6]. terminology (weak topology and mackey topology, swcg space): • let p∗ be the norm-dual of a real banach space p . the weakest and the strongest of those locally convex topologies on p∗ which yield p as the continuous dual are denoted by w (p∗,p) and m (p∗,p). known as the weak and the mackey topologies, on p∗ for its pairing with p , the two can be called the weak* and the mackey topologies (since m (p∗,p∗∗), the mackey topology on p∗ for its pairing with its norm-dual p∗∗, is identical to the norm topology of p∗). with w (p,p∗) denoting the weak topology of p , m (p∗,p) can be described as the topology of uniform convergence on all w (p,p∗)-compact, convex and balanced (a.k.a. circled) subsets of p [17, iv.3.2: corollary 1].4 • a real banach space p—whose norm-dual is p∗ and whose unit ball (centered at 0) is b—is called strongly weakly compactly generated (swcg) if it contains a w (p,p∗)-compact set g such that, for every w (p,p∗)-compact set c ⊂ p and every scalar � > 0, there exists an n ∈ n with c ⊆ ng + �b. (when such a g exists, it can be chosen to be convex and balanced, by krein’s theorem [17, iv.11.4].) see [18] for properties and examples of such spaces. theorem 1. (horsley-wrobel) let σ be a sigma-finite nonnegative measure on t. for a nondecreasing concave function f : l∞+ (t,σ) → r (defined and finite only on the nonnegative cone l∞+ ), the following conditions are equivalent to one another: (1) f is m ( l∞,l1 ) -continuous (on l∞+ ); (2) f is m ( l∞,l1 ) -continuous on bounded subsets (of l∞+ ); (3) f is sequentially m ( l∞,l1 ) -continuous (on l∞+ ); (4) on bounded subsets (of l∞+ ), f is continuous for tσ, the topology of convergence in the measure σ on subsets of t of finite measure. 4 here “convex” can actually be omitted because p is a banach space and one can apply krein’s theorem [17, iv.11.4]. that “circled” can be omitted is obvious [17, i.5.2]. mackey continuity of convex functions 189 proof. see [12, proofs of proposition 1 and proposition 3, with example 2 and example 4]. when the cone l∞+ is replaced by the whole space l ∞, the same holds without the monotonicity assumption. despite their formal similarity, the two cases are very different in their methods of proof, which overlap only in their standard uses of: (i) the krein-smulian theorem (for the “easy” semicontinuity part), and (ii) the equality, on bounded sets, of m ( l∞(σ),l1(σ) ) to tσ (the topology of convergence in measure). theorem 2. (delbaen-owari) let σ be a sigma-finite nonnegative measure on t. for a convex (or concave) function f : l∞(t,σ) → r that is finite everywhere (on the whole space l∞), the following conditions are equivalent to one another: (1) f is m ( l∞,l1 ) -continuous (on l∞); (2) f is m ( l∞,l1 ) -continuous on bounded subsets (of l∞); (3) f is sequentially m ( l∞,l1 ) -continuous (on l∞); (4) on bounded subsets (of l∞), f is continuous for tσ, the topology of convergence in the measure σ on subsets of t of finite measure. proof. the argument of [7, proof of proposition 1.2], where σ is assumed to be finite, is sufficient when σ is also nonatomic. it can be extended to the case an arbitrary sigma-finite σ by using two standard techniques: (i) embedding a finite measure with atoms in a nonatomic one, and (ii)replacing a sigma-finite measure by an equivalent finite one (i.e., one with a strictly positive and integrable density). conditions (1) and (2) of theorem 2 remain equivalent to each other when l∞ is replaced by any dual banach space p∗ (with “bounded” taken to mean “norm-bounded”). condition (3) is equivalent to the first two if p is swcg. theorem 3. (delbaen-orihuela-owari) let p be a real banach space and p∗ its norm-dual. for a convex (or concave) function f : p∗ → r that is finite everywhere (on the whole space p), the following two conditions are equivalent to each other: (1) f is m (p∗,p)-continuous (on p∗); 190 a.j. wrobel (2) f is m (p∗,p)-continuous on norm-bounded subsets (of p∗). when p is strongly weakly compactly generated, the above condition (2) is equivalent (whether f is convex or not) to the condition 5: (3) f is sequentially m (p∗,p)-continuous (on p∗). proof. see [6, theorem 8 and corollary 11]. k. owari has given a simpler proof that condition (2) implies condition (1): although his argument is essentially the same as that in [6], it does not involve the approximate subdifferential.6 also, the assumption in [6, theorem 8 and corollary 11] that f is a conjugate function (or, equivalently, that it is weakly* lower semicontinuous on p∗) is superfluous: it follows from the bounded mackey continuity (and the convexity) of f by the krein-smulian theorem (as is noted in [7, proposition 1.1] and in [12, proposition 1]). remarks (on the case of a dual banach lattice p∗ 6= l∞): • in such a banach lattice, order-bounded sets are norm-bounded, but not vice versa. four more questions arise, then, about m (p∗,p)-continuity of a finite convex function f on p∗: (a) does sequential continuity on order-intervals imply either (i) continuity on order-intervals, or (ii) sequential continuity (on balls and hence on all of p∗ by the banachsteinhaus theorem)? (b) does either of the last two conditions imply continuity on p∗? for dual orlicz spaces, all four conditions are equivalent by a result of delbaen and owari [7, theorem 4.5], whose analysis can be outlined as follows. • on order-intervals, sequential continuity implies continuity (i.e., “yes” to part (i) of (a)) whenever m (p∗,p) is metrizable on order-intervals. and m (p∗,p) is so not only when p is swcg, but also when p is an orlicz space lφ(t,σ)—where σ is a finite nonnegative measure on t and φ : r+ → r+ is a super-coercive (a.k.a. strict) young function that meets the ∆2-condition, and so ( lφ )∗ = lφ # , where φ# is the convex conjugate of φ [7, pp. 1052–1053]. on each order-interval of such a space, m ( lφ # ,lφ ) is equal to tσ, the globally metrizable topology of convergence in the measure σ [7, (2.1)]. 5 the corresponding part of theorem 2 is a special case of this, since l1(t,σ) is swcg when σ is sigma-finite [18, 2.3]. 6 personal communication. mackey continuity of convex functions 191 • once f##—the second convex conjugate of f for the pairing of lφ # with lφ—has been shown to equal f, that sequential m ( lφ # ,lφ ) -continuity on order-intervals implies unqualified m ( lφ # ,lφ ) -continuity is shown by using tσ in essentially the same way as in the case of p∗ = l∞: compare the proofs of [7, proposition 1.2 and theorem 4.5]. • so, to complete the proof that all four conditions are equivalent (for p = lφ), it remains to establish that indeed f## = f when f is m ( lφ # ,lφ ) -lower semicontinuous (l.s.c.) on order-intervals. this is done, in [7, theorem 4.1 and theorem 4.4], by using a new variant of komlós’s theorem [7, theorem 3.6 and corollary 3.10]—which, crucially, from a merely norm-bounded sequence (in lφ # ) produces an order-bounded and σ-almost everywhere convergent sequence of forward convex combinations. of interest in itself, in the present context the delbaen-owari variant of komlós’s theorem bridges the gap between orderand norm-boundedness.7 • the question of whether, on p∗, sequential continuity implies continuity (i.e., part (ii) of (b)) does not involve the order structure, and by theorem 3 the answer is “yes” when p is swcg. the outlined analysis of [7], which does use the order, adds the “yes” answer for p = lφ— whether it is swcg or not, which in general seems to be an unanswered question.8 • continuity on order-intervals (rather than balls) can be of use in verifying “full” continuity also when p is a reflexive banach lattice (e.g., the lebesgue space l% with 1 < % < +∞), in which case m (p∗,p) is the norm topology. • with a banach lattice p other than lφ, any extension of this analy7 in detail: since f is convex and m ( lφ # ,lφ ) -l.s.c. on order-intervals, it follows from [7, corollary 3.10]—by using ordinary sequences (rather than uncountable nets)—that f is tσl.s.c. on norm -bounded sets. a fortiori, f is m ( lφ # ,lφ ) -l.s.c. on such sets and so, being convex, it is w ( lφ # ,lφ ) -l.s.c. on norm-bounded sets. by the krein-smulian theorem [17, iv.6.4], this means that f is actually w ( lφ # ,lφ ) -l.s.c. on all of lφ # —i.e., that f## = f. 8 the space lφ is known to be swcg if yφ′(y)/φ(y) → 1 as y → +∞ or, equivalently, if 1 = %φ := lim supy→+∞ yφ ′(y)/φ(y), where φ′ is the one-sided (left or right) derivative of φ: see [6, paragraph after (20)]. (that %φ < +∞ is an equivalent form of the ∆2-condition [7, (1.1)].) but this criterion proves lφ to be swcg only when its %φ is the same as that of the swcg space l1. 192 a.j. wrobel sis would have to start with a metric for m (p∗,p) on order-intervals. but its very existence seems to be an unanswered question (when p is not swcg). this is so even though there is a relevant metrizability criterion—for order-continuous locally solid topologies, a.k.a. lebesgue topologies. namely, a lebesgue topology, t , on a vector lattice y is metrizable on order-intervals if and only if y has the countable sup property [2, 4.26]. and a locally convex-solid topology, t , on a vector lattice y is order-continuous if (and only if) the t -dual of y is contained in the order-continuous dual, y ∼n , of y [2, 3.12]. this inclusion always holds for y = p∗ and t = m (p∗,p), i.e., p ⊆ (p∼)∼n = (p ∗) ∼ n by [1, p. 331, line 8 f.b.], since the order-dual p∼ is equal to the normdual p∗ [1, theorem 9.11]. so the metrizability criterion applies to m (p∗,p) when (and only when) m (p∗,p) is locally solid (i.e., makes p∗ a topological vector lattice): on this condition, m (p∗,p) is metrizable on order-intervals if and only if p∗ has the countable sup property (or, equivalently, is of countable type). but, apart from l∞ (and dual orlicz spaces), few, if any, examples seem to be known of (nonreflexive) dual banach lattices whose mackey topologies are locally solid. 3. the bounded and convex bounded mackey topologies on a dual banach space mackey continuity on norm-bounded subsets of the norm-dual, p∗, of a real banach space p (which is condition (2) of theorem 3) can be restated as continuity for the bounded mackey topology. this is a case of [8, theorem 1 (b)], and it holds for every map (of p∗) into any topological space: it does not depend on any monotonicity or convexity properties of the map. denoted by bm (p∗,p) or bm∗ for brevity, the bounded mackey topology is the strongest topology that is equal to m (p∗,p) on every bounded subset (of p∗). the convex bounded mackey topology, denoted by cbm (p∗,p) or cbm∗ for brevity, is the strongest locally convex topology that is equal to m (p∗,p) on every bounded set. remarkably, this is also the strongest vector topology that is equal to m (p∗,p) on every bounded set: this is a case of [19, 2.2.2], a result given also in [5, i.1.4 and i.1.5 (iii)]. obviously bm∗ is at least as strong as cbm∗, which in turn is at least as strong as m∗ := m (p∗,p). the last two are actually one and the same. a corollary to theorem 3, this equality is next obtained more simply by applying grothendieck’s completeness theorem [17, iv.6.2], which is quoted for easy reference. mackey continuity of convex functions 193 theorem 4. (grothendieck) let t be a locally convex topology on a real vector space e. when additionally s is a saturated family 9 of tbounded sets covering e, the t-dual of e is complete under the s-topology (the topology of uniform convergence on every s ∈ s) if and only if every linear functional (on e) that is t-continuous on each s ∈ s is actually tcontinuous on the whole space e (i.e., is in the t-dual of e). proposition 1. let p be a real banach space, and p∗ its norm-dual. then cbm (p∗,p) = m (p∗,p). proof. apply theorem 4 to p∗ as e—with m∗ as t and the bounded subsets of p∗ as s, and hence with p as the t-dual and the norm topology of p as the s-topology—to conclude that a linear functional on p∗ is m∗continuous if it is so on bounded sets (i.e., if it is bm∗-continuous). a fortiori, it is m∗-continuous (i.e., is in p) if it is cbm∗-continuous. in other words, cbm∗ yields the same dual space as m∗ (viz., p). this proves that cbm∗ = m∗ (since cbm∗ is both locally convex and stronger than m∗). 10 as for bm (p∗,p), it is semi-linear (i.e., both vector addition and scalar multiplication are separately continuous in either variable): this is a case of [4, theorem 5], a result noted also in [8, p. 410]. it seems to be unknown whether bm∗ is linear (i.e., a vector topology), but if it were, it would be identical to cbm∗ = m∗. the delbaen-owari result, with its reliance on convexity, makes this implausible—as is put forward next. conjecture 1. for p = l1[0, 1] at least, and possibly for every nonreflexive banach space p , the topology bm (p∗,p) is strictly stronger than m (p∗,p)—or, equivalently, bm (p∗,p) is not linear. the conjecture, then, is based on what it takes to establish that a bm∗continuous r-valued function f, on a nonreflexive space p∗, is m∗-continuous: theorem 2 and theorem 3 require f to be convex, and it is hard to imagine 9 a family, s, of subsets of a locally convex space is called saturated [17, p. 81] if: (i) all subsets of every member of s belong to s, (ii) all scalar multiples of every member of s belong to s, and (iii) for each finite f ⊂ s, the closed convex circled hull of the union of f belongs to s. 10 alternatively, cooper’s special case of grothendieck’s theorem [5, i.1.17 (ii)] can be applied—to p∗ as his e, with m∗ as τ and the bounded subsets of p∗ as b, and hence with p∗∗ as e′b and cbm ∗ as his γ = γ (b,τ)—to conclude that the cbm∗-dual equals the m∗-dual (so cbm∗ = m∗). 194 a.j. wrobel (even when p∗ = l∞[0, 1]) how the convexity assumption might be disposed of entirely—as would be necessary for bm∗ to equal m∗. comment on bounded topologies: see [20] for a more detailed review of such topologies: in addition to bw∗ and bm∗, it includes the bounded weak topology and its convex variant (bw and cbw), which are studied in [9], as well as the compact weak topology and its convex variant (kw and ckw), which are introduced in [10]. acknowledgements i am obliged to keita owari for telling me of theorem 3 and for the reference [6]. i am also grateful to the referee for useful comments, and to the editor for his handling of the manuscript. references [1] c.d. aliprantis, k.c. border, “ infinite dimensional analysis ”, springer, berlin-heidelberg-new york, 2006. doi.org/10.1007/3-540-29587-9 [2] c.d. aliprantis, o. burkinshaw, “ locally solid riesz spaces with applications to economics ”, american mathematical society, providence, ri, 2003. doi.org/10.1090/surv/105 [3] t.f. bewley, existence of equilibria in economies with infinitely many commodities, j. econom. theory 4 (3) (1972), 514 – 540. doi.org/10.1016/0022-0531(72)90136-6 [4] h.s. collins, completeness and compactness in linear topological spaces, trans. amer. math. soc. 79 (1955), 256 – 280. doi.org/10.1090/s0002-9947-1955-0069386-1 [5] j.b. cooper, “ saks spaces and applications to functional analysis ”, second edition, north-holland, amsterdam, 1987. doi.org/10.1016/s0304-0208(08)72315-6 [6] f. delbaen, j. orihuela, mackey constraints for james’s 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complete metric spaces; locally compact spaces; open regular sets; partial ordered sets. msc (2020): 54e50, 54h12. 1. introduction back in the 1930’s, stefan banach and marshall stone proved one of the most celebrated results in functional analysis. the usual statement the reader can find of the banach-stone theorem is, give or take, the following: theorem. let x and y be compact hausdorff spaces and let t : c(x) → c(y ) be a surjective linear isometry. then there exist a homeomorphism τ : x → y and g ∈ c(y ) such that |g(y)| = 1 for all y ∈ y and (tf)(y) = g(y)f(τ(y)) for all y ∈ y,f ∈ c(x). this result is, however, much deeper. it allows to determine x by means of the structure of c(x), in the sense that x turns out to be homeomorphic to the set of extreme points of the unit sphere of (c(x))∗ (after quotienting by the sign). since then, similar results began to appear, as gel’fand-kolmogorov theorem [13] or the subsequent works by milgram, kaplansky or shirota, [18, 19, 22, 24]. in spite of this rapid development, after shirota’s 1952 work ∗ this research was supported in part by mincin project pid2019-103961gb-c21 and project ib20038 de la junta de extremadura. issn: 0213-8743 (print), 2605-5686 (online) c© the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.37.1.139 mailto:coco@unex.es https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 140 j. cabello sánchez –which we will discuss later– a standstill lasts until the last few years of the xxth century. then the topic forks in two different ways. on the one hand, there are mathematicians that begin to suspect that the proof of [24, theorem 6] did not work, so they began to study lattice isomorphisms between spaces of uniformly continuous functions (see, e.g., [10]). on the other hand, it begins to appear a significant amount of papers that deal with representation of isomorphisms between other spaces of functions or, in general, between subsets of c(x) and c(y ), see [2, 11, 12, 14, 17, 23]. anyhow, the papers where we find some of the most accurate results about isomorphisms of spaces of uniformly continuous functions [5, 6], lipschitz functions [4] and smooth functions [3] have something in common: the result labelled in the present paper as lemma 2.1; the interested reader should take a close look at [21], where the authors were able to unify all these results and find new ones. this lemma has been key in these works, and has recently lead to similar results, see [7, 9]. in the present paper we study lemma 2.1, generalizing it in two ways and providing a more accurate description of the isomorphisms between lattices of regular open sets in complete metric spaces. in the first part of the second section, we shall restrict ourselves to the study of complete metric spaces and order preserving bijections between arbitrary bases of their topologies. namely, we will show that given a couple of complete metric spaces, say x and y , every order preserving bijection between bases of their topologies induces a homeomorphism between dense gδ subspaces x0 ⊂ x and y0 ⊂ y –subspaces that can be endowed with (equivalent) metrics that turns them into complete spaces. later, we restrict ourselves to the bases of regular open sets on the wider class of hausdorff, regular topological spaces and show that whenever x0 ⊂ x is dense, the lattices r(x) and r(x0) of regular open subsets are naturally isomorphic and analyse some consequences of this. joining both parts we get an explicit representation of every isomorphism between lattices of regular open sets in complete metric spaces that may be considered as the main result in this paper. remark 1.1. apart from this introduction, the present paper contains section 2, where we prove the main results of the paper, and section 3, that contains some remarks and applications. remark 1.2. in this paper, x and y will always be topological spaces. we will denote the interior of a ⊂ x as intxa, unless the space x is clear by the context, in which case we will just write int a. the same way, a x or a will denote the closure of a in x. order isomorphisms 141 we will denote by r(x) the lattice of regular open subsets of x and bx will be a basis of the topology of x, please recall that an open subset u of some topological space x is regular if and only if u = int u. we say that t : bx → by is an isomorphism when it is a bijection that preserves inclusion, i.e., when t (u) ⊂t (v ) is equivalent to u ⊂ v . 2. the main result in this section we will prove our main result, theorem 2.14. actually, it is just a consequence of theorem 2.8 and proposition 2.13, but as both results are more general than theorem 2.14 we have decided to separate them. we have split the proof in several intermediate minor results. lemma 2.1. let (x,dx) and (y,dy ) be completely metrizable metric spaces and bx,by , bases of their topologies. suppose there is an isomorphism t : bx →by . then, there exist dense subspaces x0 ⊂ x and y0 ⊂ y and a homeomorphism τ : x0 → y0 such that τ(x) ∈ t (u) if and only if x ∈ u ∩x0. proof. the proof is the same as in [5, lemma 2], after endowing x and y with equivalent complete distances. remark 2.2. in the conditions of lemma 2.1, we will denote rx(x) = ⋂ x∈u∈bx t (u) , ry (y) = ⋂ y∈v∈by t−1(v ) . what the proof of [5, lemma 2] shows is that the subset x0 is dense in x, where x0 consists of the points x ∈ x for which there exists y ∈ y such that rx(x) = {y} and ry (y) = {x}. once we have that x0 is dense, it is clear that the map sending each x ∈ x0 to the only point in rx(x) is a homeomorphism. the following theorem is just a translation of the théorème fondamental in [20]. theorem 2.3. if there exists a bicontinuous, univocal and reciprocal correspondence between two given sets (inside an m-dimensional space), it is possible to determine another correspondence with the same nature between the points of two gδ sets containing the given sets, the second correspondence agreeing with the first in the points of the two given sets. 142 j. cabello sánchez a more general statement of lavrentieff’s theorem is the following, that can be found in [27, theorem 24.9]: theorem 2.4. (lavrentieff) if x and y are complete metric spaces and h is a homeomorphism of a ⊂ x onto b ⊂ y , then h can be extended to a homeomorphism h∗ of a∗ onto b∗, where a∗ and b∗ are gδ-sets in x and y , respectively, and a ⊂ a∗ ⊂ a, b ⊂ b∗ ⊂ b. as for the following theorem, the author has been unable to find alexandroff’s work [1], but hausdorff references the result in [15] as follows: theorem 2.5. (alexandroff–hausdorff, [1, 15]) every gδ subset in a complete space is homeomorphic to a complete space. remark 2.6. as can be seen in [25], every locally compact metric space is open in its completion, so the class of completely metrizable spaces includes that of locally compact metrizable spaces. combining theorems 2.4 and 2.5 with lemma 2.1 we obtain: proposition 2.7. let x and y be complete metric spaces and t : bx → by an inclusion preserving bijection. then, there exist a complete metric space z and dense gδ subspaces x1 ⊂ x, y1 ⊂ y such that z,x1 and y1 are mutually homeomorphic. of course, if z is an in proposition 2.7 then every dense gδ subset z ′ ⊂ z fulfils the same, so it is clear that there is no minimal z whatsoever. in spite of this, it is very easy to determine some maximal z. consider (x0,dz), where x0 is the subset given in lemma 2.1 and dz(x,x ′) = max { dx(x,x ′),dy ( τ(x),τ(x′) )} . (2.1) theorem 2.8. the metric dz makes x0 complete and, moreover, if z ′ embeds into both x and y respectively via φ′x and φ ′ y in such a way that φ′x(z) ∈ u if and only if φ ′ y (z) ∈t (u), then φ ′ x embeds z ′ into x0. proof. for the first part, take a dz-cauchy sequence (xn) in x0 and let yn = τ(xn) for every n. it is clear that both (xn) and (yn) are dx-cauchy and dy -cauchy, respectively, so let x = lim(xn) ∈ x,y = lim(yn) ∈ y , these limits exist because x and y are complete. it is clear that any sequence order isomorphisms 143 (x̃n) ⊂ x0 converges to x if and only if y = lim(τ(x̃n)). this readily implies that rx(x) = {y}, so x ∈ x0 and this means that (x0,dz) is complete. now we must see that every metric space z′ that embeds in both x and y is embeddable in x0, whenever the embeddings respect the isomorphism between the bases. for this, as x0 is endowed with the restriction of the topology of x and z′ is homeomorphic to φ′x(z ′) ⊂ x, the only we need is φ′x(z ′) ⊂ x0. so, suppose x ∈ φ′x(z ′)\x0 and let z ∈ (φ′x) −1(x). as z′ also embeds in y , there exists y = φ′y (z) ∈ φ ′ y (z ′) \ y0, too, with the property that x ∈ u if and only if y ∈t (u). by the very definition of x0 and y0 this means that x ∈ x0,y ∈ y0 and τ(x) = y. now, we approach proposition 2.13, the main result about regular topological spaces. for this, the following three elementary results will come in handy; all their proofs are clear. lemma 2.9. let z be a topological space and a ⊂ z. a is a regular open subset of z if and only if for every open v ⊂ z, v ⊂ a implies v ⊂ a. lemma 2.10. let x be a topological space. whenever y ⊂ x is dense and u ⊂ x is open, one has ux = u ∩y x. lemma 2.11. let x be a topological space and u,v ∈ r(x) such that u ⊂ v,u 6= v . then, there is ∅ 6= w ∈ r(x) such that w ∩ u = ∅ and w ⊂ v . remark 2.12. if in lemma 2.11 x is regular and hausdorff, then v can be taken as any open subset that contains u strictly. proposition 2.13. let x be a topological space and y ⊂ x a dense subset. then t : r(x) → r(y ), defined as t (u) = u ∩ y , is a lattice isomorphism with inverse s(v ) = int v . proof. we need to show that t and s are mutually inverse. let u ∈ r(x), the first we need to show is that t is well-defined, i.e., that t (u) = u ∩y is regular in y . let v ⊂ x an open subset such that v ∩ y ⊂ u ∩y y . then, as the closure in x preserves inclusions, we have v x = v ∩y x ⊂ u ∩y y x ⊂ ux, 144 j. cabello sánchez where the first equality holds because of lemma 2.10. taking interiors in x also preserves inclusions, so we obtain v ⊂ intx ( v x ) ⊂ intx ( u x ) = u, which readily implies that v ∩y ⊂ u ∩y and we obtain v ∩y ∈ r(y ) from lemma 2.9. it is clear that s(v ) ∈ r(x) for every v ∈ r(y ), so both maps are well-defined. furthermore, lemma 2.10 implies that, for any regular u ⊂ x, s ◦t (u) = s(u ∩y ) = intx ( u ∩y x ) = intx ( u x ) = u. as for the composition t ◦s, we have t ◦s(v ) = t ( intx ( v x )) = intx ( v x ) ∩y for any v ∈ r(y ). let w ⊂ x be an open subset for which v = w ∩y , the very definition of inherited topology implies that there exists such w . the previous equalities can be rewitten as t ◦s(w ∩y ) = t ( intx ( w ∩y x )) = t ( intx ( w x )) = intx ( w x ) ∩y, so we need w∩y = intx ( w x ) ∩y . it is clear that w∩y ⊆ intx ( w x ) ∩y , so what we need is intx ( w x ) ∩y ⊆ w ∩y . both subsets are regular in y , so if this inclusion does not hold, there would exist an open h ⊂ y h 6= ∅ , h ⊂ intx ( w x ) ∩y , h ∩ (w ∩y ) = ∅ , so, by lemma 2.11 there is an open g ⊂ x such that h = g∩y and so g∩y 6= ∅ , g∩y (∗) ⊂ intx ( w x ) ∩y , (g∩y ) ∩ (w ∩y ) = ∅ , (2.2) and this is absurd. indeed, the inclusion marked with (∗) implies that we may substitute g by g ∩ intx ( w x ) , so both inequalities in (2.2) hold for some open g ⊂ intx ( w x ) . as y is dense and g and w are open, the last equality implies that g∩w = ∅. of course, this implies g∩ intx ( w x ) = ∅, which means g = ∅ and we are done. order isomorphisms 145 now we are in conditions to state our main result: theorem 2.14. let x, y and z be complete metric spaces, φx : z ↪→ x and φy : z ↪→ y be continuous, dense, embeddings and x0 = φx(z). then, t : r(x) → r(y ) given by the composition u 7→ u ∩x0 7→ φ−1x (u ∩x0) 7→ φy (φ −1 x (u ∩x0)) 7→ int ( φy (φ −1 x (u ∩x0)) ) is an isomorphism between the lattices of open regular subsets of x and y and every isomorphism arises this way. the “every isomorphism arises this way” part is due to theorem 2.8, while the “the composition is an isomorphism” part is consequence of proposition 2.13. 3. applications and remarks in this section, we are going to show how proposition 2.13 leads to some properties of βn and conclude with a couple of examples that show that the hypotheses imposed in the main results are necessary. but first, we need to deal with an error in some outstanding work. in [4] f. cabello and the author of the present paper showed that some results in [24] were not properly proved. later in [5] the same authors proved that, even when the proof of [24, theorem 6] was incorrect, the result was true. now, we are going to explain what the error was. the following definitions and theorems can be found in [24]: definition 3.1. (definition 2) a distributive lattice with smallest element 0 satisfying wallman’s disjunction property is an r-lattice if there exists a binary relation � in l which satisfies: • if h ≥ f and f � g, then h � g. • if f1 � g1 and f2 � g2, then f1 ∧f2 � g1 ∧g2. • if f � g, then there exists h such that f � h � g. • for every f 6= 0 there exist g1 and g2 6= 0 such that g1 � f � g2. • if g1 � f � g2, then there exists h such that h∨f = g1 and h∧g2 = 0. immediately after definition 2 we find this: 146 j. cabello sánchez theorem 3.2. (theorem 1) a distributive lattice with smallest element 0 is an r-lattice if and only if it is isomorphic to a sublattice of the lattice of all regular open sets on a locally compact space x. this sublattice is an open base and its elements have compact closures. the open regular set in x associated to f ∈ l is denoted by u(f). with this notation, the next statement is: theorem 3.3. (theorem 2) let l be an r-lattice. then there exists uniquely a locally compact space x which satisfies the property in theorem 1 and where u(f) ⊃ u(g) if and only if f � g. our proposition 2.13 contradicts the uniqueness of x in the statement of theorem 2 and we may actually explicit a lattice isomorphism between r(x) and r(y ) for different compact metric spaces x and y . namely, we just need to take the simplest compactifications of r and the composition of the lattice isomorphisms predicted by proposition 2.13: example 3.4. let x = r ∪ {−∞,∞} and y = r ∪ {n}. then, t : r(x) → r(y ), defined by t (u) =   u if u ∩{−∞,∞} = ∅ , u ∩r if u ∩{−∞,∞} = {∞} , u ∩r if u ∩{−∞,∞} = {−∞} , (u ∩r) ∪{n} if {−∞,∞}⊂ u, is a lattice isomorphism whose inverse is given by s(v ) = { v if n 6∈ v, (v ∩r) ∪{−∞,∞} if n ∈ v. it seems that the problem here is that the definition of r-lattice, definition 2, does not include the relation �, but in theorem 2 and its consequences the author considers � as a unique, fixed, relation given by (l,≤). it is clear that the above spaces generate, say, different �x and �y in the isomorphic lattices r(x) and r(y ). this leads to the error already noted in [4, section 5]. actually, with the definition of r-lattice given in [24], in seems that the original purpose of the definition is lost. indeed, the relation � may be taken as ≥ in quite a few lattices. this leads to a topology where every regular open set is clopen, in section 3.1 we will see an example of a far from order isomorphisms 147 trivial topological space where this is true. given a lattice (l,≥), the relation between each possible � and the unique locally compact topological space given by theorem 2 probably deserves a closer look. anyway, if we include � in the definition, then [24, theorem 2] is true. so let us put everything in order. definition 3.5. (shirota) let (l,≤) be a distributive lattice with minl = 0 and � be a relation in l. the triple (l,≤,�) is an r-lattice if the following hold: 1. for every a 6= b ∈ l, there exists h ∈ l such that either a∧h = 0 and b∧h 6= 0 or the other way round. 2. if h ≥ f and f � g, then h � g. 3. if f1 � g1 and f2 � g2, then f1 ∧f2 � g1 ∧g2. 4. if f � g, then there exists h such that f � h � g. 5. for every f 6= 0 there exist g1 and g2 6= 0 such that g1 � f � g2. 6. if g1 � f � g2, then there exists h such that h∨f = g1 and h∧g2 = 0. with definition 3.5 everything works and this result remains valid, but it does not lead to the consequences stated there as theorems 3 to 6. theorem 3.6. (theorem 2) let (l,≤,�) be an r-lattice. then there exists uniquely a locally compact space x which satisfies the property in theorem 1 and where u(f) ⊃ u(g) if and only if f � g. remark 3.7. it is worth noting that the main result in [16] states that, in the more general setting of uniform spaces, every lattice isomorphism t : u(x) → u(y ) induces another lattice isomorphism t : u∗(x) → u∗(y ), thus leading to another proof of shirota’s theorem. 3.1. the stone-čech compactification of n. we will analyse the isomorphism given in proposition 2.13 when y = n and x = βn, the stonečech compactification of n. this is not going to lead to new results, but it seems to be interesting in spite of this. these are very different spaces, but they share the same lattice of regular open subsets. in any case, as n is discrete, every v ⊂ n is regular and this, along with proposition 2.13, implies that r(βn) = { int(u) : u ⊂ n } . 148 j. cabello sánchez as βn is regular, every open w ⊂ βn is the union of its regular open subsets, and these regular open subsets are determined by the integers they contain, so w is determined by a collection aw of subsets of n. of course, if w contains s(j) for some j ⊂ n and i ⊂ j then s(i) ⊂ w , too. this means that aw is closed for inclusions. furthermore, as j is open in βn, see [26, p. 144, subsection 3.9], if s(i) ⊂ w and s(j) ⊂ w then s(i) ∪s(j) = s(i ∪j), so s(i ∪j) ⊂ w and aw is closed for pairwise supremum. summing up, aw is an ideal of the lattice p(n) for every proper open subset ∅ ( w ( βn –and every s(j) ∈ r(βn) is closed, so “clopen” and “regular open” are equivalent in βn. it is also clear that every ideal a in p(n) defines an open wa ∈ βn as ∪{s(i) : i ∈a}, that these identifications are mutually inverse and that w ⊂ v if and only if aw ⊂ av , so each maximal ideal in p(n) defines a maximal open proper subset of βn. as βn is hausdorff, these maximal open subsets are exactly βn \{x} for some x, so each point is dually defined by a maximal ideal. in other words, every point in βn is associated to an ultrafilter in p(n). as our final comment in this just for fun remark, we have that βn is the only hausdorff compactification of n that fulfils: ♠ if j,i ⊂ n are disjoint, then their closures in the compactification are disjoint, too, although this is just a particular case of a result by čech, see [26, pp. 25-26]. 3.2. the hypotheses are minimal. in some sense, theorem 2.14 is optimal. here we see that there is no way to generalise it if we omit any of the hypotheses. remark 3.8. consider any infinite set z endowed with the cofinite topology τcof . it is clear that every pair of nonempty open subsets of z meet, so every nonempty open subset is dense in z and this implies that the only regular open subsets of z are z and ∅. of course the same applies to any uncountable set endowed with the cocountable topology τcon, so (r,τcof ) and (r,τcon) have the same regular open subsets. nevertheless, there is no way to identify homeomorphically any couple of dense subsets of r with each topology. in order to avoid this pathological behaviour we needed to consider only regular hausdorff spaces since these spaces are the only reasonable spaces for which the regular subsets comprise a base of the topology. in other words, theorem 2.14 will not extend to general topological spaces. order isomorphisms 149 remark 3.9. consider x = [0, 1] endowed with its usual topology and let y be its gleason cover. the lattices r(x) and r(y ) are canonically isomorphic, but it is well known that no point in y has a countable basis of neighbourhoods, so ⋂ x∈u∈bx t (u) = ∞⋂ n=1 t ( b(x, 1/n) ) is never a singleton. it is remarkable that [8, example 1.7.16] is the only place where the author has been able to find a statement that explicitly confirms that the gleason cover of some compact space k is the same topological space as the stone space associated to r(k), i.e., gk = st(r(k)). remark 3.10. consider a = q∩ [0, 1], b = i∩ [0, 1] and their stone-čech compactifications x = βa, y = βb. there is a lattice isomorphism between r(x) and r(y ), say t , given by the composition of the following isomorphisms: u ∈ r(x) 7→ u ∩a ∈ r(a) , u ∈ r(a) 7→ int u ∈ r ( [0, 1] ) , u ∈ r ( [0, 1] ) 7→ u ∩b ∈ r(b) , u ∈ r(b) 7→ int u ∈ r(y ) . in spite of this, it is intuitively evident that rx(x) = ⋂ x∈u∈bx t (u) = ∞⋂ n=1 t ( int ( b(x, 1/n) )) is never a singleton when x ∈ a, and ry (y) = ⋂ y∈v∈by t−1(v ) = ∞⋂ n=1 t−1 ( int ( b(y, 1/n) )) is neither a singleton for y ∈ b. it seems clear that x = at{ry (y) : y ∈ b} and y = b t{rx(x) : x ∈ a}, so there is no point in x0. remark 3.11. there is no “non-complete metric spaces” result. indeed, i and q have the obvious isomorphism between their bases of regular open subsets and they are, nevertheless, disjoint subsets in r. this means that when trying to generalise theorem 2.14 the problem may come not only from 150 j. cabello sánchez the lack of separation of the topologies as in remark 3.8, from the excess or points as in remark 3.9 or from the points in x not squaring with those in y as in remark 3.10 but also from the, so to say, lack of points in the spaces even when they are metric. acknowledgements it is a pleasure to thank professor denny h. leung for his interest in the present paper and for pointing out a relevant mistake. i also need to thank the anonymous referee for their careful reading and for some specially accurate comments. references [1] p.s. alexandroff, sur les ensembles de la première classe et les ensembles abstraits, c. r. acad. sci. paris 178 (1924), 185 – 187. 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[27] s. willard, “ general topology ”. addison-wesley publishing co., reading, mass.-london-don mills, ont., 1970. introduction the main result applications and remarks the stone-cech compactification of n. the hypotheses are minimal. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae article in press available online july 20, 2023 the fundamental theorem of affine geometry j.b. sancho de salas departamento de matemáticas, universidad de extremadura 06006 badajoz, spain jsancho@unex.es received march 7, 2023 presented by j.m.f. castillo accepted july 11, 2023 abstract: we deal with a natural generalization of the classical fundamental theorem of affine geometry to the case of non bijective maps. this extension geometrically characterizes semiaffine morphisms. it was obtained by w. zick in 1981, although it is almost unknown. our aim is to present and discuss a simplified proof of this result. key words: fundamental theorem, semiaffine morphisms, parallel morphisms. msc (2020): 51a05, 51a15. introduction a map ϕ: rn → rm between real affine spaces is an affine morphism if it has equations of the form  y1 = a11x1 + · · · + a1nxn + b1 ... ym = am1x1 + · · · + amnxn + bm with aij,bi ∈ r. its equations are polynomials of degree ≤ 1 hence, in some sense, affine morphisms are the simplest maps, apart of constant maps. if moreover ϕ is bijective, that is, n = m and det (aij) 6= 0, then ϕ is an affinity. the fundamental theorem geometrically characterizes affinities: for n ≥ 2, collineations ϕ: rn → rn (bijections transforming lines into lines) are just affinities. the fundamental theorem holds more generally for affine spaces over arbitrary fields of scalars: let a, a′ be affine spaces of dimensions ≥ 2 over division rings k, k′, respectively (of orders 6= 2). the classical fundamental theorem states that collineations a → a′ are just semiaffinities. issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) mailto:jsancho@unex.es https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 2 j.b. sancho de salas this theorem was first proved by e. kamke [10] and is collected in many textbooks (see [2, 3, 4, 15]). more information about its history can be found in [11, pp. 51 – 52]. the classical theorem is restricted to bijective maps; it leaves open the question of a geometrical characterization of non-bijective semiaffine morphisms. in [9, part i, chapter v, theorem 1], frenkel characterized injective semiaffine maps, with an associated bijective ring morphism k → k′. in 1981, w. zick obtains a general result without any injective or surjective condition. to improve frenkel’s result, he introduces a notion of morphism preserving parallelism, valid for non-injective maps, and he removes the traditional (and artificial) condition that the ring map k → k′, associated with a semilinear map, be bijective. in our opinion, zick’s result is the ultimate version of the fundamental theorem of affine geometry. unfortunately, his work hasn’t been published and it seems to be almost totally unknown (we learned about its existence from the paper [13]). our purpose is to give a simplified proof of this result and to explain its interest for the foundations of affine geometry. the geometric characterization of affine maps is only one aspect of the fundamental theorem. it has also a role in the foundations of affine geometry. there essentially exist two ways to define affine space. on the one hand, a synthetic definition using axioms based on the intuitive properties of points, lines and parallelism. on the other hand, an algebraic definition using algebraic structures such as fields and vector spaces. both definitions are equivalent, but they apparently suggest very different notions of morphism between affine spaces. its equivalence is the substance of the fundamental theorem. this article is divided into three sections. in the first one, we recall the synthetic and algebraic definitions of affine space; their equivalence is not trivial and for its proof the reader is addressed to the literature. in the second section, we explain that both definitions of affine space induce different notions of morphism: parallel morphisms in the synthetic case and semiaffine morphisms in the algebraic case. the last section contains the proof of the general version of the fundamental theorem, which states the equivalence between parallel and semiaffine morphisms. the classical version for bijective maps is obtained as a consequence. the fundamental theorem of affine geometry 3 1. the definition of affine space the synthetic point of view a synthetic definition of affine space is given by means of terms and axioms that are evident to our geometric intuition, without using of coordinates or algebraic structures. in the literature there are several of these definitions. the definition that we state below is due to o. tamaschke [16]. we prefer this definition because it emphasizes parallelism as a primitive element in the concept of affine space (for a definition where parallelism is not involved, using only incidence axioms, see [17]). definition 1.1. an affine space is a set a 6= ∅ (whose elements are named points), with a family l of subsets (named lines) endowed with an equivalence relation ‖ (named parallelism), satisfying the following axioms: a1. any two different points lie in a unique line; a2. any line has at least two points; a3. (parallel axiom) given a line l and a point p there is a unique parallel line to l passing through p; a4. (similar triangles axiom) let a,b,c be three non-collinear points and let a′,b′ be two different points such that ab ‖ a′b′. the line parallel to ac through a′ and the line parallel to bc through b′ intersect at a point c′. definition 1.2. a subset s ⊆ a is said to be a subspace when it fulfills the following conditions: (a) the line joining any two different points of s is contained in s; (b) for any line l ⊆ s and any point p ∈ s, the parallel line to l passing through p also is contained in s. condition (b) is superfluous when the lines of a have at least three points. 4 j.b. sancho de salas definition 1.3. the dimension of a non empty subspace s ⊆ a is the supremum of the naturals n such that there exists a strictly increasing sequence of subspaces ∅ 6= s0 ⊂ ···⊂ sn = s. points and lines are just subspaces of dimension 0 and 1, respectively. subspaces of dimension 2 are named planes. definition 1.4. two non-empty subspaces s and s′ are said to be parallel (we put s ‖ s′) when for any line l ⊆ s there is a parallel line l′ ⊆ s′ and, conversely, for any line l′ ⊆ s′ there is a parallel line l ⊆ s. the algebraic point of view in undergraduate courses it is usual to define affine spaces in terms of certain algebraic structures (fields or, more generally, division rings, vector spaces or group actions). definition 1.5. an affine space is a set a 6= ∅ (whose elements are named points) together with a vector space v over a division ring k and a map + : a × v → a, (p,v) 7→ p + v, such that the following axioms are satisfied: (1) (p + v1) + v2 = p + (v1 + v2) for all p ∈ a, v1,v2 ∈ v ; (2) p + v = p ⇐⇒ v = 0 for all p ∈ a, v ∈ v ; (3) given two points p, p̄ ∈ a there is a vector v ∈ v (necessarily unique) such that p̄ = p + v. an affine space (a,v, +) will be simply denoted a. note that each vector v ∈ v defines a bijective map τv : a → a, p 7→ p+v, named translation with respect to v. definition 1.5 captures the idea that an affine space is a set with a distinguished group of transformations (the group of translations) isomorphic to the additive group (v, +) of a vector space. alternatively, in the language of group actions, one may define an affine space as a set a endowed with a free and transitive action + : a ×v → a of the additive group of a vector space v . the dimension of an affine space a is defined to be the dimension of the vector space v (possibly infinite). the fundamental theorem of affine geometry 5 1.6. coordinates. let an be an affine space of finite dimension n. an affine reference is a sequence {p0,v1, . . . ,vn} where p0 ∈ a is a point (named origin of the reference) and {v1, . . . ,vn} is a basis of v . now, given a point p ∈ an we have p = p0 + v for a unique vector v ∈ v . writing v = x1v1 + · · · + xnvn, we obtain p = p0 + x1v1 + · · · + xnvn for a unique sequence of scalars x1, . . . ,xn ∈ k, named affine coordinates of p. assigning its coordinates to each point, we obtain a bijection an ∼−−−→ k × n · · ·×k. definitions 1.7. a non-empty subset s ⊆ a is a affine subspace when it is s = p + w := {p + w : w ∈ w} where p ∈ a is a point and w ⊆ v is a vector subspace. then w is said to be the direction of s. we agree that the empty subset also is a subspace. remark that a non-empty subspace s of direction w is an affine space (s,w, +). so the dimension of s is the dimension of its direction w as a vector space. points are just subspaces of dimension 0. subspaces of dimension 1 are named lines and subspaces of dimension 2, planes. definition 1.8. two non-empty subspaces s = p + w and s′ = p′ + w ′ are said to be parallel when both have the same direction: s ‖ s′ ⇐⇒ w = w ′. note that parallel subspaces have the same dimension. of course, two distinct lines are parallel if and only if they are coplanar and do not intersect. equivalence of definitions the algebraic definition 1.5 of affine space is deep and very convenient for an efficient development of affine geometry. although, in a certain sense it is not a primary definition, since it requires some motivation or explanation. there is a great gap between ordinary spatial intuition, with its informal ideas of point, line, parallelism, and an abstract definition in terms of algebraic 6 j.b. sancho de salas structures such as a field of scalars or a vector space. the emergence of these structures is a beautiful surprise, formulated as theorem 1.9 below. it is an easy exercise to check that any affine space, in the sense of the algebraic definition 1.5, fulfills the synthetic definition 1.1. the converse is not so easy. an essential role is played by desargues’s theorem, which holds in any algebraic affine space and also in any synthetic affine space of dimension ≥ 3. however, there exist synthetic affine planes where desargues’s theorem fails to hold (an easy example is the moulton plane [2]). with the exception of non-desarguesian planes, the algebraic and synthetic notions of affine space are equivalent: theorem 1.9. let a be a synthetic affine space of dimension ≥ 3 (or of dimension 2 and desarguesian). there exist, canonically associated to a, a division ring k, a k-vector space v and a map a×v +−−→ a such that: (i) (a,v, +) is an algebraic affine space; (ii) subspaces of the algebraic affine space (a,v, +) are just subspaces of the synthetic affine space a. variations of this theorem can be found in [2, 4, 16]. 2. which are the morphisms between affine spaces? the algebraic and synthetic definitions of affine space, although equivalent by theorem 1.9, suggest different definitions of “morphism” between affine spaces. we will show that the fundamental theorem states the equivalence of both notions of morphism. morphisms in the algebraic case in the case of algebraic structures, such as group, ring or vector space, the (homo)morphisms are defined to be maps preserving the structure. typically, an algebraic structure consists of some sets (and their direct products) with certain maps between them (named operations) satisfying certain identities (named axioms). a map between two structures of the same kind is said to preserve the structure when it is compatible with the operations in an obvious sense. for example, a group is a set g with operations g×g ·−→ g, g inv−−−→ g, ∗ 1−−→ g, satisfying the usual axioms. a morphism between groups ϕ: g → g ′ the fundamental theorem of affine geometry 7 is defined to be a map preserving the structure, in the sense that the following diagrams are commutative, g×g ϕ×ϕ −−−−→ g ′ ×g ′ · y y· g ϕ −−−−→ g ′ , g ϕ −−−−→ g ′ inv y yinv g ϕ −−−−→ g ′ , ∗ ∗ 1 y y1 g ϕ −−−−→ g ′ . the commutativity of the first diagram states that ϕ(g1·g2) = ϕ(g1)·ϕ(g2) for all g1,g2 ∈ g. the other two diagrams state that ϕ(g−1) = ϕ(g)−1 for all g ∈ g and ϕ(1) = 1 (in fact both follow from the former condition, due to the axioms of group). hence a map ϕ: g → g ′ preserves the structure when it fulfills the condition ϕ(g1 ·g2) = ϕ(g1) ·ϕ(g2) for all g1,g2 ∈ g, which is the standard definition of group morphism. now let us consider the case of vector spaces. a vector space is a list (v,k, ·) where v is an abelian group, k is a division ring and k ×v ·−→ v , (λ,v) 7→ λ · v, is a map satisfying certain axioms. therefore, a morphism between vector spaces (v,k, ·) and (v ′,k′, ·) should be defined by two maps φ: v → v ′, σ : k → k′, where φ is a morphism of groups, σ is a morphism of rings and the following diagram is commutative k ×v (σ,φ) −−−−→ k′ ×v ′ · y y· v φ −−−−→ v ′ that is to say, φ(λ ·v) = σ(λ) ·φ(v) for all λ ∈ k, v ∈ v . assuming that φ is not null then σ is uniquely determined by φ. so we arrive to the following definition. definition 2.1. let v , v ′ be vector spaces over division rings k, k′, respectively. a map ϕ: v → v ′ is said to be semilinear when: (a) it is additive: ϕ(v1 + v2) = ϕ(v1) + ϕ(v2) for all v1,v2 ∈ v ; (b) there is a ring morphism σ : k → k′ such that ϕ(λv) = σ(λ)ϕ(v) for all λ ∈ k, v ∈ v . we do not require that σ : k → k′ be surjective. analogously, a morphism between algebraic affine spaces (a,v, +) and (a′,v ′, +) should be defined by two maps ϕ: a → a′, ~ϕ: v → v ′, where ~ϕ is 8 j.b. sancho de salas semilinear, satisfying the commutative diagram a×v (ϕ,~ϕ) −−−−→ a′ ×v ′ + y y+ a ϕ −−−−→ a′ that is to say, ϕ(p + v) = ϕ(p) + ~ϕ(v). this equality implies that ~ϕ is determined by ϕ. so we arrive to the following definition. definition 2.2. let (a,v, +) and (a′,v ′, +) be affine spaces over division rings k and k′, respectively. a map ϕ: a → a′ is a semiaffine morphism when there is a semilinear map ~ϕ: v → v ′ such that ϕ(p + v) = ϕ(p) + ~ϕ(v) ∀p ∈ a , v ∈ v . the semilinear map ~ϕ is unique and it is named differential of ϕ. a semiaffine morphism ϕ: a → a′ is a semiaffine isomorphism or a semiaffinity when both ϕ and the ring morphism σ : k → k′ (associated to ~ϕ) are bijective. in such case the inverse map ϕ−1 : a′ → a also is a semiaffine isomorphism. the prefix semi in the terms semilinear, semiaffine, semiaffinity is deleted when k = k′ and the associated ring morphism σ : k → k is the identity. 2.3. any vector space v has an underlying structure of affine space (a = v, v, +), where the map + : a×v → a is just the addition of vectors, a×v = v ×v +−−−−→ v = a . conversely, given an affine space (a,v, +) and a fixed point p0 ∈ a we have an affine isomorphism v ∼−−−−→ a , v 7−→ p0 + v . observe that 0 7→ p0. this isomorphism supports the colloquial statement that an affine space is a vector space where we have forgotten the origin; once we fix a point p0 ∈ a as the origin we have an identification a = v . 2.4. let v and v ′ be vector spaces, hence also affine spaces, over division rings k and k′, respectively. a map ϕ: v → v ′ is a semiaffine morphism if and only if it is v ϕ −−−−→ v ′ , ϕ(v) = ~ϕ(v) + b , where ~ϕ: v → v ′ is a semilinear map and b := ϕ(0). the fundamental theorem of affine geometry 9 2.5. equations of a semiaffine morphism. let ϕ: an → a′m be a semiaffine morphism, between affine spaces of finite dimension, with associated ring morphism k → k′, x 7→ x′. given affine coordinates {x1, . . . ,xn} and {y1, . . . ,ym} of an and a′m, respectively, the equations of ϕ are  y1 = x ′ 1a11 + · · · + x ′ na1n + b1 ... ym = x ′ 1am1 + · · · + x ′ namn + bm where (aij) is the matrix of the semilinear map ~ϕ: v → v ′, and (b1, . . . ,bm) are the coordinates of b = ϕ(p0). morphisms in the synthetic case now, the synthetic definition of affine space is not algebraic as the previous structures, so that it is not evident what does it mean to say that a map ϕ: a → a′, between synthetic affine spaces, preserves the structure. let us consider the proposal of w. zick. first, we introduce the following 2.6. notation. given points p0,p1 ∈ a, let p0∨p1 be the smallest affine subspace containing p0,p1. if p0,p1 are distinct points, then p0 ∨p1 is a line. if p0 = p1 then p0 ∨p1 = p0 = p1. recall that any two parallel subspaces have equal dimension. therefore the expression (a∨ b) ‖ (c∨d) means that both a∨ b and c∨d are parallel lines or both are points (a = b and c = d). definition 2.7. (zick) a map ϕ: a → a′, between affine spaces, is a parallel morphism when (a∨ b) ‖ (c∨d) ⇒ ( ϕ(a) ∨ϕ(b) ) ‖ ( ϕ(c) ∨ϕ(d) ) for all a,b,c,d ∈ a. since the synthetic notion 1.1 of affine space is based on the relations of collinearity and parallelism, it seems reasonable at first sight to say that morphism ϕ: a → a′ preserving the structure are parallel morphisms. this intuition is confirmed by the fundamental theorem 3.9, stating the equivalence between the semiaffine and the parallel morphisms. in conclusion, theorem 1.9 and the fundamental theorem 3.9 are the mathematical formulation of the equivalence between the algebraic and synthetic points of views on affine geometry. 10 j.b. sancho de salas 3. fundamental theorem semiaffine morphisms are parallel morphisms lemma 3.1. any parallel morphism ϕ: a → a′ satisfies the property x0 ∈ x1 ∨x2 ⇒ ϕ(x0) ∈ ϕ(x1) ∨ϕ(x2) for any x0,x1,x2 ∈ a. proof. if x1 = x2 then x0 = x1 = x2 and it is clear. otherwise x0 is not x1 or x2, let us assume that x0 6= x2. we have x1 ∨ x2 = x0 ∨ x2, hence x1 ∨ x2 ‖ x0 ∨ x2, so that ϕ(x1) ∨ ϕ(x2) ‖ ϕ(x0) ∨ ϕ(x2), and then ϕ(x1) ∨ϕ(x2) = ϕ(x0) ∨ϕ(x2) 3 ϕ(x0). as a consequence, the restriction of a parallel morphism ϕ: a → a′ to a line is constant or it is an injection into a line of a′. now the next statement directly follows from the definition. 3.2. a map ϕ: a → a′, between affine spaces, is a parallel morphism if and only if it satisfies the following condition: for any two parallel lines l1,l2 ⊆ a the restrictions ϕ|l1 and ϕ|l2 are both constant or both injective, and in such case ϕ(l1) ⊆ l′1, ϕ(l2) ⊆ l ′ 2, where l′1,l ′ 2 are two parallel lines in a ′. proposition 3.3. any semiaffine morphism ϕ: a → a′ is a parallel morphism. proof. let l1 = p1 + 〈v〉, l2 = p2 + 〈v〉 be two parallel lines of a. if ~ϕ(v) = 0 then ϕ(l1) = ϕ(p1) and ϕ(l2) = ϕ(p2). if ~ϕ(v) 6= 0 then ϕ embeds the lines li = pi + 〈v〉 into the lines l′i = ϕ(pi) + 〈~ϕ(v)〉, (i = 1, 2), which are parallel. by 3.2, ϕ is a parallel morphism. parallel morphisms are semiaffine in this subsection, v , v ′ are vector spaces over division rings k, k′, respectively. lemma 3.4. ([6]) let ϕ,φ: v → v ′ be additive maps. if for any x ∈ v we have φ(x) ∈ k′·ϕ(x) and the image of ϕ contain two linearly independent vectors, then there is a scalar λ ∈ k′ such that φ = λ ·ϕ. the fundamental theorem of affine geometry 11 proof. for any x ∈ v \ ker ϕ we have φ(x) = λx ·ϕ(x) for a unique scalar λx ∈ k′. we have to show that λx does not depend on x. let x,y ∈ v \ker ϕ; we distinguish two cases. 1. ϕ(x) and ϕ(y) are linearly independent. then x,y,x+y ∈ v \ker ϕ and the equality φ(x+y) = φ(x)+φ(y) shows that λx+yϕ(x+y) = λxϕ(x)+λyϕ(y), that is to say, λx+yϕ(x)+λx+yϕ(y) = λxϕ(x)+λyϕ(y), hence λx = λx+y = λy. 2. ϕ(x) and ϕ(y) are linearly dependent. take z ∈ v such that ϕ(z) is linearly independent of both vectors. according to the former case, we have λx = λz = λy. proposition 3.5. (zick) let ϕ: v → v ′ be an additive map, such that ϕ(kx) ⊆ k′ϕ(x) for all x ∈ v , and such that the image contains two linearly independent vectors. then ϕ: v → v ′ is semilinear. proof. ([6]) given λ ∈ k we define the additive map φλ(x) := ϕ(λx). by lemma 3.4, there is a scalar σ(λ) ∈ k′ such that φλ = σ(λ)ϕ, that is to say, ϕ(λx) = σ(λ)ϕ(x). we have to check that σ : k → k′ is a ring morphism. taking x ∈ v \ ker ϕ we have ϕ((λ1 + λ2)x) = σ(λ1 + λ2)ϕ(x) and moreover ϕ((λ1 + λ2)x) = ϕ(λ1x + λ2x) = σ(λ1)ϕ(x) + σ(λ2)ϕ(x) = (σ(λ1) + σ(λ2))ϕ(x) , so that σ(λ1 + λ2) = σ(λ1) + σ(λ2). analogously we prove that σ(λ1λ2) = σ(λ1)σ(λ2). recall (see 2.3) that a vector space v also is an affine space. lemma 3.6. let ϕ: v → v ′, x 7→ x′, be a parallel morphism. if ϕ(0) = 0 and dim〈ϕ(v )〉≥ 2, then ϕ: v → v ′ is additive. proof. since ϕ transforms the parallelogram (eventually degenerated) with vertices 0,x,y,x + y into a parallelogram 0,x′,y′, (x + y)′, we have (x + y)′ = λx′ + y′ = x′ + µy′ (1) for certain λ,µ ∈ k′. 12 j.b. sancho de salas when x′ /∈ 〈y′〉 then (x + y)′ = x′ + y′, because either x′ = 0, so that we put λx′ = x′ en (1), or x′ and y′ are linearly independent (so that λ = µ = 1). the case y′ /∈ 〈x′〉 is similar. otherwise we have 〈x′〉 = 〈y′〉. by hypothesis, there exists z ∈ v such that z′ /∈ 〈x′〉 = 〈y′〉 and by (1) we also have z′ /∈ 〈(x + y)′〉. by the former case, we have (y + z)′ = y′ + z′ /∈ 〈x′〉 and x′ + y′ + z′ = x′ + (y + z)′ = (x + y + z)′ = (x + y)′ + z′ , hence x′ + y′ = (x + y)′. lemma 3.7. let ϕ: v → v ′ be a parallel morphism. if ϕ(0) = 0 then we have ϕ(kx) ⊆ k′ϕ(x) for all x ∈ v . proof. by lemma 3.1 we have ϕ(x1 ∨x2) ⊆ ϕ(x1) ∨ϕ(x2), so that ϕ(kx) = ϕ(0 ∨x) ⊆ ϕ(0) ∨ϕ(x) = 0 ∨ϕ(x) = k′ϕ(x) . proposition 3.8. let ϕ: v → v ′ 6= 0 be a parallel morphism. if the image of ϕ is not contained in an affine line, then ϕ is a semiaffine morphism, that is to say, we have ϕ(x) = ~ϕ(x) + b , where ~ϕ: v → v ′ is a semilinear map and b = ϕ(0). proof. composing ϕ with a translation we may assume that ϕ(0) = 0. the above two lemmas show that ϕ: v → v ′ fulfills the hypotheses of proposition 3.5, hence ϕ: v → v ′ is semilinear. according to 2.3 any affine space is isomorphic (an affinity) to its direction: a ' v . combining 3.3 and 3.8 we finally obtain 3.9. fundamental theorem (zick [18]) let ϕ: a → a′ 6= ∗ be a map such that the image is not contained in a line. then ϕ is a semiaffine morphism if and only if it is a parallel morphism. 3.10. case k = k′ = r. it is elementary that the only ring morphism r → r is the identity (see [15, p. 86]). so, in the case of real affine spaces, we may drop the prefix semi in the above theorem. a stronger result may be obtained: the fundamental theorem of affine geometry 13 let ϕ: a → a′ be a map, between real affine spaces, such that the image is not contained in a line. then ϕ is an affine morphism if and only if for any p0,p1,p2 ∈ a the following condition holds: p0 ∈ p1 ∨p2 ⇒ ϕ(p0) ∈ ϕ(p1) ∨ϕ(p2) . (2) this statement is an easy consequence of the following more general result (lenz [14, hilfssatz 3]): let p and p′ be real projective spaces, with dim p < ∞, and let u ⊆ p be an open set. let ϕ: u → p′ be a map satisfying (2) such that the image is not contained in a line. then ϕ is a linear map in homogeneous coordinates. lenz’s result was generalized by frank [8] for projective spaces, endowed with a linear topology, over division rings. 3.11. over arbitrary division rings, maps a → a′ satisfying condition (2) are algebraically characterized as fractional semiaffine morphisms (see [19]). a map ϕ: a → a′ is called a lineation if the image by ϕ of any three collinear points are collinear. it is a weaker condition than (2). see [5] for a version of the fundamental theorem for surjective lineations. in [1], several generalizations of the fundamental theorem are obtained, where collinearity preservation is assumed only for a finite number of directions of lines. 3.12. in the case of projective spaces, faure-frölicher [7] and havlicek [12] extended the classical fundamental theorem of projective geometry to non necessarily bijective maps. see also [6]. the classical fundamental theorem now we examine the case of bijective maps to obtain the classical version of the fundamental theorem. lemma 3.13. let v be a vector space over a division ring k with |k| 6= 2. let w ⊆ v be a subset such that: (a) 0 ∈ w ; (b) if w1,w2 ∈ w , then (1 − t)w1 + tw2 ∈ w for all t ∈ k (that is to say, w1,w2 ∈ w ⇒ w1 ∨w2 ⊆ w). then w is a vector subspace. 14 j.b. sancho de salas proof. it is enough to show that 〈w1,w2〉⊆ w whenever w1,w2 ∈ w . remark that if w ∈ w then 〈w〉 ⊆ w : for any t ∈ k we have tw = (1 − t)0 + tw ∈ w . now, given w1,w2 ∈ w , for all x,y ∈ k we have xw1,yw2 ∈ w , hence w 3 (1 − t)xw1 + tyw2 = x̄w1 + ȳw2 , where x̄ := (1 − t)x, ȳ := ty . taking t 6= 0, 1 (since |k| 6= 2), the values of x̄, ȳ are arbitrary, so that the vector x̄w1 + ȳw2 is any vector of 〈w1,w2〉. any affine subspace s ⊆ a satisfies x1,x2 ∈ s ⇒ x1 ∨x2 ⊆ s . that is to say, any affine subspace s ⊆ a contains the line joining any two different points of s. reciprocally, proposition 3.14. let a be an affine space over a division ring k, with |k| 6= 2. if a subset s ⊆ a contains the line joining any two different points of s, then s is an affine subspace. proof. fix p0 ∈ s and consider the affine isomorphism v ' a, v 7→ p0 +v. via this isomorphism, the subset s corresponds to a subset w ⊆ v fulfilling the conditions of the lemma, so that w is a vector (hence affine) subspace of v and, therefore, s is a subspace of a. definition 3.15. a bijective map ϕ: a → a′ is a collineation if the image of each line l of a is a line ϕ(l) of a′. note that the inverse of a collineation is also a collineation. theorem 3.16. let a and a′ be affine spaces of dimensions ≥ 2 over division rings k and k′, respectively, with |k|, |k′| 6= 2. a bijective map ϕ: a → a′ is a semiaffinity if and only if it is a collineation. proof. 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[19] w. zick, der satz von martin in desargues’schen affinen räumen, inst. f. math., universität hannover preprint nr. 134 (1981), 1 – 13. the definition of affine space which are the morphisms between affine spaces? fundamental theorem � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 2 (2022), 195 – 210 doi:10.17398/2605-5686.37.2.195 available online july 4, 2022 genus zero of projective symplectic groups h.m. mohammed salih, rezhna m. hussein department of mathematics, faculty of science, soran university kawa st. soran, erbil, iraq havalmahmood07@gmail.com , rezhnarwandz@gmail.com received january 17, 2022 presented by a. turull accepted may 23, 2022 abstract: a transitive subgroup g ≤ sn is called a genus zero group if there exist non identity elements x1, . . . ,xr ∈ g satisfying g = 〈x1,x2, . . . ,xr〉, ∏r i=1 xi = 1 and ∑r i=1 ind xi = 2n − 2. the hurwitz space hinr (g) is the space of genus zero coverings of the riemann sphere p1 with r branch points and the monodromy group g. in this paper, we assume that g is a finite group with psp(4,q) ≤ g ≤ aut(psp(4,q)) and g acts on the projective points of 3-dimensional projective geometry pg(3,q), q is a prime power. we show that g possesses no genus zero group if q > 5. furthermore, we study the connectedness of the hurwitz space hinr (g) for a given group g and q ≤ 5. key words: symplectic group, fixed point, genus zero group. msc (2020): 20b15, 20c33. 1. introduction a one dimensional compact manifold is called riemann surface. topologically, such surfaces are either spheres or tori which have been glued together. the number of holes so joined is called the genus. let f : x → p1 be a meromorphic function from a compact connected riemann surface x of genus g into the riemann sphere p1. for every meromorphic function there is a positive integer n such that all points have exactly n preimages. so every compact riemann surface can be made into the branched covering of p1. it is known that one of the basic strategies of the whole subject of algebraic topology is to find methods to reduce topological problem about continuous maps and spaces into pure algebraic problems about homomorphisms and groups by using the fundamental group. the points p are called the branch points of f if |f−1(p)| < n. it is well known that the set of branch points is finite and it will be denoted by b = {p1, . . . ,pr}. for q ∈ p1 \b, the fundamental group π1(p1 \b,q) is a free group which is generated by all homotopy classes of loops γi winding once around the point pi. these loops of generators γi issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.37.2.195 mailto:havalmahmood07@gmail.com mailto:rezhnarwandz@gmail.com https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 196 h.m.m. salih, r.m. hussein are subject to the single relation that γ1 · . . . · γr = 1 in π1(p1 \ b,q). the explicit and well known construction of hurwitz shows that a riemann surface x with n branching coverings of p1 is defined in the following way: consider the preimage f−1(q) = {x1, . . . ,xn}, every loop in γ in p1 \b can be lifted to n paths γ̃1, . . . , γ̃n where γ̃i is the unique path lift of γ and γ̃i(0) = xi for every i. the endpoints γ̃i(1) also lie over q. that is γ̃i(1) = xσ(i) in f −1(q) where σ is a permutation of the indices {1, . . . ,n} and it depends only on γ. thus it gives a group homomorphism φ : π1(p1 \b,q) → sn . the image of φ is called the monodromy group of f and denoted by g = mon(x,f). since x is connected, then g is a transitive subgroup of sn . thus a group homomorphism is determined by choosing n permutations xi = φ(γi), i = 1, . . . ,r and satisfying the relations g = 〈x1,x2, . . . ,xr〉, (1) r∏ i=1 xi = 1, xi ∈ g# = g\{1}, i = 1, . . . ,r, (2) r∑ i=1 ind xi = 2(n + g − 1), (3) where ind x = n − orb(x), orb(x) is the number of orbits of the group generated by x on ω where |ω| = n. equation (3) is called the riemann hurwitz formula. a transitive subgroup g ≤ sn is called a genus g group if there exist x1, . . . ,xr ∈ g satisfying (1), (2) and (3) and then we call (x1, . . . ,xr) the genus g system of g. if the action of g on ω is primitive, we call g a primitive genus g group and (x1, . . . ,xr) a primitive genus g system. a group g is said to be almost simple if it contains a non-abelian simple group s and s ≤ g ≤ aut(s). in [4], kong worked on almost simple groups whose socle is a projective special linear group. also, she gave a complete list for some almost simple groups of lie rank 2 up to ramification type in her phd thesis for genus 0, 1 and 2 system. furthermore, she showed that the almost simple groups with socle psl(3,q) do not possess genus low tuples if q ≥ 16. in [6], mohammed salih gave the classification of some almost simple groups with socle psl(3,q) up to braid action and diagonal conjugation. the symplectic group sp(n,q) is the group of all elements of gl(n,q) preserving a non-degenerate alternating form; the non degenerate leads to n being even. the projective symplectic group psp(n,q) is obtained by from sp(n,q) on factoring it by the subgroup of scalar matrices it contains (which has order at most 2) [1]. in this paper we consider a finite group g with genus zero of projective symplectic groups 197 psp(4,q) ≤ g ≤ aut(psp(4,q)) and g acts on the projective points of 3-dimensional projective geometry pg(3,q), q is a prime power. we will now describe the work carried out in this paper. in the second section we review some basic concepts and results will be used later. in the third section, we provide some basic facts for computing fixed points and generating tuples. finally, we show that g possesses no genus 0 group if psp(4,q) ≤ g ≤ aut(psp(4,q)) and g acts on the projective points of 3-dimensional projective geometry pg(3,q), q is a prime power and q > 5. furthermore, we study the connectedness of the hurwitz space g if q ≤ 5. 2. preliminary we begin by introducing some definitions and stating a few results which will be needed later. assume that g is a finite permutation group of degree n. the signature of the r-tuple x = (x1, . . . ,xr) is the r-tuple d = (d1, . . . ,dr) where di = o(xi). we assume that di ≤ dj if i ≤ j, because of the braid action on x. the following result will tell us the tuple x can not generate g, where g = pgl(4,q) or psl(4,q) if (ii), (iii) and (iv) below hold. so, setting a(d) = ∑r i=1 di−1 di , we have a(d) ≥ 85 42 . proposition 2.1. ([3]) assume that a group g acts transitively and faithfully on ω and |ω| = n. let r ≥ 2, g = 〈x1, . . . ,xr〉, ∏r i=1 xi = 1 and o(xi) = di > 1, i = 1, . . . ,r. then one of the following holds: (i) ∑r i=1 di−1 di ≥ 85 42 ; (ii) r = 4, di = 2 for each i = 1 and g ′′ = 1; (iii) r = 3 and (up to permutation) (d1,d2,d3) = (a) (3, 3, 3), (2, 3, 6) or (2, 4, 4) and g ′′ = 1; (b) (2, 2,d) and g is dihedral; (c) (2, 3, 3) and g ∼= a4; (d) (2, 3, 4) and g ∼= s4; (e) (2, 3, 5) and g ∼= a5; (iv) r = 2 and g is cyclic. for a permutation x of the finite set ω, let fix(x) denote the fixed points of x on ω and f(x) = |fix(x)| is the number of fixed points of x. note that the conjugate elements have the same number of fixed points. the following result provides a useful connection between fixed points and indices. 198 h.m.m. salih, r.m. hussein lemma 2.2. ([3]) if x is a permutation of order d on a set of size n, then ind x = n − 1 d ∑ y∈〈x〉f(y) where 〈x〉 is the cyclic group generated by x. the fixed point ratio of x is defined by fpr(x) = f(x) n . the codimension of the largest eigenspace of a linear transformation ḡ in gl(n,q) is denoted by v(ḡ). ω denotes the set of the projective points of projective geometry pg(n− 1,q) that is the set of 1-dimensional subspaces of vector space over a finite field gf(q). in this paper we take |ω| = q n−1 q−1 and n = 4, so we have |ω| = q3 + q2 + q + 1. the center of gl(n,q) is the set of all scalar matrices and denoted by z(gl(n,q)). the projective general linear group and the projective special linear group are defined by pgl(n,q) = gl(n,q) z(gl(n,q)) and psl(n,q) = sl(n,q) z(sl(n,q)) respectively, where z(sl(n,q)) = sl(n,q)∩z(gl(n,q)). they act primitively on ω. let 〈v〉 ∈ ω be a fixed point of g ∈ pgl(n,q) and let ḡ be an element in the preimage of g in gl(n,q) that fixes 〈v〉. the fixed points of g are the 1-spaces spanned by eigenvectors of ḡ. so we classify non identity elements in pgl(4,q) by their fixed points as follows: table 1: number of fixed points v(ḡ) type of eigenspaces of ḡ ∈ gl(n, q) number of fixed points of g ∈ pgl(4, q) 4 no eigenspace 0 3 one 1-dimensional eigenspace 1 3 two 1-dimensional eigenspaces 2 3 three 1-dimensional eigenspaces 3 2 one 2-dimensional eigenspace q + 1 1 one 3-dimensional eigenspace q2 + q + 1 2 one 1-dimensional and one 2-dimensional eigenspaces q + 2 2 one 2-dimensional and one 2-dimensional eigenspaces 2q + 2 1 one 1-dimensional and one 3-dimensional eigenspaces q2 + q + 2 2 one 1-dimensional, one 1-dimensional and one 2-dimensional eigenspaces q + 3 according to table 1, we have two cases. if v(ḡ) = 1, then g fixes q2 +q+ 1 or q2 + q + 2 points. otherwise, it fixes at most 2q + 2 points. from this fact, genus zero of projective symplectic groups 199 we will show that there are no genus zero systems for psl(4,q) and pgl(4,q) when q > 37. the following result is an interesting tool to compute β in the next section. lemma 2.3. (scott bound, [7]) let g ≤ gl(n,q). if a triple x = (x1,x2,x3) satisfies g = 〈x1,x2,x3〉 and x1x2x3 = 1, then v(xi) + v(xj) ≥ n where i 6= j and 1 ≤ i,j ≤ 3. in particular if n ≥ 3 and i 6= j, then v(xi) ≥ 2 or v(xj) ≥ 2. lemma 2.4. ([5]) if 1 n ∑r i=1 ∑di−1 j=1 f(xj ) di < a(d)−2, then d is not a genus zero system. 3. existence of genus zero system now, we are going to apply lemma 2.4, to exclude all signatures which do not satisfy the riemann hurwitz formula. as a result, we will obtain theorem 3.1. let f be the set of elements with q2 + q + 1 or q2 + q + 2 fixed points in pgl(4,q). so we have fpr(x) ≤   q2 + q + 2 n if x ∈ f, 2q + 2 n if x /∈ f. assume that α = q2+q+2 n and γ = 2q+2 n . combining the riemann hurwitz formula as done in [4], we obtain the following inequality a(d) ≤ 2 + � + β(α−γ) 1 −γ (4) where � = −2 n and β = ∑r i=1 |〈xi〉#∩f| di . if α = γ in inequality (4), then we obtain the following a(d) ≤ 2 + � 1 −α (5) and we have fpr(x) ≤ α. now bound β for the tuple x = (x1, . . . ,xr). following [4], if xi ∈ f , then every power of xi that are non-identity are also in f , because if any point fixed by xi, then it is also fixed by x l i. therefore, f(xi) ≤ f(xli). in this situation, there are di − 1 elements in f in 〈xi〉. if xi /∈ f, then there are φ(di) generators in 〈xi〉 where φ is the euler’s function. 200 h.m.m. salih, r.m. hussein all of these generators are not in f either, so there are at most di−φ(di)−1 elements in f in 〈xi〉. we obtain β 6 ∑ xi∈f di − 1 di + ∑ xi /∈f di −φ(di) − 1 di . notice that lemma 2.3 will tell us for the tuple of length 3, that at most one element lies in f. in pgl(4,q) and psl(4,q), α = q2+q+2 q3+q2+q+1 and γ = 2q+2 q3+q2+q+1 , � = −2 q3+q2+q+1 for every genus 0 tuples. let q ≥ 16, then using inequality (5), we have a(d) ≤ 32 15 . if r ≥ 4, then a(d) ≥ a((2, 2, 2, 3)) = 13 6 . but 32 15 < 13 6 . so the number of branch points r must be 3. now we are looking for signatures which satisfy the inequality 85 42 ≤ a(d) ≤ 32 15 . this leads d only can be (2, 3,d3) with 7 ≤ d3 ≤ 30, (2, 4,d3) with 5 ≤ d3 ≤ 8, (2, 5, 5), (2, 5, 6), (3, 3, 4), (3, 3, 5). now, we compute β for all signatures which satisfy 85 42 ≤ a(d) ≤ 32 15 . d = β 6 (2, 3,n) with 7 6 n 6 30 41 30 (2, 4,n) with 5 6 n 6 8 5 4 (2, 5,n) with 5 6 n 6 6 13 10 (3, 3,n) with 4 6 n 6 5 11 12 in the above table the maximum β is 41 30 . now set β ≤ 41 30 and q ≥ 16. we substitute them in inequality (4) and we obtain that a(d) ≤ 9064 4335 . from this, we find all signatures d, which are the following: d = β 6 (2, 3,n) with 7 6 n 6 13 5 4 (2, 4,n) with 4 6 n 6 6 5 4 (3, 3, 4) 11 12 again, we choose the maximum β in the above table which is β ≤ 5 4 . so we put β ≤ 5 4 and q ≥ 16 in inequality (4) and hence a(d) ≤ 3012 1445 . therefore, all signatures are (2, 3,d3) with 7 ≤ d3 ≤ 12, (2, 4, 5),(2, 4, 6), (3, 3, 4). finally, for each signature d we can compute β and a(d) and put in inequality (4). so we can solve it and obtaining the values of q. genus zero of projective symplectic groups 201 theorem 3.1. if pgl(4,q) or psl(4,q) possesses genus zero system, then one of the following holds: (i) q ≤ 13; (ii) d and q as shown in the following table d β a(d) q (2,3,7) 6/7 85/42 16, 17, 19, 23, 25, 27, 29, 31, 32, 37 (2,3,8) 25/24 49/24 16, 17, 19, 23, 25 (2,3,9) 8/9 37/18 16, 17 (2,3,10) 7/6 31/15 16, 17 (2,3,12) 5/4 25/12 16 (2,4,5) 21/20 41/20 16, 17, 19 (2,4,6) 5/4 25/12 16 (3,3,4) 5/4 25/12 16 the next results are devoted to compute indices of elements of order 2, 3, 4 and 5 in psl(4,q). let ed be an element of order d in g. lemma 3.2. in g = psl(4,q): (i) if 2 q, then f(e2) = 0, ind e2 = n2 or f(e2) = 2q + 2, ind e2 = n−(2q+2) 2 . (ii) if 2 | q, then f(e2) = q2 + q + 1, ind e2 = n−(q2+q+1) 2 or f(e2) = q + 1, ind e2 = n−(q+1) 2 . proof. there are at most two conjugacy classes of involutions in g. for each such class, we give a representative e2. let z be the center of sl(4,q). (i) suppose that q is even. note that, since z = {i}, we can identify g with sl(4,q). take the involution e2 =   0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1   , whose only eigenvalue is 1. the corresponding eigenspace is e1 = { (v2,v2,v3,v4) t : v2,v3,v4 ∈ gf(q) } . 202 h.m.m. salih, r.m. hussein since it has dimension 3, from table 1 we achieve that e2 has q 2 + q + 1 fixed points. therefore, ind e2 = n−(q2+q+1) 2 . as representative of the other class, we can take e2 =   0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0   , whose only eigenvalue is 1. the associated eigenspace, which has dimension 2, is e1 = { (v2,v2,v4,v4) t : v2,v4 ∈ gf(q) } . from table 1, we obtain that e2 has q + 1 fixed points, whence ind e2 = n−(q+1) 2 . (ii) suppose that q is odd. the matrix e2 =   −1 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 1   is a non central element of sl(4,q) : its projective image in g has order 2. the eigenvalues of e2 are 1 and −1. the associated eigenspaces are e1 = { (0, 0,v3,v4) t : v3,v4 ∈ gf(q) } , e−1 = { (v1,v2, 0, 0) t : v1,v2 ∈ gf(q)}. they both have dimension 2: from table 1, we get that e2 has 2q + 2 fixed points. hence, ind e2 = n−(2q+2) 2 . (iii) suppose that q ≡ 3 (mod 4) (so z = {±i}). in this case, we have another conjugacy class of involutions. take e2 =   0 −1 0 0 1 0 0 0 0 0 0 −1 0 0 1 0   and note that g2 = −i ∈ z. hence, the projective image of e2 in psl(4,q) has order 2. the characteristic polynomial of e2 is (x 2 + 1)2, which has no root in gf(q). from table 1, we deduce that ind e2 = n 2 . genus zero of projective symplectic groups 203 in g = psl(4,q), if 3 | q − 5, then there are two conjugacy classes of elements of order 3. otherwise there are four conjugacy classes of elements of order 3. lemma 3.3. in g = psl(4,q): (i) if q ≡ 2 (mod 3), then f(e3) = 0, q + 1, ind e3 = 2n3 , 2 3 (n −q − 1). (ii) if q ≡ 1 (mod 3), then f(e3) ∈ {2q + 2,q + 3,q2 + q + 2}, ind e3 ∈{ 2 3 (n − 2q − 2), 2 3 (n −q − 3), 2 3 (n −q2 −q − 2) } . (iii) if q ≡ 0 (mod 3), then f(e3) ∈ {q + 1,q2 + q + 1}, ind e3 ∈ { 2 3 (n− q − 1), 2 3 (n −q2 −q − 1) } . proof. suppose element e3 has prime order 3 in g. then all powers of e3 except the identity have the same fixed points. now ind e3 = 2 3 (q3 + q2 + q + 1 −f(e3)) and ind e3 is an integer. (i) since 3 divides (q3 + q2 + q + 1 −f(e3)), this gives f(e3) ∈ {0, 3,q + 1, 2q + 2}. next, we will show that 2q + 2 and 3 can not exist. we check only the first 2q + 2. suppose that v is an eigenvector of ē3 then ē3v = λv for some nonzero number in gf(q). so v = iv = (ē3) 3v = λ3v. so λ3 = 1. but 3 q − 1, there is no element of order 3 in gf(q), we obtain λ = 1. so all eigenvector of ē3 belong to eigenvalue 1. suppose that ē3 fixes 2q + 2 points, ē3 has two 2-dimensional eigenspaces. both of them belong to 1. we get ē3 is the identity. this is a contradiction. in similar way, proving 3 can not exist. (ii) since 3 divides (q3 + q2 + q + 1 −f(e3)), then f(e3) ∈{1,q + 3, 2q + 2,q2 + q + 2}. next we will show that 1 can not exist. since 3|q − 1, then ē3 is conjugate to one of the following:  α 0 0 0 0 β 0 0 0 0 1 0 0 0 0 1   ,   α 0 0 0 0 α 0 0 0 0 β 0 0 0 0 β   ,   α 0 0 0 0 α 0 0 0 0 α 0 0 0 0 1   ,   β 0 0 0 0 β 0 0 0 0 β 0 0 0 0 1   where α = β−1 is a fixed element of order 3. this implies that f(e3) ∈ {q + 3, 2q + 2,q2 + q + 2}. therefore, ind e3 ∈ {23 (n − 2q − 2), 2 3 (n − q − 3), 2 3 (n −q2 −q − 2)}. (iii) since 3 divides (q3 +q2 +q+ 1−f(e3)), so f(e3) ∈{1,q+ 1,q2 +q+ 1}. in similar way, proving 1 can not exist. so ind e3 ∈{23 (n −q−1), 2 3 (n −q2 − q − 1)}. 204 h.m.m. salih, r.m. hussein lemma 3.4. in psl(4,q): (i) if q ≡ 1 (mod 4), then f(e4) = f(e24) = 0 and ind e4 = 3n 4 , or f(e4) = q + 1, f(e24) = 2q + 2 and ind e4 = 3n−(4q+4) 4 . (ii) if q ≡ 0 (mod 4), then f(e4) = q + 1, f(e24) = q 2 + q + 1 and ind e4 = 3n−(q2+3q+3) 4 , or f(e4) = 1, f(e 2 4) = q + 1 and ind e4 = 3n−(q+3) 4 . (iii) if q ≡ 3 (mod 4), then f(e4) ∈ { 0, 2q + 2,q + 3,q2 + q + 2,q + 3 } , f(e24) ∈{2q + 2︸︷︷︸ 3-times ,q2 + q + 2︸︷︷︸ 2-times } and ind e4 ∈ { 3n−(6q+6) 4 , 3n−(2q+2) 4 , 3n−(4q+8) 4 , 3n−3(q2+q+2) 4 , 3n−(q2+3q+8) 4 } . proof. the proof is similar as lemma 3.3. lemma 3.5. in psl(4,q): (i) if q ≡ 1 (mod 5), then f(e5) ∈ { q + 3, 2q + 2,q2 + q + 2 } and ind e5 = 4n−4f(e5) 5 . (ii) if q ≡ 2 (mod 5) or q ≡ 3 (mod 5), then f(e5) = 0 and ind e5 = 4n5 . (iii) if q ≡ 4 (mod 5), then f(e5) ∈{0,q + 1} and ind e5 = 4n5 , 4n−4(q+1) 5 . (iv) if q ≡ 0 (mod 5), than f(e5) ∈ { 1,q + 1,q2 + q + 1 } and ind e5 = 4n−4f(e5) 5 . proof. the proof is similar as lemma 3.3. table 2: indices of some elements in psl(4,q) q 16 17 19 23 25 ind e6 3498, 3578, 3588, 3626 ind e8 4536, 4548, 4556, 4560, 4552 6324 11130, 11106, 11118 14222, 14214, 14226 ind e9 3822 4640, 4624, 4636 ind e10 3884, 3916, 3788, 3896 ind e12 3930 genus zero of projective symplectic groups 205 proposition 3.6. in psl(4,q), there is no generating tuple of genus zero if 16 ≤ q ≤ 37. proof. from theorem 3.1, we have to deal with seven possible signatures in the different groups psl(4,q). since 7 |psl(4,q)| where q = 17, 19, 31, there is no signature (2, 3, 7) in psl(4,q) and 8 |psl(4, 16)|, there is no signature (2, 3, 8) in psl(4, 16). also, 10 |psl(4, 17)|, there is no signature (2, 3, 10) in psl(4, 17). if q = 16, 23, 25, 32, 37, then f(e7) = 1 and ind e7 = 6n−6 7 . if q = 27, then f(e7) = 0, q + 1 and ind e7 = 6n 7 , 6n−6(q+1) 7 . if q = 29, then f(e7) = 2q + 2, q + 3, q2 + q + 2 and ind e7 = 6n−6(2q+2) 7 , 6n−6(q+3) 7 , 6n−6(q2+q+2) 7 . we can compute the indices of elements of order 2 and 3 by lemma 3.2 and lemma 3.3. the sum of the indices of the signature (2, 3, 7) does not fit the riemann huwrtiz formula. by using lemma 3.2, lemma 3.3, lemma 3.4 and lemma 3.5, we can compute the indices of elements of orders 2, 3, 4 and 5. on the other hand, from table 2, we can get the indices of the elements of the other orders. therefore, the sum of the indices of the given signatures do not fit the riemann hurwitz formula. this completes the proof. lemma 3.7. the groups psl(4,q) do not possess genus zero system, if q = 7, 9. proof. the corresponding tuples of the following signatures satisfy the riemann hurwitz formula (2, 3,d), d ∈ {7, 8, 9, 14, 16, 19, 24, 28, 42} and (3, 3,d), d ∈{7, 8, 9, 14}. however none of them generate the group psl(4, 7) that is, do not satisfy (3). also, the associated tuple of the signature (2, 4, 6) fits the riemann hurwitz formula. it is not satisfied (3). the following gap codes can be used to show that there is no tuples satisfying the riemann hurwitz formula: cc:=list(conjugacyclasses(group),representative);; n:=degreeaction(group);; ind:=list(cc,x->n-length(orbits(group(x),[1..n])));; ss:=elements(ind);; s:=difference(ss,[ss[1]]);; poss:=restrictedpartitions(2n-2,s); lemma 3.8. the groups psl(4,q) do not possess genus zero system if q = 8, 11, 13. 206 h.m.m. salih, r.m. hussein proof. the proof is a straightforward computation. theorem 3.9. if g is the projective symplectic group with psp(4,q) ≤ g ≤ aut(psp(4,q)), q > 5, then g does not possess genus zero system. proof. since psp(4,q) is a subgroup of psl(4,q), then from theorem 3.1, proposition 3.6, lemma 3.7 and lemma 3.8, we show that the group psl(4,q) does not possess genus zero system. so is psp(4,q), as desired. 4. connected components of the hurwitz space the details of the following can be found in [6]. the computation shows that there are exactly 165 braid orbits of g. the degree and the number of the branch points are given in table 3. furthermore, we discuss the connectedness of the hurwitz space for these groups. our main result is theorem 4.1, which gives the complete classification of primitive genus 0 systems of g. theorem 4.1. up to isomorphism, there exist exactly 5 primitive genus zero groups g with psp(4,q) ≤ g ≤ aut(psp(4,q)) for q ≤ 5. the corresponding primitive genus zero groups are enumerated in tables 5, 6 and 4. this will be done by both the proof in algebraic topology and calculations of gap (groups, algorithms, programming) software [2]. proposition 4.2. if g = psp(4, 4).2 and |ω| = 85a, then hinr (g,c) is connected. proof. since we have just one braid orbit for all types c and the nielsen classes n(c) are the disjoint union of braid orbits. from [6, proposition 2.4], we obtain that the hurwitz space hinr (g,c) is connected. the proof of the following proposition is similar as proposition 4.2. proposition 4.3. if g = psp(4, 3) is the projective symplectic group and r > 3, then hinr (g,c) is connected. proposition 4.4. if g = psp(4, 4) and |ω| = 120a, then hinr (g,c) is disconnected. genus zero of projective symplectic groups 207 proof. since we have more than one braid orbits for some types c and the nielsen classes n(c) are the disjoint union of braid orbit. we obtain from [6, proposition 2.4] that the hurwitz space hinr (g,c) is disconnected. the proof of the following proposition is similar as proposition 4.4. proposition 4.5. if g = psp(4, 5) and |ω| = 156, then hinr (g,c) is disconnected. acknowledgements the authors would like thank the referees, whose comments and suggestions helped to improve the manuscript. references [1] j.d. dixon, b. mortimer, “ permutation groups ”, graduate text in mathematics, 163, springer-verlag, new york, 1996. [2] the gap group, gap – groups, algorithms, and programming, version 4.9.3, 2018. http://www.gap-system.org [3] r.m. guralnick, j. thompson, finite groups of genus zero, j. algebra 131 (1) (1990), 303 – 341. [4] x. kong, genus 0, 1, 2 actions of some almost simple groups of lie rank 2, phd thesis, wayne state university, 2011. [5] k. magaard, monodromy and sporadic groups, comm. algebra 21 (12) (1993), 4271 – 4297. [6] h.m. mohammed salih, hurwitz components of groups with socle psl (3,q), extracta math. 36 (1) (2021), 51 – 62. [7] l.l. scott, matrices and cohomology, ann. of math. (2) 105 (3) (1977), 473 – 492. http://www.gap-system.org 208 h.m.m. salih, r.m. hussein appendix table 3: genus zero groups: number of components number of connected components degree number of group up to isomorphism number of ramification types with r = 3 with r = 4 with r = 5 total 27 2 49 7 15 1 23 36 2 18 19 5 24 40a 2 20 39 4 43 40b 2 15 26 2 28 45 2 11 18 2 20 120a 1 1 4 4 85a 1 1 1 1 156a 1 2 22 22 total 13 117 136 28 1 165 table 4: genus zero systems for projective symplectic groups degree group ramification type n.o l.o ramification type n.o l.o 120a psp(4, 4) (2a,4b,5e) 4 1 85a psp(4, 4).2 (2c,4b,15a) 1 1 156a psp(4, 5) (2b,4b,5b) 11 1 (2b,4b,5a) 6 1 genus zero of projective symplectic groups 209 table 5: genus zero systems for psp(4, 3) degree ramification type n.o l.o ramification type n.o l.o (2a,5a,6b) 1 1 (2a,5a,6a) 1 1 (2a,6f,9b) 3 1 (2a,6f,9a) 3 1 (2a,6f,12b) 1 1 (2a,6f,12a) 1 1 (2a,6d,9b) 1 1 (2a,6d,12b) 1 1 27 (2a,6c,9a) 1 1 (2a,6c,12a) 1 1 (2a,4b,9b) 1 1 (2a,4b,9a) 1 1 (2a,4a,9b) 3 1 (2a,4a,9a) 3 1 (2a,4a,12b) 3 1 (2a,4a,12a) 3 1 (2a,2a,2a,6b) 1 9 (2a,2a,2a,6a) 1 9 (2b,4b,9b) 3 1 (2b,4b,9a) 3 1 (2b,4b,12b) 3 1 (2b,4b,12a) 3 1 36 (2b,6b,5a) 1 1 (2b,6b,9b) 1 1 (2b,4a,9b) 1 1 (2b,4a,9a) 1 1 (2b,2b,2b,6b) 1 9 (2b,2b,2b,6a) 1 9 (2b,6b,5a) 1 1 (2b,6b,9b) 1 1 40a (2b,6a,5a) 1 1 (2b,6a,9b) 1 1 (2b,4a,5a) 1 1 (2b,4a,9b) 1 1 (2b,4a,9b) 1 1 (2b,4a,9a) 1 1 (2b,6c,5a) 1 1 (2b,6b,5a) 1 1 40b (2b,5a,12b) 1 1 (2b,5a,12a) 1 1 (2b,5a,9b) 1 1 (2b,5a,9a) 1 1 (2b,4a,9a) 1 1 (2b,4a,9b) 1 1 45 (2b,6b,5a) 1 1 (2b,6a,5a) 1 1 210 h.m.m. salih, r.m. hussein table 6: genus zero systems for psp(4, 3) : 2 degree ramification type n.o l.o ramification type n.o l.o (2c,4a,10a) 1 1 (2c,5a,6a) 2 1 (2a,12b,9a) 1 1 (2d,6a,8a) 2 1 (2d,6a,10a) 2 1 (2c,4d,12a) 2 1 (2c,6g,12b) 2 1 (2c,4d,9a) 3 1 (2c,6g,10a) 2 1 (2c,6f,12a) 2 1 (2c,6f,9a) 3 1 (2c,4c,12b) 4 1 (2c,4c,8a) 6 1 (2c,4c,10a) 5 1 27 (2c,6e,12b) 3 1 (2c,6e,8a) 4 1 (2c,6e,10a) 4 1 (2d,2d,2c,6a) 1 24 (2d,2c,2c,6g) 1 48 (2d,2c,2c,4c) 1 112 (2d,2c,2c,6e) 1 78 (2d,2c,2c,4a) 1 20 (2a,2a,4a,6b) 1 1 (2c,2c,2c,4d) 1 96 (2c,2c,2c,6f) 1 108 (2a,2d,2c,12a) 1 12 (2a,2d,2c,9a) 1 18 (2a,2c,2c,12b) 1 24 (2a,2c,2c,8a) 1 32 (2a,2c,2c,10a) 1 30 (2a,2d,2c,2c,2c) 1 648 (2c,6f,5a) 2 1 (2c,4b,10a) 5 1 (2c,4b,8a) 6 1 (2c,4b,12b) 4 1 36 (2c,4a,10a) 1 1 (2b,6f,10a) 2 1 (2b,6f,8a) 2 1 (2c,2d,2d,4c) 1 112 (2c,2d,2d,4a) 1 20 (2c,2c,2d,6e) 1 24 (2d,4d,8a) 6 1 (2d,6e,12a) 1 1 (2d,6e,10a) 1 1 (2d,6d,8a) 2 1 (2d,6a,9a) 6 1 (2d,6a,5a) 4 1 40a (2d,6a,5a) 7 1 (2d,4b,9a) 3 1 (2d,4b,12b) 2 1 (2d,4a,10a) 1 1 (2c,2c,2d,6a) 1 24 (2c,2c,2d,4a) 1 20 (2c,2c,2c,4b) 1 96 (2c,2c,2c,6b) 1 234 (2d,4c,8a) 6 1 (2d,6e,5a) 7 1 (2d,6d,8a) 2 1 (2d,6c,5a) 2 1 40b (2d,4a,10a) 1 1 (2d,2d,2c,4a) 1 20 (2a,2d,2d,5a) 1 35 (2b,4d,8a) 6 1 (2b,4b,10a) 1 1 (2b,4a,9a) 3 1 (2b,4a,12a) 2 1 45 (2b,6d,5a) 2 1 (2b,2b,2d,4b) 1 20 (2b,2b,2b,4a) 1 96 introduction preliminary existence of genus zero system connected components of the hurwitz space � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 38, num. 1 (2023), 51 – 66 doi:10.17398/2605-5686.38.1.51 available online january 23, 2023 on jordan ideals with left derivations in 3-prime near-rings a. en-guady, a. boua@ department of mathematics, polydisciplinary faculty of taza sidi mohammed ben abdellah university, fez, morocco adel.enguady@usmba.ac.ma , abdelkarimboua@yahoo.fr received september 12, 2022 presented by c. mart́ınez accepted december 13, 2022 abstract: we will extend in this paper some results about commutativity of jordan ideals proved in [2] and [6]. however, we will consider left derivations instead of derivations, which is enough to get good results in relation to the structure of near-rings. we will also show that the conditions imposed in the paper cannot be removed. key words: 3-prime near-rings, jordan ideals, left derivations. msc (2020): 16n60; 16w25; 16y30. 1. introduction a right (resp. left) near-ring a is a triple (a, +, .) with two binary operations ” + ” and ”.” such that: (i) (a, +) is a group (not necessarily abelian), (ii) (a, .) is a semigroup, (iii) (r + s).t = r.t + s.t (resp. r.(s + t) = r.s + r.t) for all r; s; t ∈a. we denote by z(a) the multiplicative center of a, and usually a will be 3-prime, that is, for r,s ∈ a, ras = {0} implies r = 0 or s = 0. a right (resp. left) near-ring a is a zero symmetric if r.0 = 0 (resp. 0.r = 0) for all r ∈ a, (recall that right distributive yields 0r = 0 and left distributive yields r.0 = 0). for any pair of elements r,s ∈ a, [r,s] = rs − sr and r◦s = rs + sr stand for lie product and jordan product respectively. recall that a is called 2-torsion free if 2r = 0 implies r = 0 for all r ∈ a. an additive subgroup j of a is said to be jordan left (resp. right) ideal of a if r ◦ i ∈ j (resp. i ◦ r ∈ j) for all i ∈ j, r ∈ a and j is said to be a jordan ideal of a if r◦ i ∈ j and i◦r ∈ j for all i ∈ j, r ∈n . an additive mapping @ corresponding author issn: 0213-8743 (print), 2605-5686 (online) c© the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.38.1.51 mailto:adel.enguady@usmba.ac.ma mailto:abdelkarimboua@yahoo.fr mailto:abdelkarimboua@yahoo.fr https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 52 a. en-guady, a. boua h : a → a is a multiplier if h(rs) = rh(s) = h(r)s for all r,s ∈ a. an additive mapping d : a→a is a left derivation (resp. jordan left derivation) if d(rs) = rd(s) + sd(r) (resp. d(r2) = 2rd(r)) holds for all r,s ∈ a. the concepts of left derivations and jordan left derivations were introduced by breşar et al. in [7], and it was shown that if a prime ring r of characteristic different from 2 and 3 admits a nonzero jordan left derivation, then r must be commutative. obviously, every left derivation is a jordan left derivation, but the converse need not be true in general (see [9, example 1.1]). in [1], m. ashraf et al. proved that the converse statement is true in the case when the underlying ring is prime and 2-torsion free. the study of left derivation was developed by s.m.a. zaidi et al. in [9] and they showed that if j is a jordan ideal and a subring of a 2-torsion-free prime ring r admits a nonzero jordan left derivation and an automorphism t such that d(r2) = 2t(r)d(r) holds for all r ∈ j, then either j ⊆ z(r) or d(j) = {0}. recently, there have been many works concerning the jordan ideals of near-rings involving derivations; see, for example, [4], [5], [6], etc. for more details, in [6, theorem 3.6 and theorem 3.12], we only manage to show the commutativity of the jordan ideal, but we don’t manage to show the commutativity of our studied near-rings, hence our goal to extend these results to the left derivations. 2. some preliminaries to facilitate the proof of our main results, the following lemmas are essential. lemma 2.1. let n be a 3-prime near-ring. (i) [3, lemma 1.2 (iii)] if z ∈ z(n) \ {0} and xz ∈ z(n) or zx ∈ z(n), then x ∈ z(n). (ii) [2, lemma 3 (ii)] if z(n) contains a nonzero element z of n which z + z ∈ z(n), then (n , +) is abelian. (iii) [5, lemma 3] if j ⊆ z(n), then n is a commutative ring. lemma 2.2. ([8, theorem 3.1]) let n be a 3-prime right near-ring. if n admits a nonzero left derivation d, then the following properties hold true: (i) if there exists a nonzero element a such that d(a) = 0, then a ∈ z(n), (ii) (n , +) is abelian, if and only if n is a commutative ring. jordan ideals with left derivations 53 lemma 2.3. ([4, lemma 2.2]) let n be a 3-prime near-ring. if n admits a nonzero jordan ideal j, then j2 6= 0 for all j ∈ j \{0}. lemma 2.4. ([4, theorem 3.1]) let n be a 2-torsion free 3-prime right near-ring and j a nonzero jordan ideal of n . if n admits a nonzero left multiplier h, then the following assertions are equivalent: (i) h(j) ⊆ z(n); (ii) h(j2) ⊆ z(n); (iii) n is a commutative ring. lemma 2.5. ([5, theorem 1]) let n be a 2-torsion free 3-prime nearring and j a nonzero jordan ideal of n . then n must be a commutative ring if j satisfies one of the following conditions: (i) i◦ j ∈ z(n) for all i,j ∈ j. (ii) i◦ j ± [i,j] ∈ z(n) for all i,j ∈ j. lemma 2.6. let n be a left near-ring. if n admits a left derivation d, then we have the following identity: xyd(yn) = yxd(yn) for all n ∈ n, x,y ∈n . proof. using the definition of d. on one hand, we have d(xyn+1) = xd(yn+1) + yn+1d(x) = xynd(y) + xyd(yn) + yn+1d(x) for all n ∈ n, x,y ∈n . on the other hand d(xyn+1) = xynd(y) + yd(xyn) = xynd(y) + yxd(yn) + yn+1d(x) for all n ∈ n, x,y ∈n . comparing the two expressions, we obtain the required result. 54 a. en-guady, a. boua 3. results characterizing left derivations in 3-prime near-rings in [2], the author proved that if n is a 3-prime 2-torsion-free near-ring which admits a nonzero derivation d for which d(n) ⊆ z(n), then n is a commutative ring. in this section, we investigate possible analogs of these results, where d is replaced by a left derivation d and by integrating jordan ideals. theorem 3.1. let n be a 2-torsion free 3-prime near-ring and j be a nonzero jordan ideal of n . if n admits a left derivation d, then the following assertions are equivalent: (i) d(j) ⊆ z(n); (ii) d(j2) ⊆ z(n); (iii) n is a commutative ring or d = 0. proof. case 1: n is a 3-prime right near-ring. it is obvious that (iii) implies (i) and (ii). therefore we only need to prove (i)⇒(iii) and (ii)⇒(iii). (i)⇒(iii): suppose that z(n) = {0}, then d(j) = {0}. from lemma 2.2 (i), we get j ⊆ z(n) and by lemma 2.1 (i), we conclude that n is a commutative ring. in this case, and by using the definition of d together with the 2-torsion freeness of n , the above equation leads to jd(n) = 0 for all j ∈ j, n ∈n . (3.1) taking j◦m of j, where m ∈n in (3.1) and using it, we get jnd(n) = {0} for all n ∈n . since n is 3-prime and j 6= {0}, then d = 0. now suppose z(n) 6= {0}. by assumption, we have d(j ◦ j) ∈ z(n) for all j ∈ j, which gives (4j)d(j) ∈ z(n) for all j ∈ j, that is (d(4j))j ∈ z(n) for all j ∈ j. invoking lemma 2.1 (i) and lemma 2.2 (i) together with the 2-torsion freeness of n , we obtain j ⊆ z(n), and lemma 2.4 (i) forces that n is a commutative ring. (ii)⇒(iii): suppose that z(n) = {0}, then d(j2) = {0}, which implies j2 ⊆ z(n) by lemma 2.2 (ii), hence n is a commutative ring by lemma 2.4 (ii). now using assumption, then we have d(j2) = 0 for all j ∈ j. by the 2-torsion freeness of n , it follows jd(j) = 0 for all j ∈ j. since n is a commutative ring, we can write jnd(j) = 0 for all j ∈ j, n ∈n , which implies that jnd(j) = {0} for all j ∈ j. by the 3-primeness of n , we conclude that d(j) = {0}. using the same techniques as we have used in the proof of (i)⇒(iii) one can easily see that d = 0 . jordan ideals with left derivations 55 now suppose z(n) 6= {0}. by our hypothesis, we have d((j◦j2)j) ∈ z(n) for all j ∈ j, and by a simplification, we find d((j2 ◦ j)j) = (j2)d(4j2) for all j ∈ j : d((j2 ◦ j)j) = d((j3 + j3)j) = d(j4 + j4) = d(2j2j2) = 2j2d(j2) + j2d(2j2) = 2j2d(j2) + d(2j2)j2 = 2j2d(j2) + 2j2d(j2) = 4j2d(j2) = j2d(4j2). hence, j2d(4j2) ∈ z(n) for all j ∈ j, which implies j2d((4j)(j)) ∈ z(n) for all j ∈ j. invoking lemma 2.1 (i), then j2 ∈ z(n) or 4d(j2) = 0 for all j ∈ j. in view of the 2-torsion freeness of n together with lemma 2.2 (i), we can assure that j2 ∈ z(n) for all j ∈ j. (3.2) applying the definition of d together with our hypothesis, and (3.2), we have for all j ∈ j and x ∈n : d(xj4) = d(xj2j2) = xj2d(j2) + j2d(xj2) = xj2d(j2) + d(xj2)j2 = xj2d(j2) + xj2d(j2) + j4d(x) = j2d(j2)x + j2d(j2)x + j4d(x) = (2j2d(j2))x + j4d(x) , d(xj4) = xd(j4) + j4d(x) = x(2j2d(j2)) + j4d(x) . comparing the two expressions, we obtain x(2j2d(j2)) = (2j2d(j2))x for all j ∈ j, x ∈n . consequently, 2j2d(j2) ∈ z(n) for all j ∈ j. according to lemma 2.1 (i) and lemma 2.2 (i), that follows 2j2 ∈ z(n) for all j ∈ j, which implies (n , +) is abelian by lemma 2.1 (ii), and lemma 2.2 (ii) assures that n is a commutative ring. case 2: n is a 3-prime left near-ring. it is obvious that (iii) implies (i) and (ii). (i)⇒(iii): suppose that z(n) = {0}. using our hypothesis, then we have d(j ◦ n) = 0 for all j ∈ j, n ∈ n . applying definition of d and using our assumption with the 2-torsion freeness of n , we get jd(n) = 0 for all n ∈n . (3.3) replacing n by jnm in (3.3) and using it, then we get j2nd(m) = 0 for all j ∈ j, n,m ∈ n , which implies that j2nd(m) = {0} for all j ∈ j, m ∈ n . using lemma 2.3 together with the 3-primeness of n , it follows that d = 0. 56 a. en-guady, a. boua now assuming that z(n) 6= {0}. by lemma 2.6, we can write jnd(j) = njd(j) for all j ∈ j, n ∈n , which reduces to d(j)n [j,m] = {0} for all j ∈ j, m ∈n and by the 3-primeness of n , we conclude that j ∈ z(n) or d(j) = 0 for all j ∈ j. (3.4) suppose that there is j0 ∈ j such that d(j0) = 0. using our hypothesis, then we have d(j0(j0 ◦n)) ∈ z(n) for all n ∈ n . applying the definition of d and using our assumption, we get j0d((j0 ◦n)) ∈ z(n) for all n ∈ n . by lemma 2.1 (i), we conclude j0 ∈ z(n) or d((j0 ◦n)) = 0 for all n ∈n . (3.5) if d((j0 ◦n)) = 0 for all n ∈n , using the 2-torsion freeness of n , we get j0d(n) = 0 for all n ∈n . (3.6) replacing n by j0nm in (3.6) and using it, then we get j 2 0nd(m) = 0 for all n,m ∈ n . since d 6= 0, the 3-primeness of n gives j20 = 0, which is a contradiction with lemma 2.3. then (3.4) becomes j ⊆ z(n), which forces that n is commutative ring by lemma 2.1 (iii). (ii)⇒(iii): suppose that z(n) = {0}, then d(j2) = 0 for all j ∈ j, by the 2-torsion freeness of n , we get jd(j) = 0 for all j ∈ j. (3.7) using lemma 2.6, we can write jnd(j) = njd(j) for all j ∈ j, n ∈ n , from (3.7), we get jnd(j) = 0 for all j ∈ j, n ∈ n , which implies jnd(j) = {0} for all j ∈ j, n ∈ n and by the 3-primeness of n , we deduce that d(j) = {0}. using the same techniques as used in the proof of (i)⇒(iii), we conclude that d = 0. assuming that z(n) 6= {0}. by lemma 2.5, we can write jnd(j2) = njd(j2) for all x,y ∈n , (3.8) which implies that d(j2)n [j,m] = {0} for all j ∈ j, m ∈n . by the 3-primeness of n , we conclude that j ∈ z(n) or d(j2) = 0 for all j ∈ j. (3.9) jordan ideals with left derivations 57 if there exists j0 ∈ j such that d(j20 ) = 0, using the definition of d and the 2-torsion freeness of n , then we have j0d(j0) = 0. (3.10) by lemma 2.6, we can write j0nd(j0) = {0}. in view of the 3-primeness of n , that follows d(j0) = 0. using our hypothesis, we have d(j0(2i2)) ∈ z(n) for all i ∈ j. applying the definition of d and using our assumption, we get j0d(2i 2) ∈ z(n) for all i ∈ j. by the 2-torsion freeness of n and lemma 2.1 (i) we conclude j0 ∈ z(n) or id(i) = 0 for all i ∈ j. (3.11) if id(i) = 0 for all i ∈ j. using the same techniques as used in the proof of (ii)⇒(iii), we conclude that d = 0. then (3.9) becomes j ⊆ z(n) or d = 0. corollary 3.2. let n be a 2-torsion free 3-prime near-ring. if n admits a left derivation d, then the following assertions are equivalent: (i) d(n) ⊆ z(n); (ii) d(n2) ⊆ z(n); (iii) n is a commutative ring or d = 0. the following example proves that the 3-primeness of n in theorem 3.1 cannot be omitted. example 3.3. let r be a 2-torsion right or left near-ring which is not abelian. define n , j and d by: n =    0 0 0r 0 0 s t 0   : r,s,t, 0 ∈r   ,j =    0 0 00 0 0 p 0 0   : p, 0 ∈r   , d  0 0 0r 0 0 s t 0   =  0 0 00 0 0 0 t 0   . then n is a right or left near-ring which is not 3-prime, j is a nonzero jordan ideal of n and d is a nonzero left derivation of n which is not a derivation. it is easy to see that 58 a. en-guady, a. boua (i) d(j) ⊆ z(n). (ii) d(j2) ⊆ z(n). however, neither d = 0 nor n is a commutative ring. 4. some polynomial identities in right near-rings involving left derivations this section is motivated by [6, theorem 3.6 and theorem 3.12]. our aim in the current paper is to extend these results of jordan ideals on 3-prime near-rings admitting a nonzero left derivation. theorem 4.1. let n be a 2-torsion free 3-prime near-ring and j be a nonzero jordan ideal of n . if n admits a nonzero left derivation d and a multiplier h satisfying d(x◦ j) = h(x◦ j) for all j ∈ j, x ∈ n , then n is a commutative ring. proof. assume that d(x◦j) = h(x◦j) for all j ∈ j, x ∈n . if h = 0, the last equation becomes d(x◦j) = 0 for all j ∈ j, x ∈n . and recalling lemma 2.2 (ii), then (x◦j) ∈ z(n) for all j ∈ j, x ∈n , so n is a commutative ring by lemma 2.5 (i). now assume that h 6= 0 and d(x ◦ j) = h(x ◦ j) for all j ∈ j, x ∈ n . replacing x by xj and using the fact that (xj ◦ j) = (x◦ j)j, we get d((x◦ j)j) = h((x◦ j)j) for all i,j ∈ j, x ∈n . by the definition of d and h, we obtain (x◦ j)d(j) + jd(x◦ j) = h(x◦ j)j for all i,j ∈ j, x ∈n . replacing j by (y ◦ i), where i ∈ j, y ∈ n , in the preceding expression, we can see that (x◦ (y ◦ i))d((y ◦ i)) + (y ◦ i)d(x◦ (y ◦ i)) = h(x◦ (y ◦ i))(y ◦ i) for all, i,j ∈ j, x,y ∈n . by a simplification, we thereby obtain (y ◦ i)h(x◦ (y ◦ i)) = 0 for all i,j ∈ j, x,y ∈n . (4.1) applying h on (4.1), it follows that (y ◦ i)h(h(x◦ (y ◦ i))) = 0 for all i,j ∈ j, x,y ∈n . (4.2) jordan ideals with left derivations 59 applying d on (4.1) and recalling (4.2), we get h(x◦ (y ◦ i))h(y ◦ i) = 0 for all x,y ∈n , (4.3) which gives xh(y ◦ i)h(y ◦ i) = −h(y ◦ i)xh(y ◦ i) for all x,y ∈n . substituting xz instead of x in preceding equation and applying it, we obviously obtain xzh(y ◦ i)h(y ◦ i) = (−h(y ◦ i))xzh(y ◦ i) = x(−h(y ◦ i))zh(y ◦ i) for all x,y,z ∈n . this forces that [x, (−h(y ◦ i))]zh(y ◦ i) = 0 for all x,y,z ∈n . then [x, (−h(y◦i))]nh(y◦i) = {0} for all x,y ∈n . by the 3-primeness of n , we get (−h(y ◦ i)) ∈ z(n) for all i ∈ j, y ∈n . (4.4) substituting yi instead y in (4.4), (−h(y◦i))i ∈ z(n) for all i ∈ j, y ∈n . it follows that lemma 2.1 (i) h(y ◦ i) = 0 or i ∈ z(n) for all i ∈ j, y ∈n . (4.5) suppose that there exists an element i0 ∈ j such that h(y ◦ i0) = 0 for all y ∈n , (4.6) which implies (−i0)h(y) = h(y)i0 for all y ∈ n . replacing y by xyz in the last equation, we get (−i0)h(xyz) = h(xyz)i0 for all x,y,z ∈n , which means that (−i0)xyh(z) = x(−i0)yh(z) for all x,y,z ∈n , so [x,−i0]nh(z) = {0} for all x,z ∈ n . since h 6= 0 and n is 3-prime, we get −i0 ∈ z(n). now substituting −i0 instead i in (4.4), we obtain 60 a. en-guady, a. boua −h(y ◦ (−i0)) ∈ z(n) for all y ∈ n , which implies (−h(2y))(−i0) ∈ z(n) for all y ∈ n , using lemma 2.1 (i), we get −2h(y) ∈ z(n) for all y ∈ n or i0 = 0. thus (4.5) becomes − 2h(y) ∈ z(n) for all y ∈n or j ⊆ z(n). (4.7) case 1: if −2h(y) ∈ z(n) for all y ∈ n . replacing y by zt in the last equation, we obtain (−2h(z))t ∈ z(n) for all z,t ∈n . since n is 2-torsion free and h 6= 0, we obtain n ⊆ z(n) by lemma 2.1 (ii). which assures that n is a commutative ring by lemma 2.1 (iii). case 2: if j ⊆ z(n), then n is a commutative ring by virtue of lemma 2.1 (iii). the next result is an immediate consequence of theorem 3.1, just to take h = idn in theorem 4.1. corollary 4.2. let n be a 2-torsion free 3-prime near-ring and j be a nonzero jordan ideal of n . if n admits a nonzero left derivation d such that d(x◦ j) = x◦ j for all j ∈ j, x ∈n , then n is a commutative ring. theorem 4.3. let n be a 2-torsion free 3-prime near-ring and j be a nonzero right jordan ideal of n . if n admits a left derivation d and a nonzero multiplier h satisfying any one of the following identities: (i) d(h(j)) = {0}; (ii) d(h(j2)) = {0}; (iii) d(h(n◦ j)) = d(h([n,j])) for all j ∈ j, n ∈n ; (iv) d(h(nj)) = h(j)d(n) for all j ∈ j, n ∈n , then d = 0. proof. (i) assume that d (h(j)) = {0}. therefore, by lemma 2.2 (i) and lemma 2.4 (i), n is a commutative ring. using our hypothesis and by the 2-torsion freeness of n , we can see d(h(j)n) = 0 for all j ∈ j, n ∈ n . applying the definition of d, we obtain h(j)d(n) = 0 for all j ∈ j, n ∈n . (4.8) replacing j by j◦m, where m ∈n in (4.8) and using it, we can easily arrive at h(j)nd(n) = {0} for all n ∈ n . by the 3-primeness of n , we conclude jordan ideals with left derivations 61 that d(n) = {0} or h(j) = {0}. if h(j) = {0}, then h((j◦m)◦n)) = 0 for all j ∈ j, n,m ∈n . in view of the 2-torsion freeness of n , we get jnh(n) = {0} and by the 3-primeness of n , we obtain j = {0} or h(n) = {0}, that would contradict with our hypothesis, then d = 0. (ii) suppose that d ( h(j2) ) = {0}, according to lemma 2.2 (i) and lemma 2.4 (i), n is a commutative ring. now using our hypothesis, d(h(i(j◦n))) = 0 for all i,j ∈ j, n ∈n , by the 2-torsion freeness of n , we can see d(h(ijn)) = 0 for all i,j ∈ j, n ∈n . applying the definition of d, we obtain ih(j)d(n) = 0 for all i,j ∈ j, n ∈n . (4.9) substituting j◦m for j, where m ∈n and i◦t for j, where t ∈n in (4.9) and using it, we can easily arrive at jnh(j)nd(n) = {0} for all n ∈ n . by the 3-primeness of n , we conclude that d(n) = {0} or h(j) = {0} or j = {0}. if h(j) = {0}, using the same techniques as we have used in the proof of (i), one can easily find d = 0. (iii) suppose that d(h(n◦j)) = d(h([n,j])) for all j ∈ j, n ∈n . taking nj instead of n, we obtain d(h((n◦ j)j)) = d(h([n,j]j)) for all j ∈ j, n ∈n . using the definition of d, we get h(n◦ j)d(j) + jd(h(n◦ j)) = h([n,j])d(j) + jd(h([n,j])) for all j ∈ j, n ∈n . by a simplification, we can rewrite this equation as 2jh(n)d(j) = 0 for all j ∈ j, n ∈n . substituting zyt for n, where x,y,z ∈n in last equation, we can see 2jyh(z)td(j) = 0 for all j ∈ j, y,z,t ∈n . by the 2-torsion freeness of n , the above equation becomes jnh(z)nd(j) = {0} for all j ∈ j,z ∈ n . since n is 3-prime and h 6= 0, it follows that d(j) = {0}, which forces that d = 0 by (i). (iv) suppose that d(h(nj)) = h(j)d(n) for all j ∈ j, n ∈ n . from this equation we obtain d(nh(j)) = h(j)d(n) for all j ∈ j, n ∈n . 62 a. en-guady, a. boua using the definition of d, we have nd(h(j)) + h(j)d(n) = h(j)d(n) for all j ∈ j, n ∈n . then nd(h(j)) = 0 for all j ∈ j, n ∈n , which implies that d(h(j)) = {0} by invoking the 3-primeness of n , and consequently d = 0 by (i). the next result is an immediate consequence of theorem 3.1, just to take h = idn in theorem 4.6. corollary 4.4. let n be a 2-torsion free 3-prime near-ring and j be a nonzero right jordan ideal of n . if n admits a left derivation d and a nonzero multiplier h satisfying any one of the following identities: (i) d(j) = {0}; (ii) d(j2) = {0}; (iii) d(n◦ j) = d([n,j]) for all j ∈ j, n ∈n , (iv) d(nj) = jd(n) for all j ∈ j, n ∈n ; then d = 0. the following example proves that the 3-primeness of n in theorem 4.1 and theorem 4.3 cannot be omitted. example 4.5. let s be a 2-torsion right near ring which is not abelian. define n , j, d and h by: n =    0 0 p0 q 0 0 0 0   : p,q, 0 ∈s   , j =    0 0 00 s 0 0 0 0   : s, 0 ∈s   , d  0 0 p0 q 0 0 0 0   =  0 0 p0 0 0 0 0 0   and h  0 0 p0 q 0 0 0 0   =  0 0 00 q 0 0 0 0   . then n is a right near-ring which is not 3-prime, j is a nonzero jordan ideal of n , d is a nonzero left derivation of n , and h is a nonzero multiplier of n , such that (i) d(x◦ j) = h(x◦ j) for all j ∈ j, x ∈n ; (ii) d (h(j)) = {0}; jordan ideals with left derivations 63 (iii) d ( h(j2) ) = {0}; (iv) d(h(n◦ j)) = d(h([n,j])) for all j ∈ j, n ∈n ; (v) d(h(nj)) = h(j)d(n) for all j ∈ j, n ∈n . however, neither d = 0 nor n is a commutative ring. theorem 4.6. let n be a 2-torsion free 3-prime near-ring and j be a nonzero jordan ideal of n and let h a nonzero multiplier on n . then there is no nonzero left derivation d such that d(x ◦ j) = h([x,j]) for all j ∈ j, x ∈n . proof. assume that d(x◦ j) = h([x,j]) for all j ∈ j, x ∈n . (4.10) replacing x by j, in (4.10), we get 2d(j2) = d(j2 + j2) = d(j ◦ j) = 0 for all j ∈ j. by the 2-torsion freeness of n , we get 0 = d(j2) = 2jd(j) for all j ∈ j. (4.11) in view of the 2-torsion freeness of n , this easily yields jd(j) = 0 for all j ∈ j. (4.12) replacing x by xj in (4.10), we get d(xj ◦ j) = h([xj,j]) for all j ∈ j, x ∈n . using the fact that (xj ◦ j) = (x◦ j)j and [xj,j] = [x,j]j, we obtain d((x◦ j)j) = h([x,j]j) for all j ∈ j, x ∈n . by the definition of d, the last equation is expressible as (x◦ j)d(j) = [h([x,j]),j] for all j ∈ j, x ∈n . substituting xj instead x, it follows from (4.12) that [h([xj,j]),j] = 0 for all j ∈ j, x ∈n . (4.13) 64 a. en-guady, a. boua replacing x by d(j)x in (4.13) and using (4.12), we can easily arrive at [d(j)h(x)j2,j] = 0 for all j ∈ j, x ∈n . which reduces to d(j)h(x)j3 = 0 for all j ∈ j, x ∈n . substituting rst instead x where r,s,t ∈ n in the last equation, we get d(j)rh(s)tj3 = 0 for all j ∈ j, r,s,t ∈ n , which implies d(j)nh(s)nj3 = {0} for all j ∈ j, s ∈n . since h 6= 0 and using the 3-primeness hypothesis, it follows that d(j) = 0 or j3 = 0 for all j ∈ j. (4.14) suppose that there exists an element j0 ∈ j \ {0} such that j30 = 0. replacing j by j0 and x by xj 2 0 in (4.10) and using (4.12), then d(xj20 ◦ j0) = h([xj 2 0,j0] for all x ∈n . using our assumption, we find that d(j0xj 2 0 ) = h(−j0xj 2 0 ) for all x ∈n . by the definition of d, we get j0d(xj 2 0 ) + xj 2 0d(j0) = −j0h(x)j 2 0 for all x ∈n . in light of equation (4.12), it follows easily that j0d(xj 2 0 ) = −j0h(x)j 2 0 for all x ∈n . so, by (4.14) and (4.12), we get −j0h(x)j20 = 0 for all x ∈n . substituting rst instead x gives −j0rh(s)tj20 = 0 for all r,s,t ∈n , which implies (−j0)nh(s)nj20 = {0} for all s ∈n . since h 6= 0, by the 3-primeness of n and lemma 2.3, the preceding expression leads to j0 = 0. hence, (4.14) becomes d(j) = {0}, which leads to d = 0 by theorem 3.1 (i); a contradiction. corollary 4.7. let n be a 2-torsion free 3-prime near-ring and j be a nonzero jordan ideal of n . then there is no nonzero left derivation d such that d(x◦ j) = [x,j] for all j ∈ j, x ∈n . jordan ideals with left derivations 65 theorem 4.8. let n be a 2-torsion free 3-prime near-ring and j be a nonzero jordan ideal of n . then n admits no nonzero left derivation d such that d([x,j]) = d(x)j for all j ∈ j, x ∈n . proof. assume that d([x,j]) = d(x)j for all x ∈n , j ∈ j. (4.15) replacing x by j in (4.15), we get d(j)j = 0 for all j ∈ j. (4.16) substituting xj instead of x in (4.15), we obtain d([xj,j]) = d(xj)j for all j ∈ j, x ∈n . notice that [xj,j] = [x,j]j, the last relation can be rewritten as d([x,j]j) = (xd(j) + jd(x))j for all j ∈ j, x ∈n . the definition of d gives us [x,j])d(j) + jd([x,j]) = jd(x)j for all j ∈ j, x ∈n . using our assumption, we obviously obtain xjd(j) = jxd(j) for all j ∈ j, x ∈n . (4.17) replacing x by yt in (4.17) and invoking it, we can see that yjtd(j) = jytd(j) for all j ∈ j, y,t ∈n . the last equation gives us [y,j]nd(j) = {0} for all j ∈ j, x ∈n . by the 3-primeness of n , we get j ∈ z(n) or d(j) = 0 for all j ∈ j. (4.18) if there exists j0 ∈ j such that d(j0) = 0. using lemma 2.4, we obtain j0 ∈ z(n). in this case, (4.18) becomes j ⊆ z(n) which forces that n is a commutative ring by lemma 2.1 (i). hence (4.6) implies that d(x)j = 0 for all j ∈ j, x ∈n . replacing j by j ◦ t in the last equation, it is obvious that 2d(x)tj = 0 for all j ∈ j, t,x ∈ n . it follows from the 2-torsion freeness of n that d(x)nj = {0} for all j ∈ j, x ∈ n . by the 3-primeness of n , we conclude that d = 0 or j = {0}; a contradiction. 66 a. en-guady, a. boua 5. conclusion in this paper, we study the 3-prime near-rings with left derivations. we prove that a 3-prime near-ring that admits a left derivation satisfying certain differential identities on jordan ideals becomes a commutative ring. in comparison to some recent studies that used derivations, these results are considered more developed. in future research, one can discuss the following issues: (i) theorem 3.1, theorem 4.1, theorem 4.3 and theorem 4.6 can be proven by replacing left derivation d by a generalized left derivation. (ii) the study of 3-prime near-rings that admit generalized left derivations satisfying certain differential identities on lie ideals is another interesting work for the future. references [1] m. ashraf, n. rehman, on lie ideals and jordan left derivation of prime rings, arch. math. (brno) 36 (2000), 201 – 206. [2] h.e. bell, g. mason, on derivations in near-rings, in “near-rings and near-fields”, north holland math. stud. 137, north-holland, amsterdam, 1987, 31 – 35. [3] h.e. bell, on derivations in near-rings ii, in “nearrings, nearfields and kloops”, math. appl. 426, kluwer acad. publ., dordrecht, 1997, 191 – 197. [4] a. boua, h.e. bell, jordan ideals and derivations satisfying algebraic identities, bull. iranian math. soc. 44 (2018), 1543 – 1554. [5] a. boua, l. oukhtitei, a. raji, jordan ideals and derivations in prime near-rings, comment. math. univ. carolin. 55 (2) (2014), 131 – 139. 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[9] s.m.a. zaidi, m. ashraf, a. shakir, on jordan ideals and left (θ,θ)-derivations in prime rings, int. j. math. math. sci. 37-40 (2004), 1957 – 1964. introduction some preliminaries results characterizing left derivations in 3-prime near-rings some polynomial identities in right near-rings involving left derivations conclusion � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 2 (2020), 205 – 219 doi:10.17398/2605-5686.35.2.205 available online november 2, 2020 invariant subspace problem and compact operators on non-archimedean banach spaces m. babahmed, a. el asri department of mathematics, university of moulay ismail faculty of sciences, meknes, morocco m.babahmed@umi.ac.ma , azzedine.elasri1@usmba.ac.ma received july 25, 2020 presented by manuel gonzález accepted september 21, 2020 abstract: in this paper, the invariant subspace problem is studied for the class of non-archimedean compact operators on an infinite-dimensional banach space e over a nontrivial complete nonarchimedean valued field k. our first main result (theorem 9) asserts that if k is locally compact, then each compact operator on e possessing a quasi null vector admits a nontrivial hyperinvariant closed subspace. in the second one (theorem 17), we prove that each bounded operator on e which contains a cyclic quasi null vector can be written as the sum of a triangular operator and a compact shift operator, each one of them possesses a nontrivial invariant closed subspace. finally, we conclude that if k is algebraically closed, then every compact operator on e either has a nontrivial invariant closed subspace or is a sum of upper triangular operator and shift operator, each of them is compact and has a nontrivial invariant closed subspace. key words: invariant subspace, hyperinvariant subspace, compact operator, t-orthogonal basis, quasi null vector, triangular operator, shift operator. ams subject class. (2010): 47a15, 47s10, 46s10. 1. introduction one of the most important problem in operator theory is the invariant subspace problem, which is concerned with the existence of invariant subspaces of a bounded operator on separable infinite-dimensional banach space. this problem is one of the best-known unresolved problem in functional analysis. it is not clear exactly when the problem was formally posed. some believe that the interest aroused by this problem stems from a beurling’s paper [6] and an unpublished work by von neumann in which he had shown that compact operators on a complex hilbert space of dimension at least 2 have nontrivial invariant closed subspaces. since then, research into the existence of invariant subspaces is intensively launched. a number of authors worked on extending this result and significant progress was made from then until now. issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.35.2.205 mailto:m.babahmed@umi.ac.ma mailto:azzedine.elasri1@usmba.ac.ma https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 206 m. babahmed, a. el asri in 1954, aronszajn and smith [3] generalized the result obtained by von neumann to compact operators on an infinite-dimensional banach space over the complex field c. in 1966, bernstein and robinson [5], using nonstandard analysis, proved that each polynomially compact operator on a complex hilbert space has a nontrivial invariant closed subspace. in the same year, halmos [10] gave a proof of the same result by a similar method, but avoiding the nonstandard analysis tools. in 1968, arveson and feldman [4] proved that if t is a quasitriangular operator on a hilbert space (an operator satisfying limn‖tpn −pntpn‖ = 0 for some sequence (pn)n of orthogonal projection operators which converges strongly to the identity i), then t has a nontrivial invariant closed subspace if the closed algebra generated by t and i contains a nonzero compact operator. in 1973, pearcy and salinas [14] proved that if t is a quasitriangular operator on a hilbert space h and r(t), the norm closure of the rational functions of t, contains a nonzero compact operator, then there exists a nontrivial invariant closed subspace under all operators in r(t). in the same year, appeared the famous theorem of lomonosov [12]. this theorem generalized all the preceding results. lomonosov showed that each operator on a banach space that commutes with a nonzero compact operator has a nontrivial hyperinvariant closed subspace. he used new techniques in his proof like schauder’s fixed point theorem. then, the theorems due to von neumann, aronszajn and smith, bernstein and robinson, become direct corollaries of this theorem. the lomonosov’s theorem obtained was a more general result than anyone had even hoped to be able to prove. in the same year, hilden found a proof of lomonosov’s theorem without using schauder’s fixed point theorem, and with only the most elementary notions of functional analysis (see [16, p. 158]). in 1977, michaels [13] presented the hilden’s proof with more simplifications, making it accessible to nonspecialists. at this time, the lomonosov result seemed so strong that some authors wondered if his hypothesis (commutativity with a compact operator) could even be true for all operators. unfortunately, in 1980 hadwin, nordgren, radjavi and rosenthal [9] constructed an operator not satisfying such hypothesis. in the non-archimedean setting, in 2008 sliwa [20] proved that every infinite-dimensional non-archimedean banach space of countable type admits a bounded operator without a nontrivial invariant closed subspace. invariant subspace problem 207 in this paper, taking inspiration from hilden’s proof techniques of lomonosov’s theorem, we prove that each compact operator on an infinitedimensional banach space over a locally compact non-archimedean valued field in which there exists a quasi null vector have a nontrivial hyperinvariant closed subspace (theorem 9). we prove also that each bounded operator on infinite-dimensional non-archimedean banach space which contains a cyclic quasi null vector can be written as the sum of a triangular operator and a compact shift operator, each one of them has a nontrivial invariant closed subspace (theorem 17). 2. preliminary and auxiliary results in this section, we collect some auxiliary results of non-archimedean analysis that we need later. let k be a field, a valuation on k is a function |.| : k → [0,∞) such that for all a,b ∈ k: (i) |a| = 0 if and only if a = 0; (ii) |ab| = |a||b|; (iii) |a + b| ≤ |a| + |b| (the triangle inequality). the pair (k, |.|) is called a valued field. we frequently write k instead of (k, |.|). a valuation |.| on k is: • non-trivial if |a| 6= 1 for some a ∈ k\{0}; • non-archimedean if |a + b| ≤ max{|a|, |b|} for all a,b ∈ k; • complete if k is complete with respect to the metric d(a,b) = |a − b| induced by the valuation |.|. for fundamentals on non-archimedean valued field we refer to [15, 21]. throughout this paper, k := (k, |.|) is a non-archimedean non-trivially valued complete field. the closed unit ball and the open unit ball in k are respectively bk = {λ ∈ k : |λ| ≤ 1} and bk = {λ ∈ k : |λ| < 1}. the residue class field of k is k = bk/bk. we say that k is locally compact if each point has a compact neighbourhood. so, clearly k is locally compact if, and only if, the unit ball bk of k is compact. and we have the following characterization of a locally compact field: theorem 1. ([15, p. 9]) k is locally compact if, and only if, the valuation is discrete and the residue class field is finite. 208 m. babahmed, a. el asri the most known example of a non-archimedean valued field is the field of p-adic numbers qp which is locally compact (see [15, p. 9]). we say that k is spherically complete if any decreasing sequence of closed balls in k has a non-empty intersection (see [15, p. 4]). now let e be a vector space over k. a norm ‖.‖ on e is called nonarchimedean if it satisfies the following strong inequality: ‖x + y‖≤ max{‖x‖,‖y‖} for all x,y ∈ e. it is easy to verify that if ‖.‖ is a non-archimedean norm in e, then for x,y ∈ e with ‖x‖ 6= ‖y‖ we have ‖x + y‖ = max{‖x‖,‖y‖}. we say that e := (e,‖.‖) is a non-archimedean banach space if the norm ‖.‖ is non-archimedean on e, and it is complete with the topology induced by the metric d(x,y) = ‖x−y‖. if e contains a countable set y such that its linear hull is dense in e, i.e., [y ] = e, then we say that e is of countable type. let t ∈ (0, 1], a vector x ∈ e is t-orthogonal (orthogonal, when t = 1) to y ∈ e\{0} if d(x,ky) := dist(x,ky) ≥ t‖x‖, where dist(x,d) := infd∈d ‖x − d‖(d ⊆ e) and ky denotes a one-dimensional linear subspace generated by the element y. we note x⊥ty. by the principle of van rooij (see [15, p. 23]), if x⊥ty then ‖λx + µy‖≥ t max{‖λx‖,‖µy‖} for all λ,µ ∈ k. so, the t-orthogonality is a symmetric relation. we say that a subset x of e\{0} is t-orthogonal (orthogonal, when t = 1) system if for each x ∈ x, x⊥ty for all y ∈ [x\{x}]. clearly, a subset x = {x1,x2, . . . ,xn, . . .}⊆ e\{0} is t-orthogonal if, and only if, each finite subset of x is t-orthogonal, i.e.,∥∥∥∥∥ m∑ i=1 λixi ∥∥∥∥∥ ≥ t mmaxi=1 ‖λixi‖ for all λ1, . . . ,λm ∈ k and all x1, . . . ,xm ∈ x, where xi 6= xj for i 6= j. and {x1,x2, . . . ,xn, . . .} is orthogonal if, and only if,∥∥∥∥∥ m∑ i=1 λixi ∥∥∥∥∥ = mmaxi=1 ‖λixi‖. if a and b are two subset of e, we say that a and b are t-orthogonal if for every (a,b) ∈ a × b we have a⊥tb. then, we write a⊥tb. a sequence (xn)n in e is called a t-orthogonal (orthogonal, when t = 1) basis of e if {x1,x2, . . . ,xn, . . .} is t-orthogonal and every x ∈ e has an expansion invariant subspace problem 209 x = ∑+∞ n=1 λnxn, where λn ∈ k for every n ∈ n. the basis is orthonormal if, in addition, ‖xn‖ = 1 for all n ∈ n. for more background on non-archimedean normed space over valued field and more details we refer the reader to [21, 18]. now, we recall the following important theorem: theorem 2. ([15, p. 30]) if e is of countable type, then for each t ∈ ]0, 1[, e has a t-orthogonal basis. in addition, if k is spherically complete, e has an orthogonal basis. we call a nonempty subset x of e absolutely convex if ax + by ∈ x for all x,y ∈ x and a,b ∈ bk. let x be any subset of e, by co(x) we denote the absolutely convex hull of x, which is the smallest absolutely convex set in e that contains x. a subset x of e is said to be compactoid if for every � > 0 there exists a finite set a ⊂ e such that x ⊂ b(0,�) + co(a) (where b(0,�) = {x ∈ e : ‖x‖≤ �}). amice [2] proved that if k is locally compact, then a subset of e is compactoid if, and only if, it is precompact, then each complete compactoid subset of e is compact. it’s easy to see that if k is not locally compact, any convex set (a translation of absolutely convex set) in e containing at least two points, is not compact. an operator on e is a linear map t : e → e. by b(e) we denote the algebra of all bounded operators on e. now, we recall the definitions of completely continuous and compact operators on non-archimedean banach space. definition 3. let t be an operator on e, we say that t is: 1. completely continuous if t(b(0, 1)) has a compact closure [19, 7]; 2. compact if t(b(0, 1)) is a compactoid (see [21, p. 142]). we note that the first definition have no sense if k is not locally compact, and the two definitions coincide when k is locally compact (see [21, p. 142]). similarly as in classical hilbert space analysis, we have the following result: theorem 4. ([21, p. 142]) let t ∈ b(e). then t is compact if, and only if, for every � > 0, there exists an s ∈ b(e) such that s is a finite rank operator and ‖t −s‖≤ �. this means that the compact operators are exactly the (norm) limits of finite rank operators. the spectral theory on compact operators in classical 210 m. babahmed, a. el asri analysis can be successfully extended to the set of compact operators in nonarchimedean analysis, see [8, 17]. the spectrum σ(t) := {λ ∈ k : t − λi is not invertible} of a compact operator t is at most countable with 0 as its only possible accumulation point, so in particular it is compact (see [17, p. 17]). moreover, each nonzero element of the spectrum is an eigenvalue (see [17, p. 16]). one also has the fredholm alternative (see [17, p. 15]). and also, as in classical analysis over complex field, if k is algebraically closed then every compact operator t is spectral in the following sense: max{|λ| : λ ∈ σ(t)} = limn‖tn‖ 1 n . for more on compact operators, see [7, 8, 17, 19]. 3. definitions and main results we are now ready to state and prove the main results of this paper. definition 5. let t be an operator on a banach space e, and m a nontrivial ({0} m e) closed subspace of e. we say that m is: 1. invariant if tm ⊆ m. 2. hyperinvariant if m is invariant for each element of {t} ′ = {s ∈ b(e) : ts = st} (the commutant of t). we note that a bounded operator on a banach space is said locally quasinilpotent at a vector x if limn‖tnx‖ 1 n = 0 [1]. we point out that t can be not locally quasinilpotent on certain nonzero vector x, and lim infn‖tnx‖ 1 n = 0. in this case we shall say that x is t-quasi null vector. it turns out that this property can be a very useful tool for the study of invariant subspaces. note that even a compact operator may not have a t-quasi null vector. for example, we consider the operator t on c0(qp) (the non-archimedean banach space of null sequences on qp) defined by: ten = p nen for all n ≥ 1 , where (en)n is the standard orthogonal basis of c0(qp). it is clear that t is compact. let x = (xk)k be a nonzero element of c0(qp). then, x = ∑ k≥1 xkek and for each n ≥ 1, tnx = ∑ k≥1 xkp nkek. let r ≥ 1 be such that xr 6= 0. so ‖tnx‖ = max k≥1 |xk||pnk| and ‖tnx‖ 1 n ≥ |xr| 1 n 1 pr . therefore, t cannot have a t-quasi null vector. invariant subspace problem 211 lemma 6. let (αn)n be a real sequence such that lim infn α 1 n n = 0. then, for each real β we have lim infn β nαn = 0. proof. let β ∈ r. then for each n ≥ 1 we have (βnαn) 1 n =βα 1 n n . proposition 7. x is t-quasi null vector if, and only if, x is λt-quasi null vector for each λ ∈ k\{0}. proof. let λ ∈ k\{0}. for each n ≥ 1, ‖(λt)nx‖ 1 n = |λ|‖tnx‖ 1 n . lemma 8. ([15, p. 18]) let e := (e,‖.‖) be a normed space over a discretely valued field k. then there is an equivalent norm ‖.‖ ′ on e for which ‖.‖ ′ is solid (i.e., ‖e‖ ′ ⊆ |k|). we note that if t is an operator on (e,‖.‖) and ‖.‖ ′ is another nonarchimedean norm on e that is equivalent to ‖.‖. then, every t-quasi null vector of (e,‖.‖) is also a t-quasi null vector of (e,‖.‖ ′ ). every closed invariant (resp. hyperinvariant) subspace of t in (e,‖.‖) is closed invariant (resp. hyperinvariant) subspace of t in (e,‖.‖ ′ ). so with this argument and lemma 8, we can, without loss of generality, assume that the non-archimedean norm in e is solid. now we are ready to show that on non-archimedean banach space over a locally compact field every compact operator with a t-quasi null vector has a nontrivial hyperinvariant closed subspace. theorem 9. let (e,‖‖) be a non-archimedean banach space over a locally compact field k and t be a compact operator on e. if there exists a t-quasi null vector in e, then there exists a nontrivial hyperinvariant closed subspace of t. proof. without loss of generality, we can assume that t is an injective operator, and ‖t‖ = 1. let z be a t-quasi null vector in e. so lim infn‖tnz‖ 1 n = 0. let λ0 ∈ k be such that ‖t(λ0z)‖ > 1. and let z0 = λ0z, so ‖z0‖ = ‖t‖‖z0‖≥‖tz0‖ > 1. for each λ ∈ k, by lemma 6, lim inf n |λ|n‖tnz0‖ = |λ0| lim inf n |λ|n‖tnz‖ = 0 . for each x ∈ e, set hx := { sx : s ∈{t} ′} . 212 m. babahmed, a. el asri then, hx is an hyperinvariant subspace of t. then, so is hx for each x. since for each x ∈ e\{0}, hx 6= {0}, we have the result unless hx = e for each x ∈ e\{0}. so, assume a contradiction and suppose that we have this. let b := b(z0, 1) be a closed unit ball in e. for each x ∈ e\{0}, z0 ∈ e = hx, then there exist y ∈ hx such that ‖y − z0‖ ≤ 1. and there exists s ∈ {t} ′ such that y = sx. hence, ‖sx−z0‖≤ 1, and sx ∈ b. therefore, e\{0}⊆ ⊔ s∈{t}′ {x ∈ e : sx ∈ b}. for each s ∈ {t} ′ , let u(s,b) := {x ∈ e : sx ∈ b}. clearly u(s,b) is open in e for each s ∈{t} ′ . since 0 6∈ b, we have 0 6∈ t(b). then, t(b) ⊆ ⊔ s∈{t}′ u(s,b) . since t is compact, there exist s1, . . . ,sn ∈{t} ′ such that t(b) ⊆ n⊔ i=1 u(si,b). tz0 ∈ t(b) ⇒ there is i1 ∈{1, . . . ,n} such that tz0 ∈ u(si1,b) ⇒si1tz0 ∈ b ⇒ tsi1tz0 ∈ t(b) ⇒ there is i2 ∈ {1, . . . ,n} such that tsi1tz0 ∈ u(si2,b) ⇒ si2tsi1tz0 ∈ b. and we continue like this so much we would like. then for each k ≥ 1, there exist si1, . . . ,sik ∈{t} ′ such that siktsik−1t · · ·si1tz0 ∈ b. since the non-archimedean norm of e is solid, let λ ∈ k\{0} be such that |λ| = max1≤j≤k ‖sij‖. so, we have (λ −1sik ) · · ·(λ −1si1 )(λt) k(z0) ∈ b ⇒ ‖(λ−1sik ) · · ·(λ −1si1 )(λt) k(z0) −z0‖≤ 1. since ‖z0‖ > 1, we have ‖(λ−1sik ) · · ·(λ −1si1 )(λt) k(z0)‖ = ‖z0‖ > 1. since ‖λ−1sir‖ ≤ 1 for all 1 ≤ r ≤ k, we get ‖(λt)kz0‖ > 1. hence, |λ|k‖tkz0‖ > 1 for each k ≥ 1, which is a contraction. and the result follows. remark 10. it is known that the closure of the range of a compact operator is of countable type (see [21, p. 134]). so, if the banach space e is not of countable type, this closure is a nontrivial hyperinvariant closed subspace. invariant subspace problem 213 we note that if e is not of countable type, then every operator t on e does not have a cyclic vector (a vector x ∈ e such that [tnx, n ≥ 0] = e). and so for each nonzero x in e, [tnx, n ≥ 0] is a nontrivial closed invariant subspace of t. therefore, without loss of generality, we assume throughout to the end of this paper that e is a non-archimedean banach space of countable type. the following lemma and theorem will be needed to prove lemma 14. lemma 11. ([21, p. 66]) let f be a closed subspace of e and a ∈ e\f. then for each t ∈]0, 1[ there exist e ∈ e such that [a] + f = [e] + f and e is t-orthogonal to f. furthermore, if k is spherically complete, we can choose e to be orthogonal to f. theorem 12. ([15, p. 30]) let t ∈]0, 1] and let {xn , n ∈ n} be a torthogonal subset of e. if [xn , n ∈ n] = e, then {x1,x2, . . .} is a t-orthogonal (orthogonal, when t = 1) basis of e. lemma 13. ([15, p. 28]) let e1, . . . ,en be distinct nonzero vectors in a normed space e. let t2, t3, . . . , tn ∈]0, 1] be such that  dist(e2, [e1]) ≥ t2‖e2‖ , dist(e3, [e1,e2]) ≥ t3‖e3‖ , ... dist(en, [e1, . . . ,en−1]) ≥ tn‖en‖ . then {e1, . . . ,en} is (t2t3 · · ·tn)-orthogonal. lemma 14. let t be a bounded operator on e. suppose that there exists a t-quasi null vector in e which is cyclic. then, for each t ∈]0, 1[, there exists (en)n, a t−orthogonal basis of e such that: 1. t[e1, . . . ,en] ⊆ [e1, . . . ,en+1] for all n ≥ 1; 2. lim infn d(t(en), [e1, . . . ,en]) = 0. proof. let z be a cyclic t-quasi null vector in e. so, we have e = [z,tz,t 2z, . . . ] , (1) lim inf n ‖tnz‖ 1 n = 0 . (2) 214 m. babahmed, a. el asri let t ∈]0, 1[. for each n ≥ 1, let fn := [z,tz,t 2z, . . . ,tn−1z]. choose t2, t3, · · · ∈]0, 1[ such that t ≤ ∏ i≥2 ti. let e1 = z, tz ∈ e\f1, then by lemma 11, there exists e2 ∈ e such that, [e1,e2] = [e1,tz] = [z,tz] = f2 and e2 ⊥t2 f1 = [e1]. t 2z ∈ e\f2, then by lemma 11, again, there exists e3 ∈ e such that, [z,tz,e3] = [z,tz,t 2z] and e3 ⊥t3 [z,tz]. hence, [e3,e2,e1] = [z,tz,t 2z] and e3 ⊥t3 [e1,e2] = f2. continuing on this direction, we construct a sequence (en)n in e such that for each n ≥ 1 we have: fn = [e1, . . . ,en] = [z,tz, . . . ,t n−1z] , (3) en ⊥tn [e1, . . . ,en−1] = fn−1 . (4) then, using lemma 13 for each n ≥ 2, we conclude that (em)1≤m≤n is (t2 · · ·tn)-orthogonal, so (en)n≥1 is a t-orthogonal sequence in e, and [e1,e2, . . . ] = [z,tz,t 2z, . . . ]. hence, by theorem 12, (en)n is a t-orthogonal basis of e. and for each n ≥ 1 we have: t(fn) = t[e1, . . . ,en] = t[z,tz, . . . ,t n−1z] ⊆ [z,tz, . . . ,tnz] = fn+1 = [e1, . . . ,en+1] . now, let show that lim infn d(t(en), [e1, . . . ,en]) = 0. since for each sequence (λn)n of nonzero scalars, (λnen)n is a t-orthogonal basis of e such that for each n ≥ 1 t[λ1e1, . . . ,λnen] ⊆ [λ1e1, . . . ,λn+1en+1], without loss of generality we can assume, that the t-orthogonal basis (en)n satisfies |ρ| ≤ ‖en‖≤ 1 for all n ≥ 1 for some scalar ρ such that 0 < |ρ| < 1. let n ≥ 1, en ∈ [z,tz, . . . ,tn−1z] = fn−1 + [tn−1z]. then, there exist xn ∈ fn−1 and λn ∈ k\{0} such that en = xn + λnt n−1z. hence, ten = txn + λnt nz. so, we have: d(en,fn−1) = d(λnt n−1z,fn−1) , (5) d(ten,fn) = d(λnt nz,fn) . (6) invariant subspace problem 215 therefore, we have d(en+1,fn) = d(λn+1t nz,fn) = |λn+1|d(tnz,fn) , d(ten,fn) = |λn|d(tnz,fn) . hence, d(ten,fn) = |λn| |λn+1| d(en+1,fn) ≥ |λn| |λn+1| t‖en+1‖≥ t|ρ| |λn| |λn+1| . (7) then, by (6), we have d(λnt nz,fn) ≥ |λn| |λn+1| t|ρ| ⇒ t|ρ||λn+1| ≤ ‖t nz‖. so, by (2), lim infn( 1 |λn+1| ) 1 n = 0. since for all n ≥ 1 ( n∏ k=1 |λk| |λk+1| )1 n = ( |λ1| |λn+1| )1 n , we have lim inf n ( n∏ k=1 |λk| |λk+1| )1 n = 0. then, there exists a subsequence ( |λkj | |λkj+1| ) j of ( |λk| |λk+1| ) k such that, lim j ( |λkj| |λkj +1| ) = 0 . on the other hand, by (7), for each j ≥ 1 we have d(tekj,fkj ) = |λkj| |λkj +1| d(ekj +1,fkj ) ≤ |λkj| |λkj +1| ‖ekj +1‖≤ |λkj| |λ kj +1| . then, limj d(tekj,fkj ) = 0. and lim infn d(t(en),fn) = 0. 216 m. babahmed, a. el asri in the classical analysis, an operator t acting on complex hilbert space h is said to be triangular, if there exists a sequence {pn} of (orthogonal) projections of finite rank on h converging strongly to i and satisfying (i−pn)tpn = 0. that is equivalent, that t have an upper triangular matrix with respect to some orthonormal basis of h [11]. in the same context, in a non-archimedean banach space of countable type, we give the following definition: definition 15. let t be a bounded operator on e. we say t is a triangular operator, if there exist t ∈]0, 1] and a t−orthogonal basis (en)n of e, such that for each n ≥ 1, [e1,e2, . . . ,ekn ] is invariant for t, where (kn)n is a strictly increasing sequence in n∗. remark 16. if k is algebraically closed, every triangular operator is an upper triangular operator (have an upper triangular matrix in such a torthogonal basis). indeed, for all n ≥ 1, let fn = [e1,e2, . . . ,ekn ] and f0 = {0}. since t(fn) ⊆ fn for all n ∈ n, then by the well known result, that every operator in finite dimensional space over k is upper triangular, we have for all n the quotient operator sn defined in fn+1/fn by sn(x) = s(x) for all x ∈ fn+1/fn, is upper triangular. so obviously there is (bn)n ⊆ e\{0} such that [b1,b2, . . . ] = e with the property that t([b1, . . . ,bn]) ⊆ [b1, . . . ,bn] for all n ∈ n∗. therefore, according to the same first part of proof of lemma 14, we can construct a t-orthogonal sequence (ln)n where t ∈]0, 1[, such that [b1, . . . ,bn] = [l1, . . . , ln] for all n ∈ n∗. so t([l1, . . . ln]) ⊆ [l1, . . . , ln] for all n ∈ n∗ and (ln)n is t-orthogonal basis of e, hence we have t is upper triangular operator. now, we state our second main result. theorem 17. let t be a bounded operator on e. if there exists a cyclic t-quasi null vector in e, then t can be written as the sum of triangular operator and a compact shift operator, each one of them possesses a nontrivial invariant closed subspace. proof. let t ∈]0, 1[. by lemma 14, there exists (en)n, a t-orthogonal basis of e such that t(fn) ⊆ fn+1 and lim infn d(ten,fn) = 0, where fn := [e1, . . . ,en] for all n ≥ 1. without loss of generality, we can assume that |ρ| ≤ ‖en‖ ≤ 1 for all n ≥ 1 for some scalar ρ with 0 < |ρ| < 1. then, there exists a sequence (kj)j in n such that limj d(tekj,fkj ) = 0. invariant subspace problem 217 for each n ≥ 1, ten ∈ fn + [en+1]. so, there exist yn ∈ fn and λn ∈ k such that ten = yn + λnen+1. for each j ≥ 1, d(λkjekj +1,fkj ) ≥ t‖λkjekj +1‖≥ t|λkj||ρ| . then, limj |λkj| = 0. let mt be the matrix of t in the basis (en)n. for each n ≥ 1, let yn = λn1e1 + · · · + λ n nen, (λ n 1, . . . ,λ n n) ∈ kn. then mt is given by: mt =   λ11 λ 2 1 λ 3 1 . . . . . . λ1 λ 2 2 λ 3 2 . . . . . . 0 λ2 λ 3 3 . . . . . . 0 0 λ3 . . . . . . 0 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   . let s be the operator defined on e by: sen := { ten if n 6= kj , yn if n = kj . (∀n ≥ 1). it is clear that the matrix of s in the basis (en)n is obtained from mt by putting 0 in the place of λkj for all j ≥ 1. consider the operator u = t −s which is defined by: uen := { 0 if n 6= kj , λkjekj +1 if n = kj . (∀n ≥ 1). u is a shift operator on e, and for each x = ∑+∞ n=1 xnen, u(x) = +∞∑ j=1 λkjxkjekj +1 . we obviously have that [e1,e2, . . . ,ekj ] is invariant under s for all j. then, s is a triangular operator. it remains to show that u is compact. for each j ≥ 2, let uj := pju, where pj is a projection on fkj parallel to [ekj +1,ekj +2, . . . ]. it is clear that uj is a finite rank operator. let x = 218 m. babahmed, a. el asri ∑+∞ n=1 xnen ∈ e\{0}. for each � > 0, there exists j0 ≥ 1 such that |λkj| ≤ t ‖x‖� for all j ≥ j0. then, for each j ≥ j0, we have ‖u(x) −uj(x)‖ = ‖ +∞∑ r≥j λkrxkrekr+1‖≤ sup r≥j |λkr||xkr|‖ekj +1‖ ≤ t ‖x‖ � sup r≥j |xkr| ≤ t ‖x‖ � ‖x‖ t = �. hence, u = limj uj. therefore, u is a compact operator. as a consequence of theorem 17, we conclude the following important result. corollary 18. if k is algebraically closed, then every compact operator in e either has a nontrivial invariant closed subspace or is a sum of two compact operators, one is upper triangular and the second one is a shift. proof. let t be a compact operator in e, we assume that the set of points of the spectrum of t is equal to {0} and t has a cyclic vector (otherwise t has a nontrivial closed invariant subspace (see [17, p. 15]), then with [17, theorem 6.14], we have limn‖tn‖ 1 n = 0, so limn‖tnz‖ 1 n = 0 for all z ∈ e. this, with remark 16 and theorem 17 completes the proof. acknowledgements the authors especially thank the referees for their valuable suggestions in preparing this paper. references [1] y.a. abramovich, c.d. aliprantis, o. burkinshaw, invariant subspaces of operators on lp-spaces, j. funct. anal. 115 (2) (1993), 418 – 424. 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[21] a.c.m. van rooij, “ non-archimedean functional analysis ”, monographs and textbooks in pure and applied math. 51, marcel dekker, inc., new york, 1978. introduction preliminary and auxiliary results definitions and main results � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae article in press available online november 23, 2022 topologies, posets and finite quandles m. elhamdadi 1, t. gona 2, h. lahrani 1 1 department of mathematics and statistics, university of south florida tampa, fl 33620, u.s.a. 2 department of mathematics, university of california berkeley, ca 94720, u.s.a. emohamed@math.usf.edu , gonatushar@berkeley.edu , lahrani@usf.edu received september 27, 2022 presented by m. mbekhta accepted october 31, 2022 abstract: an alexandroff space is a topological space in which every intersection of open sets is open. there is one to one correspondence between alexandroff t0-spaces and partially ordered sets (posets). we investigate alexandroff t0-topologies on finite quandles. we prove that there is a non-trivial topology on a finite quandle making right multiplications continuous functions if and only if the quandle has more than one orbit. furthermore, we show that right continuous posets on quandles with n orbits are n-partite. we also find, for the even dihedral quandles, the number of all possible topologies making the right multiplications continuous. some explicit computations for quandles of cardinality up to five are given. key words: quandles, topology, poset. msc (2020): 54e99, 57k12. 1. introduction quandles are algebraic structures modeled on the three reidemeister moves in classical knot theory. they have been used extensively to construct invariants of knots and links, see for example [6, 8, 10]. a topological quandle is a quandle with a topology such that the quandle binary operation is compatible with the topology. precisely, the binary operation is continuous and the right multiplications are homeomorphisms. topological quandles were introduced in [11] where it was shown that the set of homomorphisms (called also the set of colorings) from the fundamental quandle of the knot to a topological quandle is an invariant of the knot. equipped with the compact-open topology, the set of colorings is a topological space. in [5] a foundational account about topological quandles was given. more precisely, the notions of ideals, kernels, units, and inner automorphism group in the context of topological quandle were introduced. furthermore, modules and quandle group bundles issn: 0213-8743 (print), 2605-5686 (online) c© the author(s) released under a creative commons attribution license (cc by-nc 3.0) mailto:emohamed@math.usf.edu mailto:gonatushar@berkeley.edu mailto:lahrani@usf.edu https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 2 m. elhamdadi, t. gona, h. lahrani over topological quandles were introduced with the purpose of studying central extensions of topological quandles. continuous cohomology of topological quandles was introduced in [4] and compared to the algebraic theories. extensions of topological quandles were studied with respect to continuous 2cocycles, and used to show differences in second cohomology groups for some specific topological quandles. nontriviality of continuous cohomology groups for some examples of topological quandles was shown. in [2] the problem of classification of topological alexander quandle structures, up to isomorphism, on the real line and on the unit circle was investigated. in [7] the author investigated quandle objects internal to groups and topological spaces, extending the well-known classification of quandles internal to abelian groups [13]. in [14] quandle modules over quandles endowed with geometric structures were studied. the author also gave an infinitesimal description of certain modules in the case when the quandle is a regular s-manifold (smooth quandle with certain properties). since any finite t1-space is discrete, the category of finite t0-spaces was considered in [12], where the point set topological properties of finite spaces were investigated. the homeomorphism classification of finite spaces was investigated and some representations of these spaces as certain classes of matrices was obtained. this article arose from a desire to better understand the analogy of the work given in [12] in the context of finite topological quandles. it turned out that: there is no t0-topology on any finite connected (meaning one orbit under the action of the inner group) quandle x that makes x into a topological quandle (theorem 4.4). thus we were lead to consider topologies on finite quandles with more than one orbit. it is well known [1] that the category of alexandroff t0-spaces is equivalent to the category of partially ordered sets (posets). in our context, we prove that for a finite quandle x with more than one orbit, there exists a unique non trivial topology which makes right multiplications of x continuous maps (proposition 4.6). furthermore, we prove that if x be a finite quandle with two orbits x1 and x2 then any continuous poset on x is biparatite with vertex set x1 and x2 (proposition 4.7). this article is organized as follows. in section 2 we review the basics of topological quandles. section 3 reviews some basics of posets, graphs and some hierarchy of separation axioms. in section 4 the main results of the article are given. section 5 gives some explicit computations based on some computer software (maple and python) of quandles up to order five. topologies, posets and finite quandles 3 2. review of quandles and topological quandles a quandle is a set x with a binary operation ∗ satisfying the following three axioms: (1) for all x in x, x∗x = x, (2) for all y,z ∈ x, there exists a unique x such that x∗y = z, (3) for all x,y,z ∈ x, (x∗y) ∗z = (x∗z) ∗ (y ∗z). these three conditions come from the axiomatization of the three reidemeister moves on knot diagrams. the typical examples of quandles are: (i) any group g with conjugation x∗y = y−1xy, is a quandle called the conjugation quandle and (ii) any group g with operation given by x ∗ y = yx−1y, is a quandle called the core quandle. let x be a quandle. for an element y ∈ x, left multiplication ly and right multiplication ry by an element y are the maps from x to x given respectively by ly(x) := y∗x and ry(x) = x∗y. a function f : (x,∗) → (x,∗) is a quandle homomorphism if for all x,y ∈ x,f(x∗y) = f(x)∗f(y). if furthermore f is a bijection then it is called an automorphism of the quandle x. we will denote by aut(x) the automorphism group of x. the subgroup of aut(x), generated by the automorphisms rx, is called the inner automorphism group of x and denoted by inn(x). if the group inn(x) acts transitively on x, we then say that x is connected quandle meaning it has only one orbit. since we do not consider topological connectedness in this article, then through the whole article, the word connected quandle will stand for algebraic connectedness. for more on quandles refer to [6, 8, 10, 3]. topological quandles have been investigated in [2, 5, 11, 4]. here we review some basics of topological quandles. definition 2.1. a topological quandle is a quandle x with a topology such that the map x × x 3 (x,y) 7−→ x ∗ y ∈ x is a continuous, the right multiplication rx : x 3 y 7−→ y ∗x ∈ x is a homeomorphism, for all x ∈ x, and x∗x = x. it is clear that any finite quandle is automatically a topological quandle with respect to the discrete topology. example 2.2. [2] let (g, +) be a topological abelian group and let σ be a continuous automorphism of g. the continuous binary operation on g given by x∗y = σ(x) + (id−σ)(y),∀x,y ∈ g, makes (g,∗) a topological quandle called topological alexander quandle. in particular, if g = r and σ(x) = tx 4 m. elhamdadi, t. gona, h. lahrani for non-zero t ∈ r, we have the topological alexander structure on r given by x∗y = tx + (1 − t)y. example 2.3. the following examples were given in [11, 5]. the unit sphere sn ⊂ rn+1 with the binary operation x∗y = 2(x·y)y−x is a topological quandle, where · denotes the inner product of rn+1. now consider λ and µ be real numbers, and let x,y ∈ sn. then λx∗µy = λ[2µ2(x ·y)y −x]. in particular, the operation ±x∗±y = ±(x∗y) provides a structure of topological quandle on the quotient space that is the projective space rpn. 3. review of topologies on finite sets, posets and graphs now we review some basics of directed graphs, posets and t0 and t1 topologies. definition 3.1. a directed graph g is a pair (v,e) where v is the set of vertices and e is a list of directed line segments called edges between pairs of vertices. an edge from a vertex x to a vertex y will be denoted symbolically by x < y and we will say that x and y are adjacent. the following is an example of a directed graph. example 3.2. let g = (v,e) where v = {a,b,c,d} and e = {b < a, c < a, a < d}. d a b c topologies, posets and finite quandles 5 definition 3.3. an independent set in a graph is a set of pairwise nonadjacent vertices. definition 3.4. a (directed) graph g = (v,e) is called biparatite if v is the union of two disjoint independent sets v1 and v2. definition 3.5. a (directed) graph g is called complete biparatite if g is bipartite and for every v1 ∈ v1 and v2 ∈ v2 there is an edges in g that joins v1 and v2. example 3.6. let v = v1 ∪ v2 where v1 = {4, 5} and v2 = {1, 2, 3}. then the directed graph g = (v,e) is complete biparatite graph. now we recall the definition of partially ordered set. definition 3.7. a partially ordered set (poset) is a set x with an order denoted ≤ that is reflexive, antisymmetric and transitive. example 3.8. for any set x, the power set of x ordered by the set inclusion relation ⊆ forms a poset (p(x),⊆) definition 3.9. two partially ordered sets p = (x,≤) and q = (x,≤′) are said to be isomorphic if there exist a bijection f : x → x′ such that x ≤ y if and only if f(x) ≤′ f(y). definition 3.10. a poset (x,≤) is connected if for all x,y ∈ x, there exists sequence of elements x = x1,x2, . . . ,xn = y such that every two consecutive elements xi and xi+1 are comparable (meaning xi < xi+1 or xi+1 < xi). notation: given an order ≤ on a set x, we will denote x < y whenever x 6= y and x ≤ y. finite posets (x,≤) can be drawn as directed graphs where the vertex set is x and an arrow goes from x to y whenever x ≤ y. for simplicity, we will not draw loops which correspond to x ≤ x. we will then use the notation (x,<) instead of (x,≤) whenever we want to ignore the reflexivity of the partial order. 6 m. elhamdadi, t. gona, h. lahrani example 3.11. . let x = z8 be the set of integers modulo 8. the map f : x → x given by f(x) = 3x − 2 induces an isomorphism between the following two posets (x,<) and (x,<′). , . definition 3.12. a chain in a poset (x,<) is a subset c of x such that the restriction of < to c is a total order (i.e. every two elements are comparable). now we recall some basics about topological spaces called t0 and t1 spaces. definition 3.13. a topological space x is said to have the property t0 if for every pair of distinct points of x, at least one of them has a neighborhood not containing the other point. definition 3.14. a topological space x is said to have the property t1 if for every pair of distinct points of x, each point has a neighborhood not containing the other point. obviously the property t1 implies the property t0. notice also that this definition is equivalent to saying singletons are closed in x. thus a t1topology on a finite set is a discrete topology. since any finite t1-space is discrete, we will focus on the category of finite t0-spaces. first we need some notations. let x be a finite topological space. for any x ∈ x, we denote ux := the smallest open subset of x containing x. it is well known [1] that the category of t0-spaces is isomorphic to the category of posets. we have x ≤ y if and only if uy ⊆ ux which is equivalent to cx ⊂ cy, where cv is the complement ucv of uv in x. thus one obtain that ux = {w ∈ x; x ≤ w} and cx = {v ∈ x; v < x}. under this correspondence of categories, the subcategory of finite posets is equivalent to the category of finite t0-spaces. through the rest of this article we will use the notation of x < y in the poset whenever x 6= y and x ≤ y. topologies, posets and finite quandles 7 4. topologies on non-connected quandles as we mentioned earlier, since t1-topologies on a finite set are discrete, we will focus in this article on t0-topologies on finite quandles. a map on finite spaces is continuous if and only if it preserves the order. it turned out that on a finite quandle with a t0-topology, left multiplications can not be continuous as can be seen in the following theorem theorem 4.1. let x be a finite quandle endowed with a t0-topology. assume that for all z ∈ x, the map lz is continuous, then x ≤ y implies lz(x) = lz(y). proof. we prove this theorem by contradiction. let x be a finite quandle endowed with a t0-topology. assume that x ≤ y and lz(x) 6= lz(y). if x = y, then obviously lz(x) = lz(y). now assume x < y, then for all a ∈ x, the continuity of la implies that a∗x ≤ a∗ y. assume that there exist a1 ∈ x such that, z1 := a1 ∗ x = la1 (x) < a1 ∗ y = la1 (y). the invertibility of right multiplications in a quandle implies that there exist unique a2 such that a2 ∗ x = a1 ∗ y hence a1 ∗ x < a2 ∗ x which implies a1 6= a2. now we have a1∗x < a2∗x ≤ a2∗y = z2. we claim that a2∗x < a2∗y. if a2∗y = a2∗x and since a2 ∗x = a1 ∗y we will have a2 ∗y = a2 ∗x = a1 ∗y hence a2 ∗y = a1 ∗y but a1 6= a2, thus contradiction. now that we have proved a2 ∗ x < a2 ∗ y, then there exists a3 such that a2 ∗y = a3 ∗x we get, a2 ∗x < a3 ∗x repeating the above argument we get, a3 ∗ x < a3 ∗ y. notice that a1,a2 and a3 are all pairwise disjoint elements of x. similarly, we construct an infinite chain, a1 ∗x < a2 ∗x < a3 ∗x < · · · , which is impossible since x is a finite quandle. thus we obtain a contradiction. we have the following corollary corollary 4.2. let x be a finite quandle endowed with a t0-topology. if c is a chain of x as a poset then any left continuous function lx on x is a constant function on c. definition 4.3. a quandle with a topology in which right multiplications (respectively left multiplications) are continuous is called right topological quandle (respectively left topological quandle). in other words, right topological quandle means that for all x,y,z ∈ x, x < y ⇒ x∗z < y ∗z. 8 m. elhamdadi, t. gona, h. lahrani and, since left multiplications are not necessarily bijective maps, left topological quandle means that for all x,y,z ∈ x, x < y ⇒ z ∗x ≤ z ∗y. theorem 4.4. there is no t0-topology on a finite connected quandle x that makes x into a right topological quandle. proof. let x < y. since x is connected quandle, there exists φ ∈ inn(x) such that y = φ(x). since x is finite, φ has a finite order m in the group inn(x). since φ is a continuous automorphism then x < φ(x) implies x < φm(x) giving a contradiction. corollary 4.5. there is no t0-topology on any latin quandle that makes it into a right topological quandle. thus theorem 4.4 leads us to consider quandles x that are not connected, that is x = x1 ∪x2 ∪ . . .xk as orbit decomposition, search for t0-topology on x and investigate the continuity of the binary operation. proposition 4.6. let x be a finite quandle with orbit decomposition x = x1 ∪{a}, then there exist unique non trivial t0-topology which makes x right continuous. proof. let x = x1 ∪{a} be the orbit decomposition of the quandle x. for any x,y ∈ x1, there exits φ ∈ inn(x) such that φ(x) = y and φ(a) = a. declare that x < a, then φ(x) < a. thus for any z ∈ x1 we have z < a. uniqueness is obvious. the t0-topology in proposition 4.6 is precisely given by x < a for all x ∈ x1. proposition 4.7. let x be a finite quandle with two orbits x1 and x2. then any right continuous poset on x is biparatite with vertex set x1 and x2. proof. we prove this proposition by contradiction. for every x1,y1 ∈ x1 such that x1 < y1. we know that there exist φ ∈ inn(x) such that φ(x1) = y1. hence, x1 < φ(x1) implies x1 < φ m(x1) = x1, where m is the order of φ in inn(x). thus we have a contradiction. topologies, posets and finite quandles 9 proposition 4.8. let x be a finite quandle with two orbits x1 and x2. then the complete bipartite graph with vertex set x1 and x2 forms a right continuous poset. proof. let x be a finite quandle with two orbits x1 and x2. if x ∈ x1 and y ∈ x2 then for every φ ∈ inn(x) we have φ(x) ∈ x1 and φ(y) ∈ x2. proposition 4.7 gives that the graph is bipartite and thus x < y. we then obtain φ(x) < φ(y) giving the result. remark 4.9. by proposition 4.8 and theorem 4.1, there is a non-trivial t0-topology making x right continuous if and only if the quandle has more than one orbit. notice that proposition 4.8 can be generalized to n-paratite complete graph. the following table gives the list of right continuous posets on some even dihedral quandles. in the table, the notation (a,b) on the right column means a < b. table 1: right continuous posets on dihedral quandles quandle posets r4 ((0,1),(2,1),(0,3),(2,3)) . ((0, 1), (0, 5), (2, 1), (2, 3), (4, 3), (4, 5)) ; r6 ((0,3), (2, 5), (4, 1)) . (2, 7), (4, 7), (6, 1), (6, 3), (0, 5), (2, 5), (4, 1), (0, 3)) ; r8 (( 0, 1), (6, 7), (4, 5), (0, 7), (2, 1), (2,3), (4, 3), (6, 5)) . ((0, 1), (6, 7), (4, 5), (2, 1), (8, 9), (2, 3), (4, 3), (8, 7), (0, 9), (6, 5)) ; r10 ((4, 7), (6, 9), (2, 9), (8, 1), (8, 5), (0, 7), (6, 3), (2, 5), (4, 1), (0, 3)) ; ((2, 7), (8, 3), (0, 5), (4, 9), (6, 1)) . notice that in table 1, the dihedral quandle r4 has only one right continuous poset ((0, 1), (2, 1), (0, 3), (2, 3)) which is complete biparatite. while the dihedral quandle r6 has two continuous posets ((0, 1), (0, 5), (2, 1), (2, 3), (4, 3), (4, 5)) and ((0, 3), (2, 5), (4, 1)) illustrated below. 10 m. elhamdadi, t. gona, h. lahrani 3 0 5 2 1 4 3 2 5 0 1 4 moreover, in table 1, for r8 the bijection f given by f(k) = 3k−2 makes the two posets isomorphic. the same bijection gives isomorphism between the first two posets of r10. the following theorem characterizes non complete biparatite posets on dihedral quandles. theorem 4.10. let r2n be a dihedral quandle of even order. then r2n has s + 1 right continuous posets, where s is number of odd natural numbers less than n and relatively non coprime with n proof. let x = r2n be the dihedral quandle with orbits x1 = {0, 2, . . . , 2n − 2} and x2 = {1, 3, . . . , 2n − 1}. for every x ∈ x2, we construct a partial order 1. the two posets <1 and 4b, then ‖p‖ = max { a, |b|, ∣∣∣14a + b∣∣∣ + 12c, |c2−4ab|4a }. case 2: c ≥ a. if a ≤ 4b, then ‖p‖ = max { a,b, ∣∣∣14a + b∣∣∣ + 12c, |c2−4ab|2c+a+4b}. if a > 4b, then ‖p‖ = max { a, |b|, ∣∣∣14a + b∣∣∣ + 12c, c2−4ab2c−a−4b}. note that if ‖p‖ = 1, then |a| ≤ 1, |b| ≤ 1, |c| ≤ 2. theorem 2.2. ([15, 16]) ext bp(2h) = exp bp(2h) = { ±y2 , ± ( x2 + 1 4 y2 ±xy ) , ± ( x2 + 3 4 y2 ) , ± [ x2 + ( c2 4 − 1 ) y2 ± cxy ] (0 ≤ c ≤ 1) , ± [ ax2 + ( a + 4 √ 1 −a 4 − 1 ) y2 ± (a + 2 √ 1 −a)xy ] (0 ≤ a ≤ 1) } . by the krein-milman theorem, a convex function (like a functional norm, for instance) defined on a convex set (like the unit ball of a finite dimensional polynomial space) attains its maximum at one extreme point of the convex set. 246 s.g. kim theorem 2.3. ([16]) let f ∈ p(2h)∗ with α = f(x2), β = f(y2), γ = f(xy). then ‖f‖ = max { |β|, ∣∣∣α + 1 4 β ∣∣∣ + |γ|,∣∣∣α + 3 4 β ∣∣∣,∣∣∣∣α + ( c2 4 − 1 ) β ∣∣∣∣ + c|γ| (0 ≤ c ≤ 1),∣∣∣∣aα + ( a + 4 √ 1 −a 4 − 1 ) β ∣∣∣∣ + (a + 2√1 −a)|γ| (0 ≤ a ≤ 1) } . proof. it follows from theorem 2.2 and the fact that ‖f‖ = sup p∈ext b ∣∣f(p)∣∣, where b := bp(2h). note that if ‖f‖ = 1, then |α| ≤ 1, |β| ≤ 1, |γ| ≤ 1 2 . remark. let p(x,y) = ax2 + by2 + cxy ∈p(2h) with ‖p‖ = 1. then the following are equivalent: (1) p is smooth; (2) −p(x,y) = −ax2 − by2 − cxy is smooth; (3) p(x,−y) = ax2 + by2 − cxy is smooth. as a consequence of the previous remark, our attention can be restricted to polynomials q(x,y) = ax2 + by2 + cxy ∈p(2h) with a ≥ 0, c ≥ 0. we are in position to prove the main result of this paper. theorem 2.4. let p(x,y) = ax2 + by2 + cxy ∈p(2h) with a ≥ 0, c ≥ 0, ‖p‖ = 1. then p is a smooth point of the unit ball of p(2h) if and only if one of the following mutually exclusive conditions holds: (1) a = 0 , 0 < |b| < 1 ; (2) a = 1 , b = −3 4 , 1 4 , c < 1 ; (3) a = 1 , −1 < b < −3 4 , b− c 2 > −5 4 , c 2 4 − b < 1 ; (4) a = 1 , −3 4 < b < 1 4 ; (5) a = 1 , 1 4 ≤ b , b + c 2 < 3 4 ; (6) 0 < a < 1 , b = 0 ; (7) 0 < a < 1 , c ≤ a , 0 6= 4b < a ; (8) 0 < a < 1 , 0 < c ≤ a < 4b ; smooth 2-homogeneous polynomials 247 (9) 0 < a < 1 , 4b = a < c ; (10) 0 < a < 1 , 0 6= 4b < a < c , c 6= a + 2 √ 1 −a ; (11) 0 < a < 1 , a < 4b , a < c . proof. let q(x,y) = ax2 + by2 + cxy ∈ p(2h) with a ≥ 0, c ≥ 0 and ‖q‖ = 1. case 1: a = 0. note that if b = 0 or ±1, then q is not smooth. in fact, if b = 0, then q = 2xy. for j = 1, 2, let fj ∈p(2h)∗ be such that f1 ( x2 ) = 1 4 , f1 ( y2 ) = 1 , f1(xy) = 1 2 , f2 ( x2 ) = 0 = f2 ( y2 ) , f2(xy) = 1 2 . by theorem 2.3, fj(q) = 1 = ‖fj‖ for j = 1, 2. thus q is not smooth. if b = ±1, then p = ±y2. for j = 1, 2, let fj ∈p(2h)∗ be such that f1 ( x2 ) = ± 1 4 , f1 ( y2 ) = ±1 , f1(xy) = ± 1 2 , f2 ( x2 ) = 0 = f2(xy) , f2 ( y2 ) = ±1 . by theorem 2.3, fj(q) = 1 = ‖fj‖ for j = 1, 2. thus q is not smooth. claim: if a = 0, 0 < |b| < 1, then q is smooth. without loss of generality, we may assume that 0 < b < 1. by theorem 2.1, 1 = ‖q‖ = b + 1 2 c. thus c = 2(1 − b), so 0 < c < 2. let f ∈ p(2h)∗ be such that f(q) = 1 = ‖f‖. notice that 1 = bβ + cγ. we will show that α = 1 4 , β = 1, γ = 1 2 . since 0 < b < 1, 0 < c < 2, we can choose δ > 0 such that 0 < 2(1 − b) + t = c + t < 2 , 0 < b− 1 2 t < 1 , for all t ∈ (−δ,δ). let qt(x,y) = ( b − 1 2 t ) y2 + (c + t)xy for all t ∈ (−δ,δ). by theorem 2.1, ‖qt‖ = 1 for all t ∈ (−δ,δ). for all t ∈ (−δ,δ), 1 = bβ + cγ ≥ f(qt) = ( b− 1 2 t ) β + (c + t)γ , which shows that t ( γ − 1 2 β ) ≤ 0, for all t ∈ (−δ,δ). thus γ = 1 2 β. since 1 = f(q) = bβ + cγ = 2γ, we have β = 1, γ = 1 2 . by theorem 2.3, 1 ≥ 248 s.g. kim ∣∣∣α + 14β∣∣∣ + |γ| = ∣∣∣α + 14∣∣∣ + 12, so − 3 4 ≤ α ≤ 1 4 . (1) by theorem 2.3, for 0 ≤ c̃ ≤ 1, 1 ≥ ∣∣∣∣α + ( c̃2 4 − 1 )∣∣∣∣ + c̃2 = − ( α + ( c̃2 4 − 1 )) + c̃ 2 , which implies that 4α ≥ sup 0≤c̃≤1 (2c̃− c̃2) = 1 . (2) by (1) and (2), α = 1 4 . therefore, q is smooth. case 2: a = 1. if b = −1, then q = x2 −y2. for j = 1, 2, let fj ∈p(2h)∗ be such that f1(x 2) = 1 , f1(y 2) = 0 = f1(xy) , f2(x 2) = 0 = f2(xy) , f2(y 2) = −1 . by theorem 2.3, fj(q) = 1 = ‖fj‖ for j = 1, 2. hence, q is not smooth. claim: if ( a = 1, b = −3 4 , 1 4 , c < 1 ) , ( a = 1, −1 < b < 1 4 , b 6= −3 4 ) or( a = 1, 1 4 ≤ b, b + c 2 < 3 4 ) , then q is smooth. note that if a = 1,b = −3 4 , then c ≤ 1. note also that if a = 1,b = −3 4 ,c = 1, then q is not smooth. suppose that a = 1, b = −3 4 , c < 1. let f ∈ p(2h)∗ be such that f(q) = 1 = ‖f‖. then 1 = α − 3 4 β + cγ. we will show that α = 1, β = γ = 0. since 0 ≤ c < 1 and by theorem 2.1, we can choose δ > 0 such that ‖ru‖ = ‖sv‖ = 1 for all u,v ∈ (−δ,δ), where ru(x,y) = x 2 − 3 4 y2 + (c + u)xy , sv(x,y) = x 2 − ( 3 4 + v ) y2 + cxy ∈p(2h) . it follows that, for all u,v ∈ (−δ,δ), 1 = α− 3 4 β + cγ ≥ f(ru) = α− 3 4 β + (c + u)γ , 1 = α− 3 4 β + cγ ≥ f(sv) = α− ( 3 4 + v ) β + cγ , smooth 2-homogeneous polynomials 249 which shows that α = 1, β = γ = 0. therefore, q is smooth. by a similar argument, if a = 1, b = 1 4 , c < 1, then q is smooth. suppose that a = 1, −1 < b < 1 4 , b 6= −3 4 . let a = 1, −1 < b < −3 4 . we will show that c < 1. if not, then 1 ≤ c ≤ 2. by theorem 2.1, b− c 2 ≥ −5 4 , c2−4b 2c−1−4b ≤ 1, which shows that c = 1, b ≥− 3 4 . this is a contradiction. hence, by theorem 2.1, b− c 2 ≥−5 4 , c 2 4 − b ≤ 1. we claim that if a = 1 , −1 < b < − 3 4 , b− c 2 > − 5 4 , c2 4 − b < 1 , then q is smooth. let f ∈ p(2h)∗ be such that f(q) = 1 = ‖f‖. then, 1 = α + bβ + cγ. we will show that α = 1, β = γ = 0. by theorem 2.1, we can choose δ > 0 such that ‖ru‖ = ‖sv‖ = 1 for all u,v ∈ (−δ,δ), where ru(x,y) = x 2 + by2 + (c + u)xy , sv(x,y) = x 2 + (b + v)y2 + cxy ∈p(2h) . thus α = 1, β = γ = 0. therefore, q is smooth. note that if a = 1 , −1 < b < − 3 4 , b− c 2 ≥− 5 4 , c2 4 − b = 1 , then q is not smooth letting fj ∈p(2h)∗ be such that f1(x 2) = 1 , f1(y 2) = 0 = f1(xy) , f2(x 2) = − c2 4 , f2(y 2) = −1 , f2(xy) = c 2 . thus x2 + (c 2 4 − 1)y2 + cxy (0 ≤ c ≤ 1) is not smooth. note also that if a = 1 , −1 < b < − 3 4 , b− c 2 = − 5 4 , c2 4 − b ≤ 1 , then q is not smooth letting fj ∈p(2h)∗ be such that f1(x 2) = 1 , f1(y 2) = 0 = f1(xy) , f2(x 2) = − 1 4 , f2(y 2) = −1 , f2(xy) = 1 2 . 250 s.g. kim let a = 1, −3 4 < b < 1 4 . we will show that q is smooth. first, suppose that −3 4 < b < 0. since ‖q‖ = 1, by theorem 2.1, we have 0 ≤ c ≤ 1. let f ∈ p(2h)∗ be such that f(q) = 1 = ‖f‖. then 1 = α + bβ + cγ. we will show that α = 1, β = 0 = γ. since −3 4 < b < 0, by theorem 2.1, we can choose δ > 0 such that ‖ru‖ = ‖sv‖ = 1 for all u,v ∈ (−δ,δ), where ru(x,y) = x 2 + (b + u)y2 + cxy , sv(x,y) = x 2 + by2 + (c + v)xy ∈p(2h) . thus α = 1, β = 0 = γ. hence, q is smooth. suppose that c = 1. then 1 = α + γ, α ≥ 0, γ ≥ 0. by theorem 2.3, 1 ≥ sup 0≤ã≤1 ãα + (ã + 2 √ 1 − ã)γ = sup 0≤ã≤1 2 √ 1 − ã(1 −α) + ã = 1 + (1 −α)2 , which implies that α = 1. therefore, α = 1, β = 0 = γ. we have shown that if 0 < c ≤ 1, then q is smooth. suppose that c = 0. since 1 = α + bβ, β = 0, we have α = 1. by theorem 2.3, 1 ≥ ∣∣∣α + 14β∣∣∣+ |γ| = 1 + γ, which shows that γ = 0. hence, q is smooth. suppose that 0 ≤ b < 1 4 . since ‖q‖ = 1, by theorem 2.1, 0 ≤ c ≤ 1. let f ∈ p(2h)∗ be such that f(q) = 1 = ‖f‖. we will show that α = 1, β = 0 = γ. since 1 = f(q) = α + bβ + cγ, we have α > 0. indeed, if α ≤ 0, then 1 ≤ bβ + cγ ≤ b|β| + c|γ| < 1 4 + 1 2 = 3 4 , which is a contradiction. we also claim that α + 1 4 β ≥ 0. if not, then α < 1 4 |β| ≤ 1 4 , which implies that 3 4 < 1 −α = bβ + cγ ≤ b|β| + c|γ| < 3 4 , which is a contradiction. note that α + bβ = 1 − cγ ≥ 1 − c|γ| ≥ 1 − c 2 ≥ 1 2 . by theorem 2.3, α + 1 4 β + |γ| = ∣∣∣α + 1 4 β ∣∣∣ + |γ| ≤ 1 = α + bβ + cγ ≤ α + bβ + c|γ| , smooth 2-homogeneous polynomials 251 which shows that ( 1 4 − b ) β ≤ (c− 1)|γ| ≤ 0 . hence, β ≤ 0. by theorem 2.3, for all 0 ≤ c̃ ≤ 1, it follows that α + ( 1 − c̃2 4 ) |β| + c̃|γ| = ∣∣∣∣α + ( c̃2 4 − 1 ) β ∣∣∣∣ + c̃|γ| ≤ 1 = α + bβ + cγ ≤ α + bβ + c|γ| = α− b|β| + c|γ| , which implies that( 1 − c̃2 4 + b ) |β| ≤ (c− c̃)|γ| (0 ≤ c̃ ≤ 1) . thus ( 1 − c2 4 + b ) |β| = lim c̃→c− ( 1 − c̃2 4 + b ) |β| ≤ lim c̃→c− (c− c̃)|γ| = 0 , so β = 0. since 1 = f(q) = α + cγ, we have γ ≥ 0. by theorem 2.3, ãα + ( ã + 2 √ 1 − ã ) γ ≤ 1 = α + cγ (0 ≤ ã ≤ 1) , which implies that (ã− c + 2 √ 1 − ã)γ ≤ (1 − ã)α (0 ≤ ã ≤ 1) . (3) if c < 1, then (1 − c)γ = lim ã→1− ( ã− c + 2 √ 1 − ã ) γ ≤ lim ã→1− (1 − ã)α = 0 , so γ = 0. therefore, α = 1, β = 0. suppose that c = 1. by (3), (ã− 1 + 2 √ 1 − ã)γ ≤ (1 − ã)α (0 ≤ ã ≤ 1) , which implies that 2γ = lim ã→1− ( 2 − √ 1 − ã ) γ ≤ ( lim ã→1− √ 1 − ã ) α = 0 , so γ = 0. therefore, α = 1, β = 0 = γ. hence, q is smooth. 252 s.g. kim suppose that a = 1, 1 4 ≤ b. since ‖q‖ = 1, we have b+ c 2 ≤ 3 4 . if b+ c 2 = 3 4 , then q is not smooth letting fj ∈p(2h)∗ be such that f1(x 2) = 1 4 , f1(y 2) = 1 , f1(xy) = 1 2 , f2(x 2) = 1 , f2(y 2) = 0 = f2(xy) . let b + c 2 < 3 4 . note that if b = 1 4 , then q = x2 + 1 4 y2 + cxy for 0 ≤ c < 1. let f ∈p(2h)∗ be such that f(q) = 1 = ‖f‖. then α = 1, β = 0 = γ. thus q is smooth. suppose that a = 1, 1 4 < b. let f ∈p(2h)∗ be such that f(q) = 1 = ‖f‖. then α = 1, β = 0 = γ. thus q is smooth. case 3: 0 < a < 1. suppose that b = 0. we will show that c > a. if not, then ‖q‖ < 1, which is a contradiction. hence, c > a. we claim that q is smooth. let f ∈ p(2h)∗ be such that f(q) = 1 = ‖f‖. we will show that α = 1 c2 , β = 4(1−a) c2 , γ = 2(c−1) c2 . note that 1 4 a + 1 2 c < 1, 0 < c < 2. we may choose δ > 0 such that ‖ru‖ = ‖sv‖ = 1 for all u,v ∈ (−δ,δ), where ru(x,y) = ( a + u(2 − 2c−u) ) x2 + (c + u)xy , sv(x,y) = ( a + 4(a− 1)v 1 − 4v ) x2 + vy2 + cxy ∈p(2h) . then γ = 2(c− 1)α, β = 4(1 −a)α. it follows that 1 = aα + cγ = c(2 − c)α + c(2c− 2)α = c2α, proving that α = 1 c2 , β = 4(1−a) c2 , γ = 2(c−1) c2 . thus q is smooth. suppose that b 6= 0. let c ≤ a. suppose that c ≤ a ≤ 4b. notice that if a = 4b, then ‖q‖ < 1. hence, q is not smooth. suppose that a < 4b. then, 0 < b ≤ 1. if b = 1, then ‖q‖ > 1, which is impossible. we claim that if c = a, 0 < b < 1, then q is smooth. let 0 < b < 1. by theorem 2.1, 1 = ‖q‖ = 3 4 a + b. therefore, q = ax2 + ( 1 − 3 4 a ) y2 + axy for 0 < a < 1. let f ∈ p(2h)∗ be such that f(q) = 1 = ‖f‖. then 1 = aα + ( 1 − 3 4 a ) β + aγ. we will show that α = 1 4 , β = 1, γ = 1 2 . we can smooth 2-homogeneous polynomials 253 choose δ > 0 such that ‖ru‖ = ‖sv‖ = 1 for all u,v ∈ (−δ,δ), where ru(x,y) = ax 2 + ( 1 − 3a 4 + u ) y2 + (a− 2u)xy , sv(x,y) = (a− 2v)x2 + ( 1 − 3a 4 ) y2 + (a + v)xy ∈p(2h) . then β = 2γ, γ = 2α. therefore, α = 1 4 , β = 1, γ = 1 2 . thus q is smooth. notice that if 0 = c < a < 4b, then q is not smooth letting fj ∈ p(2h)∗ be such that f1(x 2) = 1 4 = f2(x 2) , f1(y 2) = 1 = f2(y 2) , f1(xy) = 1 2 , f2(xy) = 0 . claim: if 0 < c < a < 4b, then q is smooth. by theorem 2.1, 1 = ‖q‖ = 1 4 a + b + 1 2 c. thus 0 < b < 1. let f ∈p(2h)∗ be such that f(q) = 1 = ‖f‖. we will show that α = 1 4 ,β = 1,γ = 1 2 . we choose δ > 0 such that ‖ru,v‖ = 1 for all u,v ∈ (−δ,δ), where ru,v(x,y) = (a + u)x 2 + (b + v)y2 + ( c− 1 2 u− 2v ) xy ∈p(2h) . thus α = 1 4 , β = 1, γ = 1 2 . therefore, q is smooth. claim: if c ≤ a, 4b < a, then q is smooth. suppose that c = a, 4b < a. by theorem 2.1, 1 = ‖q‖ = ∣∣∣14a + b∣∣∣ + 12a. notice that 1 4 a + b < 0. thus q = ax2 + ( 1 4 a− 1 ) y2 + axy for 0 < a < 1. we will show that q is smooth. let f ∈ p(2h)∗ be such that f(q) = 1 = ‖f‖. we will show that α = −1 4 , β = −1, γ = 1 2 . choose 0 < δ < 1 such that 0 < a + 2v < a + v < 1 , (a + v)2 − 4a ( 1 4 a− 1 ) 2(a + v) −a− 4 ( 1 4 a− 1 ) < 1 254 s.g. kim for all v ∈ (−δ, 0). let rv = (a + 2v)x 2 + ( 1 4 a− 1 ) y2 + (a + v)xy for v ∈ (−δ, 0). by theorem 2.1, 1 = ‖rv‖. thus γ ≥ −2α. choose 0 < δ1 < 1 such that 0 < a + v < 1 , a2 − 4(a + v) ( 1 4 a− 1 − 1 4 v ) 2a− (a + v) − 4 ( 1 4 a− 1 − 1 4 v ) < 1 for all v ∈ (−δ1, 0). let sv = (a + v)x 2 + ( 1 4 a− 1 − 1 4 v ) y2 + axy for v ∈ (−δ1, 0). by theorem 2.1, 1 = ‖sv‖. thus α ≥ 14β. choose 0 < δ2 < 1 such that (a + 2v)2 − 4a ( 1 4 a− 1 + v ) 2(a + 2v) −a− 4 ( 1 4 a− 1 + v ) < 1 for all v ∈ (0,δ2). let wu = ax 2 + ( 1 4 a− 1 + u ) y2 + (a + 2u)xy for u ∈ (0,δ2). by theorem 2.1, 1 = ‖wu‖. thus β ≤ −2γ. let β = −1 + � for some 0 ≤ � < 1. by theorem 2.3, it follows that 1 ≥ sup 0≤c≤1 ∣∣∣∣α + ( c2 4 − 1 ) (−1 + �) ∣∣∣∣ + cγ = sup 0≤c≤1 − 1 4 (c− 2γ)2 + γ2 −γ + 5 4 + � ( 1 a − 5 4 + c2 4 ) ≥ max { γ2 −γ + 5 4 + � ( 1 a − 5 4 + (2γ)2 4 ) , − 1 4 (1 − 2γ)2 + γ2 −γ + 5 4 + � ( 1 a − 1 )} = max {( γ − 1 2 )2 + 1 + � ( 1 a − 5 4 + γ2 ) , 1 + � ( 1 a − 1 )} ≥ 1 + � ( 1 a − 1 ) ≥ 1 , smooth 2-homogeneous polynomials 255 which shows that � = 0 = (γ − 1 2 )2. thus α = −1 4 , β = −1, γ = 1 2 . hence, q is smooth. suppose that c < a, 4b < a. note that −1 ≤ b < 0. if b = −1, then q = ax2 −y2. we will show that it is smooth. let f ∈p(2h)∗ be such that f(q) = 1 = ‖f‖. notice that α = 0, β = −1, γ = 0. hence, q is smooth. let −1 < b < 0. then c > 0. claim: 1 = |c2−4ab| 4a = c 2−4ab 4a . first, suppose that 1 4 a ≥ |b|. then ∣∣1 4 a+b ∣∣+ 1 2 c = 1 4 a+b+ 1 2 c < a < 1. by theorem 2.1, 1 = ‖q‖ = |c 2−4ab| 4a . let 1 4 a < |b|. notice that ∣∣1 4 a + b ∣∣ + 1 2 c < c2+4a|b| 4a , so 1 = |c2−4ab| 4a = c 2−4ab 4a . suppose that 0 < c < 1. we will show that q is smooth. let f ∈ p(2h)∗ be such that f(q) = 1 = ‖f‖. we will show that α = − c 2 4a2 , β = −1, γ = c 2a . we choose δ > 0 such that ‖rv‖ = ‖sw‖ = 1 for all v,w ∈ (−δ,δ), where rv(x,y) = ( a− av 1 + b + v ) x2 + (b + v)y2 + cxy , sw(x,y) = ax 2 + ( b + w(2c + w) 4a ) y2 + (c + w)xy ∈p(2h) . notice that β = a 1+b α, γ = − c 2a β. therefore, α = − c 2 4a2 , β = −1, γ = c 2a . hence, q is smooth. suppose that c = 0. then q = ax2 −y2 for 0 < a < 1, which is smooth. suppose that c > a. claim: if c > a = 4b, then q is smooth. notice that q = ax2 + a 4 y2 + (2 − a)xy. let f ∈ p(2h)∗ be such that f(q) = 1 = ‖f‖. by the previous arguments, α = 1 4 ,β = 1,γ = 1 2 . thus q is smooth. claim: if c > a > 4b, c 6= a + 2 √ 1 −a, then q is smooth. by theorem 2.1, −1 < b < 1 4 , 0 < c < 2. notice that∣∣∣∣14a + b ∣∣∣∣ + 12c < 1 and c 2 − 4ab 2c−a− 4b = 1 , or c2 − 4ab 2c−a− 4b < 1 and ∣∣∣∣14a + b ∣∣∣∣ + 12c = 1 . 256 s.g. kim first, suppose that ∣∣1 4 a + b ∣∣ + 1 2 c < 1, c 2−4ab 2c−a−4b = 1. let f ∈ p( 2h)∗ be such that f(q) = 1 = ‖f‖. we will show that α = (c−4b) 2 (2c−a−4b)2 , β = 4(c−a)2 (2c−a−4b)2 , γ = 2(c−a)(c−4b) (2c−a−4b)2 . we may choose δ > 0 such that 0 < 1 − 4b− 4v , 0 < a + 4(a− 1)v 1 − 4b− 4v < 1 , −1 < b + v < 1 4 , 4(b + v) < a + 4(a− 1)v 1 − 4b− 4v < c, ∣∣∣∣14 ( a + 4(a− 1)v 1 − 4b− 4v ) + b + v ∣∣∣∣ + 12c < 1 for all v ∈ (−δ,δ). let rv(x,y) = ( a + 4(a− 1)v 1 − 4b− 4v ) x2 + (b + v)y2 + cxy for all v ∈ (−δ,δ). by theorem 2.1, ‖rv‖ = c2 − 4 ( a + 4(a−1)v 1−4b−4v ) (b + v) 2c− ( a + 4(a−1)v 1−4b−4v ) − 4(b + v) = 1 for all v ∈ (−δ,δ). notice that β = 4(1 −a) 1 − 4b α. (4) we may choose � > 0 such that −1 < b + w(2c− 2 + w) 4(a− 1) < 1 4 , 4 ( b + w(2c− 2 + w) 4(a− 1) ) < a < c + w < 2 ,∣∣∣∣14a + b + w(2c− 2 + w)4(a− 1) ∣∣∣∣ + 12 (c + w) < 1 for all w ∈ (−�,�). let sw(x,y) = ax 2 + ( b + w(2c− 2 + w) 4(a− 1) ) y2 + (c + w)xy for all w ∈ (−�,�). by theorem 2.1, ‖sw‖ = (c + w)2 − 4a ( b + w(2c−2+w) 4(a−1) ) 2(c + w) −a− 4 ( b + w(2c−2+w) 4(a−1) ) = 1 smooth 2-homogeneous polynomials 257 for all w ∈ (−�,�). notice that γ = (c−1) 2(1−a)β and by (4), γ = 2(c−1) 1−4b α. it follows that 1 = aα + bβ + cγ = α ( a + 4b(1 −a) 1 − 4b + 2c(c− 1) 1 − 4b ) = α ( 2c−a− 4b 1 − 4b ) , which implies that α = 1−4b 2c−a−4b and 1−4b 2c−a−4b = (c−4b)2 (2c−a−4b)2 . therefore, α = (c− 4b)2 (2c−a− 4b)2 , β = 4(c−a)2 (2c−a− 4b)2 , γ = 2(c−a)(c− 4b) (2c−a− 4b)2 . thus q is smooth. suppose that c 2−4ab 2c−a−4b < 1, ∣∣1 4 a + b ∣∣ + 1 2 c = 1. note that 1 4 a + b 6= 0. first, suppose that 1 4 a + b > 0. let f ∈p(2h)∗ be such that f(q) = 1 = ‖f‖. we will show that α = 1 4 , β = 1, γ = 1 2 . we choose δ > 0 such that ru(x,y) = (a + u)x 2 + ( b− 1 4 u ) y2 + cxy , sv(x,y) = ax 2 + ( b− v 2 ) y2 + (c + v)xy ∈p(2h) for all u,v ∈ (−δ,δ). notice that γ = 1 2 β, γ = 2α. thus α = 1 4 , β = 1, γ = 1 2 . hence, q is smooth. next, suppose that 1 4 a + b < 0. let f ∈p(2h)∗ be such that f(q) = 1 = ‖f‖. by the previous argument, α = −1 4 , β = −1, γ = 1 2 . thus q is smooth. suppose that c > a > 4b, c = a + 2 √ 1 −a. we will show that q is not smooth. by theorem 2.1, 1 = ‖q‖ ≥ ( a+2 √ 1−a )2 −4ab 2 ( a+2 √ 1−a ) −a−4b . thus −1 < b ≤ a+4 √ 1−a 4 − 1 < 0, so 1 4 a + b < 0. since 1 ≥ ∣∣∣∣14a + b ∣∣∣∣ + 12c = − ( 1 4 a + b ) + 1 2 c, which implies that b ≥ a+4 √ 1−a 4 − 1, so b = a+4 √ 1−a 4 − 1 and q = ax2 + ( a + 4 √ 1 −a 4 − 1 ) y2 + ( a + 2 √ 1 −a ) xy (0 < a < 1) . for j = 1, 2, let fj ∈p(2h)∗ be such that f1 ( x2 ) = − 1 4 , f1 ( y2 ) = −1 , f1(xy) = 1 2 , f2 ( x2 ) = ( 2 − √ 1 −a )2 4 , f2 ( y2 ) = 1 −a, f2(xy) = √ 1 −a ( 2 − √ 1 −a ) 2 . 258 s.g. kim clearly fj(q) = 1 = ‖f1‖ for j = 1, 2. we claim that ‖f2‖ = 1. indeed, for p = a′x2 + b′y2 + c′xy ∈p(2h), we have δ(2−√1−a 2 , √ 1−a )(p) = p(2 −√1 −a 2 , √ 1 −a ) = a′ ( 2 − √ 1 −a 2 )2 + b′ (√ 1 −a )2 + c′ ( 2 − √ 1 −a 2 )√ 1 −a = f2(p) , which implies that f2 = δ ( 2− √ 1−a 2 , √ 1−a ). thus ‖f2‖ = ∥∥∥∥δ(2−√1−a 2 , √ 1−a )∥∥∥∥ ≤ ∥∥∥∥ ( 2 − √ 1 −a 2 , √ 1 −a )∥∥∥∥ h( 1 2 ) = 1 . since f2(q) = 1, ‖f2‖ = 1. therefore, q is not smooth. claim: if c > a, a < 4b, then q is smooth. by theorem 2.1, 0 < b < 1, 0 < c < 2. let f ∈ p(2h)∗ be such that f(q) = 1 = ‖f‖. by the previous arguments, α = 1 4 , β = 1, γ = 1 2 . thus q is smooth. therefore, we complete the proof. acknowledgements the author is thankful to the referee for the careful reading and considered suggestions leading to a better presented paper. references [1] r.m. aron, m. klimek, supremum norms for quadratic polynomials, arch. math. (basel) 76 (2001), 73 – 80. [2] y.s. choi, h. ki, s.g. kim, extreme polynomials and multilinear forms on l1, j. math. anal. appl. 228 (1998), 467 – 482. [3] y.s. choi, s.g. kim, the unit ball of p(2l22), arch. math. (basel) 71 (1998), 472 – 480. [4] y.s. choi, s.g. kim, extreme polynomials on c0, indian j. pure appl. math. 29 (1998), 983 – 989. [5] y.s. choi, s.g. kim, smooth points of the unit ball of the space p(2l1), results math. 36 (1999), 26 – 33. [6] y.s. choi, s.g. kim, exposed points of the unit balls of the spaces p(2l2p) (p = 1, 2,∞), indian j. pure appl. math. 35 (2004), 37 – 41. smooth 2-homogeneous polynomials 259 [7] s. dineen, “complex analysis on infinite dimensional spaces”, springerverlag, london, 1999. [8] b.c. grecu, g.a. muñoz-fernández, j.b. seoane-sepúlveda, the unit ball of the complex p(3h), math. z. 263 (2009), 775 – 785. [9] r.b. holmes, “geometric functional analysis and its applications”, springer-verlag, new york-heidelberg, 1975. 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[19] r.a. ryan, b. turett, geometry of spaces of polynomials, j. math. anal. appl. 221 (1998), 698 – 711. introduction results � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 1 (2022), 57 – 74 doi:10.17398/2605-5686.37.1.57 available online march 9, 2022 on formal power series over topological algebras m. weigt 1, i. zarakas 2 1 department of mathematics, nelson mandela university summerstrand campus (south), port elizabeth, 6031, south africa 2 department of mathematics, hellenic military academy, athens, 19400, greece martin.weigt@mandela.ac.za , weigt.martin@gmail.com , gzarak@hotmail.gr received april 21, 2021 presented by d. dikranjan accepted february 9, 2022 abstract: we present a general survey on formal power series over topological algebras, along with some perspectives which are not easily found in the literature. key words: topological algebra, formal power series, algebra homomorphism. msc (2020): 46h05, 46h10, 46h30. 1. introduction in our research course in [17], we came across the following question: let a[τ] be a gb∗-algebra, δ a closed derivation of a with domain d(δ), x ∈ d(δ) and 0 6= λ ∈ c such that (λ1−x)−1 exists in a. does it follow that (λ1−x)−1 lies in the domain of the derivation? the latter question is answered in the affirmative in the case of c∗-algebras by bratelli and robinson in [7], and more generally for pro-c∗-algebras in [17]. the basic tool used in the proof of bratelli and robinson’s result is the neumann series which is complemented with the use of an analytic continuation argument. given that a gb∗ algebra is an algebra of possibly unbounded operators, the use of neumann series in the gb∗-algebra setting fails and consequently the bratelli-robinson proof cannot be carried over. we were thus led to ask ourselves whether it might be possible to answer our initial question, which for brevity we shall call the domain problem, with the use of formal power series. in particular, the following two questions naturally arise: (i) does there exist a unital injective algebra homomorphism φ̃ : d(δ) → d(δ)[[x]], with φ̃(x) = x? (ii) does there exist a unital homomorphism ψ : c[[x]] → d(δ) such that ψ(x) = x? an affirmative answer to any of the above questions could have the potential to place (λ1−x)−1 inside the domain of the derivation (see section 3 for details). issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.37.1.57 mailto:martin.weigt@mandela.ac.za mailto:weigt.martin@gmail.com mailto:gzarak@hotmail.gr https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 58 m. weigt, i. zarakas however, observe that the existence of such a unital homomorphism in question ii would imply that x is an element of d(δ) such that spd(δ)(x) ⊆ {0} (see section 3). although this is a severe restriction, this motivates the following more general questions: (i’) let a[τ] be a topological algebra and let x ∈ a. does there exist a unital injective algebra homomorphism φ̃ : a → a[[x]], with φ̃(x) = x? (ii’) let a[τ] be a topological algebra and let x ∈ a. does there exist a unital homomorphism ψ : c[[x]] → a such that ψ(x) = x? these two more general questions are interesting for their own sake, and are the main focus of this paper. in the current expository paper, we visit some well-known results of relevance to our purposes in answering the two more general questions in the previous paragraph, which refer to formal power series on banach and fréchet algebras. the literature is rich in papers which deal with formal power series on topological algebras and what’s more with their intriguing interplay with derivations. indicatively, we refer the reader to [16, 4, 8]. moreover, we present certain byproducts obtained along the way, for the case of generalized b∗-algebras, which give stimulus for further reserach on the topic, and which appear to be interesting for their own sake. 2. preliminaries throughout, all algebras are assumed to be over the complex numbers, i.e., all algebras are assumed to be c-algebras. a topological algebra a[τ] is an algebra which is also a topological vector space and has the property that multiplication is separately continuous. in some topological algebras, such as fréchet algebras, multiplication is not only separately continuous, but jointly continuous [10]. by a fréchet algebra, we mean a complete metrizable topological algebra. a topological algebra is said to be a locally convex algebra if it is also a locally convex space. an m-convex algebra is a locally convex algebra a[τ] for which the topology τ on a is defined by a family of seminorms {pγ : pγ ∈ γ} such that pγ(xy) ≤ pγ(x)pγ(y) for all x,y ∈ a and for all γ ∈ γ. for every γ ∈ γ, let nγ = {x ∈ a : pγ(x) = 0}. then a/nγ is a normed algebra with respect to the norm ṗγ(x + nγ) = pγ(x) for all x ∈ a and a = lim←−γ a/nγ. the completion aγ of a/nγ is therefore a banach algebra for all γ ∈ γ. if a is, in addition, complete, then a = lim←−γ aγ up to a topological and algebraic on formal power series 59 ∗-isomorphism, which is called the arens-michael decomposition of the complete m-convex algebra a. for details, we refer to [10]. a c∗-algebra is a banach ∗-algebra a such that ‖x∗x‖ = ‖x‖2 for all x ∈ a, where ‖ ·‖ denotes the submultiplicative norm on a defining the topology on a. definition 2.1. ([2]) let a[τ] be a unital topological ∗-algebra and let b∗a (denoted by b ∗ if no confusion arises as to which algebra is being considered) be a collection of subsets b of a satisfying the following: (i) b is absolutely convex, closed and bounded; (ii) 1 ∈ b, b2 ⊆ b and b∗ = b. for every b ∈ b∗, denote by a[b] the linear span of b, which is a normed algebra under the gauge function ‖ · ‖b of b. if a[b] is complete for every b ∈b∗, then a[τ] is called pseudo-complete. an element x ∈ a is called bounded, if there exists 0 6= λ ∈ c such that the set {(λx)n : n = 1, 2, 3, . . .} is a bounded subset of a. we denote by a0 the set of all bounded elements in a. a unital topological ∗-algebra a[τ] is called symmetric if, for every x ∈ a, the element (1 + x∗x)−1 exists and belongs to a0. definition 2.2. ([2]) a symmetric pseudo-complete locally convex ∗-algebra a[τ], such that the collection b∗ has a greatest member, denoted by b0, is called a gb ∗-algebra over b0. every c∗-algebra is a gb∗-algebra. an example of a gb∗-algebra, which generally need not be a c∗-algebra, is a pro-c∗-algebra. by a pro-c∗-algebra, we mean a complete topological ∗-algebra a[τ] for which the topology τ is defined by a directed family of c∗-seminorms. hence every pro-c∗-algebra is m-convex, and is therefore an inverse limit of c∗-algebras. an example of a gb∗-algebra which is not a pro-c∗-algebra is the locally convex ∗-algebra lω([0, 1]) = ∩p≥1lp([0, 1]) defined by the family of seminorms { ‖ ·‖p : p ≥ 1 } , where ‖ ·‖p is the lp-norm on lp([0, 1]) for all p ≥ 1. a formal power series in the indeterminate x over a unital algebra a is an expression of the form ∑∞ n=0 anx n, where an ∈ a for all n ∈ n. the set a[[x]] of all formal power series over a is an algebra with respect to addition and scalar multiplication defined in the obvious way, and multiplication defined as the usual cauchy product as with complex power series. this is an immediate consequence of [12, proposition 5.8]. 60 m. weigt, i. zarakas we assume throughout this paper that all our algebras are unital, i.e., have an identity element 1, unless stated otherwise. if a denotes an algebra, and x ∈ a, then the symbol spa(x) denotes the set { λ ∈ c : λ1 −x is not invertible in a } throughout. 3. main results a derivation of an algebra a is a linear mapping δ : d(δ) → a such that δ(xy) = xδ(y) + δ(x)y for all x,y ∈ a, where d(δ) denotes the domain of δ and is a subalgebra of a. if, in addition, a is equipped with an involution ∗, then we say that δ is a ∗-derivation if d(δ) is a ∗-subalgebra of a (i.e., d(δ) is closed under the involution ∗), and δ(x∗) = δ(x)∗ for all x ∈ a. for a topological algebra a[τ], a ∗-derivation δ : d(δ) → a is said to be a closed ∗-derivation if d(δ) is dense in a, and if (xα) is a net in a with xα → x ∈ a and δ(xα) → y ∈ a, then x ∈ d(δ) and y = δ(x). let a[τ] be a gb∗-algebra, δ a closed ∗-derivation of a with 1 ∈ d(δ), x ∈ d(δ), λ ∈ c −{0} such that (λ1 − x)−1 ∈ a. as already outlined in the introduction, in regards to the domain problem for a gb∗-algebra, the following two questions emerge: (i) does there exist a unital injective algebra homomorphism φ̃ : d(δ) → d(δ)[[x]], with φ̃(x) = x? (ii) does there exist a unital homomorphism ψ : c[[x]] → d(δ) such that ψ(x) = x? let us first assume that question (i) has an affirmative answer, and assume also that c[[x]] ⊆ φ̃(d(δ)). let x ∈ d(δ). then, since 0 ∈ sp φ̃(d(δ)) (x) ⊆ spc[[x]](x) = {0} and φ̃ is an injective algebra homomorphism, we get that spd(δ)(x) = {0}. so, for our initially assumed λ 6= 0 such that (λ1 − x)−1 exists in a, we conclude that (λ1 −x)−1 ∈ d(δ). the condition spd(δ)(x) = {0} is a severe restriction. observe that the condition c[[x]] ⊆ φ̃(d(δ)) is equivalent to the condition that φ̃(d(δ))∩c[[x]] = c[[x]], and this is explored in propositions 3.1 and 3.2 below. on formal power series 61 now let us assume that question (ii) has a positive answer. then, for the initially assumed non-zero λ ∈ c, we have that 1 λ ∞∑ k=0 ( 1 λ x )k = ( λ ( 1 − 1 λ x ))−1 = (λ1 −x)−1. therefore, λ1 − x is invertible in c[[x]] for all λ ∈ c − {0}. given the assumption of the existence of the unital homomorphism ψ described in question (ii), we consequently have that (λ1 −x)−1 ∈ d(δ). it is worth noting that once we assume the validity of questions (i) or (ii), with the additional assumption that c[[x]] ⊆ φ̃(d(δ)) in question i, the domain problem can be immediately answered by straightforward algebraic considerations which hold for any algebra. it is therefore expected that the intrinsic nature of the topology of a gb∗-algebra is to play its role in answering questions (i) and (ii). in question ii, however, the existence of such a unital homomorphism implies that x is an element of d(δ) having the property that spd(δ)(x) = spd(δ)(ψ(x)) ⊆ spc[[x]](x) = {0}. although the restrictions above are severe, questions i and ii motivate the following more general questions, which appear to be interesting for their own sake. (i’) let a[τ] be a topological algebra and let x ∈ a. does there exist a unital injective algebra homomorphism φ̃ : a → a[[x]], with φ̃(x) = x? (ii’) let a[τ] be a topological algebra and let x ∈ a. does there exist a unital homomorphism ψ : c[[x]] → a such that ψ(x) = x? we start with some considerations with respect to question (i). in [8, theorem 11.2], it is shown that if θ is an algebra homomorphism of a fréchet algebra into the formal power series algebra c[[x]], then θ is either continuous or a surjection. in this regard, we have the following result. proposition 3.1. let a[τ] be a topological algebra with identity element 1 and let x ∈ a such that (1 −x)−1 exists in a. assume that φ̃ : a → a[[x]] is a unital algebra homomorphism such that φ̃(a)∩c[[x]] is a fréchet locally convex algebra in some topology τ1. let ψ be the identity map of c[[x]] restricted to φ̃(a)∩c[[x]]. let τ ′ denote the topology of convergence on c[[x]] with respect to all coefficients in c[[x]]. consider the following statements. 62 m. weigt, i. zarakas (i) the map ψ is τ1 − τ ′ discontinuous. (ii) φ̃(a) ∩ c[[x]] is not m-convex with respect to τ1 and is τ ′-closed in c[[x]]. (iii) φ̃(a) ∩ c[[x]] is not a q-algebra with respect to τ1 and is τ ′-closed in c[[x]]. then (i) holds if and only if (ii) holds, and (i) implies (iii). proof. (i) ⇒ (ii) : if ψ is τ1 − τ ′ discontinuous, then ψ is surjective [8], i.e., φ̃(a) ∩ c[[x]] = c[[x]]. therefore φ̃(a) ∩ c[[x]] is τ ′-closed in c[[x]], and is commutative, m-convex, fréchet with respect to the topology τ ′, and noetherian (this is due to c[[x]] being fréchet, commutative, m-convex and noetherian with respect to the topology τ ′ [9]). since ψ is discontinuous, it is now immediate from [9, theorem 2.5] that φ̃(a) ∩ c[[x]] is not m-convex with respect to τ1. (ii) ⇒ (i) : assume that (ii) holds, and suppose that ψ is τ1−τ ′ continuous. then, since ψ is a surjective homomorphism onto its range φ̃(a)∩c[[x]], which is fréchet with respect to τ ′ and τ1, it follows from the open mapping theorem that ψ is a τ1−τ ′ topological isomorphism. this cannot be since φ̃(a)∩c[[x]] is m-convex with respect to τ ′, and φ̃(a)∩c[[x]] is not m-convex with respect to τ1. therefore ψ is τ1 − τ ′ discontinuous. (i) ⇒ (iii) : if ψ is τ1−τ ′ discontinuous, then, as in the proof of (i) ⇒ (ii), it follows that φ̃(a) ∩ c[[x]] is not m-convex with respect to τ1. by [18, theorem 13.17], φ̃(a) ∩c[[x]] is not a q-algebra with respect to τ1. consider a map φ̃ as in question (i’). the following proposition depicts a certain positioning of the image of φ̃, and the proof shows that the map ψ in proposition 3.1 is always τ1 − τ ′ continuous, and therefore ψ is not surjective. this demonstrates that φ̃(a) ∩ c[[x]] does not satisfy condition (ii) of proposition 3.1. proposition 3.2. let a[τ] be a unital topological algebra with identity 1. let x ∈ a and φ̃ : a → a[[x]] a unital algebra homomorphism with φ̃(x) = x, such that φ̃(a) ∩ c[[x]] is a fréchet algebra in some topology τ1. then φ̃(a) ∩c[[x]] is properly contained in c[[x]]. proof. let ψ := id|φ̃(a)∩c[[x]] : φ̃(a) ∩c[[x]] → c[[x]]. we have that φ̃(λ01 + λ1x + · · · + λnxn) = λ01 + λ1x + · · · + λnxn. on formal power series 63 therefore c[x] ⊂ φ̃(a)∩c[[x]]. hence, (φ̃(a)∩c[[x]],τ1) is a fréchet algebra which is a subalgebra of c[[x]] containing c[x]. then, by [8, theorem 11.2], the map ψ is τ1 −τ ′ continuous, where τ ′ is the topology of convergence with respect to all coefficients in c[[x]]. hence, by [8, p. 3], we take that ψ is not onto and hence φ̃(a) ∩c[[x]] is properly contained in c[[x]]. corollary 3.3. let a[τ] be a unital topological algebra with identity 1 and δ : d(δ) → a a closed derivation of a such that 1 ∈ d(δ). let x ∈ d(δ) and φ̃ : d(δ) → d(δ)[[x]] a unital algebra homomorphism with φ̃(x) = x, such that φ̃(d(δ)) ∩ c[[x]] is a fréchet algebra in some topology τ1. then φ̃(d(δ)) ∩c[[x]] is properly contained in c[[x]]. the following result is described in [16, first paragraph of p. 2145] and we give its proof for sake of completeness. proposition 3.4. let a be a unital algebra and x ∈ a. if θ : a → a[[x]] is a unital injective algebra homomorphism of a into the algebra a[[x]] of all formal power series over a with indeterminate x such that θ(x) = x, then x is not invertible in a and ∩∞n=1x na = {0}. proof. we first show that x is not invertible in a[[x]]. observe that x = ∑∞ k=0 akx k, where ak = 0 for all k 6= 1 and a1 = 1. if we suppose that x is invertible in a[[x]], then there is an element ∑∞ k=0 bkx k in a[[x]] such that x ( ∞∑ k=0 bkx k ) = ( ∞∑ k=0 akx k )( ∞∑ k=0 bkx k ) = ∞∑ k=0 ( k∑ r=0 arbk−r ) xk = 1 . hence, ∑k r=0 arbk−r = 0 for all k ≥ 1 and a0b0 = 1. this last equation is in contradiction with the fact that a0 = 0, thus x is not invertible in a[[x]]. so, since θ(x) = x and θ is unital, we conclude that x is not invertible in a. let now a ∈ ⋂∞ n=1 x na. then, for all n ∈ n, there exists bn ∈ a such that a = xnbn. hence, θ(a) = θ(x nbn) = θ(x) nθ(bn) = x nθ(bn), for all n ∈ n. then, xθ(b1) = x2θ(b2), implying that θ(b1) = θ(b2) = 0. thus, θ(a) = xθ(b1) = 0. so, ⋂∞ n=1 x na ⊂ ker θ = {0}. therefore a = 0, implying that ⋂∞ n=1 x na = {0}. the following corollary is a direct consequence of proposition 3.4. it gives us necessary conditions under which question (i) in the beginning of this section has a negative answer. 64 m. weigt, i. zarakas corollary 3.5. let δ : d(δ) → a be a closed ∗-derivation of a gb∗algebra a such that 1 ∈ d(δ). let x ∈ d(δ) such that (1 − x)−1 exists in a. if x is invertible in a, or ∩∞n=1x na 6= {0}, then there is no injective unital algebra homomorphism φ̃ : d(δ) → d(δ)[[x]] with φ̃(x) = x. the proof of the following result is similar to that of [16, theorem 3.10], and we only give a sketch of the proof. proposition 3.6. let a[τ] be a unital commutative topological algebra, and let x ∈ a have the following properties: (i) there exists a derivation d : a → a such that d(x) = 1 and ∞∑ k=0 (−1)kdk(a)xk k! τ-converges for all a ∈ a, (ii) ⋂∞ n=0 x na = {0}. then there exists a unital injective algebra homomorphism φ̃ : a → a[[x]] such that φ̃(x) = x. proof. by hypothesis we can form the map θ : a → a by θ(a) = ∞∑ k=0 (−1)kdk(a)xk k! for all a ∈ a. it is easily seen that θ is a unital algebra homomorphism. also, the kernel of θ is xa. define the map φ̃ : a → a[[x]] by φ̃(a) = ∞∑ n=0 θ(dn(a)) n! xn for all a ∈ a. since the kernel of θ is xa and d(xn) = nxn−1, for all n ∈ n, we get that φ̃ is injective. lastly, it is easily seen that φ̃(x) = x. proposition 3.6 is an optimal result as far as sufficient conditions are concerned: condition (i) of proposition 3.6 cannot be dropped. the reason is that if condition (ii) is sufficient on its own, then it would imply that our map exists if x is nilpotent (for x nilpotent gives us condition (ii)). this cannot be, since φ̃(x) = x and x is not nilpotent in a[[x]]. so condition (i) is required on formal power series 65 in addition to condition (ii). also, the condition d(x) = 1 in (i) implies that x is not nilpotent: for if x is nilpotent and x 6= 0 (which we assume, since the case x = 0 is trivial), there is a smallest natural number m > 1 such that xm = 0. hence 0 = d(xm) = mxm−1 6= 0, a contradiction. furthermore, x not nilpotent and condition (ii) of proposition 3.6 imply that x cannot have the property that 0 6= axm ⊂ axm+1 (see [4, corollary 2]). this is not in contradiction to [16, proposition 3.10]. also, recall from proposition 3.4 that condition (ii) is a necessary condition for the existence of our map. lastly, condition (i) is not redundant: no commutative c∗-algebra has a derivation d with d(x) = 1. corollary 3.7. let δ : d(δ) → a be a closed unbounded ∗-derivation with 1 ∈ d(δ), where a[τ] is a commutative gb∗-algebra. let x ∈ d(δ) such that (1 −x)−1 ∈ a. suppose that there exists a derivation d : d(δ) → d(δ) such that d(x) = 1 and ∞∑ k=0 (−1)kdk(a)xk k! τ-converges to an element in d(δ) for all a ∈ d(δ). if also ⋂∞ n=0 x na = {0}, then there exists a map φ̃ : d(δ) → d(δ)[[x]] which is a unital injective algebra homomorphism such that φ̃(x) = x. remarks 3.8. (i) [16, theorem 3.10] is stated for a commutative radical fréchet m-convex algebra. it is interesting to note that the conditions of proposition 3.6 above ensure that one does not require a[τ] to be a commutative radical fréchet m-convex algebra. in the proof of [16, theorem 3.10], one requires a[τ] to be a commutative radical fréchet m-convex algebra in order to apply [16, proposition 3.7 and lemma 3.9], which is to ensure that condition (i) of proposition 3.6 holds in some form. (ii) the results obtained in [16], and therefore propositions 3.4, 3.6 (above) and 3.9 (below), stem from the paper [15], in which m.p. thomas proved that the image of a derivation of a commutative banach algebra is contained in the jacobson radical of the banach algebra (commutative singer-wermer conjecture). based on proposition 3.6 and [16, lemma 3.9], the following result can be recorded. proposition 3.9. let a[τ] be a fréchet m-convex algebra and {‖.‖i} a sequence of seminorms defining the topology τ. let x ∈ a and d : a → a 66 m. weigt, i. zarakas be a derivation such that d(x) = 1. suppose that for every i ∈ n there is mi ∈ n such that xmi ∈ ⋂∞ n=0 x na ‖.‖i , where ⋂∞ n=0 x na ‖.‖i denotes the closure of ⋂∞ n=0 x na with respect to the seminorm ‖.‖i. then there exists an algebra homomorphism φ̃ : a → a0[[x]], where a0 = a/ ⋂+∞ n=0 x na, such that φ̃(x) = x. proof. based on the proof of [16, theorem 3.10] there exists an injective algebra homomorphism φ : a/ ⋂∞ k=0 x ka → a0[[x]] such that φ ( a + ∞⋂ k=0 xka ) = ∞∑ n=0 θ(dn(a) + ⋂∞ k=0 x ka) n! xn, where θ : a/ ⋂∞ k=0 x ka → a/ ⋂∞ k=0 x ka is the following algebra homomorphism: θ ( a + ∞⋂ k=0 xka ) = ∞∑ n=0 (−1)ndn(a)xn n! + ∞⋂ k=0 xka. then, by composing φ with the natural quotient map π :a→a/ ⋂∞ k=0 x ka, we get the desired algebra homomorphism φ̃. the fact that φ̃(x) = x can be easily seen through the following considerations: φ̃(x) = (φ◦π)(x) = φ ( x + +∞⋂ k=0 xka ) = +∞∑ n=0 θ(dn(x) + ⋂+∞ k=0 x ka) n! xn. since d(x) = 1, we get that dn(x) = 0 for every n ≥ 2. therefore, (φ◦π)(x) = θ ( x + +∞⋂ k=0 xka ) + θ ( 1 + +∞⋂ k=0 xka ) ·x . moreover, θ ( x + +∞⋂ k=0 xka ) = +∞∑ n=0 (−1)ndn(x)xn n! + +∞⋂ k=0 xka = (x−x) + +∞⋂ k=0 xka = +∞⋂ k=0 xka, on formal power series 67 and θ ( 1 + +∞⋂ k=0 xka ) = +∞∑ n=0 (1)ndn(1)xn n! + +∞⋂ k=0 xka = 1 + +∞⋂ k=0 xka. so, φ̃(x) = (φ◦π)(x) = +∞⋂ k=0 xka + ( 1 + +∞⋂ k=0 xka ) ·x = x , given that on a0(= a/ ⋂+∞ k=0 x ka), the elements ⋂+∞ k=0 x ka and 1 + ⋂+∞ k=0 x ka are the zero and the unit element respectively. with respect to a possible strategy for answering question (ii) in the beginning of this section, we note the following two routes, denoted by (1) and (2) in what follows. (1) in [3, theorem 2], g.r. allan proved the following result for a commutative banach algebra. theorem 3.10. let a be a commutative banach algebra with identity and let x be in the jacobson radical of a such that 0 6= axm ⊂ axm+1 for some m ≥ 1. then there is a unital injective algebra homomorphism ψ : c[[x]] → a such that ψ(x) = x. the proof of theorem 3.10 in [3] is entirely algebraic, except in the application of the arens-calderon theorem, which relies on a being a banach algebra. the arens-calderon theorem is given by the following theorem. theorem 3.11. ([11, lemma 3.2.8]) let a be a commutative banach algebra with identity and jacobson radical r. if n ∈ n and a0,a1, . . . ,an ∈ a are such that a0 ∈ r and a1 is invertible in a, then there exists y ∈ r such that ∑n k=0 aky k = 0. the proof of theorem 3.11, as given in [11], relies strongly on analytic functions of several complex variables and the implicit function theorem. if we could extend the arens-calderon theorem (theorem 3.11) to commutative fréchet algebras, then we will have extended theorem 3.10 to the case where a is a fréchet algebra, with exactly the same proof as that of [3, theorem 68 m. weigt, i. zarakas 2]. therefore question (ii’) would attain a positive answer in the setting of commutative fréchet algebras. in theorem 3.10, the condition that x is in the radical of a is strong if a is semi-simple, especially if x 6= 0. recall that a sequentially complete locally convex algebra is pseudo-complete, and hence has an abundance of banach subalgebras (see [1, proposition 4.2]). this motivates the following observation. corollary 3.12. let a[τ] be a commutative unital fréchet locally convex algebra and x ∈ a. assume that there is a unital banach subalgebra b of a (in some norm) such that x ∈ b and 0 6= bxm ⊂ bxm+1 for some m ≥ 1, and such that x is in the radical of b. then there is an embedding ψ : c[[x]] → b ⊆ a such that ψ(x) = x. if, in the above corollary, b is a commutative unital radical banach subalgebra, then the condition x in the radical of b becomes more realistic. for example, a famous theorem of b.e. johnson [13, theorem 4] asserts that there are at least some commutative banach algebras having at least one closed (hence banach) radical subalgebra. incidentally, this theorem of johnson was used by m.p. thomas in his proof of the commutative singer-wermer conjecture: he only had to settle the conjecture for commutative radical banach algebras. one word of caution: not all fréchet jacobson semi-simple algebras have closed (hence fréchet) radical subalgebras. for instance, a c∗-algebra is semisimple, and every norm closed ∗-subalgebra is again a c∗-algebra, and hence semi-simple. an extension of theorem 3.10 to fréchet m-convex algebras is given in [4, theorem 7]. (2) the following theorem is a special case of [9, lemma 2], and is the holomorphic functional calculus for complete barrelled m-convex algebras. theorem 3.13. let a[τ] be a commutative complete barelled m-convex topological algebra with weakly compact character space. let a ∈ a. then there exists a continuous homomorphism g : o(spa(a)) → a, such that g(f) = a, where o(spa(a)) denotes the algebra of all analytic functions on a neighbourhood of spa(a) and f(z) = z for all z ∈ c. the proof of theorem 3.13 relies on [5, theorem 5.1], which is the silovarens-calderon theorem for banach algebras. in what follows, we give a proof on formal power series 69 of theorem 3.13 for the case where all banach algebras in the arens-michael decomposition of a are semi-simple. proof. by the arens-michael decomposition of a we have that a[τ] is topologically isomorphic to the inverse limit lim←−γ aγ, where all aγ are semisimple by assumption. the space o(spa(a)) can be represented as the following inductive limit o(spa(a)) = lim→ h(ω) where h(ω) stands for the holomorphic functions on the open subset ω of c and ω runs over all open subsets of c which contain the weakly compact subset spa(a). the topology considered on each h(ω) is that of the uniform convergence on compact subsets of ω. the topology on o(spa(a)) is the inductive limit topology on o(spa(a) induced by the h(ω)’s. let us fix an index γ ∈ γ. we consider an open subset ω of c which contains spa(a). since spa(a) = ⋃ γ∈γspaγ (aγ) [10], we have that ω contains spaγ (aγ) for all γ ∈ γ. by the arens-calderón theorem [6, corollary ii.20.6], there exists a homomorhism g γ ω : h(ω) −→ aγ which is continuous with respect to the topology of uniform convergence on compact subsets of ω, it extends the natural homomorhism of the complex polynomials p(c) into aγ and g γ ω(f) = aγ, for f(z) = z, z ∈ c. by the proof of [6, corollary ii.20.6], we recall that if f ∈ h(ω), and, say, g γ ω(f) = lγ ∈ aγ, then φγ(lγ) = f(φγ(aγ)), for all φγ ∈m(aγ), where m(aγ) is the character space of aγ. we then define the map gγ : o(spa(a)) −→ aγ as follows: let f be a function in o(spa(a)). hence there is an open neighborhood ω of spa(a) such that f ∈ h(ω). we then define gγ(f) := g γ ω(f). we show that the map gγ is well-defined: for this, consider an open subset ω′ containing spa(a) such that f ∈ h(ω′). then for every φγ ∈ m(aγ), we have that φγ ( g γ ω(f) ) = f ( φγ(aγ) ) = φγ ( g γ ω′(f) ) . 70 m. weigt, i. zarakas hence, since aγ is semisimple, we get that g γ ω(f) = g γ ω′(f). hence, the map gγ is well-defined. moreover, if iω is the canonical embedding of h(ω) into o(spa(a)), we clearly have that gγ ◦ iω = g γ ω . therefore, for every open subset ω containing spa(a) we have that gγ ◦ iω is continuous. so, by [14, (6.1), page 54] the map gγ : o(spa(a)) → aγ is continuous for every γ ∈ γ. next, we consider the map g : o(spa(a)) −→ a, g(f) := ( gγ(f) ) γ∈γ . in order to show that the map g is well-defined, we need to show that πγδ(gδ(f)) = gγ(f), γ ≤ δ, f ∈ o(spa(a)), where πγ δ : aδ → aγ are the ‖.‖-continuous connecting maps of the inverse system. we fix one open neighborhood, say ω, of the set spa(a). by the preceding paragraphs, we have that gγ(f) = g γ ω(f) = lγ, gδ(f) = g δ ω(f) = lδ. so, we shall show that πγ δ(lδ) = lγ. we recall that a = (aλ)λ∈γ. since ω is an open neighborhood of spaδ (aδ), by the proof of [6, theorem ii.20.5], we have that there are aδ2,aδ3, . . . ,aδn ∈ aδ and fδ a holomorphic function on an open neighborhood of the polydisc γδ = { z ∈ cn : |zk| ≤ 1 + pδ(aδk), k = 1, . . .n } (in the above writing of γδ, aδ1 is identified with aδ), such that f(φδ(aδ)) = fδ ( φδ(aδ),φδ(aδ2), . . . ,φδ(aδn ) ) , φδ ∈m(aδ) . (3.1) moreover, by the proof of [6, theorem ii.20.5], we have that the series∑ k αk a k1 δ a k2 δ2 · · ·a kn δn is ‖.‖δ-convergent to the element lδ and for every φδ ∈m(aδ), φδ(lδ) = fδ ( φδ(aδ),φδ(aδ2), . . . ,φδ(aδn ) ) . (3.2) now, given that πγδ is a ‖.‖-continuous linear map, we have that πγ δ (∑ k αk a k1 δ a k2 δ2 · · ·aknδn ) = ∑ k αk a k1 γ a k2 γ2 · · ·aknγn on formal power series 71 ‖.‖γ-converges to an element in aγ, say ωγ. for any φγ ∈ m(aγ) we have that φγ(ωγ) = φγ ( πγ δ (∑ k αka k1 δ a k2 δ2 · · ·aknδn )) = φγ ( πγδ(lδ) ) = fδ ( (φγ ◦πγ δ)(aδ), . . . , (φγ ◦πγ δ)(aδn ) ) = f ( (φγ ◦πγ δ)(aδ) ) = f ( φγ(aγ) ) , where the second equality in the above string of relations derives from relation (3.2) and from the fact that φγ ◦ πγ δ ∈ m(aδ), and the third equality results from relation (3.1) above. by the proof of [6, corollary ii.20.6], we have that for every φγ ∈m(aγ): φγ(lγ) = f ( φγ(aγ) ) . therefore, we have that φγ(ωγ) = φγ(lγ) for every φγ ∈m(aγ). since aγ is semisimple, we get that ωγ = lγ, i.e., πγ δ(lδ) = lγ. the map g is clearly a homomorphism, such that g(f) = ( gγ(f) ) γ = (aγ)γ = a, for f(z) = z, z ∈ c . moreover, if by πγ we denote the canonical projections πγ : a → aγ, then we have that πγ ◦ g = gγ and we have already seen that gγ is continuous. hence, by [14, (5.2), p. 51], we get that g is continuous. let now a be a commutative complete barelled locally m-convex algebra which is also a q-algebra. then, by [9, lemma 0.1], a[τ] has weakly compact character space. let x ∈ a, and f(z) = z for all z ∈ c. then f is an entire function and thus analytic on spa(x), that is f ∈ o(spa(x)). define a map φ1 : c[x] →o(spa(x)) by φ1 ( n∑ i=1 λix i ) = g , where g(z) = n∑ i=1 λiz i. then φ1 is a homomorphism. by theorem 3.13 there is a continuous homomorphism φ2 : o(spa(x)) → a such that φ2(f) = x. let ψ0 = φ2 ◦ φ1. 72 m. weigt, i. zarakas then, ψ0(x) = x. if φ1 is continuous when c[x] is equipped with the coordinatewise topology coming from c[[x]], then ψ0 is continuous and hence extends via continuity to an algebra homomorphism ψ : c[[x]] → a such that ψ(x) = x. for the case of a general commutative algebra a, the i-adic topology on a with respect to an ideal i of a plays an important role in embedding c[[x]] into a as the following result shows. recall that a subset u of a is open with respect to the i-adic topology if and only if for all x ∈ u, there exists n ∈ n such that x + in ⊆ u. theorem 3.14. let a be a unital commutative algebra, i an ideal in a and x ∈ i. if a is i-adic complete and hausdorff with respect to the i-adic topology, then there is a unital homomorphism ψ : c[[x]] → a such that ψ(x) = x. proof. let an = a/i n, for all n ∈ n. by [12, theorem 5.5], there is a unital homomorphism ψ0,m : c[x] → am such that ψ0,m(x) = x + im. then ψ0,m ( n∑ i=0 λix i ) = n∑ i=0 λi(x + i m)i for all m ∈ n. this induces the map ψ̃0,m : c[x]/〈x〉−→ am for all m ∈ n, where 〈x〉 is the two-sided ideal of c[x] generated by x. since a is, by hypothesis, i-adic complete and hausdorff, by taking the inverse limit of the afforementioned maps ψ̃0,m, we get the desired mapping ψ (see [12, theorem 5.5] for details). remark 3.15. note that the ideal i in theorem 3.14 cannot be the whole algebra a. for then, there would exist a unital algebra homomorphism ψ : c[[x]] → a such that ψ(x) = 1, i.e., ψ(1 − x) = 0. the latter equality contradicts the fact that 1 −x is invertible in c[[x]]. lemma 3.16. let a be a unital commutative algebra and i an ideal of a such that a is hausdorff, has jointly continuous multiplication and is advertibly complete with respect to the i-adic topology. if x ∈ a, then (1 − x)−1 exists and is in a. on formal power series 73 proof. let ã be the completion of a with respect to the i-adic topology. given that a has jointly continuous multiplication with respect to the i-adic topology, ã is a hausdorff topological algebra. by theorem 3.14, there exists a unital homomorphism ψ : c[[x]] → ã such that ψ(1 −x) = 1 −x. since (1−x)−1 exists in c[[x]], we get that (1−x)−1 exists in ã. therefore, since a is advertibly complete in the i-adic topology, by [10, proposition 6.2] we conclude that (1 −x)−1 ∈ a. acknowledgements this work is based on part of the research for which the first named author received financial support from the national research foundation of south africa (nrf). the authors thank the referees for their careful reading of the manuscript, along with helpful comments which improved the quality of the manuscript. references [1] g.r. allan, a spectral theory for locally convex algebras, proc. london math. soc. 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[18] w. zelazko, metric generalizations of banach algebras, rozprawy mat. 47 (1965), 70 pp. introduction preliminaries main results � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 38, num. 1 (2023), 105 – 123 doi:10.17398/2605-5686.38.1.105 available online may 31, 2023 the character variety of one relator groups a. cavicchioli, f. spaggiari dipartimento di scienze fisiche, informatiche e matematiche universitá di modena e reggio emilia, via campi 213/b, 41125 modena, italy alberto.cavicchioli@unimore.it , fulvia.spaggiari@unimore.it received october 26, 2022 presented by a. turull accepted april 23, 2023 abstract: we consider some families of one relator groups arising as fundamental groups of 3dimensional manifolds, and calculate their character varieties in sl(2, c). then we give simple geometrical descriptions of such varieties, and determine the number of their irreducible components. our paper relates to the work of baker-petersen, qazaqzeh and morales-marcén on the character variety of certain classes of one relator groups, but we use different methods based on the concept of palindrome presentations of given groups. key words: finitely generated group, torus link, torus bundle, character variety, sl(2, c) representation, kauffman bracket skein module. msc (2020): 20c15, 57m25, 57m27. 1. introduction let g be a finitely presented group. a representation of g is a group homomorphism from g to sl(2,c). two representations are said to be conjugate if they differ by an inner automorphism of sl(2,c). a representation is reducible if it is conjugate to a representation into upper triangular matrices. otherwise, the representation is called irreducible. the character variety of g is the set of conjugacy classes of representations of g into sl(2,c). the character variety of g is a closed algebraic subset of cn for some n (see [8, 17]). the character variety of the fundamental group of any hyperbolic 3-manifold contains some topological informations about the structure of the given manifold (see [8, 25]). a general equation form for such character varieties does not exist in the literature. however, they have been calculated for many classes of (hyperbolic) 3–manifolds. representations of two-bridge knot groups have been investigated in [3, 11, 23]. character varieties of pretzel links and twisted whitehead links have been determined in [27]. recursive formulas for the character varieties of twist knots can be found in [13]. a very different method to determine the issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.38.1.105 mailto:alberto.cavicchioli@unimore.it mailto:fulvia.spaggiari@unimore.it https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 106 a. cavicchioli, f. spaggiari character variety of twist knot groups has been proposed in [5]. the results are obtained by using special presentations of the knot groups, whose relators are palindromes (see [4]). this means that the relators read the same forwards or backwards as words in the generators. in this paper we propose a method to determine the character variety of a class of torus links which is different to that developed in [21]. our method reduces the computations presented in the quoted paper, and permits to give an easy geometrical description of the character varieties of these torus links. using such a description we also give simplified proofs of some algebraic results obtained in [21]. the method is then applied to the fundamental group of once-punctured torus bundles. such manifolds can be obtained by (n+ 2, 1) dehn filling on one boundary component of the whitehead link (wl) exterior. using the concept of palindrome word, we give a geometrical description of the character varieties of such torus bundles. this relates to the main result of [1], using very different techniques for computing character varieties. as a further new result, we then derive the character varieties of another family of bordered 3-manifolds, arising from (6n + 2, 2n + 1) dehn filling on one boundary component of the wl exterior. 2. technical preliminaries we think of sl(2,c) as the 2 × 2 complex matrices of determinant 1 in the set of 2×2 complex matrices m(2,c). it is known that every matrix a ∈ m(2,c) splits as the direct sum of a scalar multiple of the identity matrix plus a trace zero matrix. in particular, we can write a = a+ + a− = αi2 + a −, with σ(a) = 2 α and σ(a−) = 0, where σ(a) denotes the trace of the matrix a and i2 denotes the 2 × 2 identity matrix. so we can write a = α + a−. for a,b ∈m(2,c), set a+ = α, b+ = β, (a− b−)+ = γ, where α, β and γ represent complex numbers or scalar diagonal matrices depending on the context. we define two families of polynomials, which naturally arise from computing the n-th powers of a matrix a ∈ sl(2,c). write a = α + a− as above, and an = fn(α) + gn(α) a −, (2.1) where σ(a) = 2 α ∈ c. the polynomial fn can be expressed in terms of gn and gn−1. the character variety of one relator groups 107 lemma 2.1. with the above notations, we have fn(α) = αgn(α) − gn−1(α). (2.2) proof. since (a−)2 = α2 − 1 from [5, lemma 2.1(3)], it follows that an = aan−1 = (α + a−) [fn−1(α) + gn−1(α)a −] = αfn−1(α) + (α 2 − 1)gn−1(α) + [fn−1(α) + αgn−1(α)]a−. equating this formula and (2.1) yields fn(α) = αfn−1(α) + (α 2 − 1)gn−1(α) (2.3) and gn(α) = fn−1(α) + αgn−1(α). (2.4) multiplying (2.4) by α, we get αgn(α) = αfn−1(α) + α 2gn−1(α). using the last expression, we can eliminate αfn−1(α) from (2.3), that is, fn(α) = αgn(α) −α2gn−1(α) + (α2 − 1)gn−1(α), which gives (2.2). moreover, we can derive the recursive expressions of fn and gn. lemma 2.2. the families of polynomials {fn} and {gn} are defined by the recurrence formulas gn(α) = 2 αgn−1(α) − gn−2(α) (2.5) and fn(α) = 2 αfn−1(α) − fn−2(α) (2.6) for every n ≥ 1, with the initial values g−1(α) = −1 and g0(α) = 0, f−1(α) = α and f0(α) = 1, respectively. proof. substituting the expression of fn−1 from (2.2) into (2.4) yields gn(α) = αgn−1(α) −gn−2(α) + αgn−1(α), 108 a. cavicchioli, f. spaggiari which gives (2.5). multiplying by α the formula of fn−1 from (2.2), we get αfn−1(α) = α 2gn−1(α) −αgn−2(α). using the last expression, we can eliminate α2gn−1(α) from (2.3), that is, fn(α) = 2 αfn−1(α) + αgn−2(α) −gn−1(α). by (2.5) written for n− 1, we get fn(α) = 2 αfn−1(α) + αgn−2(α) − [2 αgn−2(α) −gn−3(α)] = 2 αfn−1(α) −αgn−2(α) + gn−3(α) = 2 αfn−1(α) − [αgn−2(α) −gn−3(α)]. this implies (2.6) as the expression inside the brackets is precisely fn−2(α) by (2.2). lemma 2.3. the following identities g2n(α) = 1 + gn−1(α) gn+1(α) and 2 gn(α) α − g2n(α) = [gn+1(α) − 1] [1 − gn−1(α)] hold. proof. the first formula is proved by induction on n. if n = 0, 1, 2, then g20 = 1 + g−1 g1 = 0, g 2 1 = 1 + g0 g2 = 1, and g 2 2 (α) = 1 + g1 g3 = 4α 2, respectively, as g−1 = −1, g0 = 0, g1 = 1, g2(α) = 2α, and g3(α) = 4α2 − 1. using the inductive hypothesis and (2.5), we get 1 + gn−1(α) gn+1(α) = 1 + gn−1(α) [2αgn(α) −gn−1(α)] = 1 + 2αgn(α) gn−1(α) − g2n−1(α) = 1 + 2αgn(α) gn−1(α) − 1 − gn−2(α) gn(α) = gn(α)[2αgn−1(α) − gn−2(α)] = g2n(α). for the second equality, we have [gn+1(α) − 1] [1 − gn−1(α)] = gn+1(α) − gn+1(α) gn−1(α) + gn−1(α) − 1 = gn+1(α) + 1 − g2n(α) − 1 + gn−1(α) = 2 αgn(α) − gn−1(α) − g2n(α) + gn−1(α) = 2 αgn(α) − g2n(α) the character variety of one relator groups 109 by using the first equality of the statement and the recursive formula of gn(α) in (2.5). the polynomials {gn} are related to the n-th chebyshev polynomial of the first kind sn(x) (see [14]), that is, gn(α) = sn−1(2 α). furthermore, we also have gn(α) = fn(2 α), where fn denotes the n-th fibonacci polynomial (see, for example, [1, 26]). finally, gn relates with the hilden-lozano-montesinos polynomial pn (see [10]) by the formula gn+1(α) = pn(2α). further algebraic properties of polynomials fn and gn have been described in [5, proposition 2.3]. through the paper we also need the following result: lemma 2.4. let {a,b} be a set of generators of a 2-generator group g, and let ρ be an irreducible representation of g into sl(2,c). setting a = ρ(a) and b = ρ(b), the set b = {i2,a−,b−, (a−b−)−} is a basis for the 4-dimensional vector space m(2,c). for a proof see, for example, [12, lemma 1.2]. furthermore, we implicitly use the well-known fact that a representation of a group with two generators a and b is determined by the traces of these generators and of their product ab (see, for example, [9]). 3. torus links let c(2n) denote the rational link in conway’s normal form (see [15, p. 24]), which is the torus link depicted in figure 1. it is the closure of the braid σ2n1 , where σ1 is the standard generator of the braid group b2 on two strands. equivalently, it is the closure of the braid (σ2n−1 σ2n−2 · · · σ1)2 with σ1, σ2, . . . ,σ2n−1 being the standard generators of the braid group b2n on 2n strands. note that the torus link c(2n) is given by t(2n, 2) according to rolfsen’s notation [24]. theorem 3.1. the character variety of the torus link c(2n), n ≥ 1, is defined by the equation (ab − ba) gn(α) = 0. the first factor determines the character variety for abelian representations into sl(2,c), and the second factor determines the character variety for nonabelian representations of the link group gn. 110 a. cavicchioli, f. spaggiari figure 1: the torus link c(2n), n ≥ 1. proof. let gn denote the fundamental group of the exterior of c(2n) in the oriented 3-sphere s3, i.e., gn = π1(s3\c(2n)). the group gn admits the finite presentation 〈a,b : (ab)n = (ba)n〉. we provide a geometric interpretation of the generators of gn by representing them in figure 1. setting u = ab and v = b (hence a = uv−1 and b = v), we get the finite presentation 〈u,v : unv = vun〉. sending u and v to the matrices a and b, respectively, the last relation gives anb = ban in sl(2,c). for n ≥ 1, we have anb = [fn(α) + gn(α)a −] (β + b−) = βfn(α)i2 + βgn(α)a − + fn(α)b − + gn(α)a −b− and ban = (β + b−) [fn(α) + gn(α)a −] = βfn(α)i2 + βgn(α)a − + fn(α)b − + gn(α)b −a−. computing the difference gives anb − ban = (a−b− − b−a−) gn(α) hence anb − ban = (ab − ba) gn(α) as a−b− − b−a− = ab − ba. this produces the defining relations of the character variety of c(2n) (or gn). the techniques used in the above proof are different from those employed by qazaqzeh in [21, theorem 1.2]. for a given representation ρ of the group the character variety of one relator groups 111 gn = 〈a,b : (ab)n = (ba)n〉 into sl(2,c), the cited author denotes by tr(x) the trace of ρ(x), for any word x in the generators a and b. then tr(a), tr(b) and tr(ab) are abbreviated by t1, t2 and t3, respectively. his result states that the defining polynomial of the character variety of gn is given by tr ( (ab)na−1b−1 ) − tr ( (ba)n−1 ) = (t23 + t 2 2 + t 2 1 − t3t2t1 − 4) sn−1(t3), where the first (resp. second) factor on the right side determines the character variety for abelian (resp. nonabelian) representations. here sk(x) is the kth chebyshev polynomial of the first kind, defined recursively by s0(x) = 1, s1(x) = x and sk(x) = xsk−1(x) − sk−2(x). the proof of this formula is given by induction on n, using the trace identities and the recursive definition of the chebyshev polynomials. the same elementary methods in the proof of theorem 3.1 can be used to obtain the defining polynomial of the character variety of a class of torus knots from [20] and the characters of certain families of one relator groups from [18, 19, 22]. namely, the authors in [18, 19] consider the group g = 〈x,y : xm = yn〉 with m and n nonzero integers, and compute the number of irreducible components of the character variety of g in sl(2,c). a defining polynomial of the sl(2,c) character variety of the torus knot of type (m, 2) has been provided by oller-marcén in [20]. recurrence formulas based on (generalized) fibonacci polynomials have been proposed in [26, theorem 7 and theorem 11] to derive homfly polynomials (and hence alexander-conway polynomials and jones polynomials) of torus links c(2n). generalized fibonacci polynomials can be related to our classes of polynomials {fn} and {gn}, as remarked above. for n = 1, gn(α) = 1, hence the equation in theorem 3.1 reduces to ab = ba, which determines the character variety for abelian representations into sl(2,c). so in the sequel, we discuss the case c(2n+2) with n being ≥ 1. theorem 3.1 directly gives an easy geometrical description of the character variety of such torus links. theorem 3.2. in the complex 3-space (x,y,z) the character variety for nonabelian representations of the torus link c(2n + 2) consists of the union of n horizontal planes of the form zk = 2 cos[kπ/(n + 1)], for 1 ≤ k ≤ n. proof. we set z = 2 α = σ(a), x = 2 β = σ(b) = σ(b−1), and y = σ(ab−1). from the relation ab−1 = (α + a−) (β − b−) = αβ i2 + β a− − αb− − a−b−, 112 a. cavicchioli, f. spaggiari it follows that y = σ(ab−1) = 2 αβ − 2 γ as σ(a−) = σ(b−) = 0 and σ(a−b−) = 2 γ. the roots of the second factor gn(z/2) = 0 are given by zk = 2 cos[kπ/(n + 1)] for any 1 ≤ k ≤ n. see [5, proposition 2.3(9)] and [10, proposition 1.3]. using the chesebro formula for gn+1(α) (see [7]), we can give a different expression for the defining equation in theorem 3.2. corollary 3.3. in the complex 3-space (x,y,z) the character variety for nonabelian representations of the torus link c(2n + 2) is defined by the equation [ (z + √ z2 − 4)n+1 − (z − √ z2 − 4)n+1 2n+1 √ z2 − 4 ] = 0 for −2 < z < 2 (real number). to illustrate geometrically the support of the character variety in theorem 3.2 and corollary 3.3 we explicitly discuss the cases n = 1, . . . , 5. if n = 1, there is one horizontal plane of the form z = 2 cos(π/2) = 0 from theorem 3.2. the equation of the second factor in corollary 3.3 becomes z = 0. if n = 2, there are two horizontal planes with equations z = 2 cos(π/3) = 1 and z = 2 cos(2π/3) = −1 (see theorem 3.2). the equation of the second factor in corollary 3.3 becomes z2 − 1 = 0. if n = 3, there are three horizontal planes with equations z = 2 cos(π/4) =√ 2, z = 2 cos(π/2) = 0, and z = 2 cos(3π/4) = − √ 2. the equation of the second factor in corollary 3.3 becomes z(z2 − 2) = 0. if n = 4, there are four horizontal planes with equations z = 2 cos(π/5) = (1 + √ 5)/2, z = 2 cos(2π/5) = ( √ 5 − 1)/2, z = 2 cos(3π/5) = (1 − √ 5)/2, and z = 2 cos(4π/5) = (−1 − √ 5)/2. the equation of the second factor in corollary 3.3 becomes z4−3z2 +1 = 0, which has the four roots ±(1± √ 5)/2, as requested. if n = 5, there are five horizontal planes with equations z = 2 cos(π/6) =√ 3, z = 2 cos(π/3) = 1, z = 2 cos(π/2) = 0, z = 2 cos(2π/3) = −1, and z = 2 cos(5π/6) = − √ 3. the equation of the second factor in corollary 3.3 becomes z(z2 − 1)(z2 − 3) = 0, which has the above roots. the character variety of one relator groups 113 as remarked in [24, example 10], the genus of the torus link c(2n + 2) is n, which precisely coincides with the number of horizontal planes in the character variety of c(2n + 2), i.e., the degree of the polynomial gn+1(z/2). since the character varieties of gn and gm have different number of irreducible components if n 6= m, we derive the following well-known result (see [21, corollary 1.3]). corollary 3.4. the groups gn and gm are isomorphic if and only if n = m. note that corollary 3.4 also follows from the theory of seifert manifolds since the torus link complement c(2n) is a seifert fiber space with one exceptional fiber. let m be an oriented compact 3-manifold. then the kauffman bracket skein module k(m) of m is defined to be the quotient of the module freely generated by equivalence classes of framed links in m over z[t,t−1] by the smallest submodule containing kauffman relations (see [2] for more details). the topological meaning of this module has been explained in [2] for t = −1. more precisely, setting t = −1 and tensoring such a module with c produces a natural algebra structure, denoted k−1(m), over c. furthermore, this algebra is canonically isomorphic to the coordinate ring of the character variety of π1(m) after factoring it by its nilradical (see [2, theorem 10]). then theorem 3.2 allows to give a simplified proof of theorem 1.4 from [21]. theorem 3.5. let m denote the exterior of c(2n + 2), n ≥ 1, in the oriented 3-sphere, k(m) the kauffman bracket skein module of m, and n the (t + 1)-torsion submodule of k(m). then the quotient k(m)/n is a free module over z[t,t−1] with a basis b = {xiyjzk : i,j ≥ 0, 0 ≤ k ≤ n}, where x, y, and z represent the conjugacy classes of uv−1, v, and u in the presentation 〈u,v : unv = vun〉 of π1(m), respectively. proof. by theorem 3.2 the coordinate ring of the character variety of π1(m) admits the basis b (over c) indicated in the statement. in fact, the horizontal planes z = 2 cos[kπ/(n + 1)], 1 ≤ k ≤ n, plus the neutral element for k = 0, give n + 1 conjugacy classes of the statement. by [21] the quotient of k−1(m) over its nilradical is isomorphic (over c) to k−1(m). hence b is linearly independent (over c) in k−1(m). then it is a basis for k(m)/n. for a description of k(m), when m is the exterior of a 2-bridge link, we refer to [16]. 114 a. cavicchioli, f. spaggiari 4. once-punctured torus bundles let us consider the once-punctured torus bundles with tunnel number one, that is, the once-punctured torus bundles that arise from filling one boundary component of the whitehead link (wl) exterior. see figure 2. figure 2: a planar projection of the whitehead link. the character varieties of such manifolds have been determined in [1]. using the concept of palindrome word, we compute the defining polynomials of these character varieties with different techniques with respect to [1]. up to homeomorphism, the monodromy of the once-punctured torus bundle mn = (t × i)/qn is qn = τc1 τn+2c2 , where c1 and c2 are curves forming a basis for the fiber t (a torus) and τc means a right-handed dehn twist about the curve c. here i = [0, 1]. the manifold mn can be obtained by (n + 2, 1) dehn filling on one boundary component of the wl exterior, and it is the exterior of a certain genus one fibered knot in the lens space l(n + 2, 1). it is known that mn is hyperbolic if and only if |n| > 2, contains an essential torus (i.e., is toroidal) if and only if |n| = 2, and is a seifert fiber space if and only if |n| ≤ 1. see, for example, [1, lemma 2.8]. by [1, lemma 2.5], the fundamental group π1(mn) is isomorphic to γn = 〈a,b : a−n = b−1 ab2 ab−1 〉. (4.1) we provide a geometric interpretation of the generators of γn by representing them in figure 2. we choose meridians µ0, µ1 and longitudes λ0, λ1 on the oriented components k0, k1 of wl, respectively, (see figure 2) such the character variety of one relator groups 115 that [µi,λi] = 1, for i = 0, 1, and λi ∼ 0 in s3\ki. then we have µ0 = a−1, µ1 = a 2ba−1, λ0 = xab −1a−2 and λ1 = az, where x and z are represented in figure 2. the wirtinger presentation of the group π(wl) = π1(s3\wl) has generators a, b, x, y and z and relations ya−1 = aba−1, z = ab−1a−1ba−1, yx = a2ba−1y and xz = a−1x. then we obtain the relation xz = a−1b−1ab2 after doing the appropriate elimination. eliminating x = b−1ab2, y = ab and z = ab−1a−1ba−1 yields a finite presentation for π(wl) with generators a and b and relation b−1ab2ab−1a−1ba−1 = a−1b−1ab2. (4.2) a presentation for γn can be obtained from that of π(wl) by adding the surgery relation µ −(n+2) 0 λ0 = 1 (4.3) where µ0 = a −1 and λ0 = xab −1a−2 = b−1ab2ab−1a−2. substituting these formulas into (4.3) gives an+2b−1ab2ab−1a−2 = 1 hence anb−1ab2ab−1 = 1 which is equivalent to the relation in (4.1). now (4.2) is a consequence of the relation in (4.1), so it can be dropped. in fact, we have the following sequences of tietze transformations: (b−1ab2ab−1)a−1ba−1 = a−1b−1ab2, a−na−1ba−1 = a−1b−1ab2, a−n = b−1ab2ab−1, which is the relation of γn. theorem 4.1. for every n ∈ z, let mn be the once-punctured torus bundle of tunnel number one, and γn = π1(mn). in the complex plane (x,z), the defining equation of the character variety of γn is given by [gn+1(z/2) − 1] [x2 − 1 + gn−1(z/2)] = 0. in the hyperbolic case |n| > 2, the character variety for nonabelian representations of γn (or mn) consists of the hyperelliptic curve given by x2 + gn−1(z/2) − 1 = 0 116 a. cavicchioli, f. spaggiari and a finite number of horizontal lines (counted with their multiplicities) of the form z = zk, where zk is a root of the equation gn+1(z/2) − 1 = 0. proof. from the relation in (4.1), or equivalently ba−nb = ab2a, sending a and b to the matrices a and b, respectively, gives the relation in sl(2,c) b a−n b = ab2 a which is palindrome in the left and right sides. set a = α + a− and b = β + b−. as a direct application of the cayley-hamilton theorem, the formula a−n = fn(α) − gn(α) a− holds. by direct calculations on palindrome words, it follows b a−n b = q0 i2 + q1 a − + q2 b − where q0 = (2 β 2 − 1) fn(α) − 2 β γ gn(α) , q1 = −gn(α) , q2 = 2 β fn(α) − 2 γ gn(α) , with a+ = α, b+ = β and (a−b−)+ = γ, i.e., σ(a) = 2 α, σ(b) = 2 β and σ(a−b−) = 2 γ. as above, by direct computations on palindromes, we have ab2 a = q ′ 0 i2 + q ′ 1 a − + q ′ 2 b − where q ′ 0 = (2 α 2 − 1) (2 β2 − 1) + 4 αβ γ , q ′ 1 = 2 α (2 β 2 − 1) + 4 β γ , q ′ 2 = 2 β . equating qi = q ′ i, i = 0, 1, 2, gives the defining polynomials of the character variety for γn (or mn). from q2 = q ′ 2 we derive an expression of γ in terms of α and β. so the representation (up to conjugacy) is only determined by the traces σ(a) = 2α and σ(b) = 2β. substituting the cited expression of γ into q1 = q ′ 1 yields the defining equation of the character variety. in fact, q0 = q ′ 0 the character variety of one relator groups 117 is a consequence of the other equations. thus the character variety of γn has equation g2n(α) + 2 α (2 β 2 − 1) gn(α) + 4 β2 [fn(α) − 1] = 0. we can express fn(α) in terms of gn(α) and gn−1(α). multiply out gives the equation g2n(α) + 2 α (4 β 2 − 1) gn(α) − 4 β2 [gn−1(α) + 1] = 0. set z = 2 α ∈ c and x = 2 β ∈ c. then we get g2n(z/2) + z (x 2 − 1) gn(z/2) − x2 [gn−1(z/2) + 1] = 0 or, equivalently, g2n(z/2) + [gn(z/2) z − gn−1(z/2) − 1] x 2 − gn(z/2) z = 0 hence g2n(z/2) + [gn+1(z/2) − 1] x 2 − gn(z/2) z = 0. by lemma 2.3, the defining equation of the character variety of γn is given by the first formula in the statement. the last sentence of the theorem follows from [5, proposition 2.3]. since gn(z/2) = fn(z), theorem 4.1 relates to theorem 5.1 from baker and petersen [1] in the sense that we obtain a similar hyperelliptic curve. more precisely, these authors prove that if |n| > 2, then there is a unique canonical component of the sl(2,c) character variety of mn, and it is birational to the hyperelliptic curve given by w2 = −ĥn(y) ̂̀n(y) in the complex plane (w,y), where the polynomials ĥn and ̂̀n are specific factors of fibonacci polynomials. if n is not congruent to 2 (mod 4), this is the only component of the sl(2; c) character variety which contains the characters of an irreducible representation. if n ≡ 2 (mod 4), there is an additional component which is isomorphic to c. if n is not equal to −2, all the components consisting of characters of reducible representations are isomorphic to affine conics (including lines) and consist of characters of abelian representations. however, the methods used by the cited authors (based on the invariant theory) are similar to those developed by qazaqzeh in [21] for the class of torus links. to illustrate geometrically the support of the character variety in theorem 4.1 we explicitly discuss the hyperbolic cases n = 3, . . . , 6. 118 a. cavicchioli, f. spaggiari if n = 3, the equation x2 + g2(z/2) − 1 = 0 becomes x2 + z − 1 = 0 as g2(α) = 2α = z. furthermore, the equation g4(z/2) − 1 = 0 becomes z3 − 2z − 1 = (z + 1) (z2 −z − 1) = 0 as g4(α) = 8α 3 − 4α = z3 − 2z. then, in the complex plane (x,z), the character variety of γ3 (or m3) consists of the parabola z = 1 −x2 and the union of three horizontal lines with equations z = −1 and z = (1 ± √ 5)/2. if n = 4, the equation x2 + g3(z/2) − 1 = 0 becomes x2 + z2 − 2 = 0 as g3(α) = 4α 2−1 = z2−1. furthermore, the equation g5(z/2)−1 = 0 becomes z4 − 3z2 = z2(z2 − 3) as g5(α) = 16α4 − 12α2 + 1 = z4 − 3z2 + 1. so, in the complex plane (x,z), the character variety of γ4 (or m4) consists of the ellipse x2 + z2 = 2 and the union of four (counted with their multiplicities) horizontal lines with equations z = 0 (counted twice) and z = ± √ 3. if n = 5, the equation x2 + g4(z/2) − 1 = 0 becomes x2 +z3−2z−1 = 0 or, equivalently, x2 + (z + 1) (z2 −z−1) = 0. the equation g6(z/2)−1 = 0 becomes z5 − 4z3 + 3z − 1 = 0 as g6(α) = 32 α 5 − 32 α3 + 6 α = z5 − 4 z3 + 3 z. thus, in the complex plane (x,z), the character variety of γ5 (or m5) consists of the elliptic cubic (in fact, the newton divergent parabola) of equation x2 = −z3 + 2z + 1 and the union of five horizontal lines with equations of the form z = zk, where zk is a root of z5 − 4z3 + 3z − 1 = (z2 + z − 1) (z3 −z2 − 2z + 1) = 0. from the first factor we get z1,2 = (−1 ± √ 5)/2. the equation z3 −z2 − 2z + 1 = 0 becomes x3 + px + q = 0 with p = −7 3 and q = 7 27 by using the transformation z = x+ 1 3 . since ∆ = q2 4 + p3 27 = − 49 108 < 0, there are three real roots x1 = 2a, x2 = −a−b √ 3 and x3 = −a + b √ 3, where a + ib = 3 √ w and w = −q 2 + i √ ∆. if n = 6, the equation x2 + g5(z/2) − 1 = 0 becomes x2 + z4 − 3z2 = 0. the equation g7(z/2) − 1 = 0 becomes z6 − 5z4 + 6z2 − 2 = (z2 − 1) (z4 − 4z2 + 2) = 0 as g7(α) = 64 α 6 − 80 α4 + 24 α2 − 1 = z6 − 5z4 + 6z2 − 1. the character variety of one relator groups 119 thus, in the complex plane (x,z), the character variety of γ6 consists of the hyperelliptic quartic x2 = −z4 + 3z2 and the union of six horizontal lines with equations of the form z = zk, where zk takes on the values ±1 and ± √ 2 ± √ 2. 5. cusped manifolds from dehn fillings for every n ≥ 0, let nn be the one-cusped 3-manifold obtained by performing a (6n+ 2, 2n+ 1) dehn filling on one boundary component of the wl exterior, leaving the other component open. see figure 3. it is known that nn is hyperbolic for every n ≥ 1. among all fillings of one cusp of the whitehead exterior we focus on the (6n+ 2, 2n+ 1) fillings since their fundamental group has a simple palindrome presentation. see (5.1) below. however, the proposed techniques for computing character varieties of such manifolds can also be applied in the general case. by [6, proposition 4.1] the fundamental group π1(nn) is isomorphic to λn = 〈a, b : aba = (b3 a−3)2n b3 〉 = 〈a, b : aba = b3 (a−3 b3)2n 〉. (5.1) we provide a geometric interpretation of the generators of λn in figure 3. figure 3: another planar projection of the whitehead link. theorem 5.1. for every n ≥ 0, let nn be the one-cusped 3-manifold obtained by (6n + 2, 2n + 1) dehn filling on one boundary component of the 120 a. cavicchioli, f. spaggiari wl exterior, and let λn = π1(nn). in the complex 3-space (x,y,z), the character variety of the group λn (or nn) is determined by the equations y + (z2 − 1)g2n(δ) = 0 , (x2 − 1)g2n+1(δ) − 1 = 0 , where δ is given by 2δ = x3z3 − 2xz3 − 2x3z −x2y z2 + x2y + y z2 + 5xz −y. proof. from the relation in (5.1), sending a and b to the matrices a and b, respectively, gives the relation in sl(2,c) ab a = (b3 a−3)2n b3, which is palindrome in the left and right sides. by direct computations on palindromes, we obtain ab a = q̄0 i2 + q̄1 a − + q̄2 b − where q̄0 = (2α 2 − 1)β + 2αγ , q̄1 = 2αβ + 2γ , q̄2 = 1 , with a+ = α, b+ = β and (a−b−)+ = γ, as usual. define l = b3a−3. then l = δ + l−, where σ(l) = 2δ. we get l = p0i2 + p1a − + p2b − + p3a −b− where p0 = (4α 3 − 3α)(4β3 − 3β) − 2(4α2 − 1)(4β2 − 1)γ , p1 = −(4α2 − 1)(4β3 − 3β) , p2 = (4α 3 − 3α)(4β2 − 1) , p3 = (4α 2 − 1)(4β2 − 1) . since σ(a−) = σ(b−) = 0 and σ(a−b−) = 2γ, we obtain δ = p0 + γp3. (5.2) the character variety of one relator groups 121 it follows that l− = −γp3i2 + p1a− + p2b− + p3a−b− (5.3) and l−b− = (β2 − 1)p2i2 + (β2 − 1)p3a− −γp3b− + p1a−b−. (5.4) using (5.3) and (5.4) we obtain (b3 a−3)2n b3 = l2nb3 = [f2n(δ) + g2n(δ) l −] [4β3 − 3β + (4β2 − 1)b−] = q̄ ′ 0i2 + q̄ ′ 1a − + q̄ ′ 2b − where q̄ ′ 0 = (4β 3 − 3β)f2n(δ) + [(4β2 − 1)(β2 − 1)p2 − (4β3 − 3β)γp3]g2n(δ) , q̄ ′ 1 = −(4α 2 − 1)g2n(δ) , q̄ ′ 2 = (4β 2 − 1)f2n(δ) + [(4β3 − 3β)p2 − (4β2 − 1)γp3]g2n(δ) . by (2.2) and (5.2) and using the above expressions of p2 and p3 in terms of α and β, the polynomial q̄ ′ 2 becomes q̄ ′ 2 = (4β 2 − 1)[δg2n(δ) −g2n−1(δ)] + (4β2 − 1)δg2n(δ) = 2δ(4β2 − 1)g2n(δ) − (4β2 − 1)g2n−1(δ) = (4β2 − 1) g2n+1(δ). by lemma 2.4, equating q̄i = q̄ ′ i, i = 0, 1, 2, gives the equations of the character variety of the group λn. we see that q̄0 = q̄ ′ 0 is a consequence of the other two equations. we set z = 2α = σ(a), x = 2β = σ(b), and y = σ(ab) = 2αβ + 2γ. solving α, β and γ as functions of x, y and z and substituting into p0 and p3, equation (5.2) becomes the formula of 2δ given in the statement. expressing q̄1 = q̄ ′ 1 and q̄2 = q̄ ′ 2 in terms of x, y and z yields the first two equations in the statement of the theorem. acknowledgements work performed under the auspices of the scientific group g.n.s.a.g.a. of the c.n.r (national research council) of italy and partially supported by the mur (ministero dell’universitá e della ricerca) of italy within the project strutture geometriche, combinatoria e loro applicazioni, and by a research grant far 2022 of the university of modena and reggio emilia. the authors would like to thank the anonymous referee for his/her useful comments and suggestions, which improved the final version of the paper. 122 a. cavicchioli, f. spaggiari references [1] k.l. baker, k.l. petersen, character varieties of once-punctured torus bundles with tunnel number one, internat. j. math. 24 (6) (2013), 1350048, 57 pp. [2] d. bullock, rings of sl2(c)-characters and the kauffman bracket skein module, comment. math. helv. 72 (1997), 521 – 542. 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[22] k. qazaqzeh, the character variety of a family of one-relator groups, internat. j. math. 23 (1) (2012), 1250015, 12 pp. [23] r. riley, nonabelian representations of 2-bridge knot groups, quart. j. math. oxford ser. (2) 35 (1984), no. 138, 191 – 208. [24] d. rolfsen, “knots and links”, mathematics lecture series 7, publish or perish inc., berkeley, ca, 1976. [25] p.h. shalen, representations of 3-manifold groups, in “handbook of geometric topology”, north-holland, amsterdam, 2002, 955 – 1044. [26] k. tasköprü, i̇. altintas, homfly polynomials of torus links as generalized fibonacci polynomials, electron. j. combin. 22 (4) (2015), paper p4.8, 17 pp. [27] a.t. tran, character varieties of (−2, 2m + 1, 2n)-pretzel links and twisted whitehead links, j. knot theory ramifications 25 (2) (2016), 1650007, 16 pp. introduction technical preliminaries torus links once-punctured torus bundles cusped manifolds from dehn fillings � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 1 (2020), 1 – 20 doi:10.17398/2605-5686.35.1.1 available online february 17, 2020 refinements of kantorovich type, schwarz and berezin number inequalities m. garayev 1, f. bouzeffour 1, m. gürdal 2, c.m. yangöz 2 1 department of mathematics, college of science, king saud university p.o. box 2455, riyadh 11451, saudi arabia 2 suleyman demirel university, department of mathematics, 32260, isparta, turkey mgarayev@ksu.edu.sa , fbouzaffour@ksu.edu.sa , gurdalmehmet@sdu.edu.tr , mustafa yangoz@hotmail.com received june 11, 2019 presented by pietro aiena accepted january 28, 2020 abstract: in this article, we use kantorovich and kantorovich type inequalities in order to prove some new berezin number inequalities. also, by using a refinement of the classical schwarz inequality, we prove berezin number inequalities for powers of f(a), where a is self-adjoint operator on the hardy space h2(d) and f is a positive continuous function. some related questions are also discussed. key words: reproducing kernel hilbert space, berezin symbol, berezin number, kantorovich type inequality, c∗-module. ams subject class. (2010): primary 47a63; secondary 26d15, 47b10. 1. introduction, notation and preliminaries in 1948, l.v. kantorovich [24] proved the following inequality 〈ax,x〉 〈 a−1x,x 〉 ≤ (λ1 + λn) 2 4λ1λn (1.1) where x = (x1, . . . ,xn) is a unit vector in cn and a is an n×n positive-definite matrix with eigenvalues λ1 ≥ ···≥ λn > 0. the kantorovich inequality is still valid for an operator a acting on an infinite dimensional hilbert space h with mi ≥ a ≥ mi > 0 as follows: 〈ax,x〉 〈 a−1x,x 〉 ≤ (m + m) 2 4mm (x ∈ h, ‖x‖ = 1). replacing x by a 1/2x ‖a1/2x‖ in the above inequality, we get the following equivalent form of kantorovich inequality:〈 a2x,x 〉 ≤ (m + m) 2 4mm 〈ax,x〉2 (x ∈ h, ‖x‖ = 1). (1.2) issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.35.1.1 mailto:mgarayev@ksu.edu.sa mailto:fbouzaffour@ksu.edu.sa mailto:gurdalmehmet@sdu.edu.tr mailto:mustafa_yangoz@hotmail.com https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 2 f. bouzeffour, m. garayev, m. gürdal, c.m. yangöz the kantorovich inequality is a useful tool in numerical analysis and statistics for establishing the rate of convergence of the method of steepest descent. during the past decades several formulations, extensions or refinements of the kantorovich inequality in various settings have been introduced by many mathematicians; see, for instance, moslehian [34] and references therein. the first generalization of the kantorovich inequality is due to greub and rheinboldt [20]. they proved that if a is a bounded linear operator on h (i.e., a ∈b(h)) such that mi ≥ a ≥ mi > 0, then 〈x,x〉2 ≤〈ax,x〉 〈 a−1x,x 〉 ≤〈x,x〉2 (m + m) 2 4mm (1.3) for any x ∈ h. they also showed that inequality (1.3) is equivalent to 〈ax,ax〉〈bx,bx〉≤ 〈ax,bx〉2 (mm ′ + mm′) 2 4mm′mm ′ , when b is a selfadjoint operator permutable with a, x ∈ h and m ′i ≥ b ≥ m′i > 0. in what follows, strang [38] generalized inequality (1.3) by proving that if a ∈b(h) is invertible, ‖a‖ = m and ∥∥a−1∥∥ = 1 m , then ∣∣(〈ax,x〉〈y,a−1y〉)∣∣ ≤ (m + m)2 4mm 〈x,x〉〈y,y〉 (1.4) for all x,y ∈ h and that the bound is the best possible. for further generalization of the kantorovich inequality and recent developments of the operator kantorovich inequality, we refer the readers to the excellent book of furuta [14] and moslehian’s paper [34]. note that dragomir [10] gave several kantorovich type inequalities involving norms and numerical radii for operators on a hilbert space. also, garayev [19] and başaran et al. [6] applied the kantorovich inequality to get berezin number inequalities for the operators acting on the reproducing kernel hilbert space. in the present article, we use some kantorovich type inequalities to prove some new inequalities for the berezin number of operators. also, we apply a refinement of classical schwarz inequality due to dragomir [11] to prove berezin number inequalities between some powers of f(a), where f : j → [0, +∞) is a positive continuous function, where j ⊂ [0, +∞) and a is a selfadjoint operator on the hardy space h2(d) with spectrum in j. before giving our results, we need to some definitions and notations. throughout this paper, b(h) stand for the banach algebra of all bounded linear operators acting on a hilbert space (h,〈., .〉). an operator a ∈b(h) is refinements of kantorovich type 3 said to be positive if 〈ax,x〉≥ 0 for all x ∈ h. moreover, if a is invertible, we call it strictly positive and write a > 0. a reproducing kernel hilbert space is a hilbert space h = h(ω) of complex-valued functions on a (nonempty) set ω, which has the property that point evaluation f → f(λ) is continuous on h for all λ ∈ ω. then the riesz representation theorem guarantees that for every λ ∈ ω there is a unique element kλ = k(.,λ) ∈ h such that f(λ) = 〈f,kλ〉 for all f ∈ h. the function kλ is called the reproducing kernel of h and the function k̂λ := kλ ‖kλ‖h is the normalized reproducing kernel in h (see aronzajn [3]). for any operator a ∈ b(h), its berezin symbol ã is defined by (see berezin [7, 8]) ã(λ) := 〈 ak̂λ, k̂λ 〉 , λ ∈ ω . the berezin number of a is defined as ber(a) := sup{|µ| : µ ∈ ber(a)} , where ber(a) = range ( ã ) = { ã(λ) : λ ∈ ω } is the so-called berezin set of operator a (see karaev [25]). the numerical range and numerical radius of the operator a is defined, respectively, by w(a) := {〈ax,x〉 : x ∈ h and ‖x‖ = 1} and w(a) := sup{|〈ax,x〉| : x ∈ h and ‖x‖ = 1} . clearly, ∣∣∣ã(λ)∣∣∣ ≤‖a‖, and more precisely, ber(a) ≤ w(a) ≤‖a‖. note that the celebrated berger-halmos-pearcy inequality for the powers of the operator a ∈ b(h) is the following (see halmos [22] and pearcy [37], and references therein): w(an) ≤ w(a)n, n = 1, 2, . . . . since ber(a) ≤ w(a), it is natural to ask about the same inequalities for the berezin number ber(a), i.e., is it true that ber(an) ≤ ber(a)n for any integer n ≥ 1? however, coburn proved in his paper [9] that for some concrete operator x on the bergman space l2a(d) of analytic functions on d, ber(x 2) > ber(x)2, which shows that berger-halmos-pearcy theorem fails for ber(a). but, still it 4 f. bouzeffour, m. garayev, m. gürdal, c.m. yangöz is interesting to investigate inverse estimates ber(a)n ≤ c ber(a)n, n ≥ 1, and also other type berezin number inequalities between the powers of operators. in this article, motivated mainly by the paper [34], we will prove some new inequalities for the berezin number of powers of operators by using kantorovich and kantorovich type inequalities and also a refinement of schwarz inequality due to dragomir [10]. for the related results, the reader can see in [4, 5, 13, 15, 16, 17, 18, 21, 25, 26, 30, 32, 33, 36, 40, 41, 42, 44] 2. some corollaries of kantorovich inequalities in this section, we give some immediate corollaries of several kantorovich type inequalities which entail new inequalities for berezin numbers of some operator classes. first, we start with strang inequality (1.4). proposition 1. if a ∈ b(h(ω)) is invertible, ‖a‖ = m and ∥∥a−1∥∥ = m−1, then ber(a) ber(a−1) ≤ (m + m) 2 4mm . (2.1) proof. in fact, by putting in inequality (1.4) x = k̂λ, y = k̂µ (λ,µ ∈ ω), we have ∣∣∣〈ak̂λ, k̂λ〉∣∣∣∣∣∣〈k̂µ,a−1k̂µ〉∣∣∣ ≤ (m + m)2 4mm , or equivalently, ∣∣∣ã (λ)∣∣∣ ∣∣∣ã−1∗ (µ)∣∣∣ = ∣∣∣ã (λ)∣∣∣∣∣∣ã−1 (µ)∣∣∣ ≤ (m + m)2 4mm for all λ,µ ∈ ω. this implies immediately the required inequality (2.1). the following is immediate from the inequality (1.2). proposition 2. if a ∈ b(h(ω)) is an operator such that mi ≥ a ≥ mi > 0, then ber ( a2 ) ≤ (m + m) 2 4mm ber(a)2. the next result follows from furuta’s inequality [13] which is an extension of the kantorovich inequality. refinements of kantorovich type 5 proposition 3. if a,b ∈ b(h) are positive operators, a ≥ b > 0 and mi ≥ b ≥ mi > 0, then( m m )p−1 ber (ap) (2.2) ≥ (p− 1)p−1 pp ( (mp −mp)p (m −m) (mmp −mmp)p−1 ) ber (ap) ≥ ber (bp) holds for each p ≥ 1; the constant k+(m,m,p) = (p− 1)p−1 pp ( (mp −mp)p (m −m) (mmp −mmp)p−1 ) is called the ky fan-furuta constant in the literature (see, for instance, [13] and its references). proof. in fact, by the well-known furuta’s inequality( m m )p−1 ap ≥ k+(m,m,p)ap ≥ bp for each p ≥ 1 , or equivalently,( m m )p−1 〈apx,x〉≥ k+(m,m,p)〈apx,x〉≥ 〈bpx,x〉 for all x ∈ h. in particular, for x = k̂λ and for all λ ∈ ω, we obtain that( m m )p−1 ãp(λ) ≥ k+(m,m,p)ãp(λ) ≥ b̃p(λ) . then, by taking supremum in these inequalities, we obtain the desired inequality (2.2). the following is a corollary of furuta’s result in ([14, theorem 1.1]). proposition 4. let a ∈b(h) be a self-adjoint operator on a reproducing kernel hilbert space h = h(ω) satisfying mi ≥ a ≥ mi > 0. then ber(ap) ≤ (mmp −mmp) (q − 1) (m −m) ( (q − 1) (mp −mp) q (mmp −mmp) )q ber(a)q under anyone of the following conditions (i) and (ii) respectively: 6 f. bouzeffour, m. garayev, m. gürdal, c.m. yangöz (i) mp−1q ≤ m p−mp m−m ≤ m p−1q holds for real numbers p > 1 and q < 0; (ii) mp−1q ≤ m p−mp m−m ≤ m p−1q holds for real numbers p < 0 and q < 0. first, we cite the following lemma [14] to give a proof of this proposition. lemma 1. let h(t) be defined by h(t) = 1 tq ( k + k −k m −m (t−m) ) (2.3) on [m,m] (m > m > 0), where q is any real number such that q 6= 0, 1 and k and k are any real numbers. then h(t) has the following upper bound on [m,m]: (mk −mk) (q − 1)(m −m) ( (q − 1)(k −k) q(mk −mk) )q (2.4) where m,m,k,k and q in (2.4) satisfy anyone of the following conditions (i) and (ii) respectively: (i) k > k, k m > k m and k m q ≤ k−k m−m ≤ k m q holds for a real number q > 1; (ii) k < k, k m < k m and k m q ≤ k−k m−m ≤ k m q holds for a real number q < 0. proof. it is elementary that h′(t0) = 0 when t0 = q q − 1 mk −mk k −k and it turns out that to satisfies the required condition t0 ∈ [m,m] and also to gives the upper bound (2.4) of h(t) on the segment [m,m] under any one of the conditions (i) and (ii) respectively. proof of proposition 4. since f(t) is a real-valued continuous convex function on [m,m], we have f(t) ≤ f(m) + f(m) −f(m) m −m (t−m) for any t ∈ [m,m] . (2.5) then by passing to the operational calculus of positive operator a in (2.5) since m ≥ 〈 ak̂λ, k̂λ 〉 ≥ m, we obtain for every λ ∈ ω that 〈 f(a)k̂λ, k̂λ 〉 ≤ f(m) + f(m) −f(m) m −m (〈 ak̂λ, k̂λ 〉 −m ) , refinements of kantorovich type 7 or equivalently f̃(a)(λ) ≤ f(m) + f(m) −f(m) m −m ( ã(λ) −m ) . (2.6) multiplying ( ã(λ) )−q on both sides of (2.6), we get ( ã(λ) )−q f̃(a)(λ) ≤ h(t) , (2.7) where h(t) = ( ã(λ) )−q ( f(m) + f(m) −f(m) m −m )( ã(λ) −m ) . then we obtain that f̃(a)(λ) ≤ ( max m≤t≤m h(t) )( ã(λ) )q . (2.8) substituting k = f(m) and k = f(m) in [34, theorem 1.1], then (i) and (ii) in theorem 1.1 just correspond to (i) and (ii) in lemma 1. we have from (2.8) that ber(f(a)) ≤ ( max m≤t≤m h(t) ) (ber(a))q. (2.9) now put f(t) = tp for p /∈ [0, 1] in (2.9). since f(t) is a real-valued continuous convex function on [m,m], mp > mp and mp−1 > mp−1 hold for any p > 1, that is, f(m) > f(m) and f(m) m > f(m) m for any p > 1 and also mp < mp and mp−1 < mp−1 hold for any p < 0, that is f(m) < f(m) and f(m) m < f(m) m for any p < 0 respectively. whence the proof is complete by (2.9). our next result, associated with hölder-mccarthy and kantorovich inequalities, is the following. proposition 5. let a be positive operator on a hilbert space h = h(ω) satisfying mi ≥ a ≥ mi > 0. then the following inequality holds for every λ ∈ ω: (i) in case p > 1: ber(a)p ≤ ber(ap) ≤ k+(m,m) ber(a)p, where k+(m,m) = (p− 1)p−1 pp (mp −mp)p (m −m) (mmp −mmp)p−1 . 8 f. bouzeffour, m. garayev, m. gürdal, c.m. yangöz (ii) in case p < 0: ber(a)p ≤ ber(ap) ≤ k−(m,m) ber(a)p, where k−(m,m) = (mmp −mmp) (p− 1)(m −m) ( (p− 1)(mp −mp) p(mmp −mmp) ) . proof. as f(t) = tp is a convex function for p /∈ [0, 1], (i) and (ii) proposition 4 hold in case p /∈ [0, 1] and q 6= p, so that the inequalities of the right-hand sides of (i) and (ii) hold by proposition 4 and ones of the left-hand sides of (i) and (ii) follow by hölder-mccarthy inequality [14]. corollary 1. let a ∈ b(h) be a positive operator on a reproducing kernel hilbert space h = h(ω) such that mi ≥ a ≥ mi > 0. then: (i) supλ∈ω [( ã(λ) )p ã−1(λ) ] ≤ pp (p + 1) p+1 (m + m) p+1 mm ; (ii) ber(a2) ≤ pp (p + 1) p+1 (m + m) p+1 (mm) p (ber(a)) p+1 for any p ∈ [ m m , m m ] . proof. (i) in (ii) of proposition 4, we have only to put p = −1 and replacing q by −p for p > 0. (ii) in (i) in proposition 4, we have only to put p = 2 and replacing q by p + 1 for p > 0. this proves the corollary. 3. a berezin number inequality via the variance-covariance inequality following [34], we give some necessary concepts and notations. the notion of semi-inner product c∗-module is a generalization of that of semi-inner product space in which the semi-inner product takes its values in a c∗-algebra instead of the field of complex numbers. we can define a semi-norm on a semiinner product (x ,〈., .〉) over a c∗-algebra a by ‖x‖ = ‖〈x,x〉‖ 1 2 , where the latter norm denotes that of a. a pre-hilbert a-module (or an inner-product module) is a semi-inner product module in which ‖.‖ defined as above is a norm. if this norm is complete then x is called a hilbert c∗-module. each c∗algebra a can be regarded as a hilbert a-module via 〈a,b〉 = a∗b (a,b ∈a). when x is a hilbert c∗-module, we denote by b(x) the c∗-algebra of all adjointable operators on x . for every x ∈ x the absolute value of x is defined by |x| = 〈x,x〉1/2 ∈ a. some standard references for c∗-modules are [2, 29, 31]. refinements of kantorovich type 9 in this section, we present some kantorovich inequalities for berezin symbols of operators involving unital positive linear mappings and the operator geometric mean in the framework of semi-inner product c∗-modules and get some new berezin number inequalities. let x,y,z,x1, . . . ,xn be arbitrary elements of a semi-inner product a-module (x ,〈., .〉). in [34], the authors studied the covariance covz(x,y) := ‖z‖2 〈x,y〉−〈x,z〉〈z,y〉 and the variance varz(x) := covz(x,x) , and proved that [covz(xi,xj)] ∈ mn(a) is positive, or equivalently ‖z‖2 [〈xi,xj〉] ≥ [〈xi,z〉〈z,xj〉] ; (3.1) this is called generalized covariance-variance inequality. in particular, by the cauchy-schwarz inequality for the semi-inner product covz(., .), the covariance-variance inequality holds covz(x,y)covz(x,y) ∗ ≤‖varz(y)‖varz(x) . let a be a c∗-algebra and b be a c∗-subalgebra of a. a linear mapping φ : a → b is called a (right) multiplier if φ(ab) = φ(a)b (a ∈ a, b ∈ b). if φ is a positive multiplier, any semi-inner product a-module x becomes a semi-inner product b-module with respect to [x,y]φ := φ(〈x,y〉) (x,y ∈x) . by (3.1), it holds ‖φ(〈z,z〉)‖ [φ(〈xi,xj〉)] ≥ [φ(〈xi,z〉)φ(〈z,xj〉)] for all z,x1, . . . ,xn ∈x . in the sequel, we will assume that the a-module x is a reproducing kernel hilbert space over some suitable set ω with the normalized reproducing kernel k̂λ(z) = kλ(z) ‖kλ‖χ , λ,z ∈ ω. so, if we fix a normalized reproducing kernel k̂λ ∈x and take operators a and b in b(x), then we can define the λ-covariance of a,b and λ-variance of a by λ-cov(a,b) = φ (〈 ak̂λ,bk̂λ 〉) − φ (〈 ak̂λ, k̂λ 〉) φ (〈 k̂λ,bk̂λ 〉) 10 f. bouzeffour, m. garayev, m. gürdal, c.m. yangöz and λ-var(a) = λ− cov(a,a) , respectively, similar to enomoto [12]. our idea is mainly related to the work of umegaki [43] and moslehian [34] where, in particular, the proofs of several known inequalities for hilbert space operators are unified. lemma 2. let a be a unital c∗-algebra and b be a unital c∗-subalgebra of a such that both algebras have the same unit e. let x = x(ω) be a a-module consisting from the reproducing kernel hilbert space of complexvalued functions on ω with the normalized reproducing kernel k̂λ. let a1,a2 ∈ b(x) be two operators satisfying mii ≤ ai ≤ imi for some scalars mi,mi (i = 1, 2). then sup λ∈ω ‖λ-cov(a1,a2)‖≤ 1 4 (m1 −m1)(m2 −m2) . proof. it can be easily seen that (m −c)(c −m) ≤ ( m −m 2 )2 for any self-adjoint operator c of a unital c∗-algebra with spectrum in [m,m], due to ( c − m+m 2 )2 ≥ 0. hence λ-var(a1) = λ-cov(a1,a1) = φ (〈 a21k̂λ, k̂λ 〉) − φ (〈 a1k̂λ, k̂λ 〉)2 = (( m1e− φ (〈 a1k̂λ, k̂λ 〉)) φ (〈 a1k̂λ, k̂λ 〉) −m1e ) − φ (〈 (m1 −a1)(a1 −m1)k̂λ, k̂λ 〉) ≤ ( m1 − φ (〈 a1k̂λ, k̂λ 〉))( φ (〈 a1k̂λ, k̂λ 〉) −m1 ) ≤ 1 4 (m1 −m1)2 (by (m1 −a1)(a1 −m1) ≥ 0, the selfadjointness of φ (〈 a1k̂λ, k̂λ 〉) and the positivity of the berezin symbol ã1 of a positive operator a1). so, we have that λ-var(a1) ≤ 1 4 (m1 −m1)2, refinements of kantorovich type 11 and similarly λ-var(a2) ≤ 1 4 (m2 −m2)2 for all λ ∈ ω. from these inequalities, by using the covariance-variance inequality, we have for all λ ∈ ω that ‖λ-cov(a1,a2)‖2 ≤‖λ-var(a1)‖‖λ-var(a2)‖ ≤ 1 16 (m1 −m1)2(m2 −m2)2, which gives the required inequality. our next result is the “λ-parametrization” variant of moslehian’s result ([34, theorem 2.2]) about generalization of kantorovich inequality (see also [41]). theorem 1. let a be a unital c∗-algebra and b be a unital c∗-subalgebra of a such that both have the same unit e. let x = x(ω) be a amodule consisting of the reproducing kernel hilbert space of complex-valued functions on ω with reproducing kernel kλ. let a ∈ b(x) be an operator satisfying mi ≤ a ≤ mi for some scalars 0 < m < m. then sup λ∈ω ∣∣∣φ (ã(λ)) φ (ã−1(λ))∣∣∣ ≤ (m + m)2 4mm . (3.2) proof. put a1 = a, a2 = a −1, m1 = m, m1 = m, m2 = m −1, m2 = m −1 in lemma 2 to get ∣∣∣φ(e) − φ (〈ak̂λ, k̂λ〉) φ (〈a−1k̂λ, k̂λ〉)∣∣∣ ≤ (m −m)2 4mm , that is ∣∣∣φ(e) − φ (ã(λ)) φ (ã−1(λ))∣∣∣ ≤ (m −m)2 4mm , whence ∣∣∣φ (ã(λ)) φ (ã−1(λ))∣∣∣ ≤ 1 + (m −m)2 4mm = (m + m)2 4mm for all λ ∈ ω, which yields (3.2). 12 f. bouzeffour, m. garayev, m. gürdal, c.m. yangöz corollary 2. let φ : b(x) → b(x) be a unital positive linear map. if a ∈ b(x) is an operator satisfying 0 < mi ≤ a ≤ mi for some scalars m,m then sup λ∈ω [ ã(λ)ã−1(λ) ] ≤ (m + m)2 4mm . (3.3) proof. the proof is immediate from theorem 1. indeed, take a = b = c. the only positive linear mapping φ : c → c is the identity mapping. for any λ ∈ ω and k̂λ = kλ‖kλ‖, we obtain (3.3) from (3.2). 4. a kantorovich inequality via the operator geometric mean and a berezin number inequality in this section, we use λ-parametrization for proving a generalization of the kantorovich inequality in the context of hilbert c∗-modules which can be viewed as extension of theorem 1 and an inequality due to nakamoto and nakamura [35]. recall that for positive invertible elements a,b ∈a, we can use the following characterization of operator mean due to ando [1] as follows a]b = max { x ∈a : x = x∗, [ a x x b ] ≥ 0 } , where a]b = a 1 2 ( a− 1 2 ba− 1 2 )1 2 a 1 2 . this is easily obtained from a = (a]b)b−1(a]b) and the fact that a ≥ xb−1x∗ if and only if [ a x x∗ b ] ≥ 0, where x ∈a. theorem 2. let a, b be unital c∗-algebras, x = x(ω) be a reproducing kernel hilbert space with reproducing kernel kλ which is an a-module and a ∈ b(x) such that mi ≥ a ≥ mi > 0 for some scalars m,m. then, for every f ∈ x for which 〈f,f〉 is invertible and every positive linear mapping φ : a→b it holds: sup λ∈ω φ (〈 k̂λ, k̂λ 〉) ≤ sup λ∈ω φ ( ã(λ) ) ] φ ( ã−1(λ) ) ≤ m + m 2 √ mm sup λ∈ω φ (〈 k̂λ, k̂λ 〉) . (4.1) proof. the proof is based in the similar proof of [34, theorem 3.1]. indeed, first note that for any λ ∈ ω, 〈 k̂λ, k̂λ 〉 is invertible. therefore, for a := refinements of kantorovich type 13 mm 〈 a−1k̂λ, k̂λ 〉 and b := 〈 ak̂λ, k̂λ 〉 , a ≥ m 〈 k̂λ, k̂λ 〉 and b ≥ m 〈 k̂λ, k̂λ 〉 , so a and b are positive and invertible. since φ is positive and unital, φ(a) and φ(b) are also positive and invertible. observe now that m −a and 1 m −a−1 are positive commuting elements of the c∗-algebra b(x), which implies that (m −a)( 1 m −a−1) ≥ 0. hence mma−1 + a ≤ (m + m) . then, for every λ ∈ ω, mm 〈 a−1k̂λ, k̂λ 〉 + 〈 ak̂λ, k̂λ 〉 ≤ (m + m) 〈 k̂λ, k̂λ 〉 from which we have mmφ (〈 a−1k̂λ, k̂λ 〉) + φ (〈 ak̂λ, k̂λ 〉) ≤ (m + m)φ (〈 k̂λ, k̂λ 〉) . 5. refinement of schwartz inequality and berezin number inequality in his paper [11], dragomir obtained some new improvements of classical schwarz inequality in complex hilbert space h as follows. lemma 3. let x,y,e ∈ h with ‖e‖ = 1. then we have the following refinement of schwarz inequality ‖x‖2 ‖y‖2 −|〈x,y〉|2 ≥  det   |〈x,e〉| ( ‖x‖2 −|〈x,e〉|2 )1 2 |〈y,e〉| ( ‖y‖2 −|〈y,e〉|2 )1 2     2 . (5.1) recall that for a ∈b(h), its crawford number c(a) is defined by c(a) := inf {|〈ax,x〉| : x ∈ h and ‖x‖ = 1} . in [39], the authors introduced the numbers b̃(a) and c̃(a) defined by c̃(a) := inf λ∈ω ∣∣∣ã(λ)∣∣∣ and b̃(a) := inf λ∈ω ∣∣∣ã(λ)∣∣∣∥∥∥ak̂λ∥∥∥. 14 f. bouzeffour, m. garayev, m. gürdal, c.m. yangöz it is easy to see that b̃(a) ≤ 1 and c(a) ≤ c̃(a) ≤ ber(a). these, and other related numerical characteristics, are useful in studying bounded linear operators (see [39] and its references). note that the hardy space h2 = h2(d) over the disc d = {z ∈ c : |z| < 1} is the rkhs with the normalized reproducing kernel k̂λ(z) := (1−|λ|2) 1/2 1−λz , λ,z ∈ d. for definitions and more facts about this space, we recommend the book of hoffman [23]. in this section, by applying lemma 3, we prove berezin number inequality. namely, we prove the following theorem. theorem 3. if f,g : j → (0, +∞) are continuous functions on some segment j ⊂ [0, +∞), then (i) f̃2(a)(λ) + g̃2(b)(µ) + 2f̃4(a)(λ)g̃4(b)(µ)+ + 2f̃3(a)(λ)g̃3(b)(µ) ( i −f2(a) )1 2̃ (λ) ( i −g2(b) )1 2̃ (µ) ≥ g̃2(b)(µ)f̃4(a)(λ) + g̃4(b)(µ)f̃2(a)(λ) for any self-adjoint operators a,b ∈ b(h2) with spectra in j such that i −f2(a) and i −g2(b) are positive and for all λ,µ ∈ d; (ii) ber(f2(a)) + ber(f4(a))2 + ber(f3(a))2 ( ber (( i −f2(a) )1 2 ))2 ≥ sup λ∈d [ f̃2(a)(λ)f̃4(a)(λ) ] . (iii) if ∥∥i −f2(a)∥∥ < 1, then ber(f4(a))2 + ber(f3(a))2 ≥ ber(f2(a)) ( c̃(f4(a) − 1 ) . proof. (i) we set d := { (λ,µ) ∈ d×d : re(λµ) = 1 2 } . putting now h = h2(d) (the hardy space), x = k̂λ, y = k̂µ and e = i in the formula (5.1) with (λ,µ) ∈d, we have refinements of kantorovich type 15 1 − ∣∣∣〈k̂λ, k̂µ〉∣∣∣2 ≥  det   〈 k̂λ,i 〉 ( 1 − ∣∣∣〈k̂λ,i〉∣∣∣2)12〈 k̂µ,i 〉 ( 1 − ∣∣∣〈k̂µ,i〉∣∣∣2)12     2 =  det   ( 1 −|λ|2 ) ( 1 − ( 1 −|λ|2 ))1 2( 1 −|µ|2 ) ( 1 − ( 1 −|µ|2 ))1 2     2 =  det   ( 1 −|λ|2 )1 2 |λ|( 1 −|µ|2 )1 2 |µ|     2 = ( |µ| ( 1 −|λ|2 )1 2 −|λ| ( 1 −|µ|2 )1 2 )2 = |λ|2 + |µ|2 − 2 |λ|2 |µ|2 − 2 |λ| |µ| ( 1 −|λ|2 )1 2 ( 1 −|µ|2 )1 2 , hence 1 − ∣∣∣∣∣∣∣ 〈( 1 −|λ|2 )1 2 1 −λz , ( 1 −|µ|2 )1 2 1 −µz 〉∣∣∣∣∣∣∣ 2 ≥ |λ|2 + |µ|2 − 2 |λ|2 |µ|2 − 2 |λ| |µ| ( 1 −|λ|2 )1 2 ( 1 −|µ|2 )1 2 that is 1 − ( 1 −|λ|2 )( 1 −|µ|2 ) ∣∣1 −λµ∣∣2 ≥ |λ|2 + |µ|2 − 2 |λ|2 |µ|2 − 2 |λ| |µ| ( 1 −|λ|2 )1 2 ( 1 −|µ|2 )1 2 , and thus 1 − 2 re(λµ) + |λ|2 + |µ|2 1 − 2 re(λµ) + |λ|2 |µ|2 ≥ |λ|2 + |µ|2 − 2 |λ|2 |µ|2 − 2 |λ| |µ| ( 1 −|λ|2 )1 2 ( 1 −|µ|2 )1 2 . 16 f. bouzeffour, m. garayev, m. gürdal, c.m. yangöz since (λ,µ) ∈d, we have that 1 − 2 re(λµ) = 0, and hence |λ|2 + |µ|2 |λ|2 |µ|2 ≥ |λ|2 + |µ|2 − 2 |λ|2 |µ|2 − 2 |λ| |µ| ( 1 −|λ|2 )1 2 ( 1 −|µ|2 )1 2 whence |λ|2 + |µ|2 + 2 |λ|4 |µ|4 + 2 |λ|3 |µ|3 ( 1 −|λ|2 )( 1 −|µ|2 )1 2 ≥ |λ|4 |µ|2 + |λ|2 |µ|4 for any pair (λ,µ) ∈ d. we let |λ| = a > 0, |µ| = b > 0. since (λ,µ) ∈ d is arbitrary, the numbers a and b are also arbitrary. therefore there exist x,y ∈ j such that f(x) = a and g(y) = b. then, we have from the last inequality that f2(x) + g2(y) + 2f4(x)g4(y) + 2f3(x)g3(y) (( 1 −f2(x) )( 1 −g2(y) ))1 2 ≥ f4(x)g2(y) + f2(x)g4(y) for all x,y ∈ j. from this inequality, by using kian’s method [27], we have f2(a) + g2(y) + 2f4(a)g4(y) + 2f3(a)g3(y) (( 1 −f2(a) )( 1 −g2(y) ))1 2 ≥ f4(a)g2(y) + f2(a)g4(y) (5.2) for all y ∈ j and for any self adjoint operator a ∈b(h2) with σ(a) ⊂ j such that i −f2(a) ≥ 0. it follows from (5.2) that〈 f2(a)k̂λ, k̂λ 〉 + g2(y) + 2g4(y) 〈 f4(a)k̂λ, k̂λ 〉 + 2g3(y) 〈 f3(a)k̂λ, k̂λ 〉(( 1 −g2(y) )1 2 〈( i −f2(a) )1 2 k̂λ, k̂λ 〉) ≥ g2(y) 〈 f4(a)k̂λ, k̂λ 〉 + g4(y) 〈 f2(a)k̂λ, k̂λ 〉 , or equivalently, f̃2(a)(λ) + g2(y) + 2g4(y)f̃4(a)(λ) + 2g3(y)f̃3(a)(λ) ( 1 −g2(y) )1 2 ( i −f2(a) )1 2̃ (λ) ≥ g2(y)f̃4(a)(λ) + g4(y)f̃2(a)(λ), for each λ ∈ d. refinements of kantorovich type 17 now, by applying the functional calculus for a self-adjoint operator b ∈b(h2) with spectrum in j with respect to the variable y, we get f̃2(a)(λ) + 〈 g2(b)k̂µ, k̂µ 〉 + 2f̃4(a)(λ) 〈 g4(b)k̂µ, k̂µ 〉 + 2f̃3(a)(λ) 〈 g3(b)k̂µ, k̂µ 〉 ( i −f2(a) )1 2̃ (λ) ( i −g2(b) )1 2̃ (µ) ≥ g̃2(b)(µ)f̃4(a)(λ) + g̃4(b)(µ)f̃2(a)(λ) that is f̃2(a)(λ) + g̃2(b)(µ) + 2f̃4(a)(λ)g̃4(b)(µ) + 2f̃3(a)(λ)g̃3(b)(µ) ( i −f2(a) )1 2̃ (λ) ( i −g2(b) )1 2̃ (µ) ≥ g̃2 (b) (µ) f̃4 (a) (λ) + g̃4 (b) (µ) f̃2 (a) (λ) (5.3) for all λ,µ ∈ d and every pair of self-adjoint operators a,b ∈ b(h2) with spectra in j such that i −f2(a) and i −g2(b) are positive. this proves (i). (ii) we particularly obtain from inequality (5.3) for b = a, g = f and µ = λ that 2f̃2(a)(λ) + 2 [ f̃4(a)(λ) ]2 + 2 [ f̃3(a)(λ) ]2 [( i −f2(a) )1 2̃ (λ) ]2 ≥ 2f̃2(a)(λ)f̃4(a)(λ) and thereby, we obtain that ber(f2(a)) + ( ber(f4(a)) )2 + ( ber(f3(a)) )2 ( ber(i −f2(a)) 1 2 )2 ≥ sup λ∈d [ f̃2(a)(λ)f̃4(a)(λ) ] , (5.4) which proves (ii). (iii) is an immediate consequence of inequality (5.4). indeed, the operator f2(a) is invertible because ∥∥i −f2(a)∥∥ < 1. then ∥∥∥f2(a)k̂λ∥∥∥ ≥ 1∥∥∥(f2(a))−1∥∥∥ ∥∥∥k̂λ∥∥∥ = ∥∥∥(f2(a))−1∥∥∥−1 18 f. bouzeffour, m. garayev, m. gürdal, c.m. yangöz and ber ( i −f2(a) )1/2 ≤ ∥∥i −f2(a)∥∥ < 1, hence 〈 f2(a)k̂λ,f 2(a)k̂λ 〉1/2 = 〈( f2(a) )∗ f2(a)k̂λ, k̂λ 〉1/2 = 〈 f4(a)k̂λ, k̂λ 〉 (since f2(a) is self-adjoint) = f̃4(a)(λ) ≥ ∥∥∥(f2(a))−1∥∥∥−1 , which means that c̃(f4(a)) ≥ ∥∥∥(f2(a))−1∥∥∥−1 > 0. therefore, we obtain from (5.4) the desired inequality, which proves (iii). note that the berezin symbol is not multiplicative, i.e., ãb(λ) 6= ã(λ)b̃(λ) in general, see kiliç [28]. acknowledgements the authors thank to the referee for his/her useful remarks. also, the first and second authors would like to extend his sincere appreciation to the deanship of scientific research at king saud university for its funding of this research through the research group project no. rgpvpp-323. this paper was supported by süleyman demirel university project no fyl-2018-6962. references [1] t. ando, c.-k. li, r. mathias, geometric means, linear algebra appl. 385 (2004), 305 – 334. 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[44] f. zhang, equivalence of the wielandt inequality and the kantorovich inequality, linear multilinear algebra 48 (3) (2001), 275 – 279. introduction, notation and preliminaries some corollaries of kantorovich inequalities a berezin number inequality via the variance-covariance inequality a kantorovich inequality via the operator geometric mean and a berezin number inequality refinement of schwartz inequality and berezin number inequality � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 38, num. 1 (2023), 67 – 84 doi:10.17398/2605-5686.38.1.67 available online may 3, 2023 on lie ideals satisfying certain differential identities in prime rings b. dhara 1, s. ghosh 2,∗, g.s. sandhu 3 1 department of mathematics, belda college, belda paschim medinipur, 721424, w.b., india 2 department of mathematics, jadavpur university, kolkata-700032, w.b., india 3 department of mathematics, patel memorial national college 140401 rajpura, india basu dhara@yahoo.com , mathsourav99@gmail.com , gurninder rs@pbi.ac.in received january 20, 2023 presented by c. mart́ınez accepted march 28, 2023 abstract: let r be a prime ring of characteristic not 2, l a nonzero square closed lie ideal of r and let f : r → r, g : r → r be generalized derivations associated with derivations d : r → r, g : r → r respectively. in this paper, we study several conditions that imply that the lie ideal is central. moreover, it is shown that the assumption of primeness of r can not be removed. key words: prime ring, derivation, generalized derivation, lie ideal. msc (2020): 16n60, 16w25, 16r50. 1. introduction throughout this paper r denotes an associative prime ring with center z(r). a ring r is called a prime ring if for any a, b ∈ r, arb = (0) implies either a = 0 or b = 0. the symbol [x, y]=xy − yx stands for the commutator operator for x, y ∈ r and the symbol x ◦ y = xy + yx stands for the anticommutator operator for x, y ∈ r. an additive subgroup l of r is said to be a lie ideal of r if [x, r] ∈ l for any r ∈ r and x ∈ l. a lie ideal l is said to be square closed if u2 ∈ l for all u ∈ l. a map d : r → r is called a derivation, if d(x + y) = d(x) + d(y) and d(xy) = d(x)y + xd(y) holds for all x, y ∈ r. an additive map f : r → r is said to be a generalized derivation of r, if there exists a derivation d : r → r such that f (xy) = f (x)y + xd(y) holds for all x, y ∈ r. obviously, every ∗the second author expresses his thanks to the university grants commission, new delhi for its jrf awarded to him. the grant no. is ugc-ref.no.1156/(csir-ugc net dec. 2018) dated 24.07.2019. issn: 0213-8743 (print), 2605-5686 (online) c© the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.38.1.67 mailto:basu_dhara@yahoo.com mailto:mathsourav99@gmail.com mailto:gurninder_rs@pbi.ac.in https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 68 b. dhara, s. ghosh, g.s. sandhu derivation is a generalized derivation of r, but the converse is not necessarily true. several authors have studied many identities in prime and semiprime rings, involving derivations and generalized derivations, that imply commutativity of the ring. we refer the reader to [1, 2, 7, 8, 9, 11, 12, 15, 16, 18], where further references can be found. the identities (i) f (xy) −xy ∈ z(r) for all x, y ∈ i, (ii) f (xy) + xy ∈ z(r) for all x, y ∈ i, (iii) f (xy) −yx ∈ z(r) for all x, y ∈ i, (iv) f (xy) + yx ∈ z(r) for all x, y ∈ i, (v) f (x)f (y) −xy ∈ z(r) for all x, y ∈ i, (vi) f (x)f (y) + xy ∈ z(r) for all x, y ∈ i, when r is a prime ring, f is a generalized derivation of r associated with a non-zero derivation d and i is a non-zero two-sided ideal of r, were studied by ashraf et al. in [2], proving that any of them implies the commutativity of r. in a similar way, the identities (i) f (x)f (y)−yx ∈ z(r) and (ii) f (x)f (y)+ yx ∈ z(r) for all x, y in some suitable subset of r were studied in [9] by dhara et al. recently, tiwari et al. (see [18]) studied identities involving three summands and again obtained the commutativity of the prime ring r. identities involving the commutator and the anti-commutator have also been considered. in [4], bell and daif proved that if u is a nonzero right ideal of a semiprime ring r and d is a nonzero derivation of r such that [d(x), d(y)] = [x, y] for all x, y ∈ u, then u ⊆ z(r). ashraf et al. (see [3]) got the commutativity of a prime ring r satisfying any one of the following conditions: (i) d(x) ◦f (y) = 0 for all x, y ∈ i, (ii) [d(x), f (y)] = 0 for all x, y ∈ i, (iii) d(x) ◦f (y) = x◦y for all x, y ∈ i, (iv) d(x) ◦f (y) + x◦y = 0 for all x, y ∈ i, (v) [d(x), f (y)] = [x, y] for all x, y ∈ i, (vi) [d(x), f (y)] + [x, y] = 0 for all x, y ∈ i, (vii) d(x)f (y) ±xy ∈ z(r) for all x, y ∈ i, on lie ideals satisfying certain differential identities 69 where i is a nonzero ideal of r and f is a generalized derivation of r associated with a nonzero derivation d. in [13], shuliang studied the above identities for a square closed lie ideal l of a prime ring r and obtained that either d = 0 or l ⊆ z(r) and in [10], dhara et al. studied the above identities in a semiprime ring. in this paper we consider the following identities: (i) f (u) ◦v ±d(u) ◦f (v) ±u◦v = 0 for all u, v ∈ l, (ii) [f (u), v] ± [d(u), f (v)] ± [u, v] = 0 for all u, v ∈ l, (iii) f ([u, v]) ± [d(u), f (v)] ± [u, v] = 0 for all u, v ∈ l, (iv) f (u◦v) ± [d(u), f (v)] ±u◦v = 0 for all u, v ∈ l, (v) f (u)g(v) ±d(u)f (v) ±uv ∈ z(r) for all u, v ∈ l, (vi) g(uv) ±d(u)f (v) ±f (vu) = 0 for all u, v ∈ l, (vii) f (uv) ±f (v)f (u) ±uv ∈ z(r) for all u, v ∈ l, where l is a square closed lie ideal in a prime ring r and f is a generalized derivation of r associated with a derivation d. 2. preliminaries in this paper r will denote always a prime associative ring of characteristic not equal to 2. this implies that for any element x in r, 2x = 0 implies x = 0. let l be a square closed lie ideal of r. thus u2 ∈ l for all u ∈ l. now for u, v ∈ l, uv + vu = (u + v)2 − u2 − v2 ∈ l and by definition of lie ideal uv −vu ∈ l. combining these two we get 2uv ∈ l for all u, v ∈ l. the following identities shall be used very frequently throughout: (1) [xy, z] = x[y, z] + [x, z]y for all x, y, z ∈ r, (2) [x, yz] = y[x, z] + [x, y]z for all x, y, z ∈ r, (3) (x◦yz) = (x◦y)z −y[x, z] = y(x◦z) + [x, y]z for all x, y, z ∈ r, (4) (xy ◦z) = x(y ◦z) − [x, z]y = (x◦z)y + x[y, z] for all x, y, z ∈ r. moreover in order to prove our results, we need the following facts: lemma 2.1. ([5, lemma 2]) if l 6⊆ z(r) is a lie deal of r, then cr(l) = z(r). lemma 2.2. ([5, lemma 4]) if l 6⊆ z(r) is a lie ideal of r and alb = 0, then either a = 0 or b = 0. 70 b. dhara, s. ghosh, g.s. sandhu lemma 2.3. ([5, lemma 5]) if d is a nonzero derivation of r and l a lie ideal of r such that d(l) = (0), then l ⊆ z(r). lemma 2.4. ([9, lemma 2.5]) let l be a nonzero lie ideal of r and v = {u ∈ l | d(u) ∈ l}. then v is also a nonzero lie ideal of r. moreover, if l is noncentral, then v is also noncentral. lemma 2.5. ([14, theorem 5]) let d be a nonzero derivation of r and l a nonzero lie ideal of r such that [u, d(u)] ∈ z(r) for all u ∈ l. then l ⊆ z(r). lemma 2.6. ([17, lemma 2.6]) let l 6⊆ z(r) be a lie ideal of r and a, b ∈ r such that one of a, b is in l. if aub + bua = 0 for all u ∈ l, then aub = bua = 0 for all u ∈ l. consequently, a = 0 or b = 0. lemma 2.7. let d be a nonzero derivation of r and l a nonzero square closed lie ideal of r. suppose that v = {u ∈ l | d(u) ∈ l}. if [d(u), d(v)] = 0 for all u ∈ l and v ∈ v , then l ⊆ z(r). proof. if l ⊆ z(r), we are done. thus on contrary, we assume that l 6⊆ z(r). by lemma 2.4, v is also noncentral lie ideal of r such that v ⊆ l. we have that [d(u), d(v)] = 0 (1) for all u ∈ l and v ∈ v . replacing u by 2uv and then using char (r) 6= 2, we have [d(u)v + ud(v), d(v)] = 0 (2) for all u ∈ l and v ∈ v . by using (1), we have d(u)[v, d(v)] + [u, d(v)]d(v) = 0 (3) for all u ∈ l and v ∈ v . replacing u by 2wu in (3) and then using it, we have d(w)u[v, d(v)] + [w, d(v)]ud(v) = 0 (4) for all u, w ∈ l and v ∈ v . in particular, d(v)u[v, d(v)] + [v, d(v)]ud(v) = 0 (5) on lie ideals satisfying certain differential identities 71 for all u ∈ l and v ∈ v . invoking lemma 2.6, we find d(v)u[v, d(v)] = 0 for all v ∈ v and u ∈ u. again by lemma 2.2, d(v) = 0 or [v, d(v)] = 0. in any case we have [v, d(v)] = 0 for all v ∈ v . by lemma 2.5, v ⊆ z(r), a contradiction. the following lemmas are the particular cases of [6, theorem 1]. lemma 2.8. let d be a nonzero derivation of r and l a nonzero lie ideal of r such that u[[d(u), u], u] = 0 for all u ∈ l. then l ⊆ z(r). lemma 2.9. let f be a nonzero generalized derivation of r with associated nonzero derivation d and l a nonzero square closed lie ideal of r such that [f (u), u] = 0 for all u ∈ l. then l ⊆ z(r). 3. main results theorem 3.1. let l be a nonzero square closed lie ideal of r and f be a generalized derivation of r associated to nonzero derivation d of r. if f (u) ◦v ±d(u) ◦f (v) ±u◦v = 0 for all u, v ∈ l, then l ⊆ z(r). proof. we assume on the contrary that l 6⊆ z(r). by hypothesis, we have f (u) ◦v ±d(u) ◦f (v) ±u◦v = 0 (6) for all u, v ∈ l. replacing v by 2vw in (6), we obtain 2{f (u) ◦vw ±d(u) ◦ (f (v)w + vd(w)) ±u◦ (vw)} = 0 (7) for all u, v, w ∈ l. since characteristic of r is not 2, we have (f (u) ◦v)w −v[f (u), w] ± (d(u) ◦f (v))w ∓f (v)[d(u), w] ±d(u) ◦ (vd(w)) ± (u◦v)w ∓v[u, w] = 0 (8) for all u, v, w ∈ l. right multiplying (6) by w and then subtracting from (8), we get −v[f (u), w] ∓f (v)[d(u), w] ±d(u) ◦ (vd(w)) ∓v[u, w] = 0 (9) for all u, v, w ∈ l. 72 b. dhara, s. ghosh, g.s. sandhu substituting v by 2pv in (9) and then using characteristic of r is not 2, we obtain −pv[f (u), w] ∓f (pv)[d(u), w] ±p(d(u) ◦ (vd(w))) ±[d(u), p]vd(w) ∓pv[u, w] = 0 (10) for all u, v, w, p ∈ l. left multiplying (9) by p and then subtracting from (10), we get ∓(f (pv) −pf (v))[d(u), w] ± [d(u), p]vd(w) = 0 (11) for all u, v, w, p ∈ l. replacing w by 2wt in (11) and then using characteristic of r is not 2 and (11), we obtain ∓(f (pv) −pf (v))w[d(u), t] ± [d(u), p]vwd(t) = 0 (12) for all u, v, w, p, t ∈ l. since [q, d(s)] ∈ l, thus 2[q, d(s)]w ∈ l for all q, s, w ∈ l. replacing w by 2[q, d(s)]w in (12) and then using characteristic of r is not 2, we get ∓(f (pv) −pf (v))[q, d(s)]w[d(u), t] ± [d(u), p]v[q, d(s)]wd(t) = 0 (13) for all u, v, w, p, q, s, t ∈ l. now re-writing the relation (11), we can write ∓(f (pv) −pf (v))[q, d(s)] ± [p, d(s)]vd(q) = 0 (14) for all v, p, q, s ∈ l. by using (14) in (13), we have ∓[p, d(s)]vd(q)w[d(u), t] ± [d(u), p]v[q, d(s)]wd(t) = 0 (15) for all u, v, w, p, q, s, t ∈ l. we re-write it as ∓[d(s), p]vd(q)w[d(u), t] ± [d(u), p]v[d(s), q]wd(t) = 0 (16) for all u, v, w, p, q, s, t ∈ l. let v = {u ∈ l | d(u) ∈ l}. by lemma 2.4, v is also noncentral lie ideal of r such that v ⊆ l. thus if s ∈ v , then d(s) ∈ l. thus replacing p by 2d(s)p in (16) and then using characteristic of r is not 2 and the relation (16), we get ±[d(u), d(s)]pv[d(s), q]wd(t) = 0 (17) on lie ideals satisfying certain differential identities 73 for all u, v, w, p, q, t ∈ l and s ∈ v . by lemma 2.2, for each s ∈ v , either [d(u), d(s)] = 0 for all u ∈ l or [d(s), q]wd(t) = 0 for all w ∈ l. now we consider two subgroups t1 = { s ∈ v : [d(u), d(s)] = 0 for all u ∈ l } , t2 = { s ∈ v : [d(s), q]wd(t) = 0 for all q, w, t ∈ l } . it is very clear that t1 and t2 are two additive subgroups of v such that t1 ∪t2 = v . since an additive subgroups can not be union of its two proper subgroups, we have either t1 = v or t2 = v , that is, either [d(u), d(s)] = 0 for all u ∈ l and s ∈ v or [d(s), q]wd(t) = 0 for all q, w, t ∈ l and s ∈ v . if [d(u), d(s)] = 0 for all u, s ∈ l, then by lemma 2.7, l ⊆ z(r), a contradiction. on the other hand, if [d(s), q]wd(t) = 0 for all q, w, t ∈ l and s ∈ v , then [d(s), q]w[d(s), q] = 0 for all w ∈ l and q, s ∈ v . by lemma 2.2, [d(s), q] = 0 for all q, s ∈ v . by lemma 2.5, v ⊆ z(r), a contradiction. thus the proof of the theorem is completed. theorem 3.2. let l be a nonzero square closed lie ideal of r and f be a generalized derivation of r associated to nonzero derivation d of r. if [f (u), v] ± [d(u), f (v)] ± [u, v] = 0 for all u, v ∈ l, then l ⊆ z(r). proof. we assume on the contrary that l 6⊆ z(r). by hypothesis, we have [f (u), v] ± [d(u), f (v)] ± [u, v] = 0 (18) for all u, v ∈ l. replacing v by 2vw in (18), we obtain 2{[f (u), vw] ± [d(u), f (v)w + vd(w)] ± [u, vw]} = 0 (19) for all u, v, w ∈ l. since characteristic of r is not 2, we have [f (u), v]w + v[f (u), w] ± ([d(u), f (v)]w + f (v)[d(u), w]) ±[d(u), vd(w)] ± ([u, v]w + v[u, w]) = 0 (20) for all u, v, w ∈ l. right multiplying (18) by w and then subtracting from (20), we get v[f (u), w] ±f (v)[d(u), w]) ± [d(u), vd(w)] ±v[u, w] = 0 (21) 74 b. dhara, s. ghosh, g.s. sandhu for all u, v, w ∈ l. replacing v by 2pv in (21) and then usingcharacteristic of r is not 2, we obtain pv[f (u), w] ±f (pv)[d(u), w]) ±p[d(u), vd(w)] ±[d(u), p]vd(w) ±pv[u, w] = 0 (22) for all u, v, w, p ∈ l. left multiplying (21) by p and then subtracting from (22), we get ±(f (pv) −pf (v))[d(u), w]) ± [d(u), p]vd(w) = 0 (23) for all u, v, w, p ∈ l. using the same arguments used in the proof of theorem 3.1 we get a contradiction, which proves this theorem. theorem 3.3. let l be a nonzero square closed lie ideal of r and f be a generalized derivation of r associated to nonzero derivation d of r. if f ([u, v]) ± [d(u), f (v)] ± [u, v] = 0 for all u, v ∈ l, then l ⊆ z(r). proof. we assume on the contrary that l 6⊆ z(r). by hypothesis, we have f ([u, v]) ± [d(u), f (v)] ± [u, v] = 0 (24) for all u, v ∈ l. replacing u by 2uv in (24), we obtain 2(f ([u, v])v + [u, v]d(v) ± [d(u)v, f (v)] ± [ud(v), f (v)] ± [u, v]v) = 0 (25) for all u, v ∈ l. since characteristic of r is not 2, we have f ([u, v])v + [u, v]d(v) ± [d(u)v, f (v)] ± [ud(v), f (v)] ± [u, v]v = 0 (26) for all u, v ∈ l. right multiplying (24) by v and then subtracting from (26), we get [u, v]d(v) ±d(u)[v, f (v)] ± [ud(v), f (v)] = 0 (27) for all u, v ∈ l. replacing u by 2vu in (27) and then using characteristic of r is not 2, we obtain v[u, v]d(v) ± (d(v)u + vd(u))[v, f (v)] ± [vud(v), f (v)] = 0 on lie ideals satisfying certain differential identities 75 that is, v[u, v]d(v) ±d(v)u[v, f (v)] ±vd(u)[v, f (v)] ±v[ud(v), f (v)] ± [v, f (v)]ud(v) = 0 (28) for all u, v ∈ l. left multiplying (27) by v and then subtracting from (28), we get d(v)u[v, f (v)] ± [v, f (v)]ud(v) = 0 (29) for all u, v ∈ l. since [v, f (v)] ∈ l, thus 2u[v, f (v)] ∈ l and so 4u[v, f (v)]w ∈ l for all u, v, w ∈ l. therefore, replacing u by 4u[v, f (v)]w, where w ∈ l, in (29) and then since characteristic of r is not 2, we obtain d(v)u[v, f (v)]w[v, f (v)] ± [v, f (v)]u[v, f (v)]wd(v) = 0 (30) for all u, v, w ∈ l. right multiplying (29) by w[v, f (v)] and then subtracting from (30), we get [v, f (v)]u([v, f (v)]wd(v) −d(v)w[v, f (v)]) = 0 (31) for all u, v, w ∈ l. by lemma 2.2, for each v ∈ v , either [v, f (v)] = 0 or [v, f (v)]wd(v) − d(v)w[v, f (v)] = 0 for all w ∈ l. since the first case implies the second case, we conclude that [v, f (v)]wd(v) −d(v)w[v, f (v)] = 0 (32) for all v, w ∈ l. now (32) and (29) together implies that 2d(v)u[v, f (v)] = 0 for all u, v ∈ l. since characteristic of r is not 2, therefore d(v)u[v, f (v)] = 0 for all u, v ∈ l. by primeness of r, for each v ∈ l, it implies that either d(v) = 0 or [v, f (v)] = 0. let v be an element of l such that d(v) = 0. this gives f (uv) = f (u)v for all u ∈ l. hence, (24) gives that f ([v, u]) ± [v, u] = 0 (33) for all u ∈ l. replacing u by 2vu in (33) and using characteristic of r is not 2, we have f (v)[v, u] ±vd([v, u]) ±v[v, u] = 0 (34) 76 b. dhara, s. ghosh, g.s. sandhu for all u ∈ l. replacing u by 2vu in (34) and using characteristic of r is not 2, we have f (v)v[v, u] ±v2d([v, u]) ±v2[v, u] = 0 (35) for all u ∈ l. left multiplying (34) by v and then subtracting from (35), we obtain [f (v), v][v, u] = 0 (36) for all u ∈ l. replacing u by 2tu in (36) and then using characteristic of r is not 2 and (36), we have [f (v), v]t[v, u] = 0 for all t, u ∈ l. since r is prime, [f (v), v] = 0 or [v, u] = (0). by lemma 2.1, [v, u] = (0) implies v ∈ z(r). thus in any case, we have [f (v), v] = 0 for all v ∈ l. this implies by lemma 2.9, l ⊆ z(r), a contradiction. theorem 3.4. let l be a nonzero square closed lie ideal of r and f be a generalized derivation of r associated to nonzero derivation d of r. if f (u◦v) ± [d(u), f (v)] ±u◦v = 0 for all u, v ∈ l, then l ⊆ z(r). proof. we assume on the contrary that l 6⊆ z(r). we begin with the situation f (u◦v) ± [d(u), f (v)] ±u◦v = 0 (37) for all u, v ∈ l. replacing u by 2uv in (37), we obtain 2(f (u◦v)v + (u◦v)d(v) ± [d(u)v, f (v)] ±[ud(v), f (v)] ± (u◦v)v) = 0 (38) for all u, v ∈ l. since characteristic of r is not 2, we have f (u◦v)v + (u◦v)d(v) ± [d(u)v, f (v)] ±[ud(v), f (v)] ± (u◦v)v = 0 (39) that is f (u◦v)v + (u◦v)d(v) ± [d(u), f (v)]v ±d(u)[v, f (v)] ± [ud(v), f (v)] ± (u◦v)v = 0 (40) on lie ideals satisfying certain differential identities 77 for all u, v ∈ l. right multiplying (37) by v and then subtracting from (40), we get (u◦v)d(v) ±d(u)[v, f (v)] ± [ud(v), f (v)] = 0 (41) for all u, v ∈ l. replacing u by 2vu in (41) and then using characteristic of r is not 2, we obtain v(u◦v)d(v) ± (vd(u) + d(v)u)[v, f (v)] ±v[ud(v), f (v)] ± [v, f (v)]ud(v) = 0 (42) for all u, v ∈ l. left multiplying (41) by v and then subtracting from (42), we get d(v)u[v, f (v)] + [v, f (v)]ud(v) = 0 (43) for all u, v ∈ l. identity (43) coincides with identity (29) so arguing as in theorem 3.3 we get d(v)u[v, f (v)] = 0 for all u, v ∈ l. by primeness of r, for each v ∈ l, it implies that either d(v) = 0 or [v, f (v)] = 0. let v be an element of l such that d(v) = 0. this gives f (uv) = f (u)v for all u ∈ l. hence, (37) gives that f (v ◦u) ±v ◦u = 0 (44) for all u ∈ l. replacing u by 2vu in (44) and using characteristic of r is not 2, we have f (v)(v ◦u) ±vd(v ◦u) ±v(v ◦u) = 0 (45) for all u ∈ l. replacing u by 2vu in (45) and using characteristic of r is not 2, we have f (v)v(v ◦u) ±v2d(v ◦u) ±v2(v ◦u) = 0 (46) for all u ∈ l. left multiplying (45) by v and then subtracting from (46), we obtain [f (v), v](v ◦u) = 0 (47) for all u ∈ l. replacing u by 2wu in (47) and then using characteristic of r is not 2, we have 0 = [f (v), v](v ◦wu) = [f (v), v]w[u, v] + [f (v), v](v ◦w)u 78 b. dhara, s. ghosh, g.s. sandhu for all u, w ∈ l. since [f (v), v](v ◦ w) = 0 for all v, w ∈ l, we have [f (v), v]w[u, v] = 0 for all u, v, w ∈ l. since r is prime, [f (v), v] = 0 or [v, l] = (0). by lemma 2.1, [v, l] = (0) implies v ∈ z(r). thus in any case, we have [f (v), v] = 0 for all v ∈ l. this implies by lemma 2.9, that l ⊆ z(r), which is a contradiction. theorem 3.5. let l be a nonzero square closed lie ideal of r and f and g be two generalized derivations of r associated to derivations d(6= 0) and g of r respectively. if f (u)g(v) ± d(u)f (v) ± uv ∈ z(r) for all u, v ∈ l and d±g 6= 0, then l ⊆ z(r). proof. we assume on the contrary that l 6⊆ z(r). by hypothesis, we have f (u)g(v) ±d(u)f (v) ±uv ∈ z(r) (48) for all u, v ∈ l. replacing v by 2vw in (48), we have 2(f (u)g(v) ±d(u)f (v) ±uv)w + 2f (u)vg(w) ± 2d(u)vd(w) ∈ z(r) (49) for all u, v, w ∈ l. since characteristic of r is not 2, we have (f (u)g(v) ±d(u)f (v) ±uv)w + f (u)vg(w) ±d(u)vd(w) ∈ z(r) (50) for all u, v, w ∈ l. commuting both sides of (50) with w, we obtain [(f (u)g(v) ±d(u)f (v) ±uv)w, w] +[f (u)vg(w) ±d(u)vd(w), w] = 0 (51) for all u, v, w ∈ l. using (48), we obtain [f (u)vg(w) ±d(u)vd(w), w] = 0 (52) for all u, v, w ∈ l. replacing u by 2uw in (52) and then using characteristic of r is not 2, we have [f (u)wvg(w), w] ± [d(u)wvd(w), w] +[ud(w)v(g(w) ±d(w)), w] = 0 (53) for all u, v, w ∈ l. replacing v by 2wv in (52) and using characteristic of r is not 2, we have [f (u)wvg(w), w] ± [d(u)wvd(w), w] = 0 (54) on lie ideals satisfying certain differential identities 79 for all u, v, w ∈ l. subtracting (54) from (53), we obtain [ud(w)v(g(w) ±d(w)), w] = 0 for all u, v, w ∈ l. replacing u by 2tu and then using characteristic of r is not 2, it gives [t, w]ud(w)v(g(w) ±d(w)) = 0 (55) for all u, v, w, t ∈ l. thus for each w ∈ l, either [l, w] = (0) or d(w) = 0 or d(w) ±g(w) = 0. now the first case i.e., [l, w] = (0) implies w ∈ z(r) by lemma 2.1. then by (50), we have f (u)vg(w) ±d(u)vd(w) ∈ z(r) (56) for all u, v ∈ l. replacing u by 2ut, where t ∈ l, we get 2(f (u)tvg(w) + ud(t)vg(w) ±d(u)tvd(w) ±ud(t)vd(w)) ∈ z(r) (57) for all u, v, t ∈ l. now we replace v by 2tv in (56) and obtain 2(f (u)tvg(w) ±d(u)tvd(w)) ∈ z(r) (58) for all u, v, t ∈ l. subtracting (58) from (57) and then using characteristic of r is not 2, we have ud(t)v(g(w) ±d(w)) ∈ z(r) (59) for all u, v, t ∈ l. since g(w) ± d(w) ∈ z(r) for w ∈ z(r), we have 0 = [ud(t)v(g(w)±d(w)), r] = [ud(t)v, r](g(w)±d(w)) for all u, v, t ∈ l and r ∈ r. since center of a prime ring contains no divisor of zero, either [ud(t)v, r] = 0 for all u, v, t ∈ l and r ∈ r or g(w) + d(w) = 0. we consider the first case i.e., [ud(t)v, r] = 0 for all u, v, t ∈ l and r ∈ r. replacing u by 2su, where s ∈ l, we obtain 0 = 2[sud(t)v, r] = 2s[ud(t)v, r] + 2[s, r]ud(t)v = 2[s, r]ud(t)v for all u, v, t, s ∈ l and r ∈ r. by lemma 2.2, either [l, r] = (0) or d(l) = (0). by lemma 2.3 both cases give l ⊆ z(r), a contradiction. thus we conclude that for each w ∈ l, either d(w) = 0 or d(w)±g(w) = 0. but the sets {w ∈ l : d(w) = 0} and {w ∈ l : d(w) ± g(w) = 0} are two additive subgroups of l whose union is l. by using the same argument used in lemma 2.9 we get that either d(l) = (0) or (d ± g)(l) = (0). since l is noncentral, by lemma 2.3, either d = 0 or d±g = 0, a contradiction. 80 b. dhara, s. ghosh, g.s. sandhu theorem 3.6. let l be a nonzero square closed lie ideal of r and f and g be generalized derivations of r associated to derivations d and g of r, respectively. if d 6= 0, g(uv) ± d(u)f (v) ± f (vu) = 0 for all u, v ∈ l, then l ⊆ z(r). proof. we assume on the contrary that l 6⊆ z(r). by hypothesis, we have g(uv) ±d(u)f (v) ±f (vu) = 0 (60) for all u, v ∈ l. replacing v by 2vu in the above relation and using char(r) 6= 2, we get (g(uv) ±d(u)f (v) ±f (vu))u + uvg(u) ±d(u)vd(v) ±vud(u) = 0. (61) using (60) this gives uvg(u) ±d(u)vd(v) ±vud(u) = 0 (62) for all u, v ∈ l. again replacing v by 2uv and using characteristic of r is not 2, we have u2vg(u) ±d(u)uvd(v) ±uvud(u) = 0 (63) for all u, v ∈ l. left multiplying (62) by u and subtracting from (63), we have [d(u), u]vd(u) = 0 (64) for all u, v ∈ l. by primeness of r, for each u ∈ l, we have either [d(u), u] = 0 or d(u) = 0. thus in each case, we have [d(u), u] = 0 for all u ∈ l. now if d 6= 0, by lemma 2.9, [d(u), u] = 0 for all u ∈ l implies l ⊆ z(r), a contradiction. theorem 3.7. let l be a nonzero square closed lie ideal of r and f be a generalized derivation of r associated to the nonzero derivation d of r. if f (uv) ±f (v)f (u) ±uv ∈ z(r) for all u, v ∈ l, then l ⊆ z(r). proof. we assume on the contrary that l 6⊆ z(r). first we consider the relation f (uv) + f (v)f (u) + uv ∈ z(r) (65) for all u, v ∈ l. replacing u with 2uw in (65) and using characteristic of r is not 2, we have f (u)wv + ud(wv) + f (v)(f (u)w + ud(w)) + uwv ∈ z(r) (66) on lie ideals satisfying certain differential identities 81 for all u, v, w ∈ l. commuting with w, we have [f (u)wv, w] + [ud(wv), w] + [f (v)f (u), w]w +[f (v)ud(w) + uwv, w] = 0 (67) for all u, v, w ∈ l. from (65), we can write that [f (uv)+f (v)f (u)+uv, w] = 0 for all u, v, w ∈ l, that is, [f (v)f (u), w] = −[f (uv)+uv, w] for all u, v, w ∈ l. using this (67) reduces to [f (u)wv, w] + [ud(wv), w] − [f (uv) + uv, w]w +[f (v)ud(w) + uwv, w] = 0 (68) for all u, v, w ∈ l. replacing v by w2 in (68), we have [f (u)w3, w] + [ud(w3), w] − [f (uw2) + uw2, w]w +[f (w2)ud(w) + uw3, w] = 0 (69) that is, [uw2d(w), w] + [f (w2)ud(w), w] = 0 (70) for all u, w ∈ l. replacing u by 2wu in (68) and using characteristic of r is not 2, we have [(f (w)u + wd(u))wv, w] + [wud(wv), w] −[f (w)uv + wd(uv) + wuv, w]w + [f (v)wud(w) + wuwv, w] = 0 that is, [f (v)wud(w) + wd(u)wv + wud(wv) −wd(uv)w −wuvw + wuwv, w] +[f (w)uwv, w] − [f (w)uv, w]w = 0 for all u, v, w ∈ l. assuming v = w, we have [f (w)wud(w) + wud(w2) + wd(u)w2 −wd(uw)w, w] = 0 (71) this gives [f (w)wud(w) + wud(w2) −wud(w)w, w] = 0 (72) for all u, w ∈ l. 82 b. dhara, s. ghosh, g.s. sandhu subtracting (72) from (70), we get [uw2d(w), w] + [f (w2)ud(w), w] −[f (w)wud(w) + wud(w2) −wud(w)w, w] = 0 for all u, w ∈ l. this reduces to [uw2d(w), w] + [wd(w)ud(w), w] − [wuwd(w), w] = 0 (73) for all u, w ∈ l. now replacing u by 2wu in (73) and using characteristic of r is not 2, we get w[uw2d(w), w] + [wd(w)wud(w), w] −w[wuwd(w), w] = 0 (74) for all u, w ∈ l. left multiplying (73) by w and then subtracting from (74), we get [w[d(w), w]ud(w), w] = 0 (75) for all u, w ∈ l. again replacing u by 2uw in the above relation and using characteristic of r is not 2, we get [w[d(w), w]uwd(w), w] = 0 (76) for all u, w ∈ l. now right multiplying (75) by w and then subtracting from (76), we obtain [w[d(w), w]u[d(w), w], w] = 0 (77) and hence [w[d(w), w]uw[d(w), w], w] = 0 (78) for all u, w ∈ l. this implies w[d(w), w]uw[d(w), w]w −w2[d(w), w]uw[d(w), w] = 0 (79) for all u, w ∈ l. since l is a lie ideal of r, [d(w), w] ∈ l for all w ∈ l and so 2[d(w), w]x ∈ l, 4w[d(w), w]x ∈ l and 8uw[d(w), w]x ∈ l for all u, w, x ∈ l. hence, we can replace u with 8uw[d(w), w]x in (79) and then using characteristic of r is not 2 we obtain, w[d(w), w]uw[d(w), w]xw[d(w), w]w −w2[d(w), w]uw[d(w), w]xw[d(w), w] = 0 (80) on lie ideals satisfying certain differential identities 83 for all u, w, x ∈ l. using (79), (80) gives w[d(w), w]uw2[d(w), w]xw[d(w), w] −w[d(w), w]uw[d(w), w]wxw[d(w), w] = 0 (81) that is w[d(w), w]u[w[d(w), w], w]xw[d(w), w] = 0 (82) for all u, w, x ∈ l. this implies w[[d(w), w], w]uw[[d(w), w], w]xw[[d(w), w], w] = 0 for all u, w, x ∈ l. by primeness of r, w[[d(w), w], w] = 0 for all w ∈ l. then by lemma 2.8, l ⊆ z(r) if d 6= 0 which is a contradiction. the remaining identities can be proved in a similar way. now we present an example which shows that the primeness hypothesis in the theorems is not superfluous. example 3.1. let z be the ring of integers. consider r = {( a 0 b c ) : a, b, c ∈ z } and l = {( 0 0 b 0 ) : b ∈ z } . clearly, r is a ring under the usual addition and multiplication of matrices and l is a nonzero square closed lie ideal of r. but for( 0 0 1 0 ) r ( 0 0 0 1 ) = (0), r is not prime ring. define the maps on r as follows: f ( a 0 b c ) = ( a 0 2b 0 ) , d ( a 0 b c ) = ( 0 0 2b 0 ) ; g ( a 0 b c ) = ( a 0 b− c c ) , g ( a 0 b c ) = ( 0 0 a− c 0 ) . then f and g are generalized derivations of r associated with nonzero derivations d and g respectively. moreover, the conditions of theorem 3.3, theorem 3.4, theorem 3.5, theorem 3.6 and theorem 3.7 are satisfied with f , g and d, but l 6⊆ z(r). hence the primeness assumption can not be removed. acknowledgements the authors would like to thank the referee for providing very helpful comments and suggestions. 84 b. dhara, s. ghosh, g.s. sandhu references [1] s. ali, b. dhara, n.a. dar, a.n. khan, on lie ideals with multiplicative (generalized)-derivations in prime and semiprime rings, beitr. algebra geom., 56 (1) (2015), 325 – 337. 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[18] s.k. tiwari, r.k. sharma, b. dhara, identities related to generalized derivations on ideal in prime rings, beitr. algebra geom. 57 (4) (2016), 809 – 821. introduction preliminaries main results � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 38, num. 1 (2023), 85 – 104 doi:10.17398/2605-5686.38.1.85 available online may 4, 2023 estimating the number of limit cycles for one step perturbed homogeneous degenerate centers m. molaeiderakhtenjani, o. rabieimotlagh, h.m. mohammadinejad@ department of applied mathematics, university of birjand, birjand, iran m.molaei@birjand.ac.ir , orabieimotlagh@birjand.ac.ir , hmohammadin@birjand.ac.ir received january 15, 2023 presented by r. prohens accepted april 3, 2023 abstract: we consider a homogeneous degenerate center of order 2m + 1 and perturb it by a homogeneous polynomial of order 2m. we study the lyapunov constants around the origin to estimate the number of limit cycles. to do it, we classify the parameters and study their effect on the number of limit cycles. finally, we find that the perturbed degenerate center without any condition has at least two limit cycles, and the number of the bifurcated limit cycles could reach 2m + 3. key words: degenerate center, limit cycle, lyapunov constant. msc (2020): 34c07, 34d10, 34d08. 1. introduction the 16th hilbert problem is one of the 23 mathematical problems proposed by d. hilbert in 1900 at the second international congress of mathematical, cf. [10]. the second part of this problem is to find an upper bound for the number of limit cycles that bifurcates from planar polynomial ordinary differential systems. since then, this problem has been studied by many authors, cf. [6, 11, 15, 16]. the weakened 16th hilbert problem is a weaker version of this problem which was proposed by arnold in 1977, cf. [1]. this problem is to find an upper bound for the number of bifurcated limit cycles from the period annulus of systems near hamiltonian ones. d. hilbert conjectured that his 16th problem could approach by perturbation techniques. since then, some authors have been studying the number of limit cycles by perturbing the periodic orbits of a center. we remind that a fixed point of a system is called a center if it is surrounded by a neighborhood filled with periodic orbits. when the center is perturbed, the system may have limit cycles that bifurcate from some @ corresponding author issn: 0213-8743 (print), 2605-5686 (online) c© the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.38.1.85 mailto:m.molaei@birjand.ac.ir mailto:orabieimotlagh@birjand.ac.ir mailto:hmohammadin@birjand.ac.ir mailto:hmohammadin@birjand.ac.ir https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 86 m. molaeiderakhtenjani et al. of the periodic orbits of the center. therefore, these studies could approach mathematicians to solve the 16th hilbert problem. to this aim, some authors studied the perturbed center by applying the methods such as averaging theory, melnikov function, or poincaré map, cf. [4, 8, 14, 18]. there exist studies in which the authors used the lyapunov constant to estimate the number of limit cycles, cf. [5, 9, 17]. investigating the taylor expansion of the corresponding poincaré map is one way to compute the lyapunov constant. in this case, the coefficients of taylor expansion are the lyapunov constants, cf. [13, 19]. the idea to obtain k small amplitude limit cycles is based on imposing conditions on lyapunov constants vi such that v0 = · · · = vk−1 = 0 and vk 6= 0, and then performing a suitable perturbation to have k small limit cycles. few papers consider this problem for degenerate centers. we remind that a center of a polynomial differential system is a degenerate center if, after applying a linear change of variables and a suitable time rescale, the system can be written as ẋ = f1(x,y) and ẏ = f2(x,y), where f1(x,y) and f2(x,y) are nonlinear polynomials. authors in [2] used melnikov functions to study the number of limit cycles for the perturbation of the degenerate center ẋ = −y ( x2 + y2 2 )m , ẏ = x ( x2 + y2 2 )m , m ≥ 1. authors in [12] used the averaging method of second order to study the perturbation of the cubic degenerate center ẋ = −y ( 3x2 + y2 ) , ẏ = x ( x2 −y2 ) , and prove the existence of at most three limit cycles. in this paper, we will study the number of limit cycles for the perturbed system ξ̇ = x2m+1(ξ) + �x2m(ξ), ξ = (x,y), 0 < � � 1, (1.1) where x2m+1 = (p2m+1(x,y),q2m+1(x,y)) is the homogeneous polynomial system with p2m+1(x,y) = m∑ i=0 aix 2iy2(m−i)+1, q2m+1(x,y) = m∑ i=0 bix 2i+1y2(m−i), such that aibi < 0 for all i, and x2m = (p2m(x,y),q2m(x,y)) with p2m(x,y) = 2m∑ k=0 αkx ky2m−k, q2m(x,y) = 2m∑ k=0 βkx ky2m−k. estimating the number of limit cycles 87 the origin is a symmetric degenerate center for the unperturbed homogeneous polynomial system (1.1), cf. [13, theorem 1(ii)]. such systems are known as one step polynomial perturbations. in these systems, a homogeneous polynomial system is perturbed by another homogeneous polynomial term with one step bigger or smaller order. such systems receive attention from mathematicians and physicists due to their role in nonlinear mechanics, cf. [3, 7]. here, we aim to estimate the number of limit cycles for the perturbed system (1.1) by studying the lyapunov constants. we compute the lyapunov constants through the taylor expansion of the poincaré map p(r0,�) = r(2π,r0,�) = n∑ j=0 1 j! �j ∂jr ∂�j (2π,r0, 0) + o(� n+1) of the system (1.1). here, (r,θ) shows the polar coordinates around the origin. the origin is assumed to be a perturbed symmetric degenerate center, and r = r(θ,r0,�) is the solution of equation (1.1) such that r(0,r0,�) = r0 and r(2π,r0, 0) = r0. then, we will study the lyapunov constants by considering the coefficients of x2m, αks and βks, as the parameters. the paper is oriented as follows. in section 2, we will compute( ∂jr/∂�j ) (θ,r0, 0) for the perturbed system (1.1) and prove the equality (2.2). in section 3, we will simplify ( ∂jr/∂�j ) (θ,r0, 0) for θ = 2π and classify the parameters into two types α2k+1, β2k and α2k, β2k+1. in section 4, we will present our main results and estimate the number of limit cycles for the perturbed degenerate center (1.1) in theorem 4.5. we will see that the one step perturbed degenerate center has at least two limit cycles, and the number of limit cycles can reach 2m + 3. at last, we will consider our results for a perturbed degenerate center of order 3. 2. preliminaries consider the perturbed system (1.1). as [13, section 3], the corresponding polar perturbed system is dr dθ = rs(θ) + �r c a(ra + �b) , (2.1) where s(θ) = 〈nθ,x2m+1(θ)〉 (nθ ∧x2m+1(θ)) , a(θ) = (nθ ∧x2m+1(θ)), 88 m. molaeiderakhtenjani et al. b(θ) = (nθ ∧x2m(θ)), c(θ) = (x2m(θ) ∧x2m+1(θ)), such that nθ = (cos(θ), sin(θ)), xi(θ) = (pi(cos(θ), sin(θ)),qi(cos(θ), sin(θ)) for i = 2m, 2m+ 1. also, the notations 〈 , 〉 and (∧) are respectively the inner and the wedge product of vectors in r2, i.e., 〈(a,b) , (c,d)〉 = ac + bd, ( (a,b) ∧ (c,d) ) = ad− bc. then by applying the relation [13, (20)], we obtain ( ∂jr/∂�j ) (θ,r0, 0) for the polar perturbed system (2.1) as ∂jr ∂�j (θ,r0, 0) = j χ(θ) j−1∑ k=0 k∑ l=0 ( j − 1 k )( k l )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k dldk−l ( c a(ra + �b) ) dψ, where χ(θ) = exp (∫ θ 0 s(t)dt ) is the fundamental matrix of the homogeneous part of (2.1). also, d and d are the linear functional operators df(g(x),x) = (∂f/∂g) (g(x),x)g′(x) and df(g(x),x) = (∂f/∂x) (g(x),x), see [13, page 10] for more details. next, we consider the above relation more precisely. for l = 0, we have j χ(θ) ∫ θ 0 χ−1(ψ) ∂j−1r ∂�j−1 ( c a(ra + �b) ) dψ + j χ(θ) j−1∑ k=1 ( j − 1 k )( k 0 )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k dk ( c a(ra + �b) ) dψ = j χ(θ) ∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k ( c a(ra + �b) ) dψ + j χ(θ) j−1∑ k=1 ( j − 1 k )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k c ab (−1)kk!( r a b + � )k+1 dψ = j χ(θ) j−1∑ k=0 ( j − 1 k )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k c ab (−1)kk!( r a b + � )k+1 dψ. so for � = 0, we have j−1∑ k=0 (−1)kj χ(θ) ( j − 1 k ) k! rk+10 ∫ θ 0 ∂j−1−kr ∂�j−1−k c a2 ( b a )k 1 χ(ψ)k+2 dψ. estimating the number of limit cycles 89 for the remaining (when l ≥ 1), we have j χ(θ) j−1∑ k=1 k∑ l=1 ( j − 1 k )( k l )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k dldk−l ( c a(ra + �b) ) dψ = j χ(θ) j−1∑ k=1 k∑ l=1 ( j − 1 k )( k l )∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k c ab dl ( (−1)k−l(k − l)!( r a b + � )k−l+1 ) dψ. from the formula of faà di bruno, we have dl ( 1( r a b + � )k−l+1 ) = l=1∑ t=1 ( 1( r a b + � )k−l+1 )(t) bl,t ( ∂r ∂� , ∂2r ∂�2 , . . . , ∂l−t+1r ∂�l−t+1 ) = l=1∑ t=1 (−1)t(k − l + t)! (k − l)! ( r + � b a )k−l+t+1 bl,t ( ∂r ∂� , ∂2r ∂�2 , . . . , ∂l−t+1r ∂�l−t+1 ) . so for l ≥ 1, we get j−1∑ k=1 k∑ l=1 l∑ t=1 j χ(θ) ( j − 1 k )( k l ) (−1)k−l+t(k − l + t)! ∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k c ab 1( r + � b a )k−l+t+1 bl,t ( ∂r ∂� , ∂2r ∂�2 , . . . , ∂l−t+1r ∂�l−t+1 ) dψ. thus for � = 0, we have j−1∑ k=1 k∑ l=1 l∑ t=1 j χ(θ) ( j − 1 k )( k l ) (−1)k−l+t (k − l + t)! rk−l+t+20 ∫ θ 0 χ−1(ψ) ∂j−1−kr ∂�j−1−k c a2 ( b a )k−l 1 χ(ψ)k−l+t+2 bl,t ( ∂r ∂� , ∂2r ∂�2 , . . . , ∂l−t+1r ∂�l−t+1 ) dψ. finally, the following proposition is immediate. proposition 2.1. consider the system (2.1). we have ∂jr ∂�j (θ,r0, 0) = j−1∑ k=0 i(j,k)(θ) + j−1∑ k=1 k∑ l=1 l∑ t=1 m(j,k,l,t)(θ), (2.2) 90 m. molaeiderakhtenjani et al. where i(j,k)(θ) = (−1)kj χ(θ) ( j − 1 k ) k! rk+10 ∫ θ 0 ∂j−1−kr ∂�j−1−k c a2 ( b a )k 1 χ(ψ)k+2 dψ, (2.3) m(j,k,l,t)(θ) = (−1) k−l+tj χ(θ) ( j − 1 k )( k l ) (k − l + t)! rk−l+t+10∫ θ 0 ∂j−1−kr ∂�j−1−k c a2 ( b a )k−l 1 χ(ψ)k−l+t+2 bl,t ( ∂r ∂� , . . . , ∂l−t+1r ∂�l−t+1 ) dψ. (2.4) lemma 2.2. consider (2.3) and (2.4). we have r1: i(j, 0) + m(j,j−1,j−1, 1) = 0 for all j; r2: m(j, j−1, j−1, i + 1) + ∑j−2 k=i m(j, k, k, i) = 0 for all j, 1 ≤ i ≤ j−2. furthermore, ( ∂jr/∂�j ) (θ,r0, 0) is a j-th order homogeneous polynomial w.r.t. αks or βks. proof. the r1 and r2 prove by substituting indexes in (2.3) and (2.4), and proportional use of bl,t in appendix 5.1. for the remain, consider (2.2). the proof is obvious by induction on j and the second condition of faà di bruno’s formula. we note that the power of the parameters in b(θ) and c(θ) is one. in the following remark, we indicate the ( ∂jr/∂�j ) (θ,r0, 0) for j = 1, 2, 3, 4. remark 2.3. ∂r ∂� (θ,r0, 0) = χ(θ) ∫ θ 0 c χa2 dψ, ∂2r ∂�2 (θ,r0, 0) = − 2 χ(θ) r0 ∫ θ 0 c b χ2 a3 dψ, ∂3r ∂�3 (θ,r0, 0) = 6 χ(θ) r20 ∫ θ 0 c b2 χ3 a4 dψ + 6 χ(θ) r20 ∫ θ 0 ∂r ∂� c b χ3 a3 dψ, ∂4r ∂�4 (θ,r0, 0) = − 24χ(θ) r30 ∫ θ 0 c b3 χ4 a5 dψ − 48χ(θ) r30 ∫ θ 0 ∂r ∂� c b2 χ4a4 dψ − 24χ(θ) r30 ∫ θ 0 ( ∂r ∂� )2 c b χ4a3 dψ + 12χ(θ) r20 ∫ θ 0 ∂2r ∂�2 c b χ3a3 dψ. estimating the number of limit cycles 91 3. lyapunov constant in the following, we consider ( ∂jr/∂�j ) (θ,r0, 0) for θ = 2π. we prove some points to simplify them and apply for some j. consider the following lemma. lemma 3.1. ([13, lemma 6]) let f(θ) > 0 (< 0) be an even π-periodic function and i,j be nonnegative integers such that at least one of them is odd. then ∫ 2π 0 cosi(θ) sinj(θ) f(θ) dθ = 0. so we conclude the following lemma. lemma 3.2. let f(θ) > 0 (< 0) be an even π-periodic function and i,j be odd nonnegative integers. then∫ π 0 cosi(θ) sinj(θ) f(θ) dθ = 0. proof. we have∫ 2π 0 cosi(θ) sinj(θ) f(θ) dθ = ∫ π 0 cosi(θ) sinj(θ) f(θ) dθ + ∫ 2π π cosi(θ) sinj(θ) f(θ) dθ = ∫ π 0 cosi(θ) sinj(θ) f(θ) dθ + ∫ π 0 (−1)i+j cosi(ψ) sinj(ψ) f(ψ) dψ. then by considering lemma 3.1, we get 0 = ∫ 2π 0 cosi(θ) sinj(θ) f(θ) dθ = 2 ∫ π 0 cosi(θ) sinj(θ) f(θ) dθ. next, we consider x2m(θ) = (p2m(θ),q2m(θ)) as p2m(θ) = m∑ k=0 α2k cos 2k(θ) sin2m−2k(θ) + m−1∑ k=0 α2k+1 cos 2k+1(θ) sin2m−2k−1(θ), q2m(θ) = m∑ k=0 β2k cos 2k(θ) sin2m−2k(θ) + m−1∑ k=0 β2k+1 cos 2k+1(θ) sin2m−2k−1(θ). 92 m. molaeiderakhtenjani et al. now, by defining d2m and d ⊥ 2m as d2m(θ) = ( m−1∑ k=0 α2k+1 cos 2k+1(θ) sin2m−2k−1(θ), m∑ k=0 β2k cos 2k(θ) sin2m−2k(θ) ) , d⊥2m(θ) = ( m∑ k=0 α2k cos 2k(θ) sin2m−2k(θ), m−1∑ k=0 β2k+1 cos 2k+1(θ) sin2m−2k−1(θ) ) , we have x2m = d2m +d ⊥ 2m. this helps us to separate the parameters into two collections. one collection contains α2k+1s and β2ks, and the other collection contains α2ks and β2k+1s. thus, we can decompose b(θ) and c(θ) as the following, b(θ) : (nθ ∧d2m(θ)) + (nθ ∧d⊥2m(θ)) = b(θ) + b ⊥(θ), c(θ) : (d2m(θ) ∧x2m+1(θ)) + (d⊥2m(θ) ∧x2m+1(θ)) = c(θ) + c ⊥(θ). by considering b(θ) and c⊥(θ), we can see that the power of sin(θ) is even, and the power of cos(θ) is odd. also by considering b⊥(θ) and c(θ), we can see that the power of sin(θ) is odd, and the power of cos(θ) is even. this classification helps us to use lemma 3.1 and lemma 3.2. we note that we only use this classification when the relations could simplify. next, we introduce some particular cases. remark 3.3. let k be a nonnegative integer and consider c bk. according to c(θ) and b(θ), we can see that the multiplication of sin(θ) and cos(θ) exists in c bk, for all k. by applying an induction on k, we have the following results: if k is even, then the sin(θ) or cos(θ) in c bk has odd order. if k is odd, then the sin(θ) and cos(θ) in c bk are both have an even or both have an odd order. remark 3.4. consider (2.2) and note that (∂0r/∂�0)(θ,r0, 0) = χ(θ)r0. by substituting the suitable conditions, one of the terms of (∂jr/∂�j)(2π,r0, 0) for k = j − 1 is (−1)j−1 j! r j−1 0 ∫ 2π 0 c bj−1 aj+1χj(ψ) dψ. by applying an induction on j and according to lemma 3.1 and the above classification, this term is equal to zero when j is an odd integer number. estimating the number of limit cycles 93 in the following proposition, we use these remarks to investigate( ∂jr/∂�j ) (2π,r0, 0) for j = 1, 2, 3, 4. proposition 3.5. consider ( ∂jr/∂�j ) (θ,r0, 0) for j = 1, 2, 3, 4. we have ∂r ∂� (2π,r0, 0) = ∫ 2π 0 c + c⊥ χa2 dψ = 0, (3.1) ∂2r ∂�2 (2π,r0, 0) = −2 r0 ∫ 2π 0 c⊥b + cb⊥ χ2 a3 dψ, (3.2) ∂3r ∂�3 (2π,r0, 0) = 6 r20χ(π) ∂r ∂� (π,r0, 0) ∫ π 0 cb + c⊥b⊥ χ2 a3 dψ, (3.3) and also, ∂4r ∂�4 (2π,r0, 0) = − 24 r30 ∫ 2π 0 (cb + c⊥b⊥)(2bb⊥) + (cb⊥ + c⊥b)(b2 + b⊥ 2 ) χ4 a5 dψ + ( −24 r30χ 2(π) ( ∂r ∂� (π) )2 + 12 r20χ(π) ∂2r ∂�2 (π) )∫ π 0 cb + c⊥b⊥ χ2 a3 dψ + 48 r30χ(π) ∂r ∂� (π) ∫ π 0 c b2 χ3a4 dψ − 96 r30 ∫ π 0 ∂r ∂� c b2 χ4a4 dψ + 48 r30χ(π) ∂r ∂� (π) ∫ π 0 ∂r ∂� c b χ3a3 dψ − 48 r30 ∫ π 0 ( ∂r ∂� )2 c b χ4a3 dψ + 24 r20 ∫ π 0 ∂2r ∂�2 c b χ3a3 dψ. (3.4) where the relations (3.2), (3.3), and (3.4) are not generally equal to zero. proof. first, we note that a(θ) and χ(θ) are πperiodic functions, and also χ(2π) = 1, cf. [13]. now by considering lemma 3.1 and the above classification, we obviously obtain (3.2) and (3.3). next, we consider( ∂3r/∂�3 ) (2π,r0, 0). according to remark 3.4, we get it as ∂3r ∂�3 (2π,r0, 0) = 6 r20 ∫ 2π 0 c b χ3 a3 ∂r ∂� dψ, 94 m. molaeiderakhtenjani et al. where∫ 2π 0 ∂r ∂� c b χ3 a3 dψ = ∫ π 0 ∂r ∂� c b χ3 a3 dψ + ∫ π 0 ∂r ∂� (φ + π,r0, 0) c(φ + π) b(φ + π) χ3(φ + π) a3(φ + π) dφ. by considering c(θ), b(θ), and (∂r/∂�)(θ,r0, 0), we get c(θ + π) = −c(θ), b(θ + π) = −b(θ), and ∂r ∂� (θ + π,r0, 0) = χ(θ) χ(π) ∂r ∂� (π,r0, 0) − ∂r ∂� (θ,r0, 0). thus∫ 2π 0 ∂r ∂� c b χ3 a3 dψ = ∫ π 0 ∂r ∂� c b χ3 a3 dψ + 1 χ(π) ∂r ∂� (π,r0, 0) ∫ π 0 c b χ2 a3 dφ − ∫ π 0 ∂r ∂� (φ,r0, 0) c b χ3 a3 dφ = 1 χ(π) ∂r ∂� (π,r0, 0) ∫ π 0 (c + c⊥)(b + b⊥) χ2 a3 dφ. now by applying lemma 3.2, we have∫ 2π 0 c b χ3 a3 ∂r ∂� dψ = 1 χ(π) ∂r ∂� (π,r0, 0) ∫ π 0 cb + c⊥b⊥ χ2 a3 dψ. so, (3.3) is immediate. finally, by following the same process as above and also by considering ∂2r ∂�2 (θ + π,r0, 0) = χ(θ) χ(π) ∂2r ∂�2 (π,r0, 0) + ∂2r ∂�2 (θ,r0, 0), and remark 2.3, we obtain (3.4). finally, we note that by considering (3.2), (3.3) and (3.4), one easily concludes that they are not generally equal to zero. 4. estimating the number of limit cycles in this section, we represent our main results. theorem 4.1. consider the perturbed system (1.1). estimating the number of limit cycles 95 1) the system has at least two limit cycles. 2) the necessary condition that the system has more than two limit cycles is that the perturbed part has the two types of parameters, i.e., the parameters of d2m, α2k+1 or β2k, and the parameters of d ⊥ 2m, α2k or β2k+1, for arbitrary k. we note that for simplicity, we consider the following notation in proof. again by considering k as a nonnegative integer, assume cisj = cos2k+i(θ) sin2m−2k+j(θ). as an example α2t cs = α2t cos 2t(θ) sin2m−2t(θ), β2k c 1s−1 = β2k cos 2k+1(θ) sin2m−2k−1(θ). proof. 1) it is obvious by considering (3.1) and (3.2). 2) for this part, we consider (3.2) to find that under which necessary condition this relation could be zero. by substituting c, b, c⊥, and b⊥, we get c⊥b + cb⊥ = ( d⊥2m ∧x2m+1 ) (nθ ∧d2m) + (d2m ∧x2m+1) ( nθ ∧d⊥2m ) . then by considering nθ and x2m+1, we have c⊥b + cb⊥ as p2m+1(θ) (∑m k=0 α2kcs 1 )(∑m k=0 β2kcs ) + p2m+1(θ) (∑m−1 k=0 α2k+1c 1s )(∑m−1 k=0 β2k+1c 1s−1 ) −p2m+1(θ) (∑m k=0 β2kc 1s )(∑m−1 k=0 β2k+1c 1s−1 ) −p2m+1(θ) (∑m k=0 β2kcs )(∑m−1 k=0 β2k+1c 2s−1 ) + q2m+1(θ) (∑m k=0 α2kcs )(∑m k=0 β2kc 1s ) + q2m+1(θ) (∑m−1 k=0 α2k+1c 1s−1 )(∑m−1 k=0 β2k+1c 2s−1 ) −q2m+1(θ) (∑m k=0 α2kcs )(∑m−1 k=0 α2k+1c 1s ) −q2m+1(θ) (∑m k=0 α2kcs 1 )(∑m−1 k=0 α2k+1c 1s−1 ) . 96 m. molaeiderakhtenjani et al. as lemma 2.2, we can see that the order of parameters in each term is two. also, all terms appear as the production of parameters of d2m and d ⊥ 2m. now let 0 ≤ t ≤ m be a fixed integer number and assume m∑ k=0 α2kcs = α2tcs + m∑ k=0,k 6=t α2kcs. by considering the above relation, we can be zero (3.2) w.r.t. α2t. in this case, we find α2t = 1∫ 2π 0 γ(ψ) χ2 a3 dψ ∫ 2π 0 λ(ψ) χ2 a3 dψ, (4.1) where λ(ψ) is the summation of all terms which are not dependent on the parameter α2t (see appendix 5.2) and γ(ψ) = (sin(ψ)p2m+1(ψ) + cos(ψ)q2m+1(ψ)) ( m∑ k=0 β2kc 2ts2m−2t ) − 2q2m+1(ψ) ( m−1∑ k=0 α2k+1c 2t+1s2m−2t ) . as we can see, (4.1) is well defined if the parameters α2k+1 or β2k are not equal to zero for some k. it is worthwhile to note that we conclude the same result by considering the other parameters, α2t+1, β2t, and β2t+1, to be zero (3.2). in the following, we study the perturbed system (1.1) to estimate the number of limit cycles. for this consideration, we have two points. first, according to theorem 4.1, the perturbed system (1.1) must have the two types of parameters to study the existence of more than two limit cycles. we can directly conclude the following lemma from theorem 4.1. lemma 4.2. consider the perturbed system (1.1) and assume that the perturbed part has two types of parameters. by considering the parameter α2t as (4.1) for a fixed integer t, the perturbed system (1.1) has at least three limit cycles. in the following remark, we study the effect of assuming the specific parameter α2t as (4.1) in considering the other lyapunov constants. estimating the number of limit cycles 97 remark 4.3. consider (4.1). we obtain that the parameters α2k and β2k+1 appear in the numerator of (4.1) and the parameters α2k+1 and β2k appear in the denominator of (4.1). again we can see our classification of the parameters. now by substituting (4.1) in the other lyapunov constants, we can arrange the lyapunov constants in the form of algebraic equations. the variables of these algebraic equations are the parameters that appear in the numerator of (4.1). the other parameters, the parameters in the denominator of (4.1), use to study the existence of zero for the algebraic equations. we emphasize that we conclude the same result by considering the other parameters to be zero ( ∂2r/∂�2 ) (2π). the only difference is the change of the two types of parameters from the numerator and denominator. second, according to lemma 2.2, ( ∂jr/∂�j ) (2π,r0, 0) is a j-th order homogeneous polynomial w.r.t. αks or βks. in the following lemma, we consider the simplified ( ∂jr/∂�j ) (2π,r0, 0) for j ≥ 3. we can see that these relations have terms in which the order of a specific parameter is equal to j. lemma 4.4. for j ≥ 3, ( ∂jr/∂�j ) (2π,r0, 0) contains a parameter with an exponent of the order of j. proof. we note two points: first, the parameters α2k and β2k+1 exist in c⊥(θ) and b⊥(θ); and the parameters α2k+1 and β2k exist in c(θ) and b(θ). second, for the existence of a term with the coefficient, for example α j 2k, the term must be the j times production of c⊥(θ) or b⊥(θ). for j = 3, (3.3) simplifies as 6 r20χ(π) ∫ π 0 c + c⊥ χa2 dψ ∫ π 0 cb + c⊥b⊥ χ2 a3 dψ. we can see that the relation has ccb and c⊥c⊥b⊥. for j > 3, see the following considerations. let j be an even integer number and assume (2.3) for k = j − 2, i.e., i(j,j−2)(θ) = j χ(θ) ( j − 1 j − 2 ) (j − 2)! r j−1 0 ∫ θ 0 ∂r ∂� c a2 ( b a )j−2 1 χ(ψ)j dψ. by simplifying the i(j,j−2)(2π) even if (∂r/∂�) (π) comes out of the integral, the cbj−2 does not decompose to c, c⊥, b, and b⊥. we note that the order of b is even. the result is obvious according to remark 3.3. so the relation has terms with c2bj−2 and also c⊥2b⊥j−2. now let j be an odd integer number 98 m. molaeiderakhtenjani et al. and assume (2.4) for k = j − 1 and l = t = 2, i.e., m(j,j−1,2,2)(θ) = j χ(θ) ( j − 1 2 ) (j − 1)! r j−1 0 ∫ θ 0 c a2 ( b a )j−3 1 χ(ψ)j ( ∂r ∂� )2 dψ. by following the same consideration as i(j,j−2)(2π), we conclude the result. now we can conclude the following theorem. theorem 4.5. consider the perturbed system (1.1) and assume that the perturbed part has two types of parameters. then the number of the bifurcated limit cycles could reach 2m + 3. proof. according to theorem 4.1, the perturbed system (1.1) could have more than two limit cycles. we substitute the parameters α2t as (4.1) and study the other lyapunov constants. from lemma 2.2, remark 4.3, and lemma 4.4 each lyapunov constant, i.e., ( ∂jr/∂�j ) (2π) for all j, is an algebraic equation of order j. next, we use remark 4.3 to study the existence of zero for the algebraic equations. the number of the parameters which appear in the numerator of (4.1) is 2m + 1. so if we find the other parameters such that these algebraic equations have real roots, then the perturbed system (1.1) could have 2m + 3 limit cycles. in the following proposition, we consider the above results for the homogeneous degenerate center of order three. consider the perturbed system (1.1) for m = 1, i.e., ẋ = a0y 3 + a1x 2y + � ( α0y 2 + α1xy + α2x 2 ) , ẏ = b0xy 2 + b1x 3 + � ( β0y 2 + β1xy + β2x 2 ) . (4.2) proposition 4.6. assume the perturbed system (4.2) such that αk and βk, k = 0, 1, 2, is not equal to zero and consider the following conditions. i. let the parameters α1, β0, and β2 exist such that 1) β2l26 + β0l27 + α1l29 6= 0, 2) l311l321 6= 0. ii. define the parameter α0 = β1 (β2l21 + β0l22 + α1l25) + α2 (β2l23 + β0l24 + α1l28) β2l26 + β0l27 + α1l29 . estimating the number of limit cycles 99 respectively see appendix (5.3) and appendix (5.4) for l2is and l3is. if conditions i(1) and ii hold, then the perturbed system (4.2) has at least three limit cycles. if the conditions i(1,2) and ii hold, then the perturbed system (4.2) has at least four limit cycles. finally, if the conditions i(1,2) and ii hold, and also the parameters α1, β0, and β2 exist such that the quantic algebraic equation α42 (l424 + l434 + l444 + l464 + l4523) + α32 (l413 + l423 + l433 + l443 + l463 + l473 + l4523) + α22 (l412 + l422 + l432 + l442 + l462 + l472 + l4522) + α2 (l411 + l421 + l431 + l441 + l461 + l471 + l4521) + l410 + l420 + l430 + l440 + l460 + l470 + l4520, (4.3) has a real root, then the perturbed system (4.2) has at least five limit cycles, (the l4is are the coefficients of αi2, i = 0, 1, 2, 3, 4). proof. by applying (3.2) for the perturbed system (4.2) and simplifying it w.r.t. the parameters, we obtain it as β1β2l21 + β0β1l22 + α2β2l23 + α2β0l24 + α1β1l25 + α0β2l26 + α0β0l27 + α1α2l28 + α0α1l29, (4.4) where l2i, i = 1, . . . , 9, are the coefficients of parameters. now we consider (4.4) as a quadratic algebraic equation and suppose that i(1) holds. we can easily see that this algebraic equation has a real root α0 = β1 (β2l21 + β0l22 + α1l25) + α2 (β2l23 + β0l24 + α1l28) β2l26 + β0l27 + α1l29 . (4.5) thus according to this point that (3.3) is not generally equal to zero, the perturbed system (4.2) has at least three limit cycles. next, we study the third lyapunov constant, where we have the parameters α0 as (4.5). for simplicity in computation, we consider (4.5) as α0 = β1a + α2b, where a and b obtain by considering (4.5). by computing (3.3) for m = 1 and simplifying it w.r.t. the parameters β1 and α2, we have β31l311l321 + β 2 1 (α2l312l321 + α2l311l322 + l311l320) + β1 ( α22l313l321 + α 2 2l312l322 + α2l312l320 + l310l321 ) + α32l313l322 + α 2 2l313l320 + α2l310l322 + l310l320, (4.6) 100 m. molaeiderakhtenjani et al. where l3is are the coefficients of the parameters. we note that l3is are dependent on the parameters α1, β2, and β0. as we can see (4.6) is the cubic algebraic equation w.r.t. the parameters β1 and the parameters α2. we consider it as an algebraic equation w.r.t. β1. this equation has at least one real root if the coefficient of β1, i.e., l311l321 is not equal to zero. thus if i(1,2) and ii hold, the perturbed system (4.2) has at least four limit cycles. for the last step, we study the fourth lyapunov constant, (3.4), by substituting the parameters α0 and β1. then by simplifying it w.r.t. the parameter α2, we obtain (4.3). so if the parameters α1, β2, and β0 exist such that the quantic algebraic equation (4.3) has a real root, then the perturbed system (4.2) has at least five limit cycles. 5. appendix 5.1. the formula of faà di bruno. given two functions f and g, the generalization of the chain rule is known as faà di bruno’s theorem. dn dnx (f(g(x)) = n∑ k=1 f(k)(g(x))bn,k ( g(1)(x), g(2)(x), . . . ,g(n−k+1)(x) ) , where bn,k are the exponential bell polynomials. the partial or incomplete exponential bell polynomials are a triangular array of polynomials given by bn,k(x1,x2, . . . ,xn−k+1) =∑ n! j1! j2! . . . jn−k+1! ( x1 1! )j1(x2 2! )j2 . . . ( xn−k+1 (n−k + 1)! )jn−k+1 , where the sum is taken over all sequences j1, j2, j3, . . . , jn−k+1 of nonnegative integers such that these two conditions are satisfied: 1. j1 + j2 + j3 + · · · + jn−k+1 = k, 2. j1 + 2 j2 + 3 j3 + · · · + (n−k + 1) jn−k+1 = n. for example bk,k ( x1 ) : j1 = k, bk+1,k ( x1,x2 ) : j1 = k − 1, j2 = 1, bk+2,k ( x1,x2,x3 ) : j1 = k − 1, j2 = 0, j3 = 1, j1 = k − 2, j2 = 2, j3 = 0, estimating the number of limit cycles 101 bk+3,k ( x1,x2,x3,x4 ) : j1 = k − 1, j2 = 0, j3 = 0, j4 = 1, j1 = k − 2, j2 = 1, j3 = 1, j4 = 0, j1 = k − 3, j2 = 3, j3 = 0, j4 = 0. 5.2. the relation λ(θ) is m∑ k=0,k 6=t α2kcs { p2m+1(θ) ( m∑ k=0 β2kcs 1 ) − q2m+1(θ) (m−1∑ k=0 α2k+1c 1s ) −q2m+1(θ) (m−1∑ k=0 α2k+1c 1s ) + q2m+1(θ) ( m∑ k=0 β2kc 1s )} + p2m+1(θ) (m−1∑ k=0 α2k+1c 1s )(m−1∑ k=0 β2k+1c 1s−1 ) −p2m+1(θ) ( m∑ k=0 β2kc 1s )(m−1∑ k=0 β2k+1c 1s−1 ) −p2m+1(θ) ( m∑ k=0 β2kcs )(m−1∑ k=0 β2k+1c 2s−1 ) + q2m+1(θ) (m−1∑ k=0 α2k+1c 1s−1 )(m−1∑ k=0 β2k+1c 2s−1 ) . 5.3. the coefficients in the (4.4): l21 = −2 r0 ∫ 2π 0 −2p3(ψ) sin(ψ) cos4(ψ) χ2 a3 dψ, l22 = −2 r0 ∫ 2π 0 −2p3(ψ) sin3(ψ) cos2(ψ) χ2 a3 dψ, l23 = −2 r0 ∫ 2π 0 p3(ψ) sin(ψ) cos 4(ψ) + q3(ψ) cos 5(ψ) χ2 a3 dψ, l24 = −2 r0 ∫ 2π 0 p3(ψ) sin 3(ψ) cos2(ψ) + q3(ψ) sin 2(ψ) cos3(ψ) χ2 a3 dψ, 102 m. molaeiderakhtenjani et al. l25 = −2 r0 ∫ 2π 0 p3(ψ) sin 3(ψ) cos2(ψ) + q3(ψ) sin 2(ψ) cos3(ψ) χ2 a3 dψ, l26 = −2 r0 ∫ 2π 0 p3(ψ) sin 3(ψ) cos2(ψ) + q3(ψ) sin 2(ψ) cos3(ψ) χ2 a3 dψ, l27 = −2 r0 ∫ 2π 0 p3(ψ) sin 5(ψ) + q3(ψ) sin 4(ψ) cos(ψ) χ2 a3 dψ, l28 = −2 r0 ∫ 2π 0 −2q3(ψ) sin2(ψ) cos3(ψ) χ2 a3 dψ, l29 = −2 r0 ∫ 2π 0 −2q3(ψ) sin4(ψ) cos(ψ) χ2 a3 dψ. 5.4. the coefficients in the (4.6). consider α0 = aβ1 + α2b. we have l310 = 6 r20χ(π) ∫ π 0 ( α1β2p3(ψ) sin 2(ψ) cos3(ψ) + α1β0p3(ψ) sin 4(ψ) cos(ψ) χ2 a3 + β20p3(ψ) sin 4(ψ)(−cos(ψ)) −β22p3(ψ) cos5(ψ) χ2 a3 + −2β0β2p3(ψ) sin2(ψ) cos3(ψ) −α21q3(ψ) sin 3(ψ) cos2(ψ) χ2 a3 + α1β2q3(ψ) sin(ψ) cos 4(ψ) + α1β0q3(ψ) sin 3(ψ) + cos2(ψ) χ2 a3 ) dψ, l311 = 6 r20χ(π) ∫ π 0 ( ap3(ψ) sin4(ψ) cos(ψ) −p3(ψ) sin2(ψ) cos3(ψ) χ2 a3 + +aq3(ψ) sin3(ψ) cos2(ψ) −a2q3(ψ) sin5(ψ) χ2 a3 ) dψ, l312 = 6 r20χ(π) ∫ π 0 ( p3(ψ) sin 2(ψ) cos3(ψ) + bp3(ψ) sin4(ψ) cos(ψ) χ2 a3 + q3(ψ) sin(ψ) cos 4(ψ) − 2aq3(ψ) sin3(ψ) cos2(ψ) χ2 a3 + −2abq3(ψ) sin5(ψ) + bq3(ψ) sin3(ψ) cos2(ψ) χ2 a3 ) dψ, estimating the number of limit cycles 103 l313 = 6 r20χ(π) ∫ π 0 ( −q3(ψ) sin(ψ) cos4(ψ) −b2q3(ψ) sin5(ψ) χ2 a3 + −2bq3(ψ) sin3(ψ) cos2(ψ) χ2 a3 ) dψ, l320 = ∫ π 0 −β0p3(ψ) sin2(ψ) −β2p3(ψ) cos2(ψ) + α1q3(ψ) sin(ψ) cos(ψ) χa2 dψ, l321 = ∫ π 0 aq3(ψ) sin2(ψ) −p3(ψ) sin(ψ) cos(ψ) χa2 dψ, l322 = ∫ π 0 q3(ψ) cos 2(ψ) + bq3(ψ) sin2(ψ) χa2 dψ. references [1] v.i. arnold, loss of stability of self-oscillations close to resonance and versal deformations of equivariant vector fields, funct. anal. appl. 11 (2) (1977), 85 – 92. 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[19] h. żoladek, eleven small limit cycles in a cubic vector field, nonlinearity 8 (1995), 843 – 860. introduction preliminaries lyapunov constant estimating the number of limit cycles appendix the formula of faà di bruno. the coefficients in the (4.4): the coefficients in the (4.6). � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 2 (2022), 261 – 282 doi:10.17398/2605-5686.37.2.261 available online october 26, 2022 second derivative lipschitz type inequalities for an integral transform of positive operators in hilbert spaces s.s. dragomir 1, 2 1 mathematics, college of engineering & science victoria university, po box 14428, melbourne city 8001, australia 2 dst-nrf centre of excellence in the mathematical and statistical sciences school of computer science & applied mathematics university of the witwatersrand, johannesburg, south africa sever.dragomir@vu.edu.au , http://rgmia.org/dragomir received july 6, 2022 presented by m. maestre accepted october 5, 2022 abstract: for a continuous and positive function w (λ), λ > 0 and µ a positive measure on (0,∞) we consider the following integral transform d (w,µ) (t) := ∫ ∞ 0 w (λ) (λ + t)−1 dµ (λ) , where the integral is assumed to exist for t a positive operator on a complex hilbert space h. we show among others that, if a ≥ m1 > 0, b ≥ m2 > 0, then ‖d (w,µ) (b) −d (w,µ) (a) −d (d (w,µ)) (a) (b −a)‖ ≤‖b −a‖2 × { d(w,µ)(m2)−d(w,µ)(m1)−(m2−m1)d′(w,µ)(m1) (m2−m1)2 if m1 6= m2, 1 2 d′′ (w,µ) (m) if m1 = m2 = m, where d (d (w,µ)) is the fréchet derivative of d (w,µ) as a function of operator and d′′ (w,µ) is the second derivative of d (w,µ) as a real function. we also prove the norm integral inequalities for power r ∈ (0, 1] and a, b ≥ m > 0, ∥∥∥∥∥ ∫ 1 0 ((1 − t) a + tb)r−1 dt− ( a + b 2 )r−1∥∥∥∥∥ ≤ 124 (1 −r) (2 −r) mr−3 ‖b −a‖2 and ∥∥∥∥ar−1 + br−12 − ∫ 1 0 ((1 − t) a + tb)r−1 dt ∥∥∥∥ ≤ 112 (1 −r) (2 −r) mr−3 ‖b −a‖2 . key words: operator monotone functions, operator convex functions, operator inequalities, midpoint inequality, trapezoid inequality. msc (2020): 47a63, 47a60. issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.37.2.261 mailto:sever.dragomir@vu.edu.au http://rgmia.org/dragomir https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 262 s.s. dragomir 1. introduction consider a complex hilbert space (h,〈·, ·〉). an operator t on h is said to be positive (denoted by t ≥ 0) if 〈tx,x〉 ≥ 0 for all x ∈ h and also an operator t is said to be strictly positive (denoted by t > 0) if t is positive and invertible. a real valued continuous function f on (0,∞) is said to be operator monotone if f(a) ≥ f(b) holds for any a ≥ b > 0. in 1934, k. löwner [14] had given a definitive characterization of operator monotone functions as follows, see for instance [4, pp. 144 – 145]: theorem 1. a function f : [0,∞) → r is operator monotone in [0,∞) if and only if it has the representation f (t) = f (0) + bt + ∫ ∞ 0 tλ t + λ dµ (λ) (1.1) where b ≥ 0 and a positive measure µ on [0,∞) such that∫ ∞ 0 λ 1 + λ dµ (λ) < ∞. we recall the important fact proved by löwner and heinz that states that the power function f : (0,∞) → r, f(t) = tα is an operator monotone function for any α ∈ [0, 1], see [12]. the function ln is also operator monotone on (0,∞). for other examples of operator monotone functions, see [10, 11]. let b(h) be the banach algebra of bounded linear operators on a complex hilbert space h. the absolute value of an operator a is the positive operator |a| defined as |a| := (a∗a)1/2. it is known that [3] in the infinite-dimensional case the map f(a) := |a| is not lipschitz continuous on b(h) with the usual operator norm, i.e., there is no constant l > 0 such that ‖|a|− |b|‖≤ l‖a−b‖ for any a, b ∈b(h). however, as shown by farforovskaya in [7, 8] and kato in [13], the following inequality holds ‖|a|− |b|‖≤ 2 π ‖a−b‖ ( 2 + log ( ‖a‖ + ‖b‖ ‖a−b‖ )) (1.2) second derivative lipschitz type inequalities 263 for any a,b ∈b(h) with a 6= b. if the operator norm is replaced with hilbert-schmidt norm ‖c‖hs := (tr c∗c) 1/2 of an operator c, then the following inequality is true [1] ‖|a|− |b|‖hs ≤ √ 2‖a−b‖hs (1.3) for any a,b ∈b(h). the coefficient √ 2 is best possible for a general a and b. if a and b are restricted to be selfadjoint, then the best coefficient is 1. it has been shown in [3] that, if a is an invertible operator, then for all operators b in a neighborhood of a we have ‖|a|− |b|‖≤ a1 ‖a−b‖ + a2 ‖a−b‖2 + o ( ‖a−b‖3 ) (1.4) where a1 = ∥∥a−1∥∥‖a‖ and a2 = ∥∥a−1∥∥ + ∥∥a−1∥∥3 ‖a‖2 . in [2] the author also obtained the following lipschitz type inequality ‖f (a) −f (b)‖≤ f ′ (a)‖a−b‖ (1.5) where f is an operator monotone function on (0,∞) and a,b ≥ a > 0. one of the problems in perturbation theory is to find bounds for ‖f (a) −f (b)‖ in terms of ‖a−b‖ for different classes of measurable functions f for which the function of operator can be defined. for some results on this topic, see [5, 9] and the references therein. we have the following integral representation for the power function when t > 0, r ∈ (0, 1], see for instance [4, p. 145] tr = sin (rπ) π t ∫ ∞ 0 λr−1 λ + t dλ. (1.6) observe that for t > 0, t 6= 1, we have∫ u 0 dλ (λ + t) (λ + 1) = ln t t− 1 + 1 1 − t ln ( u + t u + 1 ) for all u > 0. by taking the limit over u →∞ in this equality, we derive ln t t− 1 = ∫ ∞ 0 dλ (λ + t) (λ + 1) , 264 s.s. dragomir which gives the representation for the logarithm ln t = (t− 1) ∫ ∞ 0 dλ (λ + 1) (λ + t) (1.7) for all t > 0. motivated by these representations, we introduce, for a continuous and positive function w(λ), λ > 0, the following integral transform d(w,µ)(t) := ∫ ∞ 0 w(λ) λ + t dµ(λ), t > 0 , (1.8) where µ is a positive measure on (0,∞) and the integral (1.8) exists for all t > 0. for µ the lebesgue usual measure, we put d(w)(t) := ∫ ∞ 0 w (λ) λ + t dλ, t > 0 . (1.9) if we take µ to be the usual lebesgue measure and the kernel wr(λ) = λ r−1, r ∈ (0, 1], then tr−1 = sin (rπ) π d (wr) (t) , t > 0 . (1.10) for the same measure, if we take the kernel wln(λ) = (λ + 1) −1, t > 0, we have the representation ln t = (t− 1)d (wln) (t), t > 0 . (1.11) assume that t > 0, then by the continuous functional calculus for selfadjoint operators, we can define the positive operator d (w,µ) (t) := ∫ ∞ 0 w (λ) (λ + t) −1 dµ (λ) , (1.12) where w and µ are as above. also, when µ is the usual lebesgue measure, then d (w) (t) := ∫ ∞ 0 w (λ) (λ + t) −1 dλ, (1.13) for t > 0. second derivative lipschitz type inequalities 265 in this paper, we show among others that, if a ≥ m1 > 0, b ≥ m2 > 0, then ‖d (w,µ) (b) −d (w,µ) (a) −d (d (w,µ)) (a) (b −a)‖ ≤‖b −a‖2×   d(w,µ)(m2)−d(w,µ)(m1)−(m2−m1)d′(w,µ)(m1) (m2−m1)2 if m1 6= m2, 1 2 d′′ (w,µ) (m) if m1 = m2 = m, where d (d(w,µ)) is the fréchet derivative of d (w,µ) as a function of operator and d′′ (w,µ) is the second derivative of d (w,µ) as a real function. we also prove the norm integral inequalities for power r ∈ (0, 1] and a,b ≥ m > 0, ∥∥∥∥∥ ∫ 1 0 ((1 − t) a + tb)r−1dt − ( a + b 2 )r−1 ∥∥∥∥∥ ≤ 1 24 (1 −r) (2 −r) mr−3 ‖b −a‖2 and ∥∥∥∥∥a r−1 + br−1 2 − ∫ 1 0 ((1 − t) a + tb)r−1dt ∥∥∥∥∥ ≤ 1 12 (1 −r) (2 −r) mr−3 ‖b −a‖2 . 2. preliminary results we have the following representation of the fréchet derivative: lemma 1. for all a > 0, d (d (w,µ)) (a) (v ) = − ∫ ∞ 0 w (λ) (λ + a) −1 v (λ + a) −1 dµ (λ) (2.1) for all v ∈ s (h), the class of all selfadjoint operators on h. proof. by the definition of d (w,µ) we have for t in a small open interval 266 s.s. dragomir around 0 that d(w,µ) (a + tv ) −d (w,µ) (a) = ∫ ∞ 0 w (λ) [ (λ + a + tv ) −1 − (λ + a)−1 ] dµ (λ) = ∫ ∞ 0 w (λ) [ (λ + a + tv ) −1 (λ + a−λ−a− tv ) (λ + a)−1 ] dµ (λ) = −t ∫ ∞ 0 w (λ) [ (λ + a + tv ) −1 v (λ + a) −1 ] dµ (λ) . therefore, lim t→0 d (w,µ) (a + tv ) −d (w,µ) (a) t = − lim t→0 ∫ ∞ 0 w (λ) [ (λ + a + tv ) −1 v (λ + a) −1 ] dµ (λ) = − ∫ ∞ 0 w (λ) [ (λ + a) −1 v (λ + a) −1 ] dµ (λ) and the identity (2.1) is obtained. the second fréchet derivative can be represented as follows: lemma 2. for all a > 0, d2(d(w,µ))(a)(v,v ) = 2 ∫ ∞ 0 w (λ) (λ + a) −1 v (λ + a) −1 v (λ + a) −1 dµ (λ) (2.2) for all v ∈ s (h). proof. we have by the definition of the fréchet second derivative that d2 (d (w,µ)) (a) (v,v ) = lim t→0 d (d (w,µ)) (a + tv ) (v ) −d (d (w,µ)) (a) (v ) t . observe, by (2.1), that we have for t in a small open interval around 0 d (d (w,µ)) (a + tv ) (v ) = − ∫ ∞ 0 w (λ) (λ + a + tv ) −1 v (λ + a + tv ) −1 dµ (λ) , second derivative lipschitz type inequalities 267 which gives that d (d (w,µ)) (a + tv ) (v ) −d (d (w,µ)) (a) (v ) = − ∫ ∞ 0 w (λ) (λ + a + tv ) −1 v (λ + a + tv ) −1 dµ (λ) + ∫ ∞ 0 w (λ) (λ + a) −1 v (λ + a) −1 dµ (λ) = ∫ ∞ 0 w(λ)× [ (λ+a)−1v (λ+at)−1 − (λ+a+tv )−1v (λ+a+tv )−1 ] dµ(λ). define for λ ≥ 0 and t as above, ut,λ := (λ + a) −1 v (λ + a) −1 − (λ + a + tv )−1 v (λ + a + tv )−1 . if we multiply both sides of ut,λ with λ + a + tv , the we get (λ + a + tv )ut,λ (λ + a + tv ) = (λ + a + tv ) (λ + a) −1 v (λ + a) −1 (λ + a + tv ) −v = ( 1 + tv (λ + a) −1 ) v ( 1 + t (λ + a) −1 v ) −v = ( v + tv (λ + a) −1 v )( 1 + t (λ + a) −1 v ) −v = v + tv (λ + a) −1 v + tv (λ + a) −1 v + t2v (λ + a) −1 v (λ + a) −1 v −v = 2tv (λ + a) −1 v + t2v (λ + a) −1 v (λ + a) −1 v = t [ 2v (λ + a) −1 v + tv (λ + a) −1 v (λ + a) −1 v ] . (2.3) if we multiply the equality by (λ + a + tv ) −1 both sides, we get for t 6= 0 ut,λ t = (λ+a+tv ) −1 [ 2v (λ+a) −1 v +tv (λ+a) −1 v (λ+a) −1 v ] × (λ + a + tv )−1 . (2.4) if we take the limit over t → 0 in, then we get lim t→0 ( ut,λ t ) = 2 (λ + a) −1 v (λ + a) −1 v (λ + a) −1 . 268 s.s. dragomir therefore, by the properties of limit under the sign of integral, we get lim t→0 d (d (w,µ)) (a + tv ) (v ) −d (d (w,µ)) (a) (v ) t = ∫ ∞ 0 w (λ) lim t→0 ( ut,λ t ) dµ (λ) = 2 ∫ ∞ 0 w (λ) (λ + a) −1 v (λ + a) −1 v (λ + a) −1 dµ (λ) and the representation (2.2) is obtained. remark 1. one may ask if the above integral representation can be extended for higher derivative. the author thinks that is possible, however the calculations are more difficult to perform and are not presented here. we have the following representation for the transform d(w,µ): theorem 2. for all a,b > 0 we have d (w,µ) (b) = d (w,µ) (a) − ∫ ∞ 0 w (λ) (λ + a) −1 (b −a) (λ + a)−1 dµ (λ) + 2 ∫ 1 0 (1 − t) [∫ ∞ 0 w (λ) (λ + (1 − t) a + tb)−1 (b −a) ×(λ + (1 − t) a + tb)−1(b −a)(λ + (1 − t)a + tb)−1 dµ(λ) ] dt. (2.5) proof. we use the taylor’s type formula with integral remainder, see for instance [6, p. 112], f (e) = f (c) + d (f) (c) (e −c) + ∫ 1 0 (1 − t) d2 (f) ((1 − t) c + te) (e −c,e −c) dt (2.6) that holds for functions f which are of class c2 on an open and convex subset o in the banach algebra b (h) and c,e ∈o. if we write (2.6) for d (w,µ) and a,b > 0, we get d (w,µ) (b) = d (w,µ) (a) + d (d (w,µ)) (a) (b −a) + ∫ 1 0 (1 − t) d2 (d (w,µ)) ((1 − t) a + tb) (b −a,b −a) dt and by the representations (2.1) and (2.2) we obtain the desired result (2.5). second derivative lipschitz type inequalities 269 3. main results we have the following lipschitz type inequality: theorem 3. assume that a ≥ m1 > 0, b ≥ m2 > 0, then ‖d (w,µ) (b) −d (w,µ) (a) −d (d (w,µ)) (a) (b −a)‖ (3.1) ≤‖b −a‖2×   d(w,µ)(m2)−d(w,µ)(m1)−(m2−m1)d′(w,µ)(m1) (m2−m1)2 if m1 6= m2 , 1 2 d′′ (w,µ) (m) if m1 = m2 = m. proof. from (2.5) we get ‖d (w,µ) (b) −d (w,µ) (a) −d (d (w,µ)) (a) (b −a)‖ ≤ 2 ∫ 1 0 (1 − t) [∫ ∞ 0 w (λ) ∥∥∥(λ + (1 − t) a + tb)−1 (b −a) (3.2) ×(λ + (1 − t) a + tb)−1 (b −a) (λ + (1 − t) a + tb)−1 ∥∥∥dµ (λ) ]dt ≤ 2‖b −a‖2 ∫ 1 0 (1 − t) (∫ ∞ 0 w (λ) ∥∥∥(λ + (1 − t) a + tb)−1∥∥∥3 dµ (λ))dt. assume that m2 > m1. then (1 − t) a + tb + λ ≥ (1 − t) m1 + tm2 + λ, which implies that ((1 − t) a + tb + λ)−1 ≤ ((1 − t) m1 + tm2 + λ)−1 , and ∥∥∥((1 − t) a + tb + λ)−1∥∥∥3 ≤ ((1 − t) m1 + tm2 + λ)−3 (3.3) for all t ∈ [0, 1] and λ ≥ 0. 270 s.s. dragomir therefore, by integrating (3.3) we derive∫ 1 0 (1 − t) (∫ ∞ 0 w (λ) ∥∥∥(λ + (1 − t) a + tb)−1∥∥∥3 dµ (λ))dt ≤ ∫ 1 0 (1 − t) (∫ ∞ 0 w (λ) ((1 − t) m1 + tm2 + λ)−3 dµ (λ) ) dt (3.4) = 1 (m2 −m1)2 ∫ 1 0 (1 − t) [∫ ∞ 0 w (λ) ((1 − t) m1 + tm2 + λ)−1 (m2 −m1) × ((1 − t) m1 + tm2 + λ)−1 (m2 −m1) × ((1 − t) m1 + tm2 + λ)−1 dµ(λ) ] dt. from (2.5) we have for m2 > m1 that d (w,µ) (m2) −d (w,µ) (m1) + (m2 −m1) ∫ ∞ 0 w (λ) (λ + m1) −2 dµ (λ) = 2 ∫ 1 0 (1 − t) [∫ ∞ 0 w (λ) ((1 − t) m1 + tm2 + λ)−1 (m2 −m1) × ((1 − t) m1 + tm2 + λ)−1 (m2 −m1) (3.5) × ((1 − t) m1 + tm2 + λ)−1 dµ(λ) ] dt. also ∫ ∞ 0 w (λ) (λ + m1) −2 dµ (λ) = −d′ (w,µ) (m1) , and then by (3.5) we get 1 2 (m2 −m1)2 [ d (w,µ) (m2) −d (w,µ) (m1) − (m2 −m1)d′ (w,µ) (m1) ] (3.6) = 1 (m2 −m1)2 ∫ 1 0 (1 − t) [∫ ∞ 0 w (λ) ((1 − t) m1 + tm2 + λ)−1 (m2 −m1) × ((1 − t)m1 + tm2 + λ)−1(m2 −m1)((1 − t)m1 + tm2 + λ)−1dµ(λ) ] dt. by utilizing (3.2) and (3.4) – (3.6) we derive (3.1). the case m2 < m1 goes in a similar way and we also obtain (3.1). second derivative lipschitz type inequalities 271 assume that m2 = m1 > 0. let � > 0. then b + � ≥ m + � > m. by the first inequality for m2 = m + � and m1 = m, we have ‖d (w,µ) (b + �) −d (w,µ) (a) −d (d (w,µ)) (a) (b + �−a)‖ (3.7) ≤‖b + �−a‖2 1 �2 [ d (w,µ) (m + �) −d (w,µ) (m) − �d′ (w,µ) (m) ] . by taylor’s expansion theorem with the lagrange remainder we have d (w,µ) (m + �) −d (w,µ) (m) − �d′ (w,µ) (m) = 1 2 �2d′′ (w,µ) (ζ�) with m + � > ζ� > m. therefore lim �→0+ 1 �2 [ d (w,µ) (m + �) −d (w,µ) (m) − �d′ (w,µ) (m) ] = 1 2 d′′ (w,µ) (m) and by taking the limit � → 0+ in (3.7) then we get∥∥d (w,µ) (b) −d (w,µ) (a)−d (d (w,µ)) (a) (b −a) ∥∥ ≤ 1 2 ‖b −a‖2 d′′ (w,µ) (m) and the second part of (3.1) is proved. the case of operator monotone function is as follows: corollary 1. assume that f : [0,∞) → r is an operator monotone function with f (0) = 0. if a ≥ m1 > 0, b ≥ m2 > 0, then∥∥f (b) b−1 −(2 −a−1b)a−1f (a) −a−1d (f) (a) (b −a)∥∥ (3.8) ≤‖b −a‖2 ×   1 (m2−m1)2 [ f(m2) m2 − f(m1) m1 − (m2 −m1) f′(m1)m1−f(m1) m21 ] if m1 6= m2 , 1 2 f′′(m)m2−2mf′(m)+2f(m) m3 if m1 = m2 = m. proof. we denote by ` the identity function ` (t) = t, t > 0. by `−1 we denote the function `−1(t) = t−1, t > 0. using these notations we have d(`,µ)(t) = f(t) t − b, t > 0 , where b ≥ 0 and µ is a positive measure on (0,∞). 272 s.s. dragomir the derivative of this function is d′ (`,µ) (t) = f ′ (t) t−f (t) t2 , t > 0 , and the second derivative d′′ (`,µ) (t) = (f ′ (t) t−f (t))′ t2 − 2t (f ′ (t) t−f (t)) t4 = (f ′′ (t) t + f ′ (t) −f ′ (t)) t2 − 2t (f ′ (t) t−f (t)) t4 = f ′′ (t) t3 − 2t2f ′ (t) + 2tf (t) t4 = f ′′ (t) t2 − 2tf ′ (t) + 2f (t) t3 . we have d (w,µ) (b) −d (w,µ) (a) −d ( `−1f ) (a)(b −a) = f (b) b−1 −f (a) a−1 − [ d ( `−1 ) (a) (b −a) f (a) + `−1 (a) d (f) (a) (b −a) ] = f (b) b−1 −f (a) a−1 + a−1 (b −a) a−1f (a) −a−1d (f) (a) (b −a) , since, by using the definition of the fréchet derivative, d ( `−1 ) (a) (b −a) = −a−1 (b −a) a−1. also d (w,µ) (m2)−d (w,µ) (m1) − (m2 −m1)d′ (w,µ) (m1) = f (m2) m2 − f (m1) m1 − (m2 −m1) f ′ (m1) m1 −f (m1) m21 . by making use of (3.1) we deduce (3.8). we consider the representation obtained from (1.9) for the operator t > 0 and the power r ∈ (0, 1], tr−1 = d (w̃r) (t) with the kernel w̃r(λ) := sin(rπ) π λr−1, r ∈ (0, 1]. second derivative lipschitz type inequalities 273 from (3.1) we get for a ≥ m1 > 0, b ≥ m2 > 0 and r ∈ (0, 1] that∥∥∥∥br−1 −ar−1 + ∫ ∞ 0 λr−1 (λ + a) −1 (b −a) (λ + a)−1 dλ ∥∥∥∥ (3.9) ≤‖b −a‖2 ×   (1−r)(m2−m1)mr−21 −m r−1 1 +m r−1 2 (m2−m1)2 if m1 6= m2 , 1 2 (1 −r) (2 −r) mr−3 if m1 = m2 = m. we have the following error bounds for operator jensen’s gap related to the n-tuple of positive operators a = (a1, . . . ,an) and probability density n-tuple p = (p1, . . . ,pn), j (a,p,d (w,µ)) := n∑ k=1 pkd (w,µ) (ak) −d (w,µ) ( n∑ k=1 pkak ) . theorem 4. assume that ai ≥ m > 0 for i ∈ {1, . . . ,n} and consider the probability density n-tuple p = (p1, . . . ,pn), then ∥∥j (a,p,d (w,µ)) ∥∥ ≤ 1 2 d′′ (w,µ) (m) n∑ k=1 pk ∥∥∥∥∥ak − n∑ j=1 pjaj ∥∥∥∥∥ 2 ≤ 1 2 d′′ (w,µ) (m) n∑ k=1 n∑ j=1 pjpk ‖ak −aj‖ 2 ≤ 1 2 d′′ (w,µ) (m) max k,j∈{1,...,n} ‖ak −aj‖ 2 . (3.10) proof. from (3.1) we get∥∥∥∥∥d(w,µ) (ak) −d (w,µ) ( n∑ j=1 pjaj ) −d (d (w,µ)) ( n∑ j=1 pjaj )( ak − n∑ j=1 pjaj )∥∥∥∥∥ ≤ 1 2 d′′(w,µ) (m) ∥∥∥∥∥ak − n∑ j=1 pjaj ∥∥∥∥∥ 2 (3.11) for all k ∈{1, . . . ,n}. 274 s.s. dragomir if we multiply this inequality by pk ≥ 0 and sum over k from 1 to n, then we get n∑ k=1 ∥∥∥∥∥pkd (w,µ) (ak) −pkd (w,µ) ( n∑ j=1 pjaj ) (3.12) −d (d (w,µ)) ( n∑ j=1 pjaj ) pk ( ak − n∑ j=1 pjaj )∥∥∥∥∥ ≤ 1 2 d′′ (w,µ) (m) n∑ k=1 pk ∥∥∥∥∥ak − n∑ j=1 pjaj ∥∥∥∥∥ 2 . by making use of the triangle inequality for norms, we also have n∑ k=1 ∥∥∥∥∥pkd (w,µ) (ak) −pkd (w,µ) ( n∑ j=1 pjaj ) −d (d (w,µ)) ( n∑ j=1 pjaj ) pk ( ak − n∑ j=1 pjaj )∥∥∥∥∥ ≥ ∥∥∥∥∥ n∑ k=1 pkd (w,µ) (ak) − n∑ k=1 pkd (w,µ) ( n∑ j=1 pjaj ) (3.13) −d (d (w,µ)) ( n∑ j=1 pjaj )( n∑ k=1 pkak − n∑ j=1 pjaj )∥∥∥∥∥ = ∥∥∥∥∥ n∑ k=1 pkd (w,µ) (ak) −d (w,µ) ( n∑ j=1 pjaj )∥∥∥∥∥. by utilizing (3.12) and (3.13) we deduce the first part of (3.10). the rest is obvious. remark 2. from (3.10) we can obtain the following norm inequalities for power r ∈ (0, 1], second derivative lipschitz type inequalities 275 ∥∥∥∥∥∥ n∑ k=1 pka r−1 k − ( n∑ k=1 pkak )r−1∥∥∥∥∥∥ (3.14) ≤ 1 2 (1 −r) (2 −r) mr−3 n∑ k=1 pk ∥∥∥∥∥ak − n∑ j=1 pjaj ∥∥∥∥∥ 2 ≤ 1 2 (1 −r) (2 −r) mr−3 n∑ k=1 n∑ j=1 pjpk ∥∥ak −aj∥∥2 ≤ 1 2 (1 −r) (2 −r) mr−3 max k,j∈{1,...,n} ∥∥ak −aj∥∥2, where ai ≥ m > 0 for i ∈ {1, . . . ,n} and the probability density n-tuple p = (p1, . . . ,pn). 4. midpoint and trapezoid inequalities we have the following midpoint norm inequality: theorem 5. if a,b ≥ m > 0 for some constant m, then∥∥∥∥ ∫ 1 0 d (w,µ) ((1 − t) a + tb) dt−d (w,µ) ( a + b 2 )∥∥∥∥ (4.1) ≤ 1 24 d′′ (w,µ) (m)‖b −a‖2 . proof. from (3.1) we have for all t ∈ [0, 1] and a,b ≥ m > 0,∥∥∥∥∥d (w,µ) ((1 − t)a + tb) −d (w,µ) ( a + b 2 ) −d (d (w,µ)) ( a + b 2 )( (1 − t) a + tb − a + b 2 )∥∥∥∥ ≤ 1 2 d′′ (w,µ) (m) ∥∥∥∥(1 − t) a + tb − a + b2 ∥∥∥∥2 (4.2) = 1 2 d′′ (w,µ) (m) ( t− 1 2 )2 ‖b −a‖2 . 276 s.s. dragomir if we integrate this inequality, we get∫ 1 0 ∥∥∥∥d (w,µ) ((1 − t) a + tb) −d (w,µ) ( a + b 2 ) −d (d (w,µ)) ( a + b 2 )( (1 − t) a + tb − a + b 2 )∥∥∥∥dt ≤ 1 2 d′′ (w,µ) (m)‖b −a‖2 ∫ 1 0 ( t− 1 2 )2 dt (4.3) = 1 24 d′′ (w,µ) (m)‖b −a‖2 . using the properties of norm and integral, we also have∫ 1 0 ∥∥∥∥d (w,µ) ((1 − t) a + tb) −d (w,µ) ( a + b 2 ) −d (d (w,µ)) ( a + b 2 )( (1 − t) a + tb − a + b 2 )∥∥∥∥dt ≥ ∥∥∥∥ ∫ 1 0 d (w,µ) ((1 − t) a + tb) dt−d (w,µ) ( a + b 2 ) (4.4) − (∫ 1 0 ( t− 1 2 ) dt ) d (d (w,µ)) ( a + b 2 ) (b −a) ∥∥∥∥ = ∥∥∥∥ ∫ 1 0 d (w,µ) ((1 − t) a + tb) dt−d (w,µ) ( a + b 2 )∥∥∥∥ . by employing (4.3) and (4.4) we derive the desired result (4.1). corollary 2. assume that f : [0,∞) → r is an operator monotone function with f (0) = 0. if a,b ≥ m > 0, then∥∥∥∥∥ ∫ 1 0 ((1 − t) a + tb)−1 f ((1 − t) a + tb) dt− ( a + b 2 )−1 f ( a + b 2 )∥∥∥∥∥ ≤ f ′′ (m) m2 − 2mf ′ (m) + 2f (m) 24m3 ‖b −a‖2 . (4.5) the proof follows by (4.1) for d (`,µ) (t) = f (t) t − b, t > 0 , where b ≥ 0 and µ is a positive measure on (0,∞). second derivative lipschitz type inequalities 277 remark 3. if a,b ≥ m > 0, then for r ∈ (0, 1] we get by (4.5) that∥∥∥∥∥ ∫ 1 0 ((1 − t) a + tb)r−1 dt− ( a + b 2 )r−1∥∥∥∥∥ ≤ 1 24 (1 −r) (2 −r) mr−3 ‖b −a‖2 . (4.6) the trapezoid norm inequality will be our concern from now on. for a continuous function f on (0,∞) and a,b > 0 we consider the auxiliary function fa,b : [0, 1] → r defined by fa,b (t) := f ((1 − t) a + tb) , t ∈ [0, 1]. we have the following representations of the derivatives: lemma 3. assume that the operator function generated by f is twice fréchet differentiable in each a > 0, then for b > 0 we have that fa,b is twice differentiable on [0, 1], dfa,b (t) dt = d (f) ((1 − t) a + tb) (b −a) , (4.7) d2fa,b (t) dt2 = d2 (f) ((1 − t) a + tb) (b −a,b −a) (4.8) for t ∈ [0, 1], where in 0 and 1 the derivatives are the right and left derivatives. proof. we prove only for the interior points t ∈ (0, 1). let h be in a small interval around 0 such that t + h ∈ (0, 1). then for h 6= 0, fa,b (t + h) −f (t) h = f ((1 − (t + h)) a + (t + h) b) −f ((1 − t) a + tb) h = f ((1 − t) a + tb + h (b −a)) −f ((1 − t) a + tb) h and by taking the limit over h → 0, we get dfa,b (t) dt = lim h→0 fa,b (t + h) −f (t) h = lim h→0 [ f ((1 − t) a + tb + h (b −a)) −f ((1 − t) a + tb) h ] = d (f) ((1 − t) a + tb) (b −a) , which proves (4.7). 278 s.s. dragomir similarly, 1 h [ dfa,b (t + h) dt − dfa,b (t) dt ] = d (f) ((1 − (t + h)) a + (t + h) b) (b −a) −d (f) ((1 − t) a + tb) (b −a) h = d (f) ((1 − t) a + tb + h (b −a)) (b −a) −d (f) ((1 − t) a + tb) (b −a) h and by taking the limit over h → 0, we get d2fa,b (t) dt2 = lim h→0 { 1 h [ dfa,b (t + h) dt − dfa,b (t) dt ]} = d2 (f) ((1 − t) a + tb) (b −a,b −a) , which proves (4.8). for the transform d (w,µ) (t) defined in the introduction, we consider the auxiliary function d (w,µ)a,b (t) : = d (w,µ) ((1 − t) a + tb) = ∫ ∞ 0 w (λ) (λ + (1 − t) a + tb)−1 dµ (λ) where a,b > 0 and t ∈ [0, 1]. corollary 3. for all a,b > 0 and t ∈ [0, 1], dd (w,µ)a,b (t) dt = d (d (w,µ)) ((1 − t) a + tb) (b −a) (4.9) = − ∫ ∞ 0 w (λ) (λ + (1 − t) a + tb)−1 (b −a) × (λ + (1 − t) a + tb)−1 dµ (λ) and d2d (w,µ)a,b (t) dt2 = d2 (d (w,µ)) ((1 − t) a + tb) (b −a,b −a) (4.10) = 2 ∫ ∞ 0 w (λ) (λ + (1 − t) a + tb)−1 (b −a) × (λ + (1 − t) a + tb)−1 (b −a) (λ + (1 − t) a + tb)−1 dµ (λ) . second derivative lipschitz type inequalities 279 we observe that if f (t) = d (w,µ) (t), t > 0, in lemma 3, then by the representations from lemma 1 and lemma 2 we obtain the desired equalities (4.9) and (4.10). we have the following identity for the trapezoid rule: lemma 4. for all a,b > 0 we have the identity d (w,µ) (a) + d (w,µ) (b) 2 − ∫ 1 0 d (w,µ) ((1 − t) a + tb) dt = ∫ 1 0 t (1 − t) [∫ ∞ 0 w (λ) (λ + (1 − t) a + tb)−1 (b −a) (4.11) × (λ + (1 − t) a + tb)−1 (b −a) (λ + (1 − t) a + tb)−1 dµ (λ) ] dt. proof. using integration by parts for the bochner integral, we have 1 2 ∫ 1 0 t (1 − t) d2d (w,µ)a,b (t) dt2 dt = 1 2 [ t (1 − t) dd (w,µ)a,b (t) dt ∣∣∣∣1 0 − ∫ 1 0 (1 − 2t) dd (w,µ)a,b (t) dt dt ] = ∫ 1 0 ( t− 1 2 ) dd (w,µ)a,b (t) dt dt = ( t− 1 2 ) d (w,µ)a,b (t) ∣∣∣∣1 0 − ∫ 1 0 d (w,µ)a,b (t) dt = 1 2 [ d (w,µ)a,b (1) + d (w,µ)a,b (0) ] − ∫ 1 0 d (w,µ)a,b (t) dt, that gives the identity d (w,µ) (a) + d (w,µ) (b) 2 − ∫ 1 0 d (w,µ) ((1 − t) a + tb) dt = 1 2 ∫ 1 0 t (1 − t) d2d (w,µ)a,b (t) dt2 dt. (4.12) 280 s.s. dragomir by (4.12) we have 1 2 ∫ 1 0 t (1 − t) d2d (w,µ)a,b (t) dt2 dt (4.13) = ∫ 1 0 t (1 − t) [∫ ∞ 0 w (λ) (λ + (1 − t) a + tb)−1 (b −a) × (λ + (1 − t) a + tb)−1 (b −a) (λ + (1 − t) a + tb)−1 dµ (λ) ] dt. by making use of (4.10) and (4.13) we deduce (4.11). we can state now the corresponding trapezoid norm inequality: theorem 6. if a,b ≥ m > 0 for some constant m, then∥∥∥∥d (w,µ) (a) + d (w,µ) (b)2 − ∫ 1 0 d (w,µ) ((1 − t) a + tb) dt ∥∥∥∥ ≤ 1 12 d′′ (w,µ) (m)‖b −a‖2 . (4.14) proof. by taking the norm in (4.11), we obtain∥∥∥∥d (w,µ) (a) + d (w,µ) (b)2 − ∫ 1 0 d (w,µ) ((1 − t) a + tb) dt ∥∥∥∥ (4.15) ≤‖b −a‖2 ∫ 1 0 t (1 − t) (∫ ∞ 0 w (λ) ∥∥∥(λ + (1 − t) a + tb)−1∥∥∥3 dµ (λ))dt. since a,b ≥ m > 0, then for λ ≥ 0 and t ∈ [0, 1], λ + (1 − t) a + tb ≥ λ + m, which implies that (λ + (1 − t) a + tb)−1 ≤ (λ + m)−1 . this implies that ∥∥∥(λ + (1 − t) a + tb)−1∥∥∥3 ≤ (λ + m)−3 for λ ≥ 0 and t ∈ [0, 1]. second derivative lipschitz type inequalities 281 by multiplying this inequality by t (1 − t) w (λ) ≥ 0 and integrating we get∫ 1 0 t (1 − t) (∫ ∞ 0 w (λ) ∥∥∥(λ + (1 − t) a + tb)−1∥∥∥3 dµ (λ))dt ≤ (∫ 1 0 t (1 − t) dt )(∫ ∞ 0 w (λ) (λ + m) −3 dµ (λ) ) = 1 6 ∫ ∞ 0 w (λ) (λ + m) −3 dµ (λ) . (4.16) taking the derivative over t twice in (1.8), we get d′′ (w,µ) (t) := 2 ∫ ∞ 0 w (λ) (λ + t) 3 dµ (λ) , t > 0 , that gives ∫ ∞ 0 w (λ) (λ + m) −3 dµ (λ) = 1 2 d′′ (w,µ) (m) and by (4.15) and (4.16) we derive (4.14). corollary 4. assume that f : [0,∞) → r is an operator monotone function with f (0) = 0. if a,b ≥ m > 0, then∥∥∥∥a−1f (a) + b−1f (b)2 − ∫ 1 0 ((1 − t) a + tb)−1 f ((1 − t) a + tb) dt ∥∥∥∥ ≤ f ′′ (m) m2 − 2mf ′ (m) + 2f (m) 12m3 ‖b −a‖2 . (4.17) the proof follows by (4.14) for d (`,µ) (t) = f (t) t − b, t > 0 , where b ≥ 0 and µ is a positive measure on (0,∞). remark 4. if a,b ≥ m > 0, then for r ∈ (0, 1] we get by (4.5) that∥∥∥∥ar−1 + br−12 − ∫ 1 0 ((1 − t) a + tb)r−1 dt ∥∥∥∥ ≤ 1 12 (1 −r) (2 −r) mr−3 ‖b −a‖2 . (4.18) 282 s.s. dragomir acknowledgements the author would like to thank the anonymous referee for valuable comments that have been implemented in the final version of the manuscript. references [1] h. araki, s. yamagami, an inequality for hilbert-schmidt norm, comm. math. phys. 81 (1981), 89 – 96. [2] r. bhatia, first and second order perturbation bounds for the operator absolute value, linear algebra appl. 208/209 (1994), 367 – 376. [3] r. bhatia, perturbation bounds for the operator absolute value. linear algebra appl. 226/228 (1995), 639 – 645. [4] r. bhatia, “ matrix analysis ”, graduate texts in mathematics, 169, springer-verlag, new york, 1997. [5] r. bhatia, d. singh, k.b. sinha, differentiation of operator functions and perturbation bounds, comm. math. phys. 191 (3) (1998), 603 – 611. [6] r. coleman, “ calculus on normed vector spaces ”, springer, new york, 2012. 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(in german) 38 (1934) 177 – 216. introduction preliminary results main results midpoint and trapezoid inequalities � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 1 (2022), 91 – 110 doi:10.17398/2605-5686.37.1.91 available online march 2, 2022 perturbation ideals and fredholm theory in banach algebras t. lukoto 1, h. raubenheimer 2 1 department of mathematics and applied mathematics, private bag x1106 university of limpopo, sovenga, 0727, south africa 2 department of mathematics and applied mathematics, university of johannesburg auckland park campus, south africa tshikhudo.lukoto@ul.ac.za , heinrichr@uj.ac.za received november 3, 2021 presented by p. aiena accepted january 10, 2022 abstract: in this paper we characterize perturbation ideals of sets that generate the familiar spectra in fredholm theory. key words: fredholm elements; index theory; perturbation ideals; (semi)regularities; riesz elements. msc (2020): 46h05, 47a53, 46j45. 1. introduction and preliminaries in 1971, lebow and schechter in [10] introduced and studied the notion of perturbation classes. they proved results that will be useful in establishing new results in this paper. the sets which will be of our interest are those in fredholm theory such as fredholm elements, weyl elements, browder elements and almost invertible fredholm elements. another goal in this paper is to identify whether the particular set of interest is a regularity or a semiregularity. the concepts of a regularity and a semiregularity were first identified by v. kordula and v. müller ([9] and [15]). this was a new axiomatic framework to spectral theory which was an improvement to the approach by w. żelazko ([18]) in the sense that there exist spectra for a single element in a banach algebra which could not be covered by the axiomatic theory of żelazko. let a be a complex banach algebra with a unit element 1a and for any λ ∈ c\{0}, simply write λ for λ ·1a. we will denote by a−1 the group of all invertible elements in a while a−1l (a −1 r ) represents the set of all left (right) invertible elements in a. we say p ∈a is an idempotent if it satisfies p = p2. the set σ(x) = σa(x) = {λ ∈ c : λ−x /∈a−1}, issn: 0213-8743 (print), 2605-5686 (online) c© the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.37.1.91 mailto:tshikhudo.lukoto@ul.ac.za mailto:heinrichr@uj.ac.za https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 92 t. lukoto, h. raubenheimer is the usual spectrum of x ∈ a and the corresponding spectral radius of x in a is denoted by r(x) = ra(x) = sup{|λ| : λ ∈ σ(x)}. it is well known that σ(x) is non-empty and a compact subset of the complex plane c. the element a ∈ a is called a quasinilpotent element if σ(a) = {0} and the set of all quasinilpotent elements in a banach algebra a is denoted by qn(a). if k ⊆ c or k ⊆ a, then we will denote by acck, the set of accumulation points of k. the topological boundary of k is denoted by ∂k and its closure is denoted by k. let m be a subset of a banach algebra a. the commutant of m is defined by comm(m) = {a ∈a : am = ma, m ∈ m}. by an ideal in a we mean a two-sided ideal. an ideal j is proper if j ( a. a maximal left (right) ideal is a proper left (right) ideal which is not contained in any proper left (right) ideal. a minimal left (right) ideal of a is a left (right) ideal j 6= {0}, such that {0} and j are the only left (right) ideals contained in j. the radical of a, denoted by rad(a), is the intersection of all maximal ideals of a. hence rad(a) is a two-sided ideal. if rad(a) = {0}, then we say that a is semisimple. if a has minimal left (right) ideals, then the smallest left (right) ideal containing all the minimal left (right) ideals is called the left (right) socle. if a has both minimal left and right ideals, and if the left and right socles of a are equal, we say that the socle of a exists and it is denoted by soc(a). let a be a banach algebra and i be an ideal in a. we say that i is an inessential ideal if, for every x ∈ i, σ(x), the spectrum of x has at most 0 as a limit point, i.e., x ∈ i ⇒ accσ(x) ⊆{0}. every inessential ideal in a banach algebra determines a fredholm theory. for a comprehensive account of the abstract fredholm theory in banach algebras, see [1, chapter 5]. next we define the concepts of a regularity and a semiregularity since they are key in this paper. definition 1.1. ([16, definition i.6.1]) let a be a banach algebra. a non-empty subset r of a is called regularity if it satisfies the following conditions: (i) if a ∈a and n ∈ n, then a ∈ r ⇔ an ∈ r ; perturbation ideals and fredholm theory 93 (ii) if a,b,c,d are mutually commuting elements of a and ac + bd = 1a, then ab ∈ r ⇐⇒ a ∈ r and b ∈ r. the following well known sets are examples of regularities (see [16, i.6, example 5]): a , a−1 , a−1l and a −1 r . in many cases it is possible to verify the axioms of a regularity by using the following criterion: theorem 1.2. ([16, theorem i.6.4]) let r be a non-empty subset of a banach algebra a satisfying ab ∈ r ⇐⇒ a ∈ r and b ∈ r (p1) for all commuting elements a,b ∈a. then r is a regularity. one can divide the definition of a regularity into two parts: definition 1.3. ([16, definition iii.23.1]) let a be a banach algebra. a non-empty subset r of a is called lower semiregularity if (i) a ∈a, n ∈ n, an ∈ r ⇒ a ∈ r ; (ii) a,b,c,d are mutually commuting elements of a satisfying ac + bd = 1a, and ab ∈ r, then a,b ∈ r . remark 1.4. ([16, remark iii.23.3]) let r be a non-empty subset of a banach algebra a satisfying a,b ∈a , ab = ba, ab ∈ r ⇒ a ∈ r and b ∈ r. (p1′) then clearly r is a lower semiregularity. definition 1.5. ([16, definition iii.23.10]) let a be a banach algebra. a non-empty subset r of a is called upper semiregularity if (i) a ∈ r, n ∈ n ⇒ an ∈ r ; (ii) a,b,c,d are mutually commuting elements of a satisfying ac + bd = 1a, and a,b ∈ r, then ab ∈ r ; (iii) r contains a neighbourhood of the unit element 1a . 94 t. lukoto, h. raubenheimer remark 1.6. a semigroup containing a neighborhood of the unit element of a is an upper semiregularity because it already satisfies conditions (i) and (ii) of definition 1.5. we can deduce that r is regularity if and only if it is both a lower semiregularity and an upper semiregularity. if a is a banach algebra and s ⊆a, then one can define in a natural way a spectrum relative to s for any a ∈a by σs(a) = {λ ∈ c : λ−a /∈s}. if s is a regularity or a semiregularity, then σs(a) has interesting properties, see ([16, theorem i.6.7, theorem iii.23.4 and theorem iii.23.18]). suppose a is a banach algebra and i is a closed ideal in a. we denote the canonical homomorphism from a to a/i by π : a → a/i and it is defined by π(x) = x + i (x ∈a). let φ(i) = {x ∈a : x + i ∈ (a/i)−1} = π−1((a/i)−1), (1.1) φl(i) = {x ∈a : x + i ∈ (a/i)−1l } = π −1((a/i)−1l ), (1.2) φr(i) = {x ∈a : x + i ∈ (a/i)−1r } = π −1((a/i)−1r ), (1.3) w(i) = {x ∈a : x = a + b with a ∈a−1 and b ∈ i}, (1.4) b(i) = {x ∈a : x = a + b with a ∈a−1, b ∈ i and ab = ba}. (1.5) the sets φ(i), φl(i) and φr(i) are called fredholm elements relative to i, left fredholm elements relative to i and right fredholm elements relative to i respectively. the elements in w(i) are called weyl elements relative to i and the elements in b(i) are called browder elements relative to i. it is clear that a−1l ⊆ φl(i) , a −1 r ⊆ φr(i) , (1.6) a−1 ⊆b(i) ⊆w(i) ⊆ φ(i) ⊆ φl(i) ∪ φr(i) . (1.7) since the sets (a/i)−1l , (a/i) −1 r and (a/i) −1 are open in the banach algebra a/i and since the natural homomorphism π : a → a/i is continuous, it follows that the sets φl(i) , φr(i) and φ(i) are open in a. also, since a−1 is an open set in a, it is easy to see that the sets w(i) and b(i) are open subsets of a. one can clearly see that if a is commutative, then w(i) = b(i). perturbation ideals and fredholm theory 95 the sets defined above generate in a natural way the spectra that are relevant in fredholm theory, i.e., if x ∈a define σφ(i)(x) = {λ ∈ c : λ−x /∈ φ(i)} (fredholm spectrum) , (1.8) σφl(i)(x) = {λ ∈ c : λ−x /∈ φl(i)} (left fredholm spectrum) , (1.9) σφr(i)(x) = {λ ∈ c : λ−x /∈ φr(i)} (right fredholm spectrum) , (1.10) σb(i)(x) = {λ ∈ c : λ−x /∈b(i)} (browder spectrum) , (1.11) σw(i)(x) = {λ ∈ c : λ−x /∈w(i)} (weyl spectrum) . (1.12) this together with equation (1.6) and equation (1.7) gives σφl(i)(x) ⊆ σl(x) , σφr(i)(x) ⊆ σr(x) , σφl(i) ∪σφr(i)(x) ⊆ σφ(i)(x) ⊆ σw(i)(x) ⊆ σb(i)(x) ⊆ σ(x) , where σl(x) and σr(x) are the left and right spectra, denoting the spectrum of x with respect to a−1l and a −1 r respectively. if i is a closed ideal of a banach algebra a, then we define the set kh(i) by kh(i) = {b ∈a : b + i ∈ rad(a/i)} = π−1(rad(a/i)) . 2. perturbation classes in 1971, lebow and schechter in [10] introduced and studied the notion of perturbation classes. in this section we highlight some of the results which will play a role in this paper. definition 2.1. let x be a complex banach space, and let s be a subset of x. the perturbation of s, denoted by p(s), is a set of all x ∈ x such x + s ∈s for all s ∈s, i.e., p(s) = {x ∈ x : x + s ∈s for all s ∈s}. we say p(s) is the set of elements of x that perturb s into itself. remark 2.2. let s be a subset of a banach space x closed under multiplication by nonzero scalars. if 0 ∈ s, then it is easy to see that p(s) ⊆ s. we refer the reader to [10, section 2] for the basic properties of the set p(s). if a is a banach algebra and s ⊆a is closed under scalar multiplication, the set p(s) is in general not an ideal. however, we will choose to call the set p(s) the perturbation ideal of s. 96 t. lukoto, h. raubenheimer proposition 2.3. let a be a banach space and suppose r ⊆a is closed under multiplication by nonzero scalars. then p(r) = p(a\r) . proof. since r is closed under multiplication by nonzero scalars, it also follows that a \ r is closed under multiplication by nonzero scalars. let a ∈ p(r) and let y ∈ a\ r. now assume that a + y ∈ r, then in view of [10, lemma 2.1], −a + (a + y) = y ∈ r which leads to a contradiction, hence a+y ∈a\r and it then follows that a ∈p(a\r). hence, p(r) ⊆p(a\r). similarly we can prove the inclusion p(a\r) ⊆p(r). the connection between the spectrum that a set in a banach algebra generate and the perturbation ideal of the set is the following: let a be a banach algebra and let s ⊆ a have the property that it is closed under multiplication by nonzero scalars. if a ∈a, then x ∈p(s) if and only if σs(a + x) = σs(a) . 3. fredholm elements in this section we investigate perturbation ideals for fredholm elements relative to some ideal i in a banach algebra. for more account on fredholm elements, we refer the reader to [1, chapter 5, section 5]. let a be a banach algebra and let i be a closed ideal in a. since a−1l and a−1r are regularities in a, similarly, (a/i) −1 l and (a/i) −1 r are regularities in the quotient algebra a/i. it then follows by [16, theorem i.6.3 (iii)] that φl(i) = π −1((a/i)−1l ) and φr(i) = π −1((a/i)−1r ) are regularities in a, hence φ(i) = φl(i) ∩ φr(i) is also a regularity in a. remark 3.1. let a be a banach algebra and let i be a closed ideal in a. then p(φ(i)), p(φl(i)) and p(φr(i)) are closed ideals: let ab ∈ φ(i)a−1 where a ∈ φ(i) and b ∈ a−1. now since a + i,b + i ∈ (a/i)−1, then (a + i)(b + i) = ab + i ∈ (a/i)−1. hence ab ∈ φ(i) and φ(i)a−1 ⊆ φ(i). in the same way, we can also show that a−1φ(i) ⊆ φ(i). from this and the fact that φ(i) is open in a and closed under multiplication by nonzero scalars, it then follows by [10, lemma 2.1 and theorem 2.4] that p(φ(i)) is a closed ideal. similarly, it can be shown that p(φl(i)) and p(φr(i)) are also closed ideals in a. perturbation ideals and fredholm theory 97 theorem 3.2. ([10, theorem 2.7]) let a be a banach algebra and let i be a closed ideal in a. then p(φ(i)) = p(φl(i)) = p(φr(i)) = π−1(rad(a/i)). proof. it follows from [10, theorem 2.5] in the quotient algebra a/i that rad(a/i) = p((a/i)−1). this implies that π−1(rad(a/i)) = π−1(p((a/i)−1)) = p(π−1((a/i)−1)) = p(φ(i)) . one can adapt the above proof to conclude that p(φl(i)) = p(φr(i)) = π−1(rad(a/i)) if one employs [10, theorem 2.6] in the quotient algebra a/i. set φ̃ = φl(i) ∪ φr(i). we first show that φ̃ is a lower semiregularity in a and then we will determine the perturbation ideal p(φ̃) of φ̃. proposition 3.3. let a be a banach algebra and let i be a closed ideal in a. then φ̃ = φl(i) ∪ φr(i) is a lower semiregularity in a. proof. by using that φl(i) = π −1((a/i)−1l ) and φr(i) = π −1((a/i)−1r ), we then have φ̃ = φl(i) ∪ φr(i) = π−1((a/i)−1l ) ∪π −1((a/i)−1r ) = π−1((a/i)−1l ∪ (a/i) −1 r ). note that (a/i)−1l ∪ (a/i) −1 r is a (p1 ′) lower semiregularity in the quotient algebra a/i, see remark 1.4. since π is a homomorphism, π−1((a/i)−1l ∪ (a/i)−1r ) is a (p1′) lower semiregularity in a. theorem 3.4. let a be a banach algebra and let i be a closed ideal in a. if φ̃ = φl(i) ∪ φr(i), then p(φ̃) = π−1(rad(a/i)) . proof. we first note that a−1 is open in a and a−1 ⊆ a−1l ∪a −1 r . now we show that ∂a−1∩(a−1l ∪a −1 r ) = ∅. if we assume ∂a−1∩(a −1 l ∪a −1 r ) 6= ∅, then in view of [16, theorem i.1.14] we arrive at a contradiction. since both 98 t. lukoto, h. raubenheimer a−1 and a−1l ∪a −1 r are closed under multiplication by nonzero scalars, it follows from [10, lemma 2.2] that p(a−1l ∪a −1 r ) ⊆ p(a−1). to prove the inclusion p(a−1) ⊆ p(a−1l ∪a −1 r ), let x ∈ p(a−1) and a ∈ a −1 l ∪a −1 r . if a ∈ a−1l , then by [10, theorem 2.5], x + a ∈ a −1 l ⊆ a −1 l ∪a −1 r . if a ∈ a−1r , then similarly x+a ∈a−1r ⊆a −1 l ∪a −1 r . if we combine our arguments, we get p(a−1) ⊆p(a−1l ∪a −1 r ), and hence p(a −1 l ∪a −1 r ) = p(a−1)=rad(a), see [10, theorem 2.5]. now in the quotient banach algebra a/i, it then follows that p((a/i)−1l ∪ (a/i) −1 r ) = rad(a/i). hence, π−1(rad(a/i)) = π−1(p((a/i)−1l ∪ (a/i) −1 r )) = p(π−1(a/i)−1l ∪π −1(a/i)−1r ) = p(φl(i) ∪ φr(i)) = p(φ̃(i)) . next we calculate the perturbation ideal of ∂φ(i), the boundary of the fredholm elements relative to an ideal i of a banach algebra a. since a−1 ∩ ∂φ(i) = ∅, ∂φ(i) is neither a lower nor an upper semiregularity, see definition 1.5 and [16, lemma 23.2]. theorem 3.5. let a be a banach algebra and let i be a closed ideal in a. then p(∂φ(i)) = π−1(rad(a/i)). proof. let r = a\∂φ(i). then φ(i) ⊆ r and ∂φ(i) ∩r = ∅. both r and ∂φ(i) are closed under scalar multiplication. since φ(i) is open, it then follows by [10, lemma 2.2] that p(r) ⊆ p(φ(i)) = π−1(rad(a/i)). in view of proposition 2.3, it follows that p(∂φ(i)) ⊆ π−1(rad(a/i)). now let x ∈ π−1(rad(a/i)) and let a ∈ ∂φ(i). so there exist sequences (an) in φ(i) and (bn) in a\φ(i) such that an → a and bn → a. by theorem 3.2 and proposition 2.3, x+an ∈ φ(i) with x+an → x+a and x+bn ∈a\φ(i) with x+bn → x+a. it then follows from this that x + a ∈ ∂φ(i). hence x ∈ p(∂φ(i)). if we combine our arguments we get that p(∂φ(i)) = π−1(rad(a/i)). in view of proposition 2.3 we also have that p(a\∂φ(i)) = π−1(rad(a/i)) . next we investigate the perturbation of φ(i), the closure of the fredholm elements relative to some closed ideal i in a banach algebra a. since φ(i) perturbation ideals and fredholm theory 99 satisfies all the conditions in definition 1.5, it is an upper semiregularity. in addition, p(φ(i)) is an ideal: to show a−1 ·φ(i) ⊆ φ(i), let ab ∈a−1 ·φ(i) with a ∈ a−1 and b ∈ φ(i). hence there is a sequence (bn) in φ(i) with bn → b. hence abn → ab with abn ∈ φ(i). this means that ab ∈ φ(i). it follows likewise that φ(i) ·a−1 ⊆ φ(i). by [10, lemma 2.3], p(φ(i)) is an ideal in a. theorem 3.6. let a be a banach algebra and let i be a closed ideal in a. then π−1(rad(a/i)) ⊆p(φ(i)) ⊆ φ(i) . proof. first we note that φ(i) = φ(i) ∪∂φ(i). let x ∈ π−1(rad(a/i)) and a ∈ φ(i). if a ∈ φ(i), by theorem 3.2, x + a ∈ φ(i). if a ∈ ∂φ(i), it follows from theorem 3.5 that x+a ∈ ∂φ(i). combining the two cases we can conclude that x + a ∈ φ(i). it then follows that π−1(rad(a/i)) ⊆ p(φ(i)). since 0 ∈ a−1 ⊆ φ(i), it follows by remark 2.2 that π−1(rad(a/i)) ⊆ p(φ(i)) ⊆ φ(i). 4. riesz elements our goal in this section is to calculate the perturbation ideal for the set of riesz elements relative to some closed ideal in a banach algebra. let i be a closed ideal in a banach algebra a. an element a ∈a is called riesz relative to i if σa/i(a + i) = {0}, i.e., a + i ∈ qn(a/i). the set of all riesz elements in a relative to the ideal i is denoted by r(a,i). whenever the algebra a is clear from the context, we will simply write r(a,i) = r(i). for properties of riesz elements we refer the reader to chapter r of the monograph [3]. it is clear that i ⊆ kh(i) ⊆r(i) . (4.1) remark 4.1. it must be noted that kh(i) is the largest ideal in a consisting of riesz elements [17, theorem 4.4]. since r(i) is in general not closed under addition or multiplication, it is in general not an ideal. to prove one of our main result, theorem 5.12, we need the following characterization of riesz elements. proposition 4.2. let i be a closed ideal in a banach algebra a and a ∈a. then a ∈r(i) if and only 1a + λa ∈ φ(i) for all λ ∈ c. 100 t. lukoto, h. raubenheimer proof. if a ∈r(i), then σa/i(a + i) = {0}. hence for all λ ∈ c\{0} 1 λ + a + i ∈ (a/i)−1 ⇒ 1a + λa ∈ φ(i) . since 1a ∈ φ(i), the statement is also true for λ = 0. conversely, let 1a + λa ∈ φ(i) for λ ∈ c. then 1a + λa + i ∈ (a/i)−1. for any λ 6= 0, we have 1a + 1 λ a + i ∈ (a/i)−1 ⇒ −λ− (a + i) ∈ (a/i)−1. now since σa/i(a + i) is a non-empty set, σa/i(a + i) = {0}, and so a ∈r(i). if i is an inessential ideal in a banach algebra a and a ∈ r(i), then σ(a) the spectrum of a is either a finite set or a sequence converging to zero [2, corollary 5.7.5]. let i be a closed ideal in a banach algebra a. our next result describes the perturbation ideal of the set of riesz elements relative to i. if i be a closed ideal in a banach algebra a, it follows from the spectral mapping theorem in the quotient algebra a/i that r(i) is closed under scalar multiplication. proposition 4.3. let a be a banach algebra and let i be a closed ideal in a. then kh(i) = π−1(rad(a/i)) = p(r(i)) . proof. it follows from the equivalence (i)⇔(iii) in [2, theorem 5.3.1] in the quotient algebra a/i that rad(a/i) = p(qn(a/i)). this implies that kh(i) = π−1(rad(a/i)) = π−1(p(qn(a/i))) = p(π−1(qn(a/i))) = p(r(i)) . remark 4.4. we note that r(i) is neither a regularity nor a semiregularity for any closed ideal i in a banach algebra a: since 1a is the unit element in a, 1a + i is the unit element in the quotient algebra a/i and so σa/i(1a + i) = {1}. hence 1a + i /∈ qn(a/i). this means that 1a /∈ r(i). our claim follows from definition 1.5 and [16, lemma iii.23.2]. perturbation ideals and fredholm theory 101 proposition 4.5. let a be a banach algebra and let i be a closed ideal in a. then a\r(i) is a lower semiregularity. proof. let a,b ∈a\r(i) with ab = ba and suppose ab ∈a\r(i). hence, ra/i(ab + i) > 0 and since ab = ba, 0 < ra/i(ab + i) ≤ ra/i(a + i) ·ra/i(b + i). this means that ra/i(a + i) > 0 and ra/i(b + i) > 0, and so a,b ∈a\r(i). in view of remark 1.4, a\r(i) is a lower semiregularity. 5. index theory in this section, we consider the set of fredholm elements relative to some trace ideal i in a banach algebra a. this enables us to define subsets of φ(i) for which we can calculate the perturbation ideal. we refer the reader to [4, 5] for background knowledge on trace ideals in a semisimple banach algebra. our main result in this section is theorem 5.12. definition 5.1. (see [4, section 2.1]) let i be an ideal in a banach algebra a. a linear function τ : i → c is called a trace if it satisfies the following two conditions: (tn) τ(p) = 1 for every rank one idempotent p ∈ i ; (tc) τ(ba) = τ(ab) for all a ∈ i and b ∈a . if i is a trace ideal of a banach algebra a, then it is possible to define an index function i defined on the set of fredholm elements relative to i as follows: definition 5.2. ([4, definition 3.3]) let a be a banach algebra and let τ be a trace on an ideal i in a. we define the index function i : φ(i) → c by i(a) := τ(aa0 −a0a) = τ([a,a0]) for all a ∈ φ(i) , where a0 ∈a satisfies aa0 − 1a ∈ i and a0a− 1a ∈ i. it has been proven that the index function defined above is well defined on φ(i) (see [4, proposition 3.4]). next we list some of the important properties of this index function. 102 t. lukoto, h. raubenheimer remark 5.3. by remark preceding [4, proposition 3.5], its should be noted that for any a ∈a−1, i(a) = 0. proposition 5.4. ([4, proposition 3.5]) let a be a banach algebra and let i be a trace ideal in a. if a,b ∈ φ(i), then i(ab) = i(a) + i(b) . to exhibit more properties of the index function, we will suppose that the trace ideal satisfies the condition soc(a) ⊆ i ⊆ kh(soc(a)). for a motivation of this assumption, see the remarks preceding [4, section 3]. definition 5.5. an idempotent p is called a left barnes idempotent for a ∈a if aa = (1 −p)a , while idempotent q is called a right barnes idempotent for a ∈a if aa = a(1 −q) . theorem 5.6. ([4, theorem 3.11]) let a be a semisimple banach algebra and let the trace ideal i satisfy soc(a) ⊆ i ⊆ kh(soc(a)). then (i) a ∈ φ(i) if and only if there exist left and right barnes idempotents p and q in soc(a) and an element a0 ∈a such that aa0 = 1 −p and a0a = 1 −q ; (ii) i(a) = τ(q) − τ(p) ∈ z. we are going to use (ii) the above theorem to extend the index function to the set φ̃ = φl(i) ∪ φr(i): if a ∈ φl(i) \ φ(i) define the index of a by i(a) = τ(q) − τ(p) = τ(q) −∞ = −∞ , and if a ∈ φr(i) \ φ(i) define the index of a by i(a) = τ(q) − τ(p) = ∞− τ(p) = ∞ . let z be a subset of the integers z and φz be the set φz = {a ∈ φ̃ : i(a) ∈ z} . perturbation ideals and fredholm theory 103 lemma 5.7. let a be a semisimple banach algebra and let the trace ideal i satisfy soc(a) ⊆ i ⊆ kh(soc(a)). then φz is an open subset of a and it is closed under multiplication by nonzero scalars. proof. consider the real numbers r with the usual topology. if z ⊂ r have the relative topology, then for each k ∈ z, the singleton {k} is an open subset of z. since z is the union of such singletons, it is open. since the index function is integer valued and continuous, see [4, proposition 3.7 (vi)] and theorem 5.6, it follows that φz = i −1(z) is an open subset in a. the second part easily follows since i(αa) = i(a) ∈ z for a ∈ φz and α ∈ c. remark 5.8. let i be a trace ideal in a banach algebra a. if z ⊂ z, then in general φz is neither an upper nor a lower semiregularity in a: if 0 /∈ z, then a−1 ∩φz = ∅, see the remark preceding proposition 5.4. in view of [16, lemma 23.2 (ii)], φz is not a lower semiregularity. if z = {0, 10} and a ∈ φz has the property that i(a) = 5, then i(a2) = 10, and hence a2 ∈ φz and a /∈ φz, showing that φz is not a lower semiregularity, see definition 1.3. if we choose z = {0, 10} and if we let a ∈ φz have the property that i(a) = 10, it then follows that i(a2) = 20 and so a2 /∈ φz. by definition 1.5, φz is not an upper semiregularity. however for φ0(i), the set of fredholm elements of index zero, we have proposition 5.9. let a be a banach algebra and let i be a trace ideal in a. then φ0(i) is an upper semiregularity. proof. since φ0(i) is a semigroup with 1a ∈ a−1 ⊆ φ0, it follows that φ0(i) is an upper semiregularity, see remark 1.6. if i is a trace ideal in a banach algebra a, then the perturbation ideal p(φz) of φz is a closed ideal in a (see [10, theorem 2.4] and lemma 5.7), as well as remark 5.3 and proposition 5.4. note that φz can be decomposed into equivalence classes (components) as follows: φz = ⋃ k∈z i−1({k}). lemma 5.10. let a be a banach algebra and let i be a trace ideal in a. if z ⊂ z and φ̃ = φl(i) ∪ φr(i), then ∂φz ∩ φ̃ = ∅ , where φz = {a ∈ φ̃ : i(a) ∈ z}. 104 t. lukoto, h. raubenheimer proof. by lemma 5.7, φz is an open set and it is closed under multiplication by nonzero scalars. also, φz ⊆ φ̃. suppose there is x ∈ ∂φz ∩ φ̃. since φz is open, φz avoids ∂φz and so x /∈ φz. let cx be the component of φ̃ that contains x. since φ̃ is an open set, cx is also an open set. hence, cx is a neighbourhood of x that avoids φz. this is a contradiction because if x ∈ ∂φz, then every neighbourhood of x contains points of φz and points not in φz. this completes the proof. remark 5.11. let a be a banach algebra with p ⊆ a. if a ∈ a belongs to some component ca of p, then a and αa belong to ca for all 0 6= α ∈ c: consider the set {a + t(αa − a) : t ∈ [0, 1]}. this set is connected and it contains a and αa. since ca is the maximal connected subset of p that contains a, it follows that {a + t(αa−a) : t ∈ [0, 1]}⊆ ca. we are now ready to prove one of our main theorems which is a banach algebra version of [10, theorem 2.8] proved by lebow and schechter in the banach algebra b(x) of bounded operators defined on a banach space x. theorem 5.12. let a be a semisimple banach algebra and i be a closed trace ideal in a such that soc(a) ⊆ i ⊆ kh(soc(a)). if φz 6= ∅, then p(φz) = π−1(rad(a/i)) . (5.1) proof. set φ̃ = φl(i) ∪ φr(i). we can deduce from [10, lemma 2.2], lemma 5.7, lemma 5.10 and theorem 3.4 that p(φz) ⊇p(φ̃) = π−1(rad(a/i)) . this proves containment in one direction of equation (5.1). now we prove the opposite inclusion. since φz 6= ∅, there exists some a ∈ φz. assume a ∈ φl(i) and let x ∈p(φz). since p(φz) is an ideal, then λax ∈p(φz) for every scalar λ. it then follows that a + λax ∈ φz for all λ ∈ c. note that a + λax = a(1a + λx) ∈ φz ⊆ φ̃ for all λ ∈ c. we claim that a and a + λax belong to the same component of φl(i): if λ → 0, then a + λax → a. since φl(i) is an open set, it follows that the component of φl(i) containing a is also an open set. hence, if |λ| is small enough, a and a + λax belong to the same component of φl(i). if |λ| is large, there exists α ∈ c with |αλ| small. hence, a + λax = 1 α (αa + αλax). by remark 5.11, a + λax and αa + αλax belong to the same component. again, by applying remark 5.11, a and αa belong to the same component of φl(i) hence we can conclude that a and perturbation ideals and fredholm theory 105 a + λax belong to the same component for all λ ∈ c. hence, for each λ ∈ c there exists bλ ∈a with (bλ +i)(a+λax+i) = 1a +i and so 1a +λx ∈ φl(i) for all λ ∈ c. in the same way as above, 1a and 1a + λx belong to the same component of a−1 for all λ ∈ c. this means that 1a + λx ∈ φ(i) for all λ ∈ c. by proposition 4.2 we get that x ∈ r(i). but kh(i) = π−1(rad(a/i)) is the largest ideal consisting of riesz elements relative to i, see remark 4.1. hence, x ∈ π−1(rad(a/i)) and so p(φz) ⊆ kh(i) = π−1(rad(a/i). if we combine our arguments we get p(φz) = π−1(rad(a/i)). if instead a ∈ φr(i), then it can be proved in the same way as above that p(φz) = π−1(rad(a/i)). if i is a closed trace ideal in a semisimple banach algebra a with soc(a) ⊆ i ⊆ kh(soc(a)), denote by φn(i) the set of fredholm elements of index n, n ∈ z. in view of lemma 5.7, φn(i) is an open subset of a which is closed under multiplication by nonzero scalars. also, by theorem 5.12, p(φn(i)) = π−1(rad(a/i)). 6. weyl and browder elements in this section we are going to investigate the perturbation ideals of the collection of weyl elements and the collection of browder elements. let i be a closed ideal in a banach algebra a. since a−1 + i is an open set in a, a−1w(i) ⊆ w(i) and w(i)a−1 ⊆ w(i), it follows from [10, theorem 2.4] that p(w(i)) is a closed ideal in a. proposition 6.1. ([14, corollary 8.1]) if i is a closed ideal in a banach algebra a, then the set of weyl elements relative to i forms an upper regularity in a. proposition 6.2. let a be a banach algebra and let i be a closed ideal in a. then rad(a) + i ⊆p(w(i)) . proof. let x ∈ rad(a) + i. then x = a + b where a ∈ rad(a) and b ∈ i. suppose y ∈w(i), i.e. y = c+d with c ∈a−1 and d ∈ i. it then follows that x + y = (a + b) + (c + d) = (a + c) + (b + d) . it is clear that a + c ∈a−1 and b + d ∈ i, hence, x ∈p(w(i)). 106 t. lukoto, h. raubenheimer if i is a closed trace ideal in a, recall that φ0(i) = {a ∈ φ(i) : i(a) = 0}. by [4, proposition 3.7] and remark 5.3 we get that w(i) ⊆ φ0(i) . corollary 6.3. let a be a semisimple banach algebra and let i be a closed trace ideal in a with soc(a) ⊆ i ⊆ kh(soc(a)). then p(w(i)) = π−1(rad(a/i)) . proof. in view of in [5, corollary 3.5], w(i) = φ0(i). this together with theorem 5.12 with z = {0}, gives p(w(i)) = p(φ0(i)) = π−1(rad(a/i)). we now turn our attention to b(i), the set of browder elements relative to i. it is clear from equations (1.4) and (1.5) that browder elements are weyl elements. remark 6.4. it should however be clear that if a is a commutative banach algebra, the set of weyl elements relative to i and the set of browder elements relative to i coincide, i.e., w(i) = b(i). in this case it then follows that the results in proposition 6.2 and corollary 6.3 will also hold if we replace w(i) by b(i). in general, the inclusions b(i)a−1 ⊆ b(i) and a−1b(i) ⊆ b(i) are not satisfied if a is a non-commutative banach algebra so it is not clear that p(b(i)) is an ideal. but b(i) is closed under nonzero scalar multiplication. the next result is a modified version of proposition 6.2. proposition 6.5. let a be a banach algebra and let i be a closed ideal in a with i ⊆ comm(a−1). then rad(a) + i ⊆p(b(i)) . proof. since i ⊆ comm(a−1), the rest of the proof follows from the proof of proposition 6.2. if i is a closed inessential ideal in a banach algebra a, [7, theorem 7.7.6] tells us that ab ∈ b(i) if and only if a,b ∈ b(i) whenever ab = ba. hence b(i) satisfies the (p1) condition of theorem 1.2. it then follows from this that b(i) is a regularity in a. perturbation ideals and fredholm theory 107 theorem 6.6. ([11, theorem 7.5]) suppose i is a closed inessential ideal in a banach algebra a. then b(i) is an open regularity in a. the next result describe the perturbation of browder elements by commuting riesz elements. theorem 6.7. ([13, theorem 5.1]) suppose i is a closed inessential ideal in a banach algebra a. if a ∈ a and x ∈ r(i) satisfy ax = xa, then a is browder if and only if a + x is browder. the next result shows that the browder spectrum with respect to an inessential ideal is invariant under the addition of commuting riesz element relative to i. corollary 6.8. ([13, corollary 5.2]) suppose i is a closed inessential ideal in a banach algebra a. if a ∈ a and x ∈ r(i) satisfy ax = xa, then σb(i)(a) = σb(i)(a + x) . in view of these observations we have corollary 6.9. suppose i is an closed inessential ideal in a banach algebra a. if r(i) ⊆ comm(b(i)), then r(i) ⊆p(b(i)) . proof. this easily follows from theorem 6.7. 7. almost invertible fredholm elements in this section we investigate perturbation ideals of almost invertible elements and perturbation ideals of almost invertible fredholm elements in a banach algebra. an element a in a banach algebra a is called almost invertible if 0 is not an accumulation point of the spectrum of a. this means, a is almost invertible if a is either invertible or 0 is an isolated point of the spectrum of a. in addition, if i is a closed ideal in a, then a is almost invertible fredholm relative to i if a is almost invertible and fredholm relative to i. we will denote the collection of all almost invertible elements in a by ai(a) and the collection of all almost 108 t. lukoto, h. raubenheimer invertible fredholm elements relative to closed ideal i by aφ(i). the spectra that these sets generate are σai(a)(a) = {λ ∈ c : λ−a is not almost invertible} = accσ(a) , σaφ(i)(a) = {λ ∈ c : λ−a is not almost invertible fredholm relative to i} = accσ(a) ∪σa/i(a + i) , for all a ∈a. if i is a closed ideal in a banach algebra a, then every almost invertible fredholm element is browder element (see [6, theorem 1] and [13, corollary 2.5]). in light of this, we have the following relationship invertible ⇒ almost invertible fredholm ⇒ browder ⇒ weyl ⇒ fredholm . remark 7.1. if i is a closed inessential ideal in a, then almost invertible fredholm elements and browder elements relative to i coincide (see [13, corollary 3.6] and [14, theorem 5.2]). we note that if i is a closed ideal in banach algebra a, then almost invertible elements (almost invertible fredholm elements relative to i) is closed under multiplication by nonzero scalars. remark 7.2. an element a in a banach algebra a is koliha-drazin invertible if and only if 0 is not the accumulation point of the usual spectrum of a (see [8, theorem 4.2]). this statement shows that the koliha-drazin invertible elements and the almost invertible elements are equal. proposition 7.3. in a banach algebra a the set ai(a) is a regularity. proof. the result follows from remark 7.2 and [12, theorem 1.2]. theorem 7.4. let a be a banach algebra. then rad(a) ⊆p(ai(a)) ⊆ai(a) . perturbation ideals and fredholm theory 109 proof. let x ∈ rad(a) and a ∈ ai(a). it follows that 0 /∈ accσ(a). by [2, theorem 5.3.1], it follows that σ(x + a) = σ(a), and so 0 /∈ accσ(x + a). hence x + a ∈ai(a). from this we can conclude that x ∈p(ai(a)). since σ(0) = {0}, it follows that 0 /∈ accσ(0), hence 0 ∈ ai(a). now by remark 2.2, we get that p(ai(a)) ⊆ ai(a). combing all our arguments we obtain the required result. theorem 7.5. let a be a banach algebra and let i be a closed ideal in a. then rad(a) ⊆p(aφ(i)) . proof. the result follows by theorem 7.4 and theorem 3.2. recall that a closed ideal i in a banach algebra a is called an s-inessential ideal if a ∈a⇒ accσ(a) ⊆ σa/i(a + i) . for these ideals one can prove proposition 7.6. ([11, proposition 7.1]) suppose a closed ideal i in a banach algebra a is s-inessential. then aφ(i) is a regularity in a. however one can prove a stronger result. theorem 7.7. ([11, theorem 7.5]) suppose i is a closed inessential ideal in a banach algebra a. then aφ(i) is an open regularity in a. theorem 7.8. suppose i is a closed inessential ideal in a banach algebra a. if r(i) ⊆ comm(aφ(i)), then r(i) ⊆p(aφ(i)) . proof. this is a consequence of remark 7.1 and theorem 6.9. acknowledgements the authors wish to thank the referee for some helpful remarks. 110 t. lukoto, h. raubenheimer references [1] p. aiena, “fredholm and local spectral theory, with applications to multipliers”, kluwer academic publishers, dordrecht, 2004. [2] b. aupetit, “a primer on spectral theory”, springer-verlag, new york, 1991. [3] b.a. barnes, g.j. murphy, m.r.f. smyth, t.t. west, “riesz and fredholm theory in banach algebras”, pitman, boston-londonmelbourne, 1982. 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[12] r.a. lubansky, koliha-drazin invertibles form a regularity, math. proc. r. ir. acad. 107 (2) (2007), 137 – 141. [13] h. du t. mouton, h. raubenheimer, more on fredholm theory relative to a banach algebra homomorphism, proc. roy. irish acad. sect. a 93 (1) (1993), 17 – 25. [14] h. du t. mouton, s. mouton, h. raubenheimer, ruston elements and fredholm theory relative to arbitrary homomorphisms, quaest. math. 34 (2011), 341 – 359. [15] v. müller, axiomatic theory of spectrum iii semiregularities, studia math. 142 (2) (2000), 159 – 169. [16] v. müller, “spectral theory of linear operators and spectral systems in banach algebras”, birkhäuser verlag, basel, 2007. [17] m.r.f. smyth, riesz theory in banach algebras, math. z. 145 (1975), 145 – 155. [18] w. żelazko, an axiomatic approach to joint spectra i, studia math. 64 (1979), 249 – 261. introduction and preliminaries perturbation classes fredholm elements riesz elements index theory weyl and browder elements almost invertible fredholm elements � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 2 (2022), 223 – 242 doi:10.17398/2605-5686.37.2.223 available online april 1, 2022 dynamics of products of nonnegative matrices s. jayaraman, y.k. prajapaty, s. sridharan indian institute of science education and research thiruvananthapuram (iiser-tvm), india sachindranathj@iisertvm.ac.in , sachindranathj@gmail.com prajapaty0916@iisertvm.ac.in , shrihari@iisertvm.ac.in received january 5, 2022 presented by j. torregrosa accepted march 4, 2022 abstract: the aim of this manuscript is to understand the dynamics of products of nonnegative matrices. we extend a well known consequence of the perron-frobenius theorem on the periodic points of a nonnegative matrix to products of finitely many nonnegative matrices associated to a word and later to products of nonnegative matrices associated to a word, possibly of infinite length. key words: products of nonnegative matrices, common eigenvectors, common periodic points, orbits of infinite matrix products. msc (2020): 15a27, 37h12, 15b48. 1. introduction given a family f of functions on a set ω, an element w0 ∈ ω is said to be a common fixed point for f if f(w0) = w0 for all f ∈ f. the existence and computation of such a point has been a topic of interest among several mathematicians, for instance see [2, 11]. of particular interest is when the collection is a multiplicative semigroup or a group m of matrices, where a more general question on the existence of common eigenvectors arises. a classic example of a multiplicative semigroup of matrices is the collection of matrices whose entries are nonnegative real numbers. in a recent work, bernik et al. [3] determined certain conditions that ensures the existence of a common fixed point and more generally the existence of a common eigenvector for such a collection m. the existence of common eigenvectors for a collection of matrices is in itself a nontrivial question and plays a major role in many problems in matrix analysis. for recent results on periods and periodic points of iterations of sub-homogeneous maps on a proper polyhedral cone, we refer the reader to [1] (for instance, see theorem 4.2) and the references cited therein. issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.37.2.223 mailto:sachindranathj@iisertvm.ac.in mailto:sachindranathj@gmail.com mailto:prajapaty0916@iisertvm.ac.in mailto:shrihari@iisertvm.ac.in https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 224 s. jayaraman, y.k. prajapaty, s. sridharan we work throughout with the field r of real numbers. let mn(r) denote the real vector space of n × n matrices. the subset of mn(r) consisting of matrices whose entries are nonnegative real numbers (such a matrix is usually called a nonnegative matrix ) is denoted by mn(r+). for any matrix a ∈ mn(r), we denote and define the spectrum, the spectral radius and the norm of a respectively, as follows: spec(a) = the set of all eigenvalues of a, some of which may be complex numbers; ρ(a) = max { |λ| : λ ∈ spec(a) } ; ‖a‖ = the operator norm of a, induced by the euclidean norm of rn. for any n < ∞, we fix a finite collection of matrices, { a1, a2, . . . , an : ar ∈ mn(r+) } and define the following discrete dynamical system: for x0 ∈ rn, define xj+1 := aωjxj , for ωj ∈ { 1, 2, . . . ,n } . (1.1) that is, from a point xj at time t = j, we arrive at the point xj+1 at time t = j + 1 in the iteration of any generic point in rn, by randomly choosing one of the matrices from the above mentioned finite collection and the action by the chosen matrix. observe that in order to achieve proper meaning to the above mentioned iterative scheme, one expects to understand nonhomogeneous products of matrices. recall that given a self map f on a topological space x, an element x ∈ x is called a periodic point of f if there exists a positive integer q such that fq(x) = x. in such a case, the smallest such integer q that satisfies fq(x) = x is called the period of the periodic point x. the starting point of this work is the following consequence of the perron-frobenius theorem, as can be found in [8, theorem b.4.7]. theorem 1.1. let a ∈ mn(r+) with ρ(a) ≤ 1. then, there exists a positive integer q such that for every x ∈ rn with (∥∥akx∥∥) k∈n bounded, we have lim k→∞ akqx = ξx , where ξx is a periodic point of a whose period divides q. the spectral radius condition in theorem 1.1 can be dispensed with, only at the cost of looking at the orbits of points in the positive cone rn+. an illustration to this effect, can be found in [2, page 321]. dynamics of matrix products 225 we are interested in a generalization of theorem 1.1, when the matrix a in the above theorem is replaced by a product of the matrices ar’s, possibly an infinite one, drawn from the finite collection of nonnegative matrices,{ a1, . . . ,an } . besides generalizing theorem 1.1 as described above, we also bring out the existence of common periodic points for the said collection of matrices. this manuscript is organized as follows: in section 2, we introduce basic notations, however only as much necessary to state the main results of this paper, namely theorem 2.1 and theorem 2.3. in section 3, we familiarise the readers with some results from the literature, on adequate conditions to impose on a collection of matrices that ensures the existence of common eigenvectors. in section 4, we prove theorem 2.1 and highlight a special case of the theorem as corollary 4.2, when the collection of matrices satisfy an additional hypothesis. we follow this with section 5 where we write a few examples, that illustrate the theorems. in section 6, we focus on words of infinite length based on the finite collection of matrices that we have considered so far and write the proof of theorem 2.3. 2. main results in this section, we introduce some notations, explain the underlying settings of the main results and state our main results of this paper. as explained in the introductory section, we fix a finite set of nonnegative matrices{ a1, . . . ,an } , n < ∞. for any finite m ∈ n and p ∈ n, we denote the set of all p-lettered words on the set of first m positive integers by σ p m := { ω = (ω1ω2 · · ·ωp) : ωr ∈ { 1, . . . ,m }} . for any p-lettered word ω := (ω1ω2 · · ·ωp) ∈ σ p n , we define the (finite) matrix product aω := aωp ×aωp−1 ×···×aω2 ×aω1. (2.1) a key hypothesis in our first theorem assumes the existence of a nontrivial set of common eigenvectors, say e = { v1,v2, . . . ,vd } for the given collection of matrices, { a1, . . . ,an } . these common eigenvectors may be vectors in rn or cn. a sufficient condition that ensures the existence of common eigenvectors for the given collection is to demand the collection to be partially commuting, quasi-commuting or a laffey pair when n = 2, or the collection to be quasi226 s. jayaraman, y.k. prajapaty, s. sridharan commuting when n ≥ 3. each of these terms is explained in section 3. define lc(e) = { α1v1 + · · · + αdvd : αj ∈ c satisfying αs1 = αs2 for all vs1 = vs2 and αj ∈ r otherwise } . (2.2) we now state our first result in this article. theorem 2.1. let { a1,a2, . . . ,an } , n < ∞, be a collection of n × n matrices with nonnegative entries, each having spectral radius 1. assume that the collection satisfies at least one of the following conditions, that ensures the existence of a nontrivial set of common eigenvectors. 1. if n = 2, then the collection is either partially commuting, quasicommuting or a laffey pair. 2. if n ≥ 3, then the collection is quasi-commuting. let e denote the set of all common eigenvectors of the collection of matrices. for any finite p, let ω ∈ σpn and aω be the matrix associated to the word ω. then, for any vector x ∈lc(e), there exists an integer qω ≥ 1 such that lim k→∞ akqωω x = ξ(x,ω) , (2.3) where ξ(x,ω) is a periodic point of aω, whose period divides qω. moreover, when p ≥ n and ω is such that for all 1 ≤ r ≤ n, there exists 1 ≤ j ≤ p such that ωj = r, the integer qω and the limiting point ξ(x,ω) ∈ rn are independent of the choice of ω. careful readers may have already observed that, subject to the spectral radius condition as found in the hypothesis of theorem 2.1, we have lc(e) ⊆{ x ∈ rn : sup ∥∥akωx∥∥ < ∞}. however, since the spectral radius of aω can not be determined in general, we are forced to only work with vectors in lc(e). nevertheless, when the collection of matrices satisfy the spectral radius condition and are simultaneously diagonalizable, the above two sets coincide and is equal to rn. we will look at examples of this kind, later in section 5. we now illustrate the case when the above set inclusion is proper. consider the following pair of non-commuting, diagonalizable nonnegative matrices both having spectral radius 1, with one common eigenvector, namely e1. a1 = 1 3  3 0 00 1 2 0 2 1   and a2 = 1 3  3 0 00 1 4 0 1 1   . dynamics of matrix products 227 consider the two-lettered word ω = 12 ∈ σ22. then, the spectrum of aω, given by { 2+ √ 3 3 , 1 , 2− √ 3 3 } has corresponding eigenvectors u = ( 0, 1 + √ 3,−1 ) , e1 , v = ( 0, 2 − √ 3,−1 ) that form a basis for r3. since the largest eigenvalue is larger than unity, we observe that the sequence ( ‖akωu‖ ) k∈n is unbounded. thus, limk→∞ akωu does not exist. further, in this case, we note that lc(e) = {α1e1 : α ∈ r} ( { x ∈ r3 : sup k ∥∥akωx∥∥ < ∞} = {α1e1 + α2v : α1,α2 ∈ r} . we now denote the interior of the nonnegative orthant of rn by (rn+) ◦, a convex cone and define the logarithm map and the exponential map, that appear frequently in nonlinear perron-frobenius theory as follows: log : (rn+) ◦ → rn and exp : rn → (rn+)◦ by log(x) = (log x1, . . . , log xn) and exp(x) = (e x1, . . . ,exn) . as one may expect, these functions act as inverses of each other in the interior of rn+. more on these functions and their uses in nonlinear perronfrobenius theory can be found in the monograph [8]. a nonnegative matrix, when viewed as a linear map on rn, preserves the partial order induced by rn+. a map f defined on a cone in r n is said to be subhomogeneous if for every λ ∈ [0, 1], we have λf(x) ≤ f(λx) for every x in the cone and homogeneous if f(λx) = λf(x) for every nonnegative λ and every x in the cone. it is then easy to verify that the function f := exp◦a◦ log is a well-defined subhomogeneous map on (rn+) ◦. we now state a corollary to theorem 2.1 for an appropriate subhomogeneous map, fω. corollary 2.2. let { a1, . . . ,an } , n < ∞ be a set of n × n matrices satisfying all the hypotheses in theorem 2.1. for any finite p, let ω ∈ σpn and aω be the matrix associated with the word ω. consider the function fω : (rn+) ◦ −→ (rn+)◦ given by fω = exp◦aω ◦ log. then, for any y = ex ∈ (rn+) ◦ where x ∈lc(e), there exists an integer q ≥ 1 such that lim k→∞ fkqω y = ηy , (2.4) where ηy is a periodic point of fω, whose period divides q. 228 s. jayaraman, y.k. prajapaty, s. sridharan our final theorem in this paper concerns the orbit of some x ∈ rn under the action of some infinitely long word, whose letters belong to { a1, . . . ,an } . in order to make our lives simpler, we shall assume that the given collection of matrices are pairwise commuting, with each matrix being diagonalizable over c. this ensures the existence of n linearly independent (over c) common eigenvectors e = { v1, . . . ,vn } for the given collection. let the first κ of these common eigenvectors correspond to eigenvalues of modulus 1 for every matrix ar, in the collection. observe, in this case that lc(e) = rn. further, for any vector x ∈ rn given by x = ∑n s=1 αsvs obeying the conditions mentioned in equation (2.2), we define the support of the vector x as i(x) = supp(x) = { 1 ≤ s ≤ n : αs 6= 0 } . we now give a brief overview of the space of infinite-lettered words on finitely many letters. according the discrete metric on the set of letters{ 1, 2, . . . ,n } using the kronecker delta function, one can topologize σ p n with the appropriate product metric. when p = ∞, notice that the basis for the topology on the space of infinite-lettered words on n symbols, namely σ∞n , is given by the cylinder sets that fixes the set of initial finite coordinates, i.e., given any ω ∈ σpn for some p ∈ z +, the corresponding cylinder set is given by[ ω1ω2 · · ·ωp ] = { τ ∈ σ∞n : τj = ωj for 1 ≤ j ≤ p } . for more details on the spaces σ p n or σ ∞ n , one may refer [6]. for any p-lettered word ω ∈ σpn , we denote by ω, the infinite-lettered word obtained by concatenating ω with itself, infinitely many times, i.e., ω = (ω ω · · ·). under the topology defined on σ∞n , one may observe that σ∞n = ⋃ p≥1 { ω : ω ∈ σpn } . we know, from theorem 2.1, that upon satisfying the necessary technical conditions, lim k→∞ a kq ω x = ξx, whenever x ∈ lc(e). thus, the following definition makes sense. let ãω := aω = (a q ω) k as k →∞ . however, since theorem 2.1 only asserts ξx to be a periodic point whose period divides q, we shall consider the map ãω : rn → (rn)q. the precise action of ãω on points in rn is given by ãω(x) = ( ξx,aωξx, . . . ,a q−1 ω ξx ) . (2.5) dynamics of matrix products 229 let τ = ( τ1 τ2 τ3 · · · ) ∈ σ∞n be any arbitrary infinite lettered word that encounters all the n letters within a finite time, say m. it is easy to observe that the sequence( (τ1 · · · τm), (τ1 · · · τm+1), · · · ) converges to τ, in the topology on σ∞n , as described above. for any p ≥ m, denote by τ [p], the infinite-lettered word (τ1 · · · τp) that occurs in the sequence, written above that converges to any given τ ∈ σ∞n . moreover, from the discussion above, we have that ãτ[p]x = ( ξx,aτ[p]ξx, . . . ,a q−1 τ[p] ξx ) . notice that the first component of the vector in (rn)q is always ξx for all p ≥ m. further, we define for every r ∈ { 1, . . . ,n } , φ(τ,r)(p) = # of ar in aτ[p]. we now state our final theorem in this paper. theorem 2.3. let { a1, . . . ,an } , n < ∞, be a collection of n×n simultaneously diagonalizable nonnegative matrices each having spectral radius at most 1. suppose τ ∈ σ∞n encounters all the n letters within a finite time, say m. then, for any x ∈ rn, there exists an increasing sequence {pγ}γ≥1 (depending on x), of positive integers and a finite collection of positive integers{ λ(r,s) } for 1 ≤ r ≤ n and 1 ≤ s ≤ κ such that n∑ r=1 λ(r,s) [ φ(τ,r) ( pγk ) − φ(τ,r) ( pγk′ )] ≡ 0 (mod q), for all s ∈ i(x) ∩{1, . . . ,κ}, where pγk and pγk′ are any two integers from the sequence {pγ}. the above result may appear to be explaining an arithmetic property in a paper that deals with random dynamical systems generated by finitely many matrices; however, the authors urge the readers to note the following. as explained earlier, we know that ãτ[p] : r n → (rn)q. observe that lim p→∞ ãτ[p]x does not necessarily exist. however, from the proof of theorem 2.3, we will obtain the following corollary: 230 s. jayaraman, y.k. prajapaty, s. sridharan corollary 2.4. for each x ∈ rn, there exists an increasing sequence {pγ}γ≥1 (depending on x) such that { ã τ[pγ] x } γ≥1 is a constant sequence and therefore, lim γ→∞ ã τ[pγ] x exists. 3. common eigenvectors for a collection of matrices a key ingredient in our main results in this work is the existence of a nontrivial set of common eigenvectors for a given collection { a1, . . . ,an } of matrices. it is a well known result that if every matrix in the collection is diagonalizable over c with the collection commuting pairwise, there is a common similarity matrix that puts all the matrices in a diagonal form. a collection of non-commuting matrices may or may not have common eigenvectors. the question as to which collections of matrices possess common eigenvectors is extremely nontrivial. in what follows, we give a brief account of this question that is essential for this work. we begin with the following definition. definition 3.1. a collection { a1, . . . ,an } of matrices is said to be quasicommuting if for each pair (r,r′) of indices, both ar and ar′ commute with their (additive) commutator [ar,ar′] := arar′ −ar′ar. a classical result of mccoy [5, theorem 2.4.8.7] says the following: theorem 3.2. let { a1, . . . ,an } be a collection of n×n matrices. the following statements are equivalent. 1. for every polynomial p(t1, . . . , tn ) in n non-commuting variables t1, . . . , tn and every r,r ′ = 1, . . . ,n, p(a1, . . . ,an )[ar,ar′] is nilpotent. 2. there is a unitary matrix u such that u∗aru is upper triangular for every r = 1, . . . ,n. 3. there is an ordering λ (r) 1 , . . . ,λ (r) n of the eigenvalues of each of the matrices ar, 1 ≤ r ≤ n such that for any polynomial p(t1, . . . , tn ) in n non-commuting variables, the eigenvalues of p(a1, . . . ,an ) are p ( λ (1) s , . . . ,λ (n) s ) , s = 1, . . . ,n. if the matrices and the polynomials are over the real field, then all calculations may be carried out over r, provided all the matrices have eigenvalues in r. it turns out that a sufficient condition that guarantees any of the above three statements is when the collection of matrices is quasi-commutative (see dynamics of matrix products 231 drazin et al. [4]). moreover, the first statement implies that the collection{ a1, . . . ,an } has common eigenvectors. there are also other classes of matrices which possess common eigenvectors. a pair (a1,a2) of matrices is said to partially commute if they have common eigenvectors. moreover, two matrices a1 and a2 partially commute iff the shemesh subspace n = n−1⋂ k,l=1 ker ([ ak1,a l 2 ]) is a nontrivial maximal invariant subspace of a1 and a2 over which both a1 and a2 commute (see shemesh [10]). the number of linearly independent common eigenvectors of the pair cannot exceed the dimension of n . a pair (a1,a2) of matrices is called a laffey pair if rank ([a1,a2]) = 1. it can be shown that such a pair of matrices partially commute, but do not commute. 4. proof of theorem 2.1 in this section, we prove theorem 2.1, after stating a theorem due to frobenius. suppose a is an irreducible matrix in mn(r+) such that there are exactly κ eigenvalues of modulus ρ(a). this integer κ is called the index of imprimitivity of a. if κ = 1, the matrix a is said to be primitive. if κ > 1, the matrix is said to be imprimitive. theorem 4.1. ([12, theorem 6.18]) let a be an irreducible nonnegative matrix with its index of imprimitivity equal to κ. if λ1, . . . ,λκ are the eigenvalues of a of modulus ρ(a), then λ1, . . . ,λκ are the distinct κ-th roots of [ρ(a)]κ. proof of theorem 2.1. recall that e = { v1, . . . ,vd } is a set of d common eigenvectors of the matrices a1, . . . ,an that satisfies arvs = λ(r,s)vs, where λ(r,s) is an eigenvalue of the matrix ar corresponding to the eigenvector vs, 1 ≤ s ≤ d. observe that for any p-lettered word ω = (ω1 · · ·ωp), we have aωvs = λ(ωp,s) · · ·λ(ω1,s)vs = λ(ω,s)vs , where λ(ω,s) = λ(ωp,s) · · ·λ(ω1,s). we now rearrange the common eigenvectors { v1, . . . ,vd } as{ v1, . . . ,vκ,vκ+1, . . . ,vd } , where κ is defined as κ = # { vs : arvs = λ(r,s)vs with ∣∣λ(r,s)∣∣ = 1 for all 1 ≤ r ≤ n} . (4.1) 232 s. jayaraman, y.k. prajapaty, s. sridharan it is possible that κ = 0, in which case, the limiting vector is the zero vector (as you may observe by the end of this proof). recall from equation (2.2) that lc(e) = { α1v1 + · · · + αdvd : αj ∈ c satisfying αs1 = αs2 for all vs1 = vs2 and αj ∈ r otherwise } . owing to the hypotheses on the spectral radius in the statement of the theorem, we have that for every x ∈lc(e), the sequence { ‖akωx‖ } k≥1 is bounded. in fact,∥∥akωx∥∥ = ∥∥α1akωv1 + · · · + αdakωvd∥∥ ≤ ∣∣α1∣∣∥∥v1∥∥ + · · · + ∣∣αd∣∣∥∥vd∥∥ . let q1, . . . ,qn be positive integers that satisfies the outcome of theorem 1.1, for the matrices a1, . . . ,an respectively. for some p > n, let ω be a p-lettered word in σ p n such that for all 1 ≤ r ≤ n, there exists 1 ≤ j ≤ p such that ωj = r. define q to be the least common multiple of the numbers {q1, . . . ,qn}. for every s ∈ { 1, . . . ,d } and r ∈ { 1, . . . ,n } , we enumerate the following possibilities that can occur for the values of λ(r,s): case 1. ( λ(r,s) )q = 1 for every r and for some s with λ(r,s) ∈ r. this implies that the corresponding eigenvector vs lies in lc(e). case 2. ( λ(r,s) )q = 1 for every r and for some s with λ(r,s) ∈ c. this implies that there exists eigenvectors vs and vs with corresponding eigenvalues conjugate to each other such that αsvs + αsvs lies in lc(e). case 3. ∣∣λ(r,s)∣∣ < 1 for some s and for some r. in this case, the iterates of vs under the map aω goes to 0; that is, lim k→∞ akωvs = 0. for any x ∈lc(e) that can be written as x = α1v1 + · · ·+ αdvd, we have lim k→∞ akqω x = α1 lim k→∞ ( λ(ω,1) )kq v1 + · · · + αd lim k→∞ ( λ(ω,d) )kq vd = α1v1 + · · · + ακvκ =: ξ(x,ω). observe that ξ(x,ω) and q are independent of the length of the word and in fact, the word ω itself. we denote ξ(x,ω) = ξx. moreover, ξx ∈ lc(e). dynamics of matrix products 233 further, for ξx = α1v1 + · · · + ακvκ and 1 ≤ r ≤ n, we have aqrr ξx = α1a qr r v1 + · · · + ακa qr r vκ = α1 ( λ(r,1) )qr v1 + · · · + ακ ( λ(r,κ) )qr vκ = ξx . since ξx is a periodic point of a1, . . . ,an with periods q1, . . . ,qn respectively, we have ξx to be a periodic point of aω with period q. we now state a corollary to theorem 2.1, where we include conditions in the hypotheses that ensures lc(e) = rn. the corollary can be proved analogously to the above theorem. however, we present a simpler proof in this case. corollary 4.2. in addition to the hypothesis of theorem 2.1, assume that the considered collection of matrices is pairwise commuting with each matrix being diagonalizable over c. then, for any x ∈ rn, the same conclusion, as in theorem 2.1 holds. proof. we first observe that the extra hypotheses in the statement of the corollary ensures that the matrices a1, . . . ,an are simultaneously diagonalizable, i.e., there exists a nonsingular matrix q such that the matrix ar = q −1drq, for dr = diag ( λ(r,1),λ(r,2), . . . ,λ(r,n) ) , where λ(r,s) are the eigenvalues of the matrix ar, arranged in non-increasing modulus. let ω ∈ σpn with corresponding matrix product aω. then, aω = q −1dωq. by theorem 4.1, we have λ q (r,s) = 1 for every eigenvalue of modulus 1. hence, for every x ∈ rn, we have lim k→∞ akqω x = lim k→∞ ( q−1dkqω q ) x = ξx , where ξx is a periodic point of ar for every 1 ≤ r ≤ n and is given by ξx = α1v1 + . . . + ακvκ, the definition of κ, as in the proof of theorem 2.1. we now show that the diagonalizability condition can not be weakened in the hypothesis of corollary 4.2. however, since the matrices in the collection { a1, . . . ,an } commute pairwise, we still obtain a collection of common eigenvectors, e = { v1, . . . ,vd } . for example, consider the following pair of non-diagonalizable, commuting matrices: a1 = [ 1 1 0 1 ] and a2 = [ 1 2 0 1 ] . 234 s. jayaraman, y.k. prajapaty, s. sridharan observe that e1 = (1, 0) t is a common eigenvector for a1 and a2, whereas e2 = (0, 1) t is a common generalized eigenvector for a1 and a2. we further note that for any word ω that contains both the letters, the orbit of e2 under aω is unbounded while that of e1 is bounded. remarks 4.3. a few remarks are in order. (a) if all the matrices a1, . . . , an are pairwise commuting nonnegative symmetric matrices, subject to the spectral radius assumption in corollary 4.2, then the periods of all the periodic points corresponding to the eigenvalues 1 and −1 for all ar’s is at most 2. hence, for a matrix product aω corresponding to a word ω, we have q = 2. (b) we have proved theorem 2.1 for a special choice of ω that contains all the n letters. suppose ω′ is any arbitrary p-lettered word. then we can take the appropriate subset of { 1, . . . , n } , whose members have been used for the writing of the word ω′ and the same result as above follows for ω′. suppose the p-lettered word ω = (rr · · ·r) for some 1 ≤ r ≤ n. then the above theorem reduces to a particular case, as one may find in [7, 9]. we now state the same as a corollary. corollary 4.4. let a be an n×n matrix with nonnegative entries that is diagonalizable over c and of spectral radius 1. then there exists an integer q ≥ 1 such that for every x ∈ rn, we have lim k→∞ aqkx = ξx, where ξx is a periodic point of a with its period dividing q. 5. examples in this section, we provide several examples that illustrate the various results, that have been proved until now. we first fix a few notations. we denote the standard basis vectors of rn by e1, . . . ,en, while in denotes the identity matrix of order n. we write the permutation matrices of order n in column partitioned form denoted by pn; for instance, we denote the 2 × 2 permutation matrix [ e2 |e1 ] by p2. the matrix of 1’s (of order n) is denoted by jn. the diagonal matrix of order n with diagonal entries d1, . . . ,dn is denoted by diag(d1, . . . ,dn). our first example is a fairly simple one and illustrates the scenario in corollary 4.4. example 5.1. consider the diagonalizable matrix a = p2 with spectral radius 1. if x = e2 ∈ r2, then observe that dynamics of matrix products 235 akx = { e2 , if k is even , e1 , if k is odd . in this example, we obtain q = 2. the next two examples illustrate corollary 4.2. the first one involves a pair of 6 × 6 commuting nonnegative matrices. example 5.2. consider a1 = [ p4 0 0 a (22) 1 ] where p4 = [ e4 |e1 |e2 |e3 ] and a (22) 1 = 1 3 [ 1 2 2 1 ] ; a2 = [ a (11) 2 0 0 a (22) 2 ] where a (11) 2 = 1 2   0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0   and a (22) 2 = 1 10 [ 3 √ 7√ 7 3 ] . it can be easily seen that the matrices a1 and a2 commute and are diagonalizable over c and therefore, are simultaneously diagonalizable. the following table gives the common eigenvectors of a1 and a2 and the corresponding eigenvalues of the matrices a1 and a2. eigenvectors v1 v2 v3 v4 v5 v6 eigenvalues of a1 1 −1 −i i 1 −1/3 eigenvalues of a2 1 −1 0 0 λ1 λ2 where λ1 = 3 + √ 7 10 and λ2 = 3 − √ 7 10 . the common eigenvectors are given by v1 = (1, 1, 1, 1, 0, 0) t , v2 = (1,−1, 1,−1, 0, 0)t , v3 = (1, i,−1,−i, 0, 0)t, v4 = (1,−i,−1, i, 0, 0)t , v5 = (0, 0, 0, 0, 1, 1) t , v6 = (0, 0, 0, 0, 1,−1)t . 236 s. jayaraman, y.k. prajapaty, s. sridharan following the lines of the proof of corollary 4.2, we consider the nonsingular matrix (written in column partitioned form) q = [ v1| · · · |v6 ] . then, for r = 1, 2 we have ar = qdrq −1, where dr is the diagonal matrix consisting of the eigenvalues of ar. looking at the table of eigenvalues, one can see that q1 = 4 and q2 = 2. consider any x ∈ r6 given by x = 6∑ i=1 αivi where α1,α2,α5,α6 ∈ r and α3,α4 ∈ c with α3 = α4. for any word aω that contains both a1 and a2 , we have lim k→∞ a4kω x = ( q lim k→∞ d4kω q −1 ) x = α1v1 + α2v2 , a periodic point of aω with period at most 2, that divides the least common multiple of q1 and q2. we now present another pair of commuting and diagonalizable matrices, this time in r7, where we exhibit a periodic point of aω, whose period is equal to the least common multiple of the relevant qi’s. example 5.3. let a1 =  i3 0 00 p2 0 0 0 d1   where d1 = diag (1 2 , 1 3 ) ; a2 =  p3 0 00 i2 0 0 0 d2   where d2 = diag (1 5 , 1 6 ) and p3 = [ e3 |e1 |e2 ] . as earlier, we write a table with the common eigenvectors and the corresponding eigenvalues for the matrices a1 and a2. eigenvectors v1 v2 v3 v4 v5 v6 v7 eigenvalues of a1 1 1 1 1 −1 1/2 1/3 eigenvalues of a2 1 ω ω 2 1 1 1/5 1/6 where ω is the cubic root of unity and the vi’s are v1 = (1, 1, 1, 0, 0, 0, 0) t , v2 = (1,ω,ω 2, 0, 0, 0, 0)t , v3 = (1,ω 2,ω, 0, 0, 0, 0)t , v4 = (0, 0, 0, 1, 1, 0, 0) t , v5 = (0, 0, 0, 1,−1, 0, 0)t , v6 = e6 , v7 = e7 . dynamics of matrix products 237 in this example, we have q1 = 2 and q2 = 3. consider x ∈ r7 given by x = ∑7 i=1 αivi where α1,α4,α5,α6,α7 ∈ r, α2,α3 ∈ c with α2 = α3 and α2,α5 being non-zero. then, for the word aω = a p1 1 a p2 2 with p1 6= 0 (mod 2) and p2 6= 0 (mod 3), we have lim k→∞ a6kω x = ξx, a periodic point of aω of period 6, the least common multiple of q1 and q2. if p1 violates the above condition, then the period of ξx is 3; if p2 violates the above condition, then the period of ξx is 2 and if both p1 and p2 violate the above conditions, then the period of ξx is 1, all three numbers being factors of the least common multiple of q1 and q2. we now write two examples in the non-commuting set up that illustrates theorem 2.1. example 5.4. let a1 = [ p4 0 0 a′1 ] where a′1 = [ 1/5 1/6 1/6 1/5 ] and p4 = [ e4 |e1 |e2 |e3 ] ; a2 =  p2 0 00 p2 0 0 0 a′2   where a′2 = [1/7 1/81/7 1/8 ] . observe that the matrices a1 and a2 do not commute, but partially commute giving rise to the existence of a set of common eigenvectors that are given by v1 = (1, 1, 1, 1, 0, 0) t , v2 = (1,−1, 1,−1, 0, 0)t , v3 = (0, 0, 0, 0, 1, 1)t . the corresponding eigenvalues of the respective matrices are given in the following table. eigenvectors v1 v2 v3 eigenvalues of a1 1 −1 1/5 + 1/6 eigenvalues of a2 1 −1 1/7 + 1/8 in this case, we obtain q1 = 4 and q2 = 2. here, lc(e) ( { x ∈ r6 : sup ∥∥∥akωx∥∥∥ < ∞} , 238 s. jayaraman, y.k. prajapaty, s. sridharan where aω contains both a1 and a2. suppose x ∈ lc(e) given by x = α1v1 + α2v2 + α3v3 for αi ∈ r. then, we have lim k→∞ a4kω x = ξx, a periodic point of aω, however with period at most 2, that divides the least common multiple of q1 and q2. as in the commuting case, we now present an example in the non-commuting setup and exhibit a periodic point for a particular choice of aω whose period is equal to the least common multiple of the appropriate qi’s. example 5.5. let a1 =  i3 0 00 p2 0 0 0 1 2 j2   and a2 =  p3 0 00 i2 0 0 0 a′2   , where a′2 = [1/3 1/41/3 1/4 ] and p3 is the permutation matrix as defined in example 5.3. in this example, a1 and a2 form a laffey pair. they have the following six common eigenvectors: v1 = ( 1,ω,ω2, 0, 0, 0, 0 )t , v2 = ( 1,ω2,ω, 0, 0, 0, 0 )t , v3 = (1, 1, 1, 0, 0, 0, 0) t , v4 = (0, 0, 0, 1, 1, 0, 0) t , v5 = (0, 0, 0, 1,−1, 0, 0)t , v6 = (0, 0, 0, 0, 0, 1, 1)t . as earlier, we write the corresponding the eigenvalues of the matrices in the following table: eigenvectors v1 v2 v3 v4 v5 v6 eigenvalues of a1 1 1 1 1 −1 1 eigenvalues of a2 ω ω 2 1 1 1 1/3 + 1/4 here, q1 = 2 and q2 = 3. let aω be a matrix product such the matrix ar occurs pr times in aω and satisfies p1 6= 0 (mod 2) and p2 6= 0 (mod 3). for any vector x = α1v1 + · · · + α6v6 with α1,α2 ∈ c satisfying α1 = α2, α3,α4,α5,α6 ∈ r and α1,α5 being non-zero, we obtain lim k→∞ a6kω x = ξx, a periodic point of aω of period 6, the least common multiple of q1 and q2. if p1 violates the above condition, then the period of ξx is 3; if p2 violates the above condition, then the period of ξx is 2 and if both p1 and p2 violate the above conditions, then the period of ξx is 1, all three numbers being factors of the least common multiple of q1 and q2. dynamics of matrix products 239 at this juncture, we write one more example that showcases the dependence of the limiting periodic point on the word ω, in the non-commuting set-up, even when the non-common eigenvectors have a bounded orbit. example 5.6. let a1 = 1 3 [ 1 2 2 1 ] and a2 = 1 5 [ 1 4 2 3 ] . it can be easily seen that a1a2 6= a2a1. the eigenvalues of a1 and a2 are 1,−1 3 and 1,−1 5 respectively. the vector (1, 1)t is a common eigenvector for a1 and a2 corresponding to the eigenvalue 1. moreover, the eigenvalues of a1a2 are 1, 1 15 (and so the same is true for a2a1). it easily follows from this that any x ∈ r2 has a bounded orbit. the eigenvector corresponding to the eigenvalue −1 3 for a1 is (1,−1)t and the eigenvector corresponding to the eigenvalue −1 5 for a2 is (2,−1)t. note that (a1a2) k − (a2a1)k = [ −α(k) α(k) −α(k) α(k) ] for k ≥ 1 , and therefore the commutator has rank 1, making this a laffey pair. it is now obvious that lim k→∞ (a1a2) k (1, 1)t = lim k→∞ (a2a1) k (1, 1)t since ( (a1a2) k − (a2a1)k ) (1, 1)t = (0, 0)t. nevertheless, ( (a1a2) k − (a2a1)k ) (2,−1)t = −3α(k)(1, 1)t whereas( (a1a2) k − (a2a1)k ) (1,−1)t = −2α(k)(1, 1)t . therefore, if x is one of the points (2,−1)t or (1,−1)t, then, lim k→∞ (a1a2) k x 6= lim k→∞ (a2a1) k x, since lim k→∞ α(k) 6= 0. it is possible to study these examples under the action of the appropriate non-homogeneous map, described in corollary 2.2. we conclude this section by describing another way of writing theorem 2.1. recall that σ∞n denotes the set of all infinite-lettered words on the set of symbols { 1, . . . ,n } . considering the cartesian product of the symbolic space σ∞n and r n, one may describe the dynamical system discussed 240 s. jayaraman, y.k. prajapaty, s. sridharan in this paper thus: given a collection { a1, . . . ,an } of n × n matrices, let t : x = σ∞n × r n → x be defined by t(τ, x) = (στ,aτ1x) where τ =( τ1τ2τ3 · · · ) and σ is the shift map defined on σ∞n by (στ)n = τn+1 for n ≥ 1. we equip x with the corresponding product topology and study t as a non-invertible map. theorem 5.7. let { a1, . . . ,an } , n < ∞, be a collection of n × n matrices that satisfy the hypotheses of theorem 2.1. suppose e denotes the set of all common eigenvectors of the collection of matrices. let τ ∈ σ∞n be any arbitrary infinite lettered word that encounters all the n letters within a finite time, say m. let { τ [p] } p≥m be a sequence of infinite-lettered words that converges to τ. let aτ[p] be the matrix associated to the p-lettered word τ [p] ∈ σpn . then, for every p ≥ m and any vector x ∈lc(e), there exists an integer q ≥ 1 such that lim k→∞ tkpq ( τ [p],x ) = ( τ [p],ξx ) , where ( τ [p],ξx ) is a periodic point of t, whose period divides the least common multiple of p and q. 6. words of infinite length we conclude this paper with this final section where we write the proof of theorem 2.3. recall that the hypotheses of theorem 2.3 and corollary 4.2 are one and the same. proof of theorem 2.3. recall from equation (2.5) that whenever x ∈ lc(e) = rn (in this case), we have ãτ[p]x = ( ξx ,aτ[p]ξx , . . . ,a q−1 τ[p] ξx ) , for p ≥ m, where m is the finite stage by when the word τ [p] encounters all the letters in { 1, 2, . . . ,n } . in general, it is not necessary that aτ[m]ξx = aτ[m+1]ξx. however, owing to ξx being a periodic point of aτ[p] for p ≥ m, whose period divides q (> 1, say), a simple application of the pigeonhole principle ensures aτ[m]ξx = aτ[m′]ξx, for some m ′ > m. we choose m′ that guarantees ãτ[m]x = ãτ[m′]x, as vectors in (r n) q . dynamics of matrix products 241 proceeding along similar lines, one obtains an increasing sequence, say {pγ} such that { ã τ[pγ] x } γ≥1 is a constant sequence of vectors in (rn)q for every x ∈ rn. thus, for any two integers pγk and pγk′ from the sequence {pγ}, we have a j τ [pγk ]ξx = a j τ [pγ k′ ]ξx for every 0 ≤ j ≤ q − 1. since ξx = κ∑ s=1 αsvs, we obtain λ( τ [pγk ] ,s ) = λ( τ [pγ k′ ] ,s ) for every s ∈ i(x) ∩{1, . . . ,κ} . this implies that for every s ∈ i(x) ∩{1, . . . ,κ}, we have n∏ r=1 λ φ(τ,r)(pγk ) (r,s) = n∏ r=1 λ φ(τ,r)(pγk′ ) (r,s) ⇐⇒ n∏ r=1 λ φ(τ,r)(pγk )−φ(τ,r)(pγk′ ) (r,s) = 1 . since the numbers λ(r,s)’s are q-th roots of unity, we obtain positive integers λ(r,s) that satisfies n∑ r=1 λ(r,s) [ φ(τ,r)(pγk) − φ(τ,r)(pγk′) ] ≡ 0 (mod q) for all s ∈ i(x) ∩ { 1, . . . ,κ } . as pointed out after the statement of theorem 2.3 in section 2 and as one may observe from the proof above, { ã τ[pγ] x } γ≥1 is constructed to be a constant sequence, thus proving corollary 2.4. acknowledgements the authors are thankful to the 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[8] b. lemmens, r.d. nussbaum, “ nonlinear perron-frobenius theory ”, cambridge tracts in mathematics, 189, cambridge university press, cambridge, 2012. [9] r.d. nussbaum, s.m. verduyn lunel, generalizations of the perronfrobenius theorem for nonlinear maps, mem. amer. math. soc. 138 (1999), no. 659, viii+98. [10] d. shemesh, common eigenvectors of two matrices, linear algebra appl. 62 (1984), 11 – 18. [11] x. wang, z. cheng, infinite products of uniformly paracontracting matrices, linear multilinear algebra 64 (2016), 856 – 862. [12] x. zhan, “ matrix theory ”, graduate studies in mathematics, 147, american mathematical society, providence, ri, 2013. introduction main results common eigenvectors for a collection of matrices proof of theorem 2.1 examples words of infinite length e extracta mathematicae vol. 32, núm. 2, 209 – 211 (2017) corrigendum to “moore-penrose inverse and operator inequalities” extracta mathematicae 30 (2015), 29 – 39 ameur seddik department of mathematics, faculty of mathematics and computer science, university of batna 2, batna, algeria seddikameur@hotmail.com presented by manuel gonzález received january 18, 2017 abstract: we correct a mistake which affect our main results, namely the proof of lema 1. the main results of the article remain unchanged. key words: closed range operator, moore-penrose inverse, normal operator, operator inequality. ams subject class. (2010): 47a30, 47a03, 47b15. the paper mentioned in the title includes the following result as lemma 1: lemma 1. let s ∈ b(h). if s is surjective or injective with closed range and satisfies the following inequality ∀x ∈ b(h), ∥∥s2x∥∥ + ∥∥xs2∥∥ ≥ 2 ∥sxs∥ , (∗) then s is normal. in the proof of this lemma, the matrix representation of the operator r2 was computed incorrectly obtaining [ s∗1s1 0 0 s∗2s2 ] , while the correct form of this matrix is r2 = [ s∗1s1 0 0 (s1s2) ∗ (s1s2) ] . since all the results of the paper are based on this lemma, we shall give here a correct proof of it. the original proof is given in two cases. the second case follows immediately from the first one. the proof of the first case is divided in six steps. the mistake is in the fourth step. for s ∈ b(h) with closed range, s+ denotes the moore-penrose inverse of s. 209 210 a. seddik proof of lemma 1. assume that s ̸= 0 and that all 2×2 matrices used in this proof are given with respect to the orthogonal direct sum h = r(s) ⊕ ker s∗. then s = [ s 1 s 2 0 0 ] . we put p = |s| , q = |s∗| , p1 = |s1| , p2 = |s2| , q1 = (s1s∗1 + s2s ∗ 2) 1 2 . so we have s∗s = p 2 = [ p 21 s ∗ 1s2 s∗2s1 p 2 2 ] , ss∗ = q2 = [ q2 1 0 0 0 ] . it is clear that q1 is invertible and q + = [ q−11 0 0 0 ] . case 1. assume that s is injective with closed range and satisfies (∗). then s+s = i, ker p = ker s = {0}, and r(p) = r(s∗s) is closed (since r(s∗) is also closed). thus ker p = {0} and r(p) = (ker p)⊥ = h. so, p is invertible. note that inequality (∗) implies the following inequality: ∀x ∈ b(h), ∥∥s2s+xs+∥∥ + ∥∥s+xs∥∥ ≥ 2 ∥∥ss+x∥∥ . (1) the proof is given in four steps. step 1. ( s2 )+ s = s+. see step 2 of the original proof. step 2. ( s2 )+ = (s+)2. see step 3 of the original proof. step 3. ker s∗ = {0} . since s is injective, then ker s∗ = {0} if and only if s 2 = 0. assume that s 2 ̸= 0. since ( s2 )+ = (s+)2, then the two operators s∗s and ss+ commute (see [1, 2]). thus p 2 = [ p 2 1 0 0 p 22 ] , hence p = [ p 1 0 0 p2 ] . since ker s∗ ̸= {0}, then σ(q2) = σ(q21)∪{0}. from the fact that σ(p 2) = σ(q2) − {0}, we have σ(p 2) = σ(q21). then σ(p 2 1 ) ∪ σ(p 22 ) = σ(q 2 1). hence σ(p 2 1 ) ⊂ σ(q21). thus σ(p1) ⊂ σ(q1). using the polar decomposition of s and s∗ in inequality (1), we obtain the following inequality: ∀x ∈ b(h), ∥∥s2s+xp −1∥∥ + ∥∥q+xq∥∥ ≥ 2 ∥∥ss+x∥∥ . by taking x = [ x 1 0 0 0 ] (resp. x = [ 0 x2 0 0 ] ), where x 1 ∈ b(r(s)) (resp. x2 ∈ b(ker s∗, r(s))) in the last inequality, and since s2s+ =[ s 1 0 0 0 ] , we deduce the two following inequalities ∀x 1 ∈ b(r(s)), ∥∥p1x1p −11 ∥∥ + ∥∥q−11 x1q1∥∥ ≥ 2 ∥x1∥ , (2) corrigendum to “moore-penrose inverse and . . . ” 211 ∀x2 ∈ b(ker s∗, r(s)), ∥∥p1x2p −12 ∥∥ ≥ 2 ∥x2∥ . (3) by taking x2 = x ⊗ y (where x ∈ (r(s))1, y ∈ ker s∗) in (3), we obtain ∀x ∈ (r(s))1, ∀y ∈ ker s∗, ∥p1x∥ ∥∥p −12 y∥∥ ≥ 2 ∥y∥ . so we have ∀x ∈ (r(s))1, ∀y ∈ (ker s∗)1 , ∥p1x∥ ≥ 2 ∥p2y∥ . thus ∥p2y∥ ≤ k2, for every y ∈ (ker s ∗)1 (where k = inf∥x∥=1 ∥p1x∥ > 0), and then ⟨ p 22 y, y ⟩ ≤ k2/4, for every y ∈ (ker s∗)1. so we obtain σ(p 2 2 ) ⊂( 0, k 2 4 ] and σ(p 21 ) ⊂ [k 2, ∞). since σ(p 1 ) ⊂ σ(q 1 ), and p 1 , q 1 satisfy the inequality (2), then using a variation of [3, theorem 3.6] (in that paper theorem 3.6 is stated with equality between the spectra but the proof is the same for inclusion between the spectra), we obtain p1 = q1. hence σ(q 2 1 ) = σ(p 21 ) = σ(p 2 1 ) ∪ σ(p 22 ). then σ(p 22 ) ⊂ σ(p 2 1 ), that is impossible since ( 0, k 2 4 ] ∩[k2, ∞) = ∅. therefore ker s∗ = {0}. step 4. s is normal. since ker s∗ = {0}, we obtain r(s) = h. so that s is invertible and satisfies the inequality (∗). hence s satisfies the following inequality ∀x ∈ b(h), ∥∥sxs−1∥∥ + ∥∥s−1xs∥∥ ≥ 2 ∥x∥ . therefore s is normal (using [4]). case 2. assume that s is surjective and satisfies (∗). then s∗ is injective with closed range and satisfies inequality (∗). from case 1, s∗ is normal. hence s is normal, and the proof is finished. references [1] r. bouldin, the pseudo-inverse of a product, siam j. appl. math. 24 (1973), 489 – 495. [2] s. izumino, the product of operators with closed range and an extension of the reverse order law, tôhoku math. j. (2) 34 (1) (1982), 43 – 52. [3] a. seddik, some results related to the corach-porta-recht inequality, proc. amer. math. soc. 129 (2001), 3009 – 3015. [4] a. seddik, on the injective norm and characterization of some subclasses of normal operators by inequalities or equalities, j. math. anal. appl. 351 (1) (2009), 277 – 284. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae article in press available online december 1, 2022 construction of hom-pre-jordan algebras and hom-j-dendriform algebras t. chtioui 1, s. mabrouk 2, a. makhlouf 3 1 university of sfax, faculty of sciences sfax, bp 1171, 3038 sfax, tunisia 2 university of gafsa, faculty of sciences gafsa, 2112 gafsa, tunisia 3 université de haute alsace, irimas département de mathématiques f-68093 mulhouse, france chtioui.taoufik@yahoo.fr , mabrouksami00@yahoo.fr , abdenacer.makhlouf@uha.fr received january 25, 2022 presented by c. mart́ınez accepted october 18, 2022 abstract: the aim of this work is to introduce and study the notions of hom-pre-jordan algebra and hom-j-dendriform algebra which generalize hom-jordan algebras. hom-pre-jordan algebras are regarded as the underlying algebraic structures of the hom-jordan algebras behind the rota-baxter operators and o-operators introduced in this paper. hom-pre-jordan algebras are also analogues of hom-pre-lie algebras for hom-jordan algebras. the anti-commutator of a hom-pre-jordan algebra is a hom-jordan algebra and the left multiplication operator gives a representation of a hom-jordan algebra. on the other hand, a hom-j-dendriform algebra is a hom-jordan algebraic analogue of a hom-dendriform algebra such that the anti-commutator of the sum of the two operations is a hom-pre-jordan algebra. key words: hom-jordan algebra, hom-pre-jordan algebra, hom-j-dendriform algebra, o-operator. msc (2020): 17a15, 17c10, 17c50. introduction in order to study periodicity phenomena in algebraic k-theory, j.-l. loday introduced, in 1995, the notion of dendriform algebra (see [9]). dendriform algebras are algebras with two operations, which dichotomize the notion of associative algebra. later the notion of tridendriform algebra were introduced by loday and ronco in their study of polytopes and koszul duality (see [8]). in 2003 and in order to determine the algebraic structure behind a pair of commuting rota-baxter operators (on an associative algebra), aguiar and loday introduced the notion of quadri-algebra [1]. we refer to this kind of algebras as loday algebras. thus, it is natural to consider the jordan algebraic analogue of loday algebras as well as their lie algebraic analogue. jordan algebras were introduced in the context of axiomatic quantum mechanics in 1932 by the physicist p. jordan and appeared in many areas of issn: 0213-8743 (print), 2605-5686 (online) c© the author(s) released under a creative commons attribution license (cc by-nc 3.0) mailto:chtioui.taoufik@yahoo.fr mailto:mabrouksami00@yahoo.fr mailto:abdenacer.makhlouf@uha.fr https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 2 t. chtioui, s. mabrouk, a. makhlouf mathematics such as differential geometry, lie theory, physics and analysis (see [3, 7, 14] for more details). the jordan algebraic analogues of loday algebras were considered. indeed, the notion of pre-jordan algebra as a jordan algebraic analogue of a pre-lie algebra was introduced in [6]. a pre-jordan algebra is a vector space a with a bilinear multiplication · such that the product x◦y = x ·y + y ·x endows a with the structure of a jordan algebra, and the left multiplication operator l(x) : y 7→ x·y defines a representation of this jordan algebra on a. in other words, the product x ·y satisfies the following identities: (x◦y) · (z ·u) + (y ◦z) · (x ·u) + (z ◦x) · (y ·u) = z · [(x◦y) ·u] + x · [(y ◦z) ·u] + y · [(z ◦x) ·u], x · [y · (z ·u)] + z · [y · (x ·u)] + [(x◦z) ◦y] ·u = z · [(x◦y) ·u] + x · [(y ◦z) ·u] + y · [(z ◦x) ·u]. in order to find a dendriform algebra whose anti-commutator is a pre-jordan algebra, hou and bai introduced the notion of j-dendriform algebra [5]. they are, also related to pre-jordan algebras in the same way as pre-jordan algebras are related to jordan algebras. they showed that an o-operator (specially a rota-baxter operator of weight zero) on a pre-jordan algebra or two commuting rota-baxter operators on a jordan algebra give a j-dendriform algebra. in addition, they considered the relationships between j-dendriform algebras and loday algebras especially quadri-algebras. hom-type algebras have been investigated by many authors. in general, hom-type algebras are a kind of algebras in which the usual identities defining the structure are twisted by homomorphisms. such algebras appeared in 1990s in examples of q-deformations of witt and virasoro algebras. motivated by these examples and their generalization, hartwig, larsson and silvestrov introduced and studied hom-lie algebras in [4]. the notion of hom-jordan algebras was first introduced by a. makhlouf in [11] with a connection to hom-associative algebras and then d. yau modified slightly the definition in [15] and established their relationships with hom-alternative algebras. we aim in this paper to introduce and study hom-pre-jordan algebras and hom-j-dendriform algebras generalizing pre-jordan algebras and j-dendriform algebras. the anti-commutator of a hom-pre-jordan algebra is a hom-jordan algebra and the left multiplication operators give a representation of this homjordan algebra, which is the beauty of such a structure. similarly, a hom-jdendriform algebra gives rise to a hom-pre-jordan algebra and a hom-jordan hom-pre-jordan and hom-j-dendriform algebras 3 algebra in the same way as a hom-dendriform algebra gives rise to a hompre-lie algebra and a hom-lie algebra (see [10]). the paper is organized as follows. in section 1, we recall some basic facts about hom-jordan algebras. in section 2, we introduce the notions of hompre-jordan algebra and bimodule of a hom-pre-jordan algebra. we provide some properties and develop some construction theorems. in section 3, we introduce the notion of hom-j-dendriform algebra and study some of their fundamental properties in terms of o-operators of pre-jordan algebras. throughout this paper k is a field of characteristic 0 and all vector spaces are over k. we refer to a hom-algebra as a tuple (a,µ,α), where a is a vector space, µ is a multiplication and α is a linear map. it is said to be regular if α is invertible. a hom-associator with respect to a hom-algebra is a trilinear map asα defined for all x,y,z ∈ a by asα(x,y,z) = (xy)α(z) − α(x)(yz). we denote for simplicity the multiplication and composition by concatenation when there is no ambiguity. 1. basic results on hom-jordan algebras in this section, we recall some basics about hom-jordan algebras introduced in [15] and introduce the notion of a representation of a hom-jordan algebra. definition 1.1. a hom-jordan algebra is a hom-algebra (a,◦,α) satisfying the following conditions x◦y = y ◦x, (1.1) asα ( x◦x,α(y),α(x) ) = 0 , (1.2) for all x,y ∈ a. remark 1.1. since the characteristic of k is 0, condition (1.2) is equivalent to the following identity (for all x,y,z,u ∈ a) x,y,z asα(x◦y,α(u),α(z)) = 0 , (1.3) or equivalently, ((x◦y) ◦α(u)) ◦α2(z) + ((y ◦z) ◦α(u)) ◦α2(x) + ((z ◦x) ◦α(u)) ◦α2(y) = α(x◦y)(α(u) ◦α(z)) + α(y ◦z)(α(u) ◦α(x)) + α(z ◦x)(α(u) ◦α(y)) . (1.4) 4 t. chtioui, s. mabrouk, a. makhlouf definition 1.2. let (a,◦,α) be a hom-jordan algebra and v be a vector space. let ρ: a → gl(v ) be a linear map and φ: v → v be an algebra morphism. then (v,ρ,φ) is called a representation (or a module) of (a,◦,α) if for all x,y,z ∈ a φρ(x) = ρ(α(x))φ, (1.5) ρ(α2(x))ρ(y ◦z)φ + ρ(α2(y))ρ(z ◦x)φ + ρ(α2(z))ρ(x◦y)φ = ρ(α(x) ◦α(y))ρ(α(z))φ + ρ(α(y) ◦α(z))ρ(α(x))φ (1.6) + ρ(α(z) ◦α(x))ρ(α(y))φ, ρ((x◦y) ◦α(z))φ2 + ρ(α2(x))ρ(α(z))ρ(y) + ρ(α2(z))ρ(α(y))ρ(x) = ρ(α(x) ◦α(y))ρ(α(z))φ + ρ(α(y) ◦α(z))ρ(α(x))φ (1.7) + ρ(α(z) ◦α(x))ρ(α(y))φ. proposition 1.1. let (a,◦,α) be a hom-jordan algebra, then (v,ρ,φ) is a representation of a if and only if there exists a hom-jordan algebra structure on the direct sum a⊕v given by (x + u) ∗ (y + v) = x◦y + ρ(x)v + ρ(y)u, ∀x,y ∈ a, u,v ∈ v. (1.8) we denote it by a nρ,φ v or simply a n v . example 1.1. let (a,◦,α) be a hom-jordan algebra. let ad: a → gl(a) be a map defined by ad(x)(y) = x◦y = y◦x, for all x,y ∈ a. then (a,ad,α) is a representation of (a,◦,α) called the adjoint representation of a. definition 1.3. let (a,◦,α) be a hom-jordan algebra and (v,ρ,φ) be a representation. a linear map t : v → a is called an o-operator of a associated to ρ if it satisfies tφ = αt, (1.9) t(u) ◦t(v) = t ( ρ(t(u))v + ρ(t(v))u ) , ∀u,v ∈ v. (1.10) an o-operator on a associated to the adjoint representation (a,ad,α) is called a rota-baxter operator of weight zero. hence, a rota-baxter operator on a hom-jordan algebra (a,◦,α) is a linear map r: a → a satisfying rα = αr, (1.11) r(x) ◦r(y) = r ( r(x) ◦y + x◦r(y) ) , ∀x,y ∈ a. (1.12) hom-pre-jordan and hom-j-dendriform algebras 5 2. hom-pre-jordan algebras in this section, we generalize the notion of pre-jordan algebra introduced in [6] to the hom case and study the relationships with hom-jordan algebras, hom-dendriform algebras and hom-pre-alternative algebras in terms of ooperators of hom-jordan algebras. 2.1. definition and basic properties definition 2.1. a hom-pre-jordan algebra is a hom-algebra (a, ·,α) satisfying, for any x,y,z,u ∈ a, the following identities [α(x) ◦α(y)] · [α(z) ·α(u)] + [α(y) ◦α(z)] · [α(x) ·α(u)] + [α(z) ◦α(x)] · [α(y) ·α(u)] (2.1) = α2(x) · [(y ◦z) ·α(u)] + α2(y) · [(z ◦x) ·α(u)] + α2(z) · [(x◦y) ·α(u)], [(x◦z) ◦α(y)] ·α2(u) + α2(x) · [α(y) · (z ·u)] + α2(z) · [α(y) · (x ·u)] (2.2) = α2(x) · [(y ◦z) ·α(u)] + α2(y) · [(z ◦x) ·α(u)] + α2(z) · [(x◦y) ·α(u)], where x◦y = x·y+y·x. when α is an algebra morphism, the hom-pre-jordan algebra (a, ·,α) will be called multiplicative. remark 2.1. equations (2.1) and (2.2) are equivalent to the following equations (for any x,y,z,u ∈ a) respectively (x,y,z,u)1α + (y,z,x,u) 1 α + (z,x,y,u) 1 α (2.3) + (y,x,z,u)1α + (x,z,y,u) 1 α + (z,y,x,u) 1 α = 0 , asα(α(x),α(y),z ·u) −asα(x ·z,α(y),α(u)) + (y,z,x,u)2α (2.4) +(y,x,z,u)2α + asα(α(z),α(y),x ·u) −asα(z ·x,α(y),α(u)) = 0 , where (x,y,z,u)1α = [α(x) ·α(y)] · [α(z) ·α(u)] −α 2(x) · [(y ·z) ·α(u)], (x,y,z,u)2α = [α(x) ·α(y)] · [α(z) ·α(u)] − [α(x) · (y ·z)] ·α 2(u). remark 2.2. any hom-associative algebra is a hom-pre-jordan algebra. 6 t. chtioui, s. mabrouk, a. makhlouf proposition 2.1. let (a, ·,α) be a hom-pre-jordan algebra. then the product given by x◦y = x ·y + y ·x (2.5) defines a hom-jordan algebra structure on a, which is called the associated hom-jordan algebra of (a, ·,α) and (a, ·,α) is also called a compatible hompre-jordan algebra structure on the hom-jordan algebra (a,◦,α). proof. let x,y,z,u ∈ a, it is easy to show that ((x◦y) ◦α(u)) ◦α2(z) + ((y ◦z) ◦α(u)) ◦α2(x) + ((z ◦x) ◦α(u)) ◦α2(y) = (α(x) ◦α(y))(α(u) ◦α(z)) + (α(y) ◦α(z))(α(u) ◦α(x)) + (α(z) ◦α(x))(α(u) ◦α(y)) if and only if l1 + l2 + l3 + l4 = r1 + r2 + r3 + r4, where l1 = x,y,z α 2(x) · [(y ◦z) ·α(u)], l2 = [(x◦y) ◦α(u)] ·α2(z) + α2(x) · [α(u) · (y ·z)] + α2(y) · [α(u) · (x ·z)], l3 = [(x◦z) ◦α(u)] ·α2(y) + α2(x) · [α(u) · (z ·y)] + α2(z) · [α(u) · (x ·u)], l4 = [(y ◦z) ◦α(u)] ·α2(x) + α2(y) · [α(u) · (z ·x)] + α2(z) · [α(u) · (y ·x)], and r1 = x,y,z (α(x) ◦α(y)) · (α(z) ·α(u)), r2 = x,y,u (α(x) ◦α(y)) · (α(u) ·α(z)), r3 = x,z,u (α(x) ◦α(z)) · (α(u) ·α(y)), r4 = y,z,u (α(y) ◦α(z)) · (α(u) ·α(x)). now using definition 2.1, we can easily see that li = ri, for i = 1, . . . , 4. example 2.1. consider the 2-dimensional vector space a generated by the basis {e1,e2} and define the multiplication · e1 e2 e1 e1 0 e2 0 0 and the linear map α(e1) = e1 , α(e2) = 0 . hom-pre-jordan and hom-j-dendriform algebras 7 then (a, ·,α) is a hom-pre-jordan algebra. according to the above proposition, the associated hom-jordan algebra (a,◦,α) is given by ◦ e1 e2 e1 2e1 0 e2 0 0 . the following conclusion can be obtained straightforwardly using the previous proposition. proposition 2.2. let (a, ·,α) be a hom-algebra. then (a, ·,α) is a hom-pre-jordan algebra if and only if (a,◦,α) defined by equation (2.5) is a hom-jordan algebra and (a,l,α) is a representation of (a,◦,α), where l denotes the left multiplication operator on a. proof. straightforward. proposition 2.3. let (a,◦,α) be a hom-jordan algebra and (v,ρ,φ) be a representation. if t is an o-operator associated to ρ, then (v,∗,φ) is a hom-pre-jordan algebra, where u∗v = ρ(t(u))v, ∀u,v ∈ v. (2.6) therefore there exists an associated hom-jordan algebra structure on v given by equation (2.5) and t is a homomorphism of hom-jordan algebras. moreover, t(v ) = {t(v)|v ∈ v}⊂ a is a hom-jordan subalgebra of (a,◦,α) and there is an induced hom-pre-jordan algebra structure on t(v ) given by t(u).t(v) = t(u∗v), ∀u,v ∈ v. (2.7) the corresponding associated hom-jordan algebra structure on t(v ) given by equation (2.5) is just a hom-jordan subalgebra of (a,◦,α) and t is a homomorphism of hom-pre-jordan algebras. proof. let u,v,w,a ∈ v and set x = t(u), y = t(v), z = t(w) and u•v = u∗v + v ∗u. note first that t(u•v) = t(u) ◦t(v). then (φ(u) •φ(v)) ∗ (φ(w) ∗φ(a)) = ρ(t(ρ(t(φ(u) •φ(v)))))ρ(t(φ(w)))φ(a) = ρ(t(φ(u)) ◦t(φ(v)))ρ(t(φ(w)))φ(a) = ρ(α(x) ◦α(y))ρ(α(z))φ(a), 8 t. chtioui, s. mabrouk, a. makhlouf φ2(u) ∗ [(v •w) ∗φ(a)] = ρ(t(φ2(u)))ρ(t(v •w))φ(a) = ρ(t(φ2(u)))ρ(t(v) ◦t(w))φ(a) = ρ(α2(x))ρ(y ◦z)φ(a), φ2(u) ∗ [φ(v) ∗ (w ∗a)] = ρ(t(φ2(u)))ρ(t(φ(v)))ρ(t(w))a = ρ(α2(x))ρ(α(y))ρ(z)a, [(u•v) •φ(w)] ∗φ2(a) = ρ(t([(u•v) •φ(w)]))φ2(a) = ρ([t(u•v) ◦t(φ(w))])φ2(a) = ρ([(t(u) ◦t(v)) ◦t(φ(w))])φ2(a) = ρ([(x◦y) ◦α(z)])φ2(a). hence, (φ(u)•φ(v)) ∗ (φ(w) ∗φ(a)) + (φ(v) •φ(w)) ∗ (φ(u) ∗φ(a)) + (φ(w) •φ(u)) ∗ (φ(v) ∗φ(a)) = ρ(α(x) ◦α(y))ρ(α(z))φ(a) + ρ(α(y) ◦α(z))ρ(α(x))φ(a) + ρ(α(z) ◦α(x))ρ(α(y))φ(a) = ρ(α2(x))ρ(y ◦z)φ(a) + ρ(α2(y))ρ(z ◦x)φ(a) + ρ(α2(z))ρ(x◦y)φ(a) + φ2(u) ∗ [(v •w) ∗φ(a)] + φ2(v) ∗ [(w •u) ∗φ(a)] + φ2(w) ∗ [(u•v) ∗φ(a)], and [(u•v)•φ(w)] ∗φ2(a) + φ2(u) ∗ [φ(w) ∗ (v ∗a)] + φ2(w) ∗ [φ(v) ∗ (u∗a)] = ρ([(x◦y) ◦α(z)])φ2(a) + ρ(α2(x))ρ(α(z))ρ(y)a + ρ(α2(z))ρ(α(y))ρ(x)a = ρ(α2(x))ρ(y ◦z)φ(a) + ρ(α2(y))ρ(z ◦x)φ(a) + ρ(α2(z))ρ(x◦y)φ(a) + φ2(u) ∗ [(v •w) ∗φ(a)] + φ2(v) ∗ [(w •u) ∗φ(a)] + φ2(w) ∗ [(u•v) ∗φ(a)]. therefore, (v,∗,φ) is a hom-pre-jordan algebra. the other conclusions follow immediately. hom-pre-jordan and hom-j-dendriform algebras 9 an obvious consequence of proposition 2.3 is the following construction of a hom-pre-jordan algebra in terms of a rota-baxter operator (of weight zero) of a hom-jordan algebra. corollary 2.1. let (a,◦,α) be a hom-jordan algebra and r be a rotabaxter operator (of weight zero) on a. then there is a hom-pre-jordan algebra structure on a given by x ·y = r(x) ◦y, ∀x,y ∈ a. proof. straightforward. example 2.2. let {e1,e2} be a basis of a 2-dimensional vector space a over k. the following product ◦ and the linear map α define, for any scalar a, a hom-jordan algebra on a: ◦ e1 e2 e1 2e1 2ae2 e2 2ae2 0 , α(e1) = e1 , α(e2) = ae2 . define the linear map r: a → a with respect to the basis {e1,e2} by r(e1) = be2 , r(e2) = 0 . then r is a rota-baxter operator on the hom-jordan algebra (a,◦,α), where a and b are parameters in k. using corollary 2.1, there is a hom-pre-jordan algebra structure, with respect the same twist map α, given by the following multiplication table · e1 e2 e1 2abe2 0 e2 0 0 . example 2.3. let {e1,e2,e3} be a basis of a 3-dimensional vector space a over k. the following product ◦ and the linear map α define the following hom-jordan algebras over k. ◦ e1 e2 e3 e1 ae1 ae2 be3 e2 ae2 ae2 b 2 e3 e3 be3 b 2 e3 0 , α(e1) = ae1, α(e2) = ae2, α(e3) = be3, 10 t. chtioui, s. mabrouk, a. makhlouf where a and b are parameters in k. let r be the operator defined with respect to the basis {e1,e2,e3} by r(e1) = λ1e3, r(e2) = λ2e3, r(e3) = 0, where λ1 and λ2 are parameters in k. then we can easily check that r is a rota-baxter operator on a. now, using corollary 2.1, there is a hom-prejordan algebra structure on a, with the same twist map and a multiplication given by x ·y = r(x) ◦y for all x,y ∈ a, that is · e1 e2 e3 e1 λ1be3 λ1 b 2 e3 0 e2 λ2be3 λ2 b 2 e3 0 e3 0 0 0 . corollary 2.2. let (a,◦,α) be a hom-jordan algebra. then there exists a compatible hom-pre-jordan algebra structure on a if and only if there exists an invertible o-operator of (a,◦,α). proof. let (a, ·,α) be a hom-pre-jordan algebra and (a,◦,α) be the associated hom-jordan algebra. then the identity map id: a → a is an invertible o-operator of (a,◦,α) associated to (a,ad,α). conversely, suppose that there exists an invertible o-operator t of (a,◦,α) associated to a representation (v,ρ,φ), then by proposition 2.3, there is a hom-pre-jordan algebra structure on t(v ) = a given by t(u) ·t(v) = t(ρ(t(u))v), for all u,v ∈ v. if we set t(u) = x and t(v) = y, then we obtain x ·y = t(ρ(x)t−1(y)), for all x,y ∈ a. it is a compatible hom-pre-jordan algebra structure on (a,◦,α). indeed, x ·y + y ·x = t ( ρ(x)t−1(y) + ρ(y)t−1(x) ) = t(t−1(x)) ◦t(t−1(y)) = x◦y. the following result reveals the relationship between hom-pre-jordan algebras, hom-pre-alternative algebras and so hom-dendriform algebras. we recall the following definitions introduced in [12, 10]. hom-pre-jordan and hom-j-dendriform algebras 11 definition 2.2. a hom-pre-alternative algebra is a quadruple (a,≺,�, α), where ≺,�: a⊗a → a and α: a → a are linear maps satisfying (x � y) ≺ α(z) −a(x) � (y ≺ z) + (y ≺ x) ≺ α(z) −a(y) ≺ (x ? z) = 0, (2.8) (x � y) ≺ α(z) −a(x) � (y ≺ z) + (x ? z) � α(y) −a(x) � (z � y) = 0, (2.9) (x ≺ y) ≺ α(z) −a(x) ≺ (y ? z) + (x ≺ z) ≺ α(y) −a(x) ≺ (z ? y) = 0, (2.10) (x ? y) � α(z) −a(x) � (y � z) + (y ? x) � α(z) −a(y) � (x � z) = 0, (2.11) for all x,y,z ∈ a, where x ? y = x ≺ y + x � y. definition 2.3. a hom-dendriform algebra is a quadruple (a,≺,�,α), where ≺,�: a⊗a → a and α : a → a are linear maps satisfying (x � y) ≺ α(z) −a(x) � (y ≺ z) = 0, (2.12) (x ≺ y) ≺ α(z) −a(x) ≺ (y ? z) = 0, (2.13) (x ? y) � α(z) −a(x) � (y � z) = 0, (2.14) for all x,y,z ∈ a, where x ? y = x ≺ y + x � y. proposition 2.4. let (a,≺,�,α) be a hom-pre-alternative algebra. then the product given by x ·y = x � y + y ≺ x, ∀x,y ∈ a, defines a hom-pre-jordan algebra structure on a. proof. let x,y,z,u ∈ a, set x?y = x ≺ y +x � y and x◦y = x·y +y ·x = x ? y + y ? x. we will just prove the identity (2.1). one has x,y,z ( [α(x) ◦α(y)] · [α(z) ·α(u)] −α2(x) · [(y ◦z) ·α(u)] ) = x,y,z ( [α(x) ◦α(y)] � [α(z) �α(u)] + [α(x) ◦α(y)] � [α(u) ≺α(z)] + [α(z) �α(y)] ≺ [α(x) ◦α(y)] + [α(u) ≺α(z)] ≺ [α(x) ◦α(y)] −α2(x) � [(y ◦z) �α(u)] −α2(x) � [α(u) ≺ (y ◦z)] − [(y ◦z) �α(u)] ≺α2(x) − [α(u) ≺ (y ◦z)] ≺α2(x) ) = x,y,z ( [(x◦y) ◦α(z)] �α2(u) + [α(u) ≺α(z)] ≺ [α(x) ◦α(y)] − [α(u) ≺ (y ◦z)] ≺α2(x) ) . 12 t. chtioui, s. mabrouk, a. makhlouf since (a,?,α) is a hom-alternative algebra (see [12]), we have x,y,z ( [(x◦y) ◦α(z)] �α2(u) ) = 0. in addition using the fact that (a,≺,�,α) is a hom-pre-alternative algebra, then we obtain x,y,z ( [(x◦y) ◦α(z)] �α2(u) + [α(u) ≺α(z)] ≺ [α(x) ◦α(y)] − [α(u) ≺ (y ◦z)] ≺α2(x) ) = 0. the identity (2.2) can be obtained similarly. since any hom-dendriform algebra is a hom-pre-alternative algebra, we obtain the following conclusion. corollary 2.3. let (a,≺,�,α) be a hom-dendriform algebra. then the product given by x ·y = x � y + y ≺ x, ∀x,y ∈ a, defines a hom-pre-jordan algebra structure on a. 2.2. bimodules and o-operators in this section, we introduce and study bimodules of hom-pre-jordan algebras. definition 2.4. let (a, ·,α) be a hom-pre-jordan algebra and v be a vector space. let l,r : a → gl(v ) be two linear maps and φ ∈ gl(v ). then (v,l,r,φ) is called a bimodule of a if the following conditions hold (for any x,y,z ∈ a): φl(x) = l(α(x))φ, φr(x) = r(α(x))φ, (2.15) l(α2(x))l(y ◦z)φ + l(α2(y))l(z ◦x)φ + l(α2(z))l(x◦y)φ = l(α(x) ◦α(y))l(α(z))φ + l(α(y) ◦α(z))l(α(x))φ (2.16) + l(α(z) ◦α(x))l(α(y))φ, l((x◦z) ◦α(y))φ2 + l(α2(x))l(α(z))l(y) + l(α2(z))l(α(y))l(x) = l(α(x) ◦α(y))l(α(z))φ + l(α(y) ◦α(z))l(α(x))φ (2.17) + l(α(z) ◦α(x))l(α(y))φ, hom-pre-jordan and hom-j-dendriform algebras 13 φ ( l(x◦y)r(z) + r(x ·z)l(y) + r(y ·z)r(x) + r(x ·z)r(y) + r(y ·z)l(x) ) = l(α2(x))r(α(z))l(y) + l(α2(y))r(α(z))r(x) + r[(x◦y)α(z)]φ2 (2.18) + l(α2(y))r(α(z))l(x) + l(α2(x))r(α(z))r(y), φ ( r(z ·y)l(x) + r(x ·y)r(z) + l(x◦z)r(y) + r(x ·y)l(z) + r(z ·y)r(x) ) = ( l(α2(x))r(z ·y) + r(α2(y))r(x◦z) (2.19) + r(α2(y))l(x◦z) + l(α2(z))r(x ·y) ) φ, φ ( l(x ·y)r(z) + r(x ·z)l(y) + r(y ·z)r(x) + l(y ·x)r(z) + r(x ·z)r(y) + r(y ·z)l(x) ) = l(α2(x))l(α(y))r(z) + r(α2(z))l(α(y))r(x) (2.20) + r(α2(z))r(α(y))r(x) + r(α2(z))l(α(y))l(x) + r[α(y) · (x ·z)]φ2 + r(α2(z))r(α(y))l(x), where x◦y = x ·y + y ·x. proposition 2.5. let (a, ·,α) be a hom-pre-jordan algebra, v be a vector space, l,r : a → gl(v ) be linear maps and φ ∈ gl(v ). then (v,l,r,φ) is a bimodule of a if and only if the direct sum a⊕v (as vector spaces) turns into a hom-pre-jordan algebra (semidirect sum) by defining the multiplication in a⊕v as (x + u) ∗ (y + v) = x ·y + l(x)v + r(y)u, ∀x,y ∈ a, u,v ∈ v. we denote it by a nα,φl,r v or simply a n v . proposition 2.6. let (v,l,r,φ) be a bimodule of a hom-pre-jordan algebra (a, ·,α) and (a,◦,α) be its associated hom-jordan algebra. then (a) (v,l,φ) is a representation of (a,◦,α), (b) (v,l + r,φ) is a representation of (a,◦,α). proof. (a) follows immediately from equations (2.16)-(2.17). for (b), by proposition 2.5, anα,φl,r v is a hom-pre-jordan algebra. consider its associated 14 t. chtioui, s. mabrouk, a. makhlouf hom-jordan algebra (a⊕v, ◦̃,α + φ), we have (x + u)◦̃(y + v) = (x + u) ∗ (y + v) + (y + v) ∗ (x + u) = x ·y + l(x)v + r(y)u + y ·x + l(y)u + r(x)v = x◦y + (l + r)(x)v + (l + r)(y)u. according to proposition 1.1, we deduce that (v,l + r,φ) is a representation of (a,◦,α). definition 2.5. let (a, ·,α) be a hom-pre-jordan algebra and (v,l,r,φ) be a bimodule. a linear map t : v → a is called an o-operator of (a, ·,α) associated to (v,l,r,φ) if tφ = αt, (2.21) t(u) ·t(v) = t ( l(t(u))v + r(t(v))u ) , ∀u,v ∈ v. (2.22) in particular, a rota-baxter operator (of weight zero) on a hom-pre-jordan algebra (a, ·,α) is a linear map r : a → a satisfying rα = αr, (2.23) r(x) ·r(y) = r ( r(x) ·y + x ·r(y) ) , ∀x,y ∈ a. (2.24) remark 2.3. if t is an o-operator of a hom-pre-jordan algebra (a, ·,α) associated to (v,l,r,φ), then t is an o-operator of its associated hom-jordan algebra (a,◦,α) associated to (v,l + r,φ). 3. hom-j-dendriform algebras in this section, we introduce the notion of hom-j-dendriform algebra and discuss the relationship with hom-pre-jordan algebras. definition 3.1. a hom-j-dendriform algebra is a quadruple (a,≺,�,α), where a is a vector space equipped with a linear map α: a → a and two products denoted by ≺,�: a⊗a → a satisfying the following identities (for any x,y,z,u ∈ a) α(x◦y) �α(z �u) + α(y ◦z) �α(x�u) + α(z ◦x) �α(y �u) = α2(x) � [(y ◦z) �α(u)] + α2(y) � [(z ◦x) �α(u)] (3.1) + α2(z) � [(x◦y) �α(u)], hom-pre-jordan and hom-j-dendriform algebras 15 α(x◦y) �α(z �u) + α(y ◦z) �α(x�u) + α(z ◦x) �α(y �u) = α2(x) � [α(y) � (z �u)] + α2(z) � [α(y) � (x�u)] (3.2) + [α(y) ◦ (z ◦x)] �α2(u), α(x◦y) �α(z ≺u) + α(x ·z) ≺α(y �u) + α(y ·z) ≺α(x�u) = α2(x) � [α(z) ≺ (y �u)] + α2(y) � [α(z) ≺ (x�u)] (3.3) + [(x◦y) ·α(z)] ≺α2(u), α(z ·y) ≺α(x�u) + α(x ·y) ≺α(z �u) + α(x◦z) �α(y ≺u) = α2(x) � [(z ·y) ≺α(u)] + α2(z) � [(x ·y) ≺α(u)] (3.4) + α2(y) ≺ [(x◦z) �α(u)], α(x◦y) �α(z ≺u) + α(x ·z) �α(y �u) + α(y ·z) ≺α(x�u) = α2(x) � [α(y) � (z ≺u)] + α2(z) ≺ [α(y) � (z �u)] (3.5) + [α(y) · (x ·z)] ≺α2(u), where x ·y = x�y + y ≺x, (3.6) x�y = x�y + x≺y, (3.7) x◦y = x ·y + y ·x = x�y + y �x. (3.8) remark 3.1. let (a,≺,�,α) be a hom-j-dendriform algebra. if ≺ := 0 then (a,�,α) is a hom-pre-jordan algebra. proposition 3.1. let (a,≺,�,α) be a hom-j-dendriform algebra. (a) the product given by equation (3.6) defines a hom-pre-jordan algebra (a, ·,α), called the associated vertical hom-pre-jordan algebra. (b) the product given by equation (3.7) defines a hom-pre-jordan algebra (a,�,α), called the associated horizontal hom-pre-jordan algebra. (c) the associated vertical and horizontal hom-pre-jordan algebras (a, ·,α) and (a,�,α) have the same associated hom-jordan algebra (a,◦,α) defined by equation (3.8), called the associated hom-jordan algebra of (a,≺,�,α). 16 t. chtioui, s. mabrouk, a. makhlouf proof. we will just prove (a). let x,y,z,u ∈ a [α(x) ◦α(y)] · [α(z) ·α(u)] + [α(y) ◦α(z)] · [α(x) ·α(u)] + [α(z) ◦α(x)] · [α(y) ·α(u)] = [α(x) ◦α(y)] � [α(z) �α(u)] + [α(y) ◦α(z)] � [α(x) �α(u)] + [α(z) ◦α(x)] � [α(y) �α(u)] + [α(x) ◦α(y)] � [α(u) ≺α(z)] + [α(x) ·α(u)] ≺ [α(y) �α(z)] + [α(y) ·α(u)] ≺ [α(x) �α(z)] + [α(y) ◦α(z)] � [α(u) ≺α(x)] + [α(z) ·α(u)] ≺ [α(y) �α(x)] + [α(y) ·α(u)] ≺ [α(z) �α(x)] + [α(z) ◦α(x)] � [α(u) ≺α(y)] + [α(x) ·α(u)] ≺ [α(z) �α(y)] + [α(z) ·α(u)] ≺ [α(x) �α(y)] = α2(x) � [(y ◦z) �α(u)] + α2(y) � [(z ◦x) �α(u)] + α2(z) � [(x◦y) �α(u)] + α2(x) � [α(u) ≺ (y �z)] + α2(y) � [α(u) ≺ (x�z)] + [(x◦y) ·α(u)] ≺α2(x) + α2(z) � [α(u) ≺ (y �x)] + α2(y) � [α(u) ≺ (z �x)] + [(z ◦y) ·α(u)] ≺α2(x) + α2(z) � [α(u) ≺ (x�y)] + α2(x) � [α(u) ≺ (z �y)] + [(z ◦x) ·α(u)] ≺α2(y) = α2(x) · [(y ◦z)·α(u)] + α2(y) · [(z ◦x)·α(u)] + α2(z) · [(x◦y)·α(u)]. similarly, we get (2.2). proposition 3.2. let (a,≺,�,α) be a hom-j-dendriform algebra. then (a,l�,r≺,α) is a bimodule of its associated horizontal hom-pre-jordan algebra (a,�,α). proof. we check equation (2.16) and equation (2.19). indeed, for any x,y,z,u ∈ a, we have l�(α 2(x))l�(y ◦z)α(u) + l�(α2(y))l�(z ◦x)α(u) + l�(α 2(z))l�(x◦y)α(u) = α2(x) � [(y ◦z) �α(u)] + α2(y) � [(z ◦x) �α(u)] + α2(z) � [(x◦y) �α(u)] = α(x◦y) �α(z �u) + α(y ◦z) �α(x�u) + α(z ◦x) �α(y �u) = l�(α(x) ◦α(y))l�(α(z))α(u) + l�(α(y) ◦α(z))l�(α(x))α(u) + l�(α(z) ◦α(x))l�(α(y))α(u). hom-pre-jordan and hom-j-dendriform algebras 17 moreover, α ( r≺(z �y)l�(x)u + r≺(x�y)r≺(z)u + l�(x◦z)r≺(y)u + r≺(x�y)l�(z)u + r≺(z �y)r≺(x)u ) = α(x�u) ≺α(z �y) + α(u≺z) ≺α(x�y) + α(x◦z) �α(u≺y) + α(z �u) ≺α(x�y) + α(u≺x) ≺α(z �y) = α(x ·u) ≺α(z �y) + α(z ·u) ≺α(x�y) + α(x◦z) �α(u≺y) = α2(x) � [α(u) ≺ (z �y)] + α2(z) � [α(u) ≺ (x�y)] + [(x◦z) ·α(u)] ≺α2(y) = l�(α 2(x))r≺(z �y)α(u) + l�(α2(z))r≺(x�y)α(u) + r≺(α 2(y))l�(x◦z)α(u) + r≺(α2(y))r≺(x◦z)α(u). other identities can be proved using similar computations. proposition 3.3. let (a,≺,�) be a j-dendriform algebra and α: a → a be an algebra morphism. then (a,≺α,�α,α) is a hom-j-dendriform algebra, where for any x,y ∈ a x≺α y = α(x) ≺α(y), x�α y = α(x) �α(y). proof. straightforward. example 3.1. let a be a three dimensional vector space with basis {e1, e2,e3}. then (a, ·) is a pre-jordan algebra, where the formal characteristic matrix is given by · e1 e2 e3 e1 e1 e2 e3 e2 −e2 e2 e3 e3 −e3 0 0 . let r: a → a be the linear map defined with respect to the basis {e1,e2,e3} by the matrix   0 r12 r130 r12 r13 0 r32 −r12   with r212 + r13r32 = 0 . 18 t. chtioui, s. mabrouk, a. makhlouf it is easy to check that r is a rota-baxter operator on a, see [13]. therefore, it induces a j-dendriform algebra structure on a given by ≺ e1 e2 e3 e1 0 r12e1 + r12e2 + r32e3 r13e1 + r13e2 −r12e3 e2 0 r32e3 −r12e3 e3 0 −r12e3 −r13e3 , � e1 e2 e3 e1 0 0 0 e2 r12e1 + r12e2 + r32e3 r32e2 2r12e3 e3 r13e1 −r13e2 + r12e3 0 2r13e3 . consider, now the linear map α: a → a defined by α(e1) = e1, α(e2) = e2, α(e3) = λe3, λ 6= 0 . it is easy to check that α is a morphism of j-dendriform algebras. then according to proposition 3.3, (a,≺α,�α,α) is a hom-j-dendriform algebra. proposition 3.4. let (a,≺,�,α) be a hom-j-dendriform algebra. define two bilinear products ≺t,�t on a by x≺t y = y ≺x, x�t y = y �x, ∀x,y ∈ a. (3.9) then (a,≺t,�t,α) is a hom-j-dendriform algebra called the transpose of a. moreover, its associated horizontal hom-pre-jordan algebra is the associated vertical hom-pre-jordan algebra (a, ·,α) of (a,≺,�,α) and its associated vertical hom-pre-jordan algebra is the associated horizontal hom-pre-jordan algebra (a,�,α) of (a,≺,�,α). proof. note first that x ·t y = x�t y + y ≺t x = x�y + x≺y = x�y, x�t y = x�t y + x≺t y = x�y + y ≺x = x ·y, x◦t y = x�t y + x≺t y + y �t x + y ≺t x = x�y + y ≺x + y �x + x≺y = x◦y. therefore we can easily check (equation (3.1))t=(equation (3.1)), (equation (3.2))t=(equation (3.2)), (equation (3.3))t=(equation (3.4)), (equation (3.4))t =(equation (3.3)) and (equation (3.5))t=(equation (3.5)). hence (a,≺t,�t,α) is a hom-j-dendriform algebra. hom-pre-jordan and hom-j-dendriform algebras 19 proposition 3.5. let (a, ·,α) be a hom-pre-jordan algebra and (v,l, r,φ) be a bimodule. let t : v → a be an o-operator of a associated to (v,l,r,φ). then there exists a hom-j-dendriform algebra structure on v given by u≺v = r(t(u))v, u�v = l(t(u))v, ∀u,v ∈ v. (3.10) therefore, there is a hom-pre-jordan algebra on v given by equation (3.6) as the associated vertical hom-pre-jordan algebra of (v,≺,�) and t is a homomorphism of hom-pre-jordan algebras. moreover, t(v ) = {t(v) |v ∈ v} ⊂ a is a hom-pre-jordan subalgebra of (a, ·,α), and there is an induced hom-j-dendriform algebra structure on t(v ) given by t(u) ≺t(v) = t(u≺v), t(u) �t(v) = t(u�v), ∀u,v ∈ v. (3.11) furthermore, its corresponding associated vertical hom-pre-jordan algebra structure on t(v ) is just the subalgebra of the hom-pre-jordan (a, ·,α), and t is a homomorphism of hom-j-dendriform algebras. proof. for any a,b,c,u ∈ v , we set t(a) = x, t(b) = y and t(c) = z. then φ(a◦ b) �φ(c�u) = (φ(a) �φ(b) + φ(b) ≺φ(a) + φ(b) �φ(a) + φ(a) ≺φ(b)) � (φ(c) �φ(u)) = ( l(t(φ(a)))φ(b) + r(t(φ(b)))φ(a) + l(t(φ(b)))φ(a) + r(t(φ(a)))φ(b) ) � l(t(φ(c)))φ(u) = l(t(l(t(φ(a)))φ(b) + r(t(φ(b)))φ(a) + l(t(φ(b)))φ(a) + r(t(φ(a)))φ(b)))l(t(φ(c)))φ(u) = l(t(φ(a)) ·t(φ(b)) + t(φ(b)) ·t(φ(a)))l(t(φ(c)))φ(u) = l(α(x) ◦α(y))l(α(z))φ(u) 20 t. chtioui, s. mabrouk, a. makhlouf and φ2(a)�[(b◦ c) �φ(u)] = φ2(a) � [(b� c + c≺ b + c� b + b≺ c) �φ(u)] = φ2(a) � l(t(l(t(b))c + r(t(c))b + l(t(c))b + r(t(b))c))φ(u) = φ2(a) � l(y ◦z)φ(u) = l(t(φ2(a)))l(y ◦z)φ(u) = l(α2(x))l(y ◦z)φ(u). hence φ(a◦ b) �φ(c�u) + φ(b◦ c) �φ(a�u) + φ(c◦a) �φ(b�u) = l(α(x) ◦α(y))l(α(z))φ(u) + l(α(y) ◦α(z))l(α(x))φ(u) + l(α(z) ◦α(x))l(α(y))φ(u) = l(α2(x))l(y ◦z)φ(u) + l(α2(y))l(z ◦x)φ(u) + l(α2(z))l(x◦y)φ(u) = φ2(a) � [(b◦ c) �φ(u)] + φ2(b) � [(c◦a) �φ(u)] + φ2(c) � [(a◦ b) �φ(u)]. therefore, equation (3.1) holds. using a similar computation, equations (3.2)– (3.5) hold. then (v,≺,�,φ) is a hom-j-dendriform algebra. the other conclusions can be checked similarly. corollary 3.1. let (a, ·,α) be a hom-pre-jordan algebra and r be a rota-baxter operator (of weight zero) on a. then the products, given by x≺y = y ·r(x), x�y = r(x) ·y, ∀x,y ∈ a define a hom-j-dendriform algebra on a with the same twist map. theorem 3.1. let (a, ·,α) be a hom-pre-jordan algebra. then there is a hom-j-dendriform algebra such that (a, ·,α) is the associated vertical hom-pre-jordan algebra if and only if there exists an invertible o-operator of (a, ·,α). proof. suppose that (a,≺,�,α) is a hom-j-dendriform algebra with respect to (a, ·,α). then the identity map id: a → a is an o-operator of (a, ·,α) associated to (a,l�,l≺,α), where, for any x,y ∈ a, l�(x)(y) = x�y and l≺(x)(y) = x≺y. hom-pre-jordan and hom-j-dendriform algebras 21 conversely, let t : v → a be an o-operator of (a, ·,α) associated to a bimodule (v,l,r,φ). by proposition 3.5, there exists a hom-j-dendriform algebra on t(v ) = a given by t(u) ≺t(v) = t(r(t(u))v), t(u) �t(v) = t(l(t(u))v), ∀u,v ∈ v. by setting x = t(u) and y = t(v), we get x≺y = t(r(x)t−1(y)) and x�y = t(l(x)t−1(y)). finally, for any x,y ∈ a, we have x�y + y ≺x = t(r(x)t−1(y)) + t(l(x)t−1(y)) = t(r(x)t−1(y) + l(x)t−1(y)) = t(t−1(x)) ·t(t−1(y)) = x ·y. lemma 3.1. let r1 and r2 be two commuting rota-baxter operators (of weight zero) on a hom-jordan algebra (a,◦,α). then r2 is a rota-baxter operator (of weight zero) on the hom-pre-jordan algebra (a, ·,α), where x ·y = r1(x) ◦y, ∀x,y ∈ a. proof. for any x,y ∈ a, we have r2(x) ·r2(y) = r1(r2(x)) ◦r2(y) = r2(r1(r2(x)) ◦y + r1(x) ◦r2(y)) = r2(r2(x) ·y + x ·r2(y)). this finishes the proof. corollary 3.2. let r1 and r2 be two commuting rota-baxter operators (of weight zero) on a hom-jordan algebra (a,◦,α). then there exists a hom-j-dendriform algebra structure on a given by x≺y = r1(y) ◦r2(x), x�y = r1r2(x) ◦y, ∀x,y ∈ a. (3.12) 22 t. chtioui, s. mabrouk, a. makhlouf proof. by lemma 3.1, r2 is rota-baxter operator of weight zero on (a, ·,α), where x ·y = r1(x) ◦y. then, applying corollary 3.1, there exists a hom-j-dendriform algebra structure on a given by x≺y = r1(y) ◦r2(x), x�y = r1r2(x) ◦y, ∀x,y ∈ a. we end this section by discussing some adjunctions between the categories of considered non-associative algebras. let homrbj be the category of rota-baxter hom-jordan algebras in which objects are quadruples of the form (a,◦,α,r). let homrbpj be the category of rota-baxter hom-pre-jordan algebras in which objects are quadruples of the form (a, ·,α,r). notice that the morphisms are defined in a natural way, that is maps which are compatible with the multiplication, the twist maps and rota-baxter operators. the category of hom-pre-jordan algebras is denoted by hompj and that of hom-j-dendriform algebras by homjdend. theorem 3.2. 1. there is an adjoint pair of functors uhp : hompj � homrbj : hp, (3.13) in which the right adjoint is given by hp(a,◦,α,r) = (a, ·,α) ∈ hompj with x ·y = r(x) ◦y (3.14) for x,y ∈ a. 2. there is an adjoint pair of functors uhd : homjdend � homrbpj : hd, (3.15) in which the right adjoint is given by hd(a, ·,α,r) = (a,≺,�,α) ∈ homjdend with x ≺ y = x ·r(y) and x � y = r(x) ·y (3.16) for x,y ∈ a. hom-pre-jordan and hom-j-dendriform algebras 23 proof. the proof is based on corollaries 2.1, 2.2, 2.3, 3.1 and proposition 3.1. the following result says that, a rota-baxter hom-pre-jordan algebra can be given a new hom-pre-jordan structure. corollary 3.3. let (a, ·,α,r) be an object in homrbpj. define a multiplication on a by x∗y = x ·r(y) + r(x) ·y for x,y ∈ a. then a′ = (a,∗,α) is a hom-pre-jordan algebra and r(x∗y) = r(x) ·r(y). remark 3.2. following [2] and considering the operads of hom-jordan algebras, hom-pre-jordan algebras and hom-j-dendriform algebras, we have that the operad of hom-jordan algebras is the successor of the operad of hom-pre-jordan algebras and the operad of hom-j-dendriform algebras is the successor of the operad of hom-pre-jordan algebras. references [1] m. aguiar, j.-l. loday, quadri-algebras, j. pure appl. algebra 191 (2004), 205 – 221. 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[9] j.-l. loday, “ dialgebras ”, lecture notes in math. 1763, springer, berlin, 2001, 7 – 66. 24 t. chtioui, s. mabrouk, a. makhlouf [10] a. makhlouf, hom-dendriform algebras and rota-baxter hom-algebras, in “ operads and universal algebra ”, nankai ser. pure appl. math. theoret. phys. 9, world sci. publ., hackensack, nj, 2012, 147 – 171. [11] a. makhlouf, hom-alternative algebras and hom-jordan algebras, int. electron. j. algebra 8 (2010), 177 – 190. [12] q. sun, on hom-prealternative bialgebras, algebr. represent. theory 19 (3) (2016), 657 – 677. [13] y. sun, z. huang, s. zhao, z. tian, classification of pre-jordan algebras and rota-baxter operators on jordan algebras in low dimensions, 2021, arxiv:2111.02035. [14] h. upmeier, jordan algebras and harmonic analysis on symmetric spaces, amer. j. math. 108 (1986), 1 – 25. [15] d. yau, hom-maltsev, hom-alternative and hom-jordan algebras, int. electron. j. algebra 11 (2012), 177 – 217. https://arxiv.org/abs/2111.02035 basic results on hom-jordan algebras hom-pre-jordan algebras definition and basic properties bimodules and o-operators hom-j-dendriform algebras e extracta mathematicae vol. 31, núm. 2, 119 – 121 (2016) spectral rank of maximal finite-rank elements in banach jordan algebras abdelaziz maouche department of mathematics and statistics faculty of science, sultan qaboos university, oman, maouche@squ.edu.om presented by jesús m.f. castillo received november 15, 2015 abstract: we give a new proof to a spectral characterisation of the spectral rank established by aupetit by replacing his deep analytic arguments by the new characterisation of the connected component of the group of invertible elements obtained by o. loos. key words: banach jordan algebra, spectral rank, maximal finite-rank element. ams subject class. (2010): 17a15, 46h70. 1. preliminaries let a be a semisimple complex unital banach jordan algebra and ω(a) its set of invertible elements. for x ∈ a we denote sp(x) = {λ : λ1 − x /∈ ω(a)} and ρa(x) = sup{|λ| : λ ∈ sp(x)} the spectrum and spectral radius of x. for each nonnegative integer m, let fm = {a ∈ a : ♯(sp ux a \ {0}) ≤ m for all x ∈ a}, where the symbol ♯k denotes the number of distinct elements in a set k ∈ c. following [1], we define the rank of an element a of a as the smallest integer m such that a ∈ fm, if it exists; otherwise the rank is infinite. in other words, rank(a) = {sup ♯(sp ux a \ {0}), x ∈ a}. if a ∈ a is a finite-rank element, then e (a) = {x ∈ a : ♯(sp ux a \ {0}) = rank(a)} is a dense open subset of a [2, theorem 2.1]. it is shown in [1] that the socle, denoted soc a, of a semisimple banach jordan algebra a coincides with the collection ∪∞m=0fm of finite-rank elements. 119 120 a. maouche we first recall a very important theorem obtained by o. loos [2], saying that the connected component of ω(a) is arcwise connected as in the case of banach algebras. theorem 1. (o. loos [2]) let a be a real or complex banach jordan algebra with unit element 1. then ω1 = {u(exp x1) · · · u(exp xn)(1) : xi ∈ a, n ≥ 1} is the connected component of 1 of the set ω of invertible elements of a. with the help of this theorem 1 we are able now to eliminate the deep and difficult analytic arguments used by aupetit to prove the next theorem. 2. the rank in banach jordan algebras theorem 2. ([1], theorem 3.1) let a be a banach jordan algebra with identity. suppose that a ∈ a and that m ≥ 0 is an integer. the following properties are equivalent: (1) ♯(sp(ux a) \ {0}) ≤ m for every x ∈ a, (2) {t ∈ c : 0 ∈ sp(y + ta)) ≤ m} for every y invertible in a, (3) ∩ t∈f sp(y + ta) ⊂ sp y for every y ∈ a and every subset f of c having m + 1 non-zero elements. proof. (1) ⇒ (2) first suppose that 0 /∈ σ(y). by the holomorphic functional calculus theorem applied to y and a branch of √ z, there exists an invertible x such that y = x2. since y + ta = x2 + ta = ux(1 + tux−1 a) we get y + ta is non-invertible if and only if − 1 t ∈ sp(ux−1 a). by hypothesis (1) this set sp(u−1x a) contains at most m non-zero points. thus (2) is proved in this situation. now {y ∈ a : 0 /∈ σ(y)} is an open subset of ω, by upper semicontinuity of the spectrum. let y ∈ ω1, then by o. loos’s theorem 1: y = uexp(x1) · · · uexp(xn)1 so y + ta = uexp(x1) · · · uexp(xn) [ 1 + t · uexp(−xn) · · · uexp(−x1)a ] . spectral rank 121 then 0 ∈ sp(y + ta) ⇐⇒ − 1 t ∈ sp(uexp(−xn) · · · uexp(−x1)a). the set sp(uexp(−xn) · · · uexp(−x1)a) contains at most m points by (1) because exp(−xi) is invertible in a. (2) ⇒ (3) if λ ∈ ε(y) then λ − y ∈ ω1, so ♯{t : λ ∈ sp(y + ta)} ≤ m, hence λ /∈ ∩t∈f sp(y + ta). (3) ⇒ (1) same as in [1]. references [1] a. bernard, spectral characterization of the socle in jordan-banach algebras, math. proc. cambridge philos. soc. 117 (3) (1995), 479 – 489. [2] o. loos, on the set of invertible elements in banach jordan algebras, results math. 29 (1) (1996), 111 – 114. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 2 (2020), 221 – 228 doi:10.17398/2605-5686.35.2.221 available online june 19, 2020 around some extensions of casas-alvero conjecture for non-polynomial functions a. cima, a. gasull, f. mañosas departament de matemàtiques, universitat autònoma de barcelona barcelona, spain cima@mat.uab.cat , gasull@mat.uab.cat , manyosas@mat.uab.cat received april 21, 2020 presented by manuel maestre accepted may 2, 2020 abstract: we show that two natural extensions of the real casas-alvero conjecture in the nonpolynomial setting do not hold. key words: polynomial, casas-alvero conjecture, zeroes of functions. ams subject class. (2010): primary: 30c15. secondary: 12d10, 13p15, 26c10. 1. introduction the casas-alvero conjecture affirms that if a complex polynomial p of degree n > 1 shares roots with all its derivatives, p(k), k = 1, 2, . . . ,n − 1, then there exist two complex numbers, a and b 6= 0, such that p(z) = b(z − a)n. notice that, in principle, the common root between p and each p(k) might depend on k. casas-alvero arrived to this problem at the turn of this century, when he was working in his paper [1] trying to obtain an irreducibility criterion for two variable power series with complex coefficients. see [2] for an explanation of the problem in his own words. although several authors have got partial answers, to the best of our knowledge the conjecture remains open. for n ≤ 4 the conjecture is a simple consequence of the wonderful gauss-lucas theorem ([6]). in 2006 it was proved in [5], by using maple, that it is true for n ≤ 8. afterwards in [6, 7] it was proved that it holds when n is pm, 2pm, 3pm or 4pm, for some prime number p and m ∈ n. the first cases left open are those where n = 24, 28 or 30. see again [6] for a very interesting survey or [3, 8] for some recent contributions on this question. adding the hypotheses that p is a real polynomial and all its n roots, taking into account their multiplicities, are real, the conjecture has a real issn: 0213-8743 (print), 2605-5686 (online) c© the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.35.2.221 mailto:cima@mat.uab.cat mailto:gasull@mat.uab.cat mailto:manyosas@mat.uab.cat https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 222 a. cima, a. gasull, f. mañosas counterpart, that also remains open. it says that p(x) = b(x−a)n for some real numbers a and b 6= 0. for this real case, the conjecture can be proved easily for n ≤ 4, simply by using rolle’s theorem. this tool does not suffice for n ≥ 5, see for instance [4] for more details, or next section. also in the real case, in [6] it is proved that if the condition for one of the derivatives of p is removed, then there exist polynomials satisfying the remaining n−2 conditions, different from b(x−a)n. the construction of some of these polynomials presented in that paper is very nice and is a consequence of the brouwer’s fixed point theorem in a suitable context. finally, it is known that if the conjecture holds in c, then it is true over all fields of characteristic 0. on the other hand, it is not true over all fields of characteristic p, see again [7]. for instance, consider p(x) = x2(x2 + 1) in characteristic 5 with roots 0, 0, 2 and 3. then p ′(x) = 2x(2x2 + 1), p ′′(x) = 12x2 + 2 = 2(x2 + 1) and p ′′′(x) = 4x and all them share roots with p . the aim of this note is to present two natural extensions of the real casasalvero conjecture to smooth functions and show that none of them holds. question 1. fix 1 < n ∈ n. let f be a class cn real function such that f(n)(x) 6= 0 for all x ∈ r, and has n real zeroes, taking into account their multiplicities. assume that f shares zeroes with all its derivatives, f(k), k = 1, 2, . . . ,n − 1. is it true that f(x) = b(f(x))n for some 0 6= b ∈ r and some f, a class cn real function, that has exactly one simple zero? notice that one of the hypotheses of the real casas-alvero conjecture can be reformulated as follows: the polynomial f shares roots with all its derivatives but one, precisely the one corresponding to its degree. trivially, this is so, because all the derivatives of order higher than n are identically zero. the second question that we consider is: question 2. fix 1 < n ∈ n. let f be a real analytic function that shares zeroes with all its derivatives but one, say f(n). is it true that f(x) = b(f(x))n for some 0 6= b ∈ r and some real analytic function f, that has exactly one simple zero? theorem a. (i) the answer to the question 1 is “yes” for n ≤ 4 and “no” for n = 5. (ii) the answer to the question 2 is already “no” for n = 2. our result reinforces the intuitive idea that casas-alvero conjecture is mainly a question related with the rigid structure of the polynomials. extensions of casas-alvero conjecture 223 2. proof of theorem a (i) the answer to question 1 is “yes” for n = 2, 3, 4 because the proof of the real casas-alvero conjecture for the same values of n, based on the rolle’s theorem and given in [4], does not uses at all that p is a polynomial. let us adapt it to our setting. since f(n) does not vanish we know that f has exactly n real zeroes, taking into account their multiplicites. moreover we know that f has to have at least a double zero, that without loss of generality can be taken as 0. next we can do a case by case study to discard all situations except that f has only a zero and it is of multiplicity n. for the sake of brevity, we give all the details only in the most difficult case, n = 4. assume, to arrive to a contradiction, that n = 4, f is under the hypotheses of question 1 and x = 0 is not a zero of multiplicity four. notice that by rolle’s theorem, for k = 1, 2, 3, each f(k) has exactly 4−k zeroes, taking into account their multiplicities. moreover, the only zero of f ′′′ must be one of the zeroes of f. if f ′′(0) = 0 and f ′′′(0) 6= 0 then f has only another zero at x = a and, without loss of generality, we can assume that a > 0. applying three times rolle’s theorem we get that f ′′′(b) = 0 for some b ∈ (0,a) which is a contradiction with the hypotheses. if f ′′(0) 6= 0 then f has two more zeroes counting multiplicities. there are three possibilities. the first one is that there is a > 0 such that f(a) = f ′(a) = 0. in this case, applying two times rolle’s theorem we obtain that there exist b,c ∈ (0,a) with f ′′(b) = f ′′(c) = 0 and they are the only zeroes of f ′′. this fact gives again a contradiction because none of them is a zero of f. the second one is that there exist a1,a2 ∈ r with 0 ∈ (a1,a2) such that f(a1) = f(a2) = 0. also in this case, by applying two times rolle’s theorem we obtain that there exist b,c ∈ (a1,a2) such that 0 ∈ (b,c) and f ′′(b) = f ′′(c) = 0 giving us the desired contradiction. lastly, assume that the other two zeroes of f are a1 and a2, with 0 < a1 < a2. by rolle’s theorem the zeroes of f ′ are 0,b1 and b2 and satisfy 0 < b1 < a1 < b2 < a2. then, since f ′′ has to have two zeroes, say c1,c2, and they satisfy 0 < c1 < b1 < c2 < b2, the hypotheses force that c2 = a1. hence the zero of f ′′′ has to be between c1 and c2 = a1, that is in particular in (0,a1), interval that contains no zero of f, arriving once more to the desired contradiction. in short, we have proved for n ≤ 4, that f(x) = xng(x), for some class cn function g, that does not vanish. hence f(x) = sign(g(0)) ( x n √ g(x) sign(g(0)) )n = b(f(x))n, 224 a. cima, a. gasull, f. mañosas where f has only one zero, x = 0, that is simple, as we wanted to prove. to find a map f for which the answer to question 1 is “no” we consider n = 5 and a configuration of zeroes of f and its derivatives proposed in [4] as the simplest one, compatible with the hypotheses of the casas-alvero conjecture and rolle’s theorem. specifically, we will search for a function f , of class at least c5, with the five zeroes 0, 0, 1,c,d, to be fixed, satisfying 0 < 1 < c < d, and moreover f ′(0) = 0, f ′′(1) = 0, f ′′′(c) = 0, f(4)(1) = 0, (2.1) and such that f(5) does not vanish. notice that f ′(0) = 0 is not a new restriction. we start assuming that f(5)(x) = r − sin(x), for some r > 1 to be determined. by imposing that conditions (2.1) hold, together with f(0) = 0, we get that f(x) = ∫ x 0 ∫ u 0 ∫ w 1 ∫ z c ∫ y 1 ( r − sin(t) ) dt dy dz dw du. some straightforward computations give that f(x) = r 120 x5 − r + cos(1) 12 x4 + 2rc− 2 sin(c) + 2 cos(1)c−rc2 12 x3 + 6 sin(c) + 2r + 9 cos(1) − 6rc + 3rc2 − 6 cos(1)c 12 x2 − 1 + cos(x). imposing now that f(1) = 0 we obtain that r = 5 ( 8 cos(1)c− 41 cos(1) − 8 sin(c) + 24 ) 4(5c2 − 10c + 4) = r(c). next we have to impose that f(c) = 0. by replacing the above expression of r in f we obtain that f(c) = g(c) 96(5c2 − 10c + 4) , where g(c) = − c2 ( 12 c4 − 369 c3 + 1437 c2 − 1708 c + 532 ) cos (1) − 8 c2 (c− 1) (c− 2)2 sin (c) + ( 480 c2 − 960 c + 384 ) cos (c) − 24 (c− 1) ( 9 c4 − 36 c3 + 24 c2 + 24 c− 16 ) . extensions of casas-alvero conjecture 225 a carefully study shows that g has exactly one real zero c1 ∈ (17/10, 19/10) = i, with c1 ≈ 1.79343096. to prove its existence it suffices to show that g ( 17 10 ) = − 99211099 500000 cos (1) − 18207 12500 sin ( 17 10 ) + 696 5 cos ( 17 10 ) + 1583211 12500 > 0, g ( 19 10 ) = − 180110481 500000 cos (1) − 3249 12500 sin ( 19 10 ) + 1464 5 cos ( 19 10 ) + 3616677 12500 < 0. by using taylor’s formula we know that for any c > 0, s−(c) < sin(c) < s+(c) and c−(c) < cos(c) < c+(c) where s±(c) = c− c3 3! + c5 5! − c7 7! + c9 9! ± c11 11! and c±(c) = 1 − c2 2! + c4 4! − c6 6! + c8 8! ± c10 10! . hence we can replace the values of the trigonometric functions in g by rational numbers to have upper or lower bounds of this function evaluated at 1, 17/10 or 19/10. for instance, 0.5403023 ≈ 1960649 3628800 = c−(1) < cos(1) < c+(1) = 280093 518400 ≈ 0.5403028. we obtain that g ( 17 10 ) >− 99211099 500000 c+ (1) − 18207 12500 s+ ( 17 10 ) + 696 5 c− ( 17 10 ) + 1583211 12500 = 3444600099561969856969 49896000000000000000000 > 0 and g ( 19 10 ) <− 180110481 500000 c− (1) − 3249 12500 s− ( 19 10 ) + 1464 5 c+ ( 19 10 ) + 3616677 12500 = − 1689627895469649855823 16632000000000000000000 < 0. 226 a. cima, a. gasull, f. mañosas to show the uniqueness of the zero in i, we will prove that g is strictly decreasing in this interval. it holds that g′(c) = t(c) cos (1) + u(c) sin (c) + v (c cos (c) + w(c), with t(c) = − c ( 72 c4 − 1845 c3 + 5748 c2 − 5124 c + 1064 ) , u(c) = − 8 ( 5 c2 − 10 c + 4 )( c2 − 2 c + 12 ) , v (c) = − 8 (c− 1) ( c4 − 4 c3 + 4 c2 − 120 ) , w(c) = − 120(9c4 − 36c3 + 36c2 − 8). by computing the sturm sequences of t,u and v we can prove that t(c) < 0, u(c) < 0 and v (c) > 0 for all c ∈ i. hence, for c ∈ i, g′(c) < t(c)c−(c) + u(c)s−(c) + v (c)c+(c) + w(c) = q(c), where q(c) = 72469 64800 c− 669211 43200 c2 + 18852329 302400 c3 − 8854991 80640 c4 + 4732471 50400 c5 − 532 15 c6 + 8 7 c7 + 191 70 c8 − 797 1890 c9 − 34 405 c10 + 1651 103950 c11 + 3533 2494800 c12 − 193 623700 c13 + 1 142560 c14 − 1 831600 c15. the sturm sequence of q shows that it has no zeroes in i. moreover, it is negative in this interval, and as a consequence, g′ is also negative, as we wanted to prove. we fix c = c1. then, r = r(c1) and f is also totally fixed. moreover, by using the same techniques we get that r = r(c1) > r(19/10) > 1 and as a consequence f(5) does not vanish. in fact, r = r(c1) ≈ 1.04591089. finally, f has one more real zero d ∈ (33/10, 34/10). in fact, d ≈ 3.32178369. this f gives our desired example, see figure 1. (ii) consider f(x) = 4x2 + π2(cos(x) − 1) that has a double zero at 0 and also vanishes at ±π/2. moreover, f ′(x) = 8x − π2 sin(x) vanishes at x = 0, f ′′(x) = 8 − π2 cos(x) has no common zeroes with f and, for any k > 1, extensions of casas-alvero conjecture 227 |f(2k)(x)| = π2|cos(x)| vanishes at x = π/2 and |f(2k−1)(x)| = π2|sin(x)| vanishes at x = 0. a similar example for n = 3 is f(x) = 4x3 − 6πx2 + π3(1 − cos(x)), that vanishes at 0,π (double zeroes) and π/2. figure 1: plot of a map f for which the answer to question 1 for n = 5 is “no”. acknowledgements the authors are supported by ministerio de ciencia, innovación y universidades of the spanish government through grants mtm201677278-p (mineco/aei/feder, ue, first and second authors) and mtm2017-86795-c3-1-p (third author). the three authors are also supported by the grant 2017-sgr-1617 from agaur, generalitat de catalunya. references [1] e. casas-alvero, higher order polar germs, j. algebra 240 (2001), 326 – 337. [2] interview to e. casas-alvero in spanish. https://www.gaussianos.com/la-conjetura-de-casas-alvero-contada -por-eduardo-casas-alvero/. https://www.gaussianos.com/la-conjetura-de-casas-alvero-contada-por-eduardo-casas-alvero/ https://www.gaussianos.com/la-conjetura-de-casas-alvero-contada-por-eduardo-casas-alvero/ 228 a. cima, a. gasull, f. mañosas [3] w. castryck, r. laterveer, m. ounäıes, constraints on counterexamples to the casas-alvero conjecture and a verification in degree 12, math. comp. 83 (2014), 3017 – 3037. [4] m. chellali, on the number of real polynomials of the casas-alvero type, j. of taibah univ. for science 9 (2015), 351 – 356. [5] g.m. d́ıaz-toca, l. gonzález-vega, on analyzing a conjecture about univariate polynomials and their roots by using maple in “ a maple conference 2006 ” (proc. of the conference; waterloo, ontario, canada, july 23–26, 2006; waterloo: maplesoft), 2006, 81 – 98. [6] j. draisma, j.p. de jong, on the casas-alvero conjecture, eur. math. soc. newsl. 80 (2011), 29 – 33. [7] h.-c. graf von bothmer, o. labs, j. schicho, c. van de woestijne, the casas-alvero conjecture for infinitely many degrees, j. algebra 316 (2007), 224 – 230. [8] s. yakubovich, polynomial problems of the casas-alvero type, j. class. anal. 4 (2014), 97 – 120. introduction proof of theorem a e extracta mathematicae vol. 32, núm. 2, 163 – 172 (2017) a symmetrical property of the spectral trace in banach algebras abdelaziz maouche department of mathematics and statistics, faculty of science sultan qaboos university, oman maouche@squ.edu.om presented by martin mathieu received march 7, 2016 abstract: our aim in this paper is to extend a symmetrical property of the trace by m. kennedy and h. radjavi for bounded operators on a banach space to the more general situation of banach algebras. the main ingredients are vesentini’s result on subharmonicity of the spectral radius and the new spectral rank and trace defined on the socle of a banach algebra by b. aupetit and h. du t. mouton. key words: banach algebra, rank, spectral additivity, trace, subharmonic function. ams subject class. (2010): primary 46h70; secondary 17a15. 1. preliminaries in [5], the authors investigate the properties of bounded operators which satisfy a certain spectral additivity condition and use their results to study lie and jordan algebras of compact operators. as a first result, they obtain a symmetric trace condition on bounded operators (see [5, lemma 3.12]). b. aupetit and h. du t. mouton proved that the spectral rank and trace as defined in [3] coincide with the classical notion of trace and rank in the case where u = l(x), the banach algebra of bounded linear operators on a banach space x. it is our aim to extend some results obtained in [5] to the general situation of banach algebras, by replacing the classical trace by the spectral one defined in [3]. let u be a semi-simple complex unital banach algebra and ω(u) its set of invertible elements. for x ∈ u we denote sp(x) = {λ : λ1 − x /∈ ω(u)} and ρu(x) = sup{|λ| : λ ∈ sp(x)} the spectrum and spectral radius of x. we denote by ŝp(x) the full spectrum of x, i.e., the polynomially convex hull of sp(x), that is the set obtained by filling the holes in sp(x). 163 164 a. maouche let α ∈ c and γ a small curve isolating α from the rest of the spectrum of a. by definition, the riesz projection associated to a and α is given by p(α, a) = 1 2πi ∫ γ (λ − a)−1dλ . using the identity (λ − a)−1 = 1 λ + 1 λ a(λ − a)−1 we obtain by integration p(α, a) = 1 2πi ∫ γ 1 λ (λ − a)−1dλ = a 2πi ∫ γ 1 λ (λ − a)−1dλ . (∗) obviously p(α, a) = 0 if α /∈ sp(a). the holomorphic functional calculus yields that the p(α, a) corresponding to different values of α are orthogonal projections (have zero product) and their sum is 1. the next result is well known, we include it to illustrate the previous definition of riesz projections and a kind of spectral additivity in a particular situation. proposition 1. let a, b be two elements of a banach algebra u such that ab = ba = 0. then sp(a + b) \ {0} = (sp(a)) ∪ (sp(b)) \ {0} . moreover, if λ0 ̸= 0 is isolated in sp(a+b) then the riesz projection associated with a + b, a and b respectively satisfy the identity p(λ0, a + b) = p(λ0, a) + p(λ0, b) . proof. for λ ̸= 0 it is easy to see from the identity, λ − (a + b) = 1 λ (λ − a)(λ − b) = 1 λ (λ − b)(λ − a) that λ − (a + b) is invertible if and only if both λ − a and λ − b are invertible. let γ be a circle centered at λ0 which separates λ0 from 0 and the rest of the spectrum of a + b. if λ ∈ γ, by the previous identity we have λ ̸= 0, λ − a and spectral additivity 165 λ−b invertible. since b = (λ−a) b λ we obtain (λ−a)−1b = b λ on γ. moreover, we have (λ − (a + b))−1 = λ(λ − b)−1(λ − a)−1 = (λ − a)−1 + [λ(λ − b)−1 − 1](λ − a)−1 = (λ − a)−1 + b(λ − b)−1(λ − a)−1 = (λ − a)−1 + (λ − b)−1(λ − a)−1b = (λ − a)−1 + b λ (λ − b)−1. now, integrating this quantity on γ and multiplying by 1 2πi , we get p(λ0, a + b) = p(λ0, a) + b 2πi ∫ γ 1 λ (λ − b)−1dλ = p(λ0, a) + p(λ0, b) by formula (*) applied to b. 2. trace and rank in banach algebras for each nonnegative integer m, let fm = {a ∈ u : #(sp(xa) \ {0}) ≤ m for all x ∈ u} , where the symbol #k denotes the number of distinct elements in a set k ⊂ c. following [3], we define the rank of an element a of u as the smallest integer m such that a ∈ fm, if it exists; otherwise the rank is infinite. in other words, rank(a) = sup x∈u #(sp(xa) \ {0}) ≤ ∞ . of course, rank(a) = sup x∈u #(sp(ax) \ {0}) . a few elementary properties of the rank taken from [3], where more details and proofs are given, are listed below: (a) #(sp(a) \ {0}) ≤ rank(a) for a in u. (b) rank(xa) ≤ rank(a) and rank(ax) ≤ rank(a) for a, x ∈ u; moreover, rank(ua) = rank(au) = rank(a) if u is invertible. 166 a. maouche (c) if a ∈ u is a finite-rank element, then e(a) = {x ∈ u : #(sp(xa) \ {0}) = rank(a)} is a dense open subset of u (see [3, theorem 2.2]). it is known that the socle, denoted soc(u), of a semisimple banach algebra u coincides with the collection ∪∞m=0fm of finite-rank elements. following [3], if a ∈ soc(u) we define the trace of a by tr(a) = ∑ λ∈sp(a) λ · m(λ, a) where m(λ, a) is the multiplicity of the spectral value λ. more details on the trace and rank in banach algebras are contained in [3], from which we recall the following results on the trace that will be used in the proof of our main result. for instance, it is shown in [3], formula (3) page 130, that the trace and rank satisfy |tr(a)| ≤ ρ(a) · rank(a) where ρ(a) is the usual spectral radius of a. proposition 2. (i) let a ∈ soc(u), b, x, y ∈ (u). then we have tr(alxlyb + blxlya) = tr(alylxb + blylxa) . in particular tr(x(ya)) = tr(y(xa)). (ii) let a ∈ soc((u)) be such that tr(au) = 0, for every u ∈ soc(u). then a = 0. (iii) if a ∈ soc(u), then ϕ(x) = tr(ax)is a bounded linear functional on (u). proof. for properties (i) and (ii), see [2, corollary 1.2 and corollary 1.3, p. 181]. for property (iii) see of [3, theorem 3.3]. theorem 1. ([3, theorem 2.6]) let a ∈ u have finite rank and λ1, . . . , λn be non-zero distinct elements of its spectrum with multiplicity m(λi, a). if p denotes the riesz projection associated with a and λ1, . . . , λn that is, p = p(λ1, a) + · · · + p(λn, a), then rank(p) = m(λ1, a) + · · · + m(λn, a). another important result that we shall use in the proof of our main result is the following theorem. spectral additivity 167 theorem 2. ([3, theorem 3.1]) let f be an analytic function from a domain d of c into the socle of a semisimple banach algebra u. then tr(f(λ)) is holomorphic on d. in what follows, an important tool will be the theory of subharmonic functions, based essentially on the celebrated result of e. vesentini: if f is an analytic function from a domain d of the complex plane into a banach algebra, then the functions λ 7→ ρ(f(λ)) and λ 7→ log ρ(f(λ)) are subharmonic (see [1, theorem 3.4.7]). we will require the following two fundamental results from the theory of subharmonic functions from ([1, theorem a.1.3 and theorem a.1.29]). theorem 3. (maximum principle for subharmonic functions) let f be a subharmonic function on a domain d of c. if there exists λ0 ∈ d such that f(λ) ≤ f(λ0) for all λ ∈ d, then f(λ) = f(λ0) for all λ in d. we state here a special case of h. cartan’s theorem (see [5] and the references given there). theorem 4. (h. cartan’s theorem) let f be a subharmonic function on a domain d of c. if f(λ) = −∞ on an open disc in d, then f(λ) = −∞ for all λ in d. to apply later h. cartan’s theorem, we shall need the maximum principle theorem for the full spectrum due to e. vesentini, where ∂k means the boundary of the compact set k. theorem 5. (spectrum maximum principle) let f be a an analytic function on a domain d of c into a banach algebra a. suppose that there exists λ0 of d such that spf(λ) ⊂ spf(λ0), for all λ ∈ d. then ∂spf(λ0) ⊂ ∂spf(λ) and ̂spf(λ0) = ŝpf(λ), for all λ ∈ d. in particular, if spf(λ0) has no interior points or if spf(λ) does not separate the plane for all λ ∈ d, then spf(λ) is constant on d. 3. elements with stable spectrum l. harris and r. kadison define spectrally additive elements in a c*algebra as follows. definition 1. an element a of a c*-algebra u is said to be spectrally additive (in u) when sp(a + b) ⊆ sp(a) + sp(b) for each b in u. 168 a. maouche this same definition may be made for elements of a (unital) banach algebra u (over c). the concept of spectral additivity was studied in the context of banach algebras with the aid of (purely algebraic) commutator results for ‘schurian algebras’ by l. harris and r. kadison. it is proved there that a is spectrally additive in u if and only if au−ua lies in the radical of u for each u in u. in particular, if u is semi-simple, as is the case when u is a c*-algebra, then a is spectrally additive if and only if it lies in the center of u. following [5], we introduce the notion of stable spectrum in a banach algebra. definition 2. let u be an element of a banach algebra u. we say that an element a has u-stable spectrum if ρ(a + λu) ≤ ρ(a) for every complex number λ. a family of elements of u is said to have u-stable spectrum if each of its elements has u-stable spectrum. remark 1. (a) for elements a and u of a complex banach algebra u, the function λ 7→ a + λu is analytic, so by vesentini’s result, the functions λ 7→ ρ(a + λu) and λ 7→ log ρ(a + λu) are subharmonic. (b) if a has u-stable spectrum, the maximum principle for subharmonic functions immediately implies that ρ(a + λu) = ρ(a) for all complex numbers. (c) if a and u have sublinear spectrum, that is, sp(a + λu) ⊆ sp(a) + λsp(u) for every complex number λ and u is quasi-nilpotent (ρ(u) = 0), then a has u-stable spectrum. lemma 1. let a and u be elements of a semi-simple complex banach algebra u. if a has u-stable spectrum, then u is quasi-nilpotent. proof. by the above remark, ρ(a + λu) = ρ(a) for all λ in c, so ρ ( λ−1a + u ) = |λ|−1ρ(a) for all non zero λ in c. thus, by subharmonicity of ρ(λ−1a + u), we get ρ(u) = lim sup λ→∞ ρ ( λ−1a + u ) = 0 . theorem 6. let a and u be elements of a semi-simple complex banach algebra u. then a has u-stable spectrum if and only if (µ − a)−1u is quasinilpotent for all µ /∈ ŝp(a). spectral additivity 169 proof. by remark 1, ρ(a + λu) = ρ(a) for all λ in c, so for µ in c with |µ| > ρ(a), both µ − a and µ − a − λu are invertible. therefore, λ−1(µ − a)−1(µ − a − λu) = λ−1 − (µ − a)−1u is invertible for all non-zero λ in c. this means that the values of the analytic function µ 7→ (µ − a)−1u for µ /∈ sp(a), are quasi-nilpotent whenever |µ| > ρ(a). consider the subharmonic function µ 7→ log(ρ(µ − a)−1u) defined for µ /∈ sp(a). since log(ρ(µ−a)−1u) = −∞ whenever |µ| > ρ(a), by h. cartan’s theorem, log(ρ(µ − a)−1u) = −∞ for all µ /∈ [ŝp(a)]. in other words, (µ − a)−1u is quasi-nilpotent for all µ /∈ [ŝp(a)]. corollary 1. let a and u be elements of a semi-simple complex banach algebra u, with sp(a) without holes. then a has u-stable spectrum if and only if sp(a + λu) ⊆ sp(a) + λsp(u) for every λ in c. proof. (⇐) clear. (⇒) suppose µ ∈ sp(a + λu), but that µ /∈ sp(a). then obviously λ is non-zero, and λ−1(µ − a)−1(µ − a − λu) = λ−1 − (µ − a)−1u is not invertible. by theorem 6, we get (µ − a)−1u is quasi-nilpotent for all µ /∈ ŝp(a); so it also holds for µ /∈ sp(a) which yields a contradiction. the next result follows from [1, theorem 3.4.14], as we can see in the following proof. lemma 2. let a and u be elements of a semi-simple complex banach algebra u. if a has u-stable spectrum and sp(a) has no interior points, then sp(a + λu) = sp(a) for all λ in c. proof. (⇐) clear by corollary 1. (⇒) by remark 1, ρ(a + λu) = ρ(a) for all λ in c, so the analytic multifunction λ → sp(a + λu) is bounded, and consequently by liouville’s theorem for analytic multivalued functions, (see [1, theorem 3.4.14]), we have ̂sp(a + λu) = ŝp(a), where ŝp(x) denotes the full spectrum of x. since sp(a) has no interior points, the result follows from theorem 5 (see the proof in [1, theorem 3.4.13]). 170 a. maouche theorem 7. let a and u be elements of a semi-simple complex banach algebra u. if a has u-stable spectrum and sp(a) has no interior points, then sp ( (1 − νu)−1a ) = sp(a) for all ν in c. proof. first suppose λ is non-zero, and that λ /∈ sp(a). by lemma 2, we have sp ( λ−1a + νu ) = sp ( λ−1a ) , and by lemma 1, u is quasi-nilpotent. these two facts imply that 1 − νu and 1 − λ−1a − νu are both invertible, and hence that λ(1 − νu)−1 ( 1 − λ−1a − νu ) = λ − (1 − νu)−1a is invertible for all ν in c. therefore, λ /∈ sp ( (1−νu)−1a ) for all ν in c. now suppose 0 /∈ sp(a). then a is invertible, implying (1 − νu)−1a is invertible, and hence by quasi-nilpotence of u, that 0 /∈ sp ( (1 − νu)−1a ) for all ν in c. we have shown that sp ( (1 − νu)−1a ) ⊆ sp(a) for all ν in c. since sp(a) has no interior points, the result follows from theorem 5. we arrive at our main result which gives a symmetric spectral trace condition on stable elements, extending [5, lemma 3.12] from the semi-simple algebra of bounded operators b(x) on a banach space x to the more general situation of a semi-simple complex banach algebra u. theorem 8. let a and b two elements of a semi-simple complex unital banach algebra u. if a is b-stable and one of a or b is of finite-rank, then tr(anb) = tr(abn) = 0 for all n ≥ 1. proof. first suppose that a is of finite rank. since b is quasi-nilpotent by lemma 1, the function ν 7→ ((1 − νb)−1a) is entire. moreover, sp ( (1 − νb)−1a ) = sp(a) for all ν in c by theorem 7. then, taking n-th powers, the function ν 7−→ ( (1 − νb)−1a )n is also entire, and sp ( (1 − νb)−1a )n = sp(an) for all ν in c. clearly, rank (( (1 − νb)−1a )n) ≤ rank(a) , so tr (( (1 − νb)−1a )n) = tr(an) spectral additivity 171 for all ν ∈ c by theorem 2, property (3) of proposition 2 and liouville’s theorem for entire functions. for |ν| < ||b||−1, we may expand ( (1 − νb)−1a )−1 as a power series in ν, (1 − νb)−1a = ∑ k≥0 bkaνk. hence ( (1 − νb)−1a )n =  ∑ k≥0 bkaνk  n . the coefficient of νk in the above expansion is bka, and for n ≥ 1, the coefficient of ν is ban + aban−1 + · · · + an−1ba. but we may expand the constant function tr (( (1 − νb)−1a )n) as a power series in ν, and the linearity of the trace implies that for n = 1, the coefficient of νk in this expansion is tr(bka), and for n ≥ 1, that the coefficient of ν is tr ( ban + aban−1 + · · · + an−1ba ) = ntr(anb) . comparing the coefficients on the left and right hand side of the equation tr (( (1 − νb)−1a )n) = tr(an) therefore gives tr(anb) = 0 for all n ≥ 1 and tr(abk) = 0 for all k ≥ 1. now suppose that b is of finite rank. the function (1 − νa)−1b is analytic, with quasi-nilpotent values by theorem 6, for 1 ν /∈ ŝp(a). taking n-th powers, the function ν 7→ ( (1 − νa)−1b )n is also analytic for all 1 ν /∈ sp(a). as above, for |ν| < ||a||−1, we may expand ((1 − νa)−1b)n as a power series in ν, ( (1 − νa)−1b )n =  ∑ k≥0 akbνk  n . for n = 1, the coefficient of νk in the above expansion is akb, and for n ≥ 1, the coefficient of ν is abn + babn−1 + · · · + bn−1a. proceeding as before, we may expand the constant function tr (( (1 − νa)−1b )n) as a power series in ν, and linearity of the trace implies that for n = 1, the coefficient of νk in this expansion is tr(akb), and for n ≥ 1, that coefficient of ν is tr ( abn + babn−1 + · · · + bn−1ab ) = ntr(abn) . comparing the coefficients of the left and right hand side of the equation tr (( (1 − νa)−1b )n) = 0 hence gives tr(akb) = 0 for all k ≥ 1, and tr(abn) = 0 for all n ≥ 1. 172 a. maouche references [1] b. aupetit, “ a primer on spectral theory ”, universitext, springer-verlag, new york, 1991. [2] b. aupetit, trace and spectrum preserving linear mappings in jordanbanach algebras, monatsh. math. 125 (1998), 179 – 187. [3] b. aupetit, h. du t. mouton, trace and determinant in banach algebras, studia math. 121 (2) (1996), 115 – 136. [4] g. braatvedt, r. brits, f. schultz, rank, trace and determinant in banach algebras: generalized frobenius and sylvester theorems, studia math. 229 (2015), 173 – 180. [5] m. kennedy, h. radjavi, spectral conditions on lie and jordan algebras of compact operators, j. funct. anal. 256 (2009), 3143 – 3157. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 1 (2020), 99 – 126 doi:10.17398/2605-5686.35.1.99 available online november 12, 2019 generalized representations of 3-hom-lie algebras s. mabrouk 1, a. makhlouf 2, s. massoud 3 1 university of gafsa, faculty of sciences gafsa, 2112 gafsa, tunisia 2 université de haute alsace, irimas-département de mathématiques 6, rue des frères lumière f-68093 mulhouse, france 3 université de sfax, faculté des sciences, sfax tunisia mabrouksami00@yahoo.fr , abdenacer.makhlouf@uha.fr , sonia.massoud2015@gmail.com received june 6, 2019 presented by consuelo mart́ınez accepted september 30, 2019 abstract: the propose of this paper is to extend generalized representations of 3-lie algebras to hom-type algebras. we introduce the concept of generalized representation of multiplicative 3hom-lie algebras, develop the corresponding cohomology theory and study semi-direct products. we provide a key construction, various examples and computation of 2-cocycles of the new cohomology. also, we give a connection between a split abelian extension of a 3-hom-lie algebra and a generalized semidirect product 3-hom-lie algebra. key words: 3-hom-lie algebra, representation, generalized representation, cohomology, abelian extension. ams subject class. (2010): 17a42, 17b10. introduction the first instances of ternary lie algebras appeared first in nambu’s generalization of hamiltonian mechanics [23], which was formulated algebraically by takhtajan [29]. the structure of n-lie algebras was studied by filippov [15] then completed by kasymov in [21]. the representation theory of n-lie algebras was first introduced by kasymov in [21]. the adjoint representation is defined by the ternary bracket in which two elements are fixed. through fundamental objects one may also represent a 3-lie algebra and more generally an n-lie algebra by a leibniz algebra [11]. the cohomology of n-lie algebras, generalizing the chevalleyeilenberg lie algebras cohomology, was introduced by takhtajan [30] in its simplest form, later a complex adapted to the study of formal deformations was introduced by gautheron [17], then reformulated by daletskii and takhtajan [11] using the notion of base leibniz algebra of an n-lie algebra. in [2, 3], the structure and cohomology of 3-lie algebras induced by lie algebras has been investigated. issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.35.1.99 mailto:mabrouksami00@yahoo.fr mailto:abdenacer.makhlouf@uha.fr mailto:sonia.massoud2015@gmail.com\ https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 100 s. mabrouk, a. makhlouf, s. massoud the concept of generalized representation of a 3-lie algebra was introduced by liu, makhlouf and sheng in [19]. they study the corresponding generalized semidirect product 3-lie algebra and cohomology theory. furthermore, they describe general abelian extensions of 3-lie algebras using maurer-cartan elements. non-abelian extensions were explored in [26]. the aim of this paper is to extend the concept of generalized representation of 3-lie algebras to hom-type algebras. the notion of hom-lie algebras was introduced by hartwig, larsson, and silvestrov in [18] as part of a study of deformations of the witt and the virasoro algebras. the n-hom-lie algebras and various generalizations of n-ary algebras were considered in [4]. in a hom-lie algebra, the jacobi identity is twisted by a linear map, called the hom-jacobi identity. in particular, representations and cohomologies of homlie algebras were studied in [25], while the representations and cohomology of n-hom-lie algebras were first studied in [1]. the paper is organized as follows. in section 1, we provide some basics about 3-hom-lie algebras, representations and cohomology. the second section includes the new concept of generalized representation of a 3-hom-lie algebra, extending to hom-type algebras the notion and results obtained in [19]. we define a corresponding semi-direct product and provide a twist procedure leading to generalized representations of 3-hom-lie algebras starting from generalized representations of 3-hom-lie algebras and algebra maps. in section 3, we construct a new cohomology corresponding to generalized representations and show examples. in the last section we discuss abelian extensions of multiplicative 3-hom-lie algebras. one recovers the results in [19] when the twist map is the identity. 1. representations of 3-hom-lie algebras the aim of this section is to recall some basics about 3-lie algebras and 3-hom-lie algebras. we refer mainly to [15] and [4]. in this paper, all vector spaces are considered over a field k of characteristic 0. definition 1.1. a 3-lie algebra is a pair (g, [·, ·, ·]) consisting of a kvector space g and a trilinear skew-symmetric multiplication [·, ·, ·] satisfying the filippov-jacobi identity: for x,y,z,u,v in g [u,v, [x,y,z]] = [[u,v,x],y,z] + [x, [u,v,y],z] + [x,y, [u,v,z]]. in this paper, we are dealing with 3-hom-lie algebras corresponding to the following definition. generalized representations of 3-hom-lie algebras 101 definition 1.2. a 3-hom-lie algebra is a triple (g, [·, ·, ·],α) consisting of a k-vector space g, a trilinear skew-symmetric multiplication [·, ·, ·] and an algebra map α : g → g satisfying the hom-filippov-jacobi identity: for x,y,z,u,v in g [α(u),α(v), [x,y,z]] = [[u,v,x],α(y),α(z)] + [α(x), [u,v,y],α(z)] + [α(x),α(y), [u,v,z]]. remark 1.3. there is more general definition of 3-hom-lie algebras which are given by a quadruple (g, [·, ·, ·],α1,α2) consisting of a k-vector space g, two linear maps α1,α2 : g → g and a trilinear skew-symmetric multiplication [·, ·, ·] satisfying the following generalized hom-filippov-jacobi identity: for x,y,z,u,v in g [α1(u),α2(v), [x,y,z]] = [[u,v,x],α1(y),α2(z)] + [α1(x), [u,v,y],α2(z)] + [α1(x),α2(y), [u,v,z]]. we get our class of 3-hom-lie algebras when α1 = α2 = α and where α is an algebra morphism. this kind of algebras are usually called multiplicative 3-hom-lie algebras. proposition 1.4. let (g, [·, ·, ·]) be a 3-lie algebra and α : g → g be a 3-lie algebra morphism. then (g, [·, ·, ·]α := α ◦ [·, ·, ·],α) is a 3-hom-lie algebra. let (g, [·, ·, ·],α) be a 3-hom-lie algebra, elements in ∧2g are called fundamental objects of the 3-hom-lie algebra (g, [·, ·, ·],α). there is a bilinear operation [·, ·]l on ∧2g, which is given by [x,y ]l = [x1,x2,y1] ∧α(y2) + α(y1) ∧ [x1,x2,y2] for all x = x1 ∧ x2 and y = y1 ∧ y2, and a linear map α on ∧2g defined by α(x) = α(x1) ∧ α(x2), for simplicity, we will write α(x) = α(x). it is well-known that (∧2g, [·, ·]l,α) is a hom-leibniz algebra [1, 31]. definition 1.5. a representation of a 3-hom-lie algebra (g, [·, ·, ·],α) on a vector space v with respect to a ∈ gl(v ) is a skew-symmetric linear map ρ : ∧2g → end(v ) such that ρ(α(x1),α(x2)) ◦a = a◦ρ(x1,x2), (1.1) ρ(α(x1),α(x2))ρ(x3,x4) −ρ(α(x3),α(x4))ρ(x1,x2) (1.2) = ( ρ([x1,x2,x3],α(x4)) −ρ([x1,x2,x4],α(x3)) ) ◦a, 102 s. mabrouk, a. makhlouf, s. massoud ρ([x1,x2,x3],α(x4)) ◦a−ρ(α(x2),α(x3))ρ(x1,x4) (1.3) = ρ(α(x3),α(x1))ρ(x2,x4) + ρ(α(x1),α(x2))ρ(x3,x4), for x1,x2,x3 and x4 in g. theorem 1.6. let (g, [·, ·, ·]) be a 3-lie algebra, (v,ρ) be a representation, α : g → g be a 3-lie algebra morphism and a : v → v be a linear map such that a ◦ ρ(x1,x2) = ρ(α(x1),α(x2)) ◦ a. then (v,ρ̃ := a ◦ ρ,a) is a representation of the 3-hom-lie algebra (g, [·, ·, ·]α := α◦ [·, ·, ·],α). proof. let xi ∈ g, where 1 ≤ i ≤ 5. then we have ρ̃([x3,x4,x5]α,α(x1))◦a− ρ̃(α(x3),α(x4))ρ̃(x5,x1) − ρ̃(α(x4),α(x5))ρ̃(x3,x1) − ρ̃(α(x5),α(x3))ρ̃(x4,x1) = a2◦(ρ([x3,x4,x5],x1) −ρ(x3,x4)ρ(x5,x1) −ρ(x4,x5)ρ(x3,x1) −ρ(x5,x3)ρ(x4,x1)) = 0. the second condition (1.2) is obtained similarly. the previous result allows to twist along morphisms a 3-lie algebra with a representation to a 3-hom-lie algebra with a corresponding representation. proposition 1.7. let (g, [·, ·, ·],α) be a 3-hom-lie algebra, v be a vector space, a ∈ gl(v ) and ρ : ∧2g → gl(v ) be a skew-symmetric linear map. then (v ; ρ,a) is a representation of 3-hom-lie algebra g if and only if there is a 3-hom-lie algebra structure (g⊕v, [·, ·, ·]ρ,αg⊕v ) on the direct sum of vector spaces g⊕v , defined by [x1 + v1,x2 + v2,x3 + v3]ρ = [x1,x2,x3] + ρ(x1,x2)v3 + ρ(x3,x1)v2 + ρ(x2,x3)v1, and αg⊕v = α + a, for all xi ∈ g, vi ∈ v , 1 ≤ i ≤ 3. the obtained 3-hom-lie algebra is denoted by g nρ v and called semidirect product. let (g, [·, ·, ·],α) be a 3-hom-lie algebra and (v,ρ,a) be a representation of g. we denote by c p α,a(g,v ) the space of all linear maps ϕ : ∧2g⊗···⊗∧2g︸︷︷︸ (p−1) ∧g → v generalized representations of 3-hom-lie algebras 103 satisfying: a◦ϕ(x1 ⊗···⊗xp−1,y) = ϕ(α(x1) ⊗···⊗α(xp−1),α(y)), for all x1, . . . ,xp−1 ∈∧2g, y ∈ g. it is called the space of p-cochains. let ϕ be a (p − 1)-cochain, the coboundary operator δρ : c p−1 α,a (g,v ) → c p α,a(g,v ) is given by (δρϕ)(x1, . . . ,xp,z) = ∑ 1≤jm+1 (−1)(j,i)(−1)(l,h)ξ(αp+q(xh1 ), . . . ,α p+q(xhm ),ψ(α q(xl1 ), . . . ,α q(xlp ), ) •α αp+q(xhm+1 ),α p+q(xhm+2 ), . . . ,α p+q(xhn−1 ),α p(ϕ(xj1, . . . ,xjq, ) •α xik+1 ),α p+q(xhn+1, . . . ,α p+q(xhr ),α p+q(x)) 1 0 6 s . m a b r o u k , a . m a k h l o u f , s . m a s s o u d + ∑ j,jq h or α /∈ nk. remark 3. for any |α| ≤ h, the map x(m) −→ x(tam) x 7−→ x(α) is r-linear and for any smooth map ϕ : m → n and any ϕ-related vector fields x ∈ x(m), y ∈ x(n), the vector fields x(α) ∈ x(tam), y (α) ∈ x(tan) are ta(ϕ) related. proposition 2. for x,y ∈ x(m), we have:[ x(α),y (β) ] = [x,y ] (α+β) for all 0 ≤ |α,β| ≤ h. proof. see [5]. prolongations of g-structures related to weil bundles 117 remark 4. the family of α-lift of vector fields is very important, because, if s and s′ are two tensor fields of type (1,p) or (0,p) on ta (m) such that, for all x1, . . . ,xp ∈ x (m), and multiindex α1, . . . ,αp, the equality s ( x (α1) 1 , . . . ,x (αp) p ) = s′ ( x (α1) 1 , . . . ,x (αp) p ) holds, then s = s′ (see [2]). 1.2. lifts of tensor fields of type (1,q). let s be a tensor field of type (1,q), we interpret the tensor s as a q-linear mapping s : tm ×m · · ·×m tm −→ tm of the bundle product over m of q copies of the tangent bundle tm. for all 0 ≤ |α| ≤ h, we put: s(α) : t(tam) ×tam · · ·×tam t(t am) −→ t(tam) with s(α) = κa,m ◦ χ (α) tm ◦ t a(s) ◦ ( κ−1a,m ×···×κ −1 a,m ) . it is a tensor field of type (1,q) on ta(m) called α-prolongation of the tensor field s from m to ta(m). in the particular case where α = 0, it is denoted by s(c) and is called complete lift of s from m to ta (m). proposition 3. the tensor s(α) is the only tensor field of type (1,q) on ta(m) satisfying s(α) ( x (α1) 1 , . . . ,x (αq) q ) = ( s(x1, . . . ,xq) )(α+α1+···+αq) for all x1, . . . ,xq ∈ x(m) and multiindex α1, . . . ,αq. proof. see [2]. for some properties of these lifts, see [2] and [3]. 1.3. lifts of tensor fields of type (0,s). we fix the linear map p : a → r, for any vector bundle (e,m,π), we consider the natural vector bundle morphism τ p a,e : t ae∗ → ( tae )∗ (see [10]) defined for any jaϕ ∈ tae∗ and jaψ ∈ tae by: τ p a,e(j aϕ)(jaψ) = p ( ja(〈ψ,ϕ〉e) ) 118 p.m.k. wamba, g.f. wankap nono, a. ntyam where 〈ψ,ϕ〉e : r k → r, z 7→ 〈ψ (z) ,ϕ (z)〉e and 〈·, ·〉e the canonical pairing. for any manifold m of dimension m, we consider the vector bundle morphism ε p a,m = [ κ−1a,m ]∗ ◦ τpa,tm : t at∗m −→ t∗tam. it is clear that the family of maps ( ε p a,m ) defines a natural transformation between the functors ta◦t∗ and t∗◦ta on the category mfm of m-dimensional manifolds and local diffeomorphisms, denoted by ε p a,∗ : t a ◦t∗ −→ t∗ ◦ta. when (a,p) is a weil-frobenius algebra (see [4]), the mapping ε p a,m is an isomorphism of vector bundles over idtam . being { x1, . . . ,xm } a local coordinate system of m, we introduce the coordinates ( xi, ẋi ) in tm, ( xi,πi ) in t∗m, (xi, ẋi,xiβ, ẋ i β) in t atm, (xi,πj,x i β,π β j ) in t at∗m, (xi,xiβ, ẋ i, ẋ i β) in ttam and (xi,xiβ,ξj,ξ β j ) in t ∗tam. we have ε p a,m ( xi,πj,x i β,π β j ) = ( xi,xiβ,ξj,ξ β j ) with   ξj = πjp0 + ∑ µ∈ba π µ j pµ , ξ β j = ∑ µ∈ba π µ−β j pµ , and pα = p (eα). let g be a tensor fields of type (0,s) on a manifold m. it induces the vector bundle morphism g] : tm ×m · · · ×m tm → t∗m of the bundle product over m of s− 1 copies of tm. we define, g(p) : t(tam) ×tam · · ·×tam t(t am) −→ t∗(tam) as g(p) = ε p a,m ◦ t a(g]) ◦ ( κ−1a,m ×···×κ −1 a,m ) . it is a tam-morphism of vector bundles, so g(p) is tensor field of type (0,s) on tam called pprolongation of g from m to tam. example 1. in a particular case, where s = 2 and locally g = gijdx i ⊗ dxj then g(p) = gijp0dx i ⊗dxj + ∑ α∈ba pα ( ∑ β∈ba g (α−β) ij ) dxi ⊗dxjβ + ∑ µ,β∈ba ( ∑ α∈ba pαg (α−β−µ) ij ) dxiµ ⊗dx j β. prolongations of g-structures related to weil bundles 119 in the particular case where a = jr0 (r k,r) and p(jr0ϕ) = 1 α! dα (ϕ(z))|z=0, then g(p) coincides with the α-prolongation of g from m to trkm defined in [13]. example 2. if ωm is a liouville 2-form on t ∗m defined in local coordinate system ( xi,ξj ) by: ωm = dx i ∧dξi, then we have: ω (p) m = p0dx i ∧dξi + ∑ α∈ba pαdx i ∧dξαi + ∑ α,β∈ba pαdx i β ∧dξ α−β i . proposition 4. the tensor field g(p) is the only tensor field of type (0,s) on ta(m) satisfying, for all x1, . . . ,xs ∈ x(m) and multiindex α1, . . . ,αs g(p) ( x (α1) 1 , . . . ,x (αs) s ) = (g(x1, . . . ,xs)) (p◦lα1+···+αs) where la : a → a is given by la(x) = ax. proof. see [5]. 2. the natural transformations ja,e : t a(fe) → f(tae) let v be a real vector space of dimension n, we denote by gl(v ) the lie group of automorphisms of v . 2.1. the embedding ja,v : t a(gl(v )) → gl(tav ). let g be a lie group and m be a m−dimensional manifold, m ≥ 1. we consider the differential action ρ : g×m → m, then the lie group tag acts to tam by the differential action taρ : tag×tam → tam. lemma 1. if the lie group g operates on m effectively, then tag operates on tam effectively by the differential action ta(ρ). proof. see [5]. let ρv : gl(v ) × v → v be the canonical action of gl(v ), then the lie group ta(gl(v )) operates effectively on the vector space tav by the induced action ta(ρv ) : t a(gl(v )) ×tav −→ tav( jaϕ,jau ) 7−→ ja(ϕ∗u) 120 p.m.k. wamba, g.f. wankap nono, a. ntyam where ϕ∗u : rk → v is defined for any z ∈ rk by: ϕ∗u(z) = ϕ(z)(u(z)). we deduce an injective map ja,v : t a(gl(v )) → gl(tav ) such that, ja,v (j ag) : tav −→ tav jaξ 7−→ ja(g ∗ ξ) . proposition 5. the map ja,v : t a(gl(v )) → gl(tav ) is an embedding of lie groups. proof. by calculation, it is clear that ja,v is a homomorphism of lie groups. remark 5. let {e1, . . . ,en} be a basis of v (dim v = n), we consider the global coordinate system of v , ( e1, . . . ,en ) , we denote by ( yij ) the global coordinate of gl(v ), for any f ∈ gl(v ), yij(f) = 〈 ei,f(ej) 〉 where 〈·, ·〉 is the duality bracket v ∗×v → r. we deduce that, the coordinate system of ta(gl(v )) is denoted by ( yij,y i j,α ) α∈ba . on the other hand, the global coordinate system of tav is ( ei,eiα ) , such that:  ei ( jau ) = ei(u(0)) , eiα ( jau ) = 1 α! dα(e i ◦u)(z) ∣∣ z=0 + ∑ β∈ba λα β β! dβ(e i ◦u)(z) ∣∣ z=0 , j au ∈ tav, the global coordinate of gl(tav ) denoted ( zij,z i,β j,α ) α,β∈ba is such that:  zij(ξ) = 〈 ei,πa,v (ξ)(ej) 〉 , z i,β j,α(ξ) = 〈 eiβ,ξ ( eαj )〉 , ξ ∈ gl(tav ) , we deduce that the local coordinate of the map ja,v is given by: ja,v ( yij,y i j,α ) =   yij 0 · · · · · · 0 ... . . . . . . ... ... . . . . . . ... ... . . . 0 · · · · yij,α · · · y i j   prolongations of g-structures related to weil bundles 121 in fact, z i,β j,α ( ja,v ( jag )) = 〈 eiβ,ja,v ( jag )( eαj )〉 = 1 β! dβ ( tα 〈 ei,g(t)(ej) 〉)∣∣ t=0 + ∑ µ∈ba λ µ β µ! dµ ( tα 〈 ei,g(t)(ej) 〉)∣∣ t=0 for any jag ∈ ta(gl(v )). 2.2. frame gauge functor on the vector bundles. we denote by vbm the category of vector bundles with m-dimensional base together with local isomorphism. let bvbm : vbm → mf and bfm : fm → mf be the respective base functors. definition 1. (see [11]) a gauge bundle functor on vbm is a covariant functor f : vbm →fm satisfying: 1. (base preservation) bfm ◦f = bvbm; 2. (locality) for any inclusion of an open vector bundle ıe|u : e|u → e, f (e|u ) is the restriction p−1e (u) of pe : e → vbm(e) over u and f ( ıe|u ) is the inclusion p−1e (u) → fe. definition 2. let g be a lie group. a principal fiber bundle is a fiber bundle (p,m,π) of standard fiber g such that: there is a fiber bundle atlas( uα,ϕα : π −1 (uα) → uα ×g ) α∈a, the family of smooth maps θαβ : uα ∩ uβ → g which satisfies the cocycle condition (θαβ(x) · θβγ(x) = θαγ(x) for x ∈ uα ∩uβ ∩uγ and θαα (x) = e) and for each x ∈ uα ∩uβ , for each g ∈ g, ϕα ◦ϕ−1β (x,g) = (x, θαβ(x) ·g) . example 3. let (e,m,π) be a vector bundle of standard fiber the real vector space v of dimension n ≥ 1. for any x ∈ m, we denote by fxe the set of all linear isomorphisms of v on ex and we set fe = ⋃ x∈mfxe, it is clear that fe is an open set of the manifold hom(m ×v, e). we denote by pe : fe → m the canonical projection. let (uα,ψα)α∈λ the fiber bundle atlas of (e,m,p), so for all x ∈ uα∩uβ and v ∈ v , ψα◦ψ−1β (x,v) = (x,θαβ(x)(v)), where θαβ : uα ∩ uβ → gl(v ) satisfies the cocycle condition. we consider 122 p.m.k. wamba, g.f. wankap nono, a. ntyam the smooth map ϕα : p −1 e (uα) → uα ×gl(v ) such that, for any x ∈ uα and fx ∈ p−1m (uα), ϕα(fx) = (x,ψα|ex ◦fx) . it is clear that, (uα,ϕα)α∈λ is the fiber bundle atlas of (fe,m,pe). as ϕβ ◦ ϕ−1α (x,f) = (x,θαβ(x) ◦f), it follows that (fe,m,pe) is a principal bundle of standard fiber, the linear lie group gl(v ). it is called the frame bundle of the vector bundle (e,m,π). remark 6. let ( u,xi ) be a local coordinate system of m, we denote by( xi,xij ) the local coordinate of fm induced by ( u,xi ) , it is such that: { xi (ξ) = xi(pe(ξ)) , xij (ξ) = 〈 dxi, (ξ(ej)) 〉 , for ξ ∈ fm and (e1, . . . ,en) is a basis of v . definition 3. φ : (p,m,p,g) → (p ′,m ′,p′,g′) is a homomorphism of principal bundles over the homomorphism of lie groups φ : g → g′ if φ : p → p ′ is smooth and satisfies for each u ∈ p , for each g ∈ g, φ(u ·g) = φ(u) ·φ(g) . the collection of principal bundles and their homomorphisms form a category, it is called the category of principal bundles and denoted by pb. in particular, it is subcategory of the category fm. example 4. let f : e1 → e2 an isomorphism of vector bundles over the diffeomorphism f : m1 → m2. the smooth map f(f) : fe1 → fe2 defined for any ϕx ∈ fxe1 by: f(f)(ϕx) = fx ◦ϕx ∈ ff(x)e1 is such that ( f,f(f) ) : (fe1,m1,pe1 ) → (fe2,m2,pe2 ) is an isomorphism of principal bundles. we obtain in particular a functor f : vbn → pb, it is a covariant functor. proposition 6. the functor f : vbn → fm is a gauge bundle functor on vbn which do not preserves the fiber product. it is called the frame gauge functor on vbn. prolongations of g-structures related to weil bundles 123 proof. the properties of gauge functor f : vbn →fm are easily verified by calculation. since do not exists an isomorphism between the lie groups gl(v1) × gl(v2) and gl(v1 ⊕ v2), it follows that the gauge functor f do not preserves the fiber product. remark 7. let (p,m,π) be a principal fiber bundle with total space p , base space m, projection π and structure group g. if {uα}α∈λ is an open covering of m, for each α ∈ λ, p giving a trivial bundle over uα, and if gαβ : uα ∩uβ → g are the transition functions of p , we express this fiber bundle by p = {uα,gαβ}. when g is a lie subgroup of a lie group g′ and j : g → g′ is the injection map, then there is a fiber bundle p ′ = {uα,j ◦gαβ} and an injection j : p → p ′ which is a bundle homomorphism i.e. j(p ·a) = j(p) ·a, for any p ∈ p and a ∈ g. 2.3. the natural embedding ja,e : t a(fe) → f(tae). we denote with (e,m,π) a vector bundle of standard fiber the real vector space v of dimension n ≥ 1. then, ( tae,tam,taπ ) is a real vector bundle of standard fiber tav , in particular the frame bundle of this vector bundle is a gl(tav )-principal ( f(tae),tam,ptae ) . on the other hand, (fe,m,pe) is a gl(v )-principal bundle, so ( ta(fe),tam,ta(pe) ) is a ta(gl(v ))principal bundle. let (uα,ψα)α∈λ a fiber bundle atlas of (e,m,π), so that( tauα,t aψα ) α∈λ is a fiber bundle atlas of ( tae,tam,taπ ) . the bundle atlas of the principal bundle (fe,m,pe) is denoted by (uα,ϕα)α∈λ where ϕα : p −1 e (uα) −→ uα ×gl(v ) g 7−→ ( pe(g), (ψα)pe(g) ◦g ) , we deduce that ( tauα,t a(ϕα) ) α∈λ is the following fiber bundle atlas of( ta(fe),tam,ta(pe) ) , ta(ϕα) : ( tape )−1 ( tauα ) −→ tauα ×ta(gl(v )) jag 7−→ ( tape(j ag),ja(ψα ·g) ) , where (ψα · g)(z) = (ψα)pe(g(z)) ◦ g(z) : v → v is a linear isomorphism, for all z ∈ rk. as ( tauα,t aψα ) α∈λ is a fiber bundle atlas of ( tae,tam,taπ ) , it follows that the fiber bundle atlas of the principal bundle ( f(tae),tam,ptae ) 124 p.m.k. wamba, g.f. wankap nono, a. ntyam is denoted by ( tauα,ϕα,a ) α∈λ where ϕα,ap −1 tae (tauα) −→ tauα ×gl(tav ) ξ 7−→ ( ptae(ξ), ( ta(ψα) ) p tae (ξ) ◦ ξ ) and ϕ−1α,a ( x̃, ξ̃ ) = ( taψα )−1 (x̃, ·) ◦ ξ̃, for any ( x̃, ξ̃ ) ∈ tauα × gl(tav ). for any α ∈ λ, we put ja,uα = ϕ −1 α,a ◦ (idtauα,ja,v ) ◦t a(ϕα) : ( tape )−1 (tauα) −→ p−1tae(t auα) and for any jag ∈ ( tape )−1 (tauα), we have: ja,uα(j ag) = ϕ−1α,a ( ja(pe ◦g),ja,v ( ja(ψα ·g) )) = ( taψα )−1 ( ja(pe ◦g), · ) ◦ ja,v ( ja(ψα ·g) ) . for β ∈ λ such that uα ∩ uβ 6= ∅, we have ja,uα ∣∣ (tape) −1(tauα∩tauβ) = ja,uβ ∣∣ (tape) −1(tauα∩tauβ) , it follows that, it exists one and only one principal fiber bundle homomorphism ja,e : t a(fe) → f(tae) such that, for any α ∈ a, ja,e ∣∣ (tape) −1 (tauα) = ja,uα. in particular, for any ξ̃ ∈ ta(fe) and ũ ∈ ta(gl(v )), ja,e ( ξ̃ · ũ ) = ja,e ( ξ̃ ) · ja,v (ũ) . theorem 1. the map ja,e : t a(fe) → f(tae) is a principal fiber bundle homomorphism over the homomorphism of lie groups ja,v : t a(gl(v )) → gl(tav ). in particular, ja,e is an embedding. proof. it is clear that, ja,e : t a(fe) → f(tae) is a principal fiber bundle homomorphism over ja,v , because for any ξ̃ ∈ ta(fe) and ũ ∈ ta(gl(v )), ja,e ( ξ̃ · ũ ) = ja,e ( ξ̃ ) · ja,v (ũ) . on the other hand, for any α ∈ a, ja,e ∣∣ (tape) −1 (tauα) = ja,uα, it follows that ja,e is an embedding. remark 8. let ( π−1(u),xi,yj ) be a fiber chart of e, then the local coordinate of fe and tae are ( p−1e (ui),x i,y j k ) and (( taπ )−1 (tau),xiα,y j α ) . prolongations of g-structures related to weil bundles 125 we deduce that, the local coordinate of ta(fe) and f(tae) are given by( ta(p−1e (ui)),x i α,y j k, y j k,α ) and ( p−1 tae ( tau ) ,xiα,y j,α k,β ) , so the local expression of ja,e is given by: ja,e ∣∣ (tape) −1 (tau) ( xiα,y j k,y j k,α ) =  xiα,   y j k 0 · · · 0 ... . . . . . . ... ... . . . 0 · · · yjk,α · · · y j k     . proposition 7. let f : e → e′ is an isomorphism of vector bundles over the diffeomorphism f : m → m ′. the following diagram ta(fe) ta(ff)) −−−−−−−→ ta(fe′) ja,e y yja,e′ f(tae) −−−−−−→ f(taf) f(tae′) commutes. proof. let (uα,ψα)α∈λ and (u ′ α,ψ ′ α)α∈λ the bundle atlas of (e,m,π) and (e′,m ′,π′) such that f(uα) = u ′ α, α ∈ λ. as f : e → e′ is an isomorphism of vector bundles over f, it follows that it exists a smooth map fα : uα×v → v such that ψ′α◦f ∣∣ π−1(uα) ◦ψ−1α (x,v) = ( f(x), fα(x,v) ) , for any (x,v) ∈ uα×v and fα(x, ·) is a linear isomorphism. it follows that, the diagram p−1e (uα) ff ∣∣ p −1 e (uα) −−−−−−−−−→ p−1e′ (u ′ α) ϕα y yϕ′α uα ×gl(v ) −−−−−−−→ f̃α u ′α ×gl(v ) commutes, and f̃α(x,g) = ( f(x), fα(x, ·) ◦g ) , for each (x,g) ∈ uα ×gl(v ). it is clear that the following diagram ( tape )−1 ( tauα ) ta(ff)∣∣(tape)−1(tauα)−−−−−−−−−−−−−−−−−−→ (tape′)−1 (tau ′α) taϕα y ytaϕ′α tauα ×ta(gl(v )) −−−−−−−−−−−−−−−→ ta ( f̃α ) tau ′α ×ta(gl(v )) 126 p.m.k. wamba, g.f. wankap nono, a. ntyam commutes. on the other hand, as the diagram following commutes tauα ×ta(gl(v )) ta ( f̃α ) −−−−−−−→ tau ′α ×ta(gl(v )) (iduα,ja,v ) y y(idu′α,ja,v ) tauα ×gl ( tav ) −−−−−−→ f̃α,a tau ′α ×gl ( tav ) with f̃α,a ( x̃, ξ̃ ) = ( taf (x̃) , fα,a(x̃, ·) ◦ ξ̃ ) where ta(ψ′α) ◦t af ∣∣ (taπ) −1 (tauα) ◦ ( taψα )−1 (x̃,v) = ( taf (x̃) , fα,a(x̃, ·) ) , it follows that( idu′α,ja,v ) ◦ta ( f̃α )( jau,jaξ ) = ( idu′α,ja,v )( taf ( jau ) ,ja ( f̃ (u, ·) ◦ ξ )) = ( taf(jau), ja,v ( ja ( f̃ (u, ·) ◦ ξ ))) . as ja,v ( ja ( f̃ (u, ·) ◦ ξ ))( jav ) = ja (( f̃ (u, ·) ◦ ξ ) ·v ) and ( f̃(u, ·) ◦ ξ ) ·v (z) = f̃ (u(z),ξ(z)(v(z))) , for any z ∈ rk, thus, f(taf) ◦ ja,uα ( jau,jaξ ) = f(taf) ( jau, ja,v ( jaξ )) = ( taf ( jau ) ,taf̃ ( jau,◦ ) ◦ ja,v ( jaξ )) . for any jav ∈ tav , as ja,v ( jaξ )( jav ) = ja (ξ ∗v) with ξ ∗ v(z) = ξ(z)(v(z)), for all z ∈ rk, we deduce that taf̃ ( jau,◦ ) ◦ ja,v ( jaξ )( jav ) = taf̃ ( jau,ja (ξ ∗v) ) = ja ( f̃ (u,ξ ∗v) ) , prolongations of g-structures related to weil bundles 127 so taf̃ ( jau,◦ ) ◦ ja,v ( jaξ )( jav ) = ja,v ( ja ( f̃ (u, ·) ◦ ξ ))( jav ) for any jav ∈ tav . more precisely, ja,u′α ◦t a ( f̃α ) = f̃α,a ◦ ja,uα, ja,e′ ∣∣ (tape′) −1 (tau′α) ◦ta(ff) = ϕ ′−1 α,a ◦ ja,u′α ◦t a ( ϕ′α ) ◦ta(ff) = ϕ ′−1 α,a ◦ ja,u′α ◦t a ( ϕ′α ◦ff ◦ϕ −1 α ) ◦ta(ϕ−1α ) = ϕ ′−1 α,a ◦ ja,u′α ◦t a ( f̃α ) ◦ta(ϕ−1α ) = ϕ ′−1 α,a ◦ f̃α,a ◦ ja,uα ◦t a(ϕ−1α ) = ( ϕ ′−1 α,a ◦ f̃α,a ◦ϕα,a ) ◦ϕ−1α,a ◦ ja,uα ◦t a(ϕ−1α ) = f(taf) ◦ ja,e ∣∣ (tape) −1 (tauα) , thus, ja,e′ ◦ta(ff) = f(taf) ◦ ja,e. let (e,m,π) be a vector bundle of standard fiber v , for any t ∈ r, we consider the linear automorphism of e, gt : e → e defined by: gt(u) = exp(t)u, for any u ∈ e. we consider the principal bundle isomorphism over idm , ϕt + f(gt) : fe → fe such that, for any x ∈ m, ϕt ∣∣ fxe : fxe −→ fxe hx 7−→ hx ◦gt . in particular, we deduce a smooth map ϕ : r × fe → fe, (t,ξ) 7→ ϕt(ξ). for any multi index α, we consider the smooth map ϕα,e : t a(fe) −→ ta(fe) ξ 7−→ taϕ(eα,ξ). then ta(pe) ◦ϕα,e = ta(pe). in particular, it is a homomorphism of principal bundle of ta(fe) in to ta(fe). proposition 8. let f : e → e′ be an isomorphism of vector bundles over the diffeomorphism f : m → m ′. then the following diagram ta(fe) ta(ff) −−−−−−−→ ta(fe′) ϕα,e y yϕα,e′ ta(fe) −−−−−−−→ ta(ff) ta(fe′) commutes. 128 p.m.k. wamba, g.f. wankap nono, a. ntyam proof. let jaξ ∈ ta(fe), we have: ϕα,e′ ◦ta(ff) ( jaξ ) = ϕα,e′ ( ja(f(f) ◦ ξ) ) = taϕ ( ja(tα),ja(f(f) ◦ ξ) ) = ja (ϕ(tα,f(f) ◦ ξ)) = ja (f(f) ◦ϕ(tα,ξ)) = ta(f(f)) ◦ϕα,e ( jaξ ) . therefore, ϕα,e′ ◦ta(ff) = ta(f(f)) ◦ϕα,e. 3. prolongations of g-structures to weil bundles 3.1. the natural embedding ja,m : t a(fm) → f(tam). let m be a smooth manifold of dimension n ≥ 1, we denote by gl(n) the lie group gl(rn) and (f(m),m,pm ) the frame bundle of the tangent vector bundle (tm,m,πm ), so that ( ta(fm),tam,ta(pm ) ) is a principal fiber bundle over the lie group ta(gl(n)). by the same way ( f(tam),tam,ptam ) is a frame bundle of the vector bundle ( t(tam),tam,πtam ) . if f : m → n is a local diffeomorphism, we denote with f(f) the principal bundle homomorphism f(tf) : fm → fn. let m be a smooth n-dimensional manifold, f(κa,m ) : f(t atm) −→ f(tam) is an isomorphism of principal bundles over idtam and ptam ◦ f(κa,m ) = ptatm , where κa,m : t a(tm) → t(tam) is the canonical isomorphism defined in [7]. we put ja,m = f(κa,m ) ◦ ja,tm : ta(fm) −→ f(tam) such that ptam ◦ja,m = ta(pm ) and ja,m (x̃·g) = ja,m (x̃)·ja,rn(g). in particular ja,m is a homomorphism of principal bundles over ja,rn. we identify tarn with the euclidian vector space rn×dim a, it follows that ta(gl(n)) is a lie subgroup of gl(n× dim a). proposition 9. let m and n be two manifolds and f : m → n be a diffeomorphism between them. then the following diagram prolongations of g-structures related to weil bundles 129 ta(fm) ta(ff) −−−−−−−→ ta(fn) ja,m y yja,n f(tam) −−−−−−−→ f(taf) f(tan) commutes. proof. let f : m → n a diffeomorphism, ja,n ◦ta(ff) = f(κa,n ) ◦ ja,tn ◦ta(ff) = f(κa,n ) ◦f(tatf) ◦ ja,tm = f ( κa,n ◦ta(tf) ) ◦ ja,tm = f(t(taf) ◦κa,m ) ◦ ja,tm = f(t(taf)) ◦f(κa,m ) ◦ ja,tm = f(taf) ◦f(κa,m ) ◦ ja,tm. we deduce that ja,n ◦ta(ff) = f ( taf ) ◦ ja,m . remark 9. let ( u,xi ) be a local coordinate on a manifold m, the local coordinate of fm is denoted by ( p−1m (u),x i,xij ) , ( tau,xi,xiα ) the local coordinate of tam , (( tapm )−1 ( tau ) ,xi,xij,x i α,x i j,α ) the local coordinate of ta(fm) and ( p−1 tam ( tau ) ,xi,xiα,x i j,x i,β j,α ) local coordinate of f(tam). the formula ja,m (x i,xij,x i α,x i j,α) =  xi, xiα,   xij 0 · · · 0 ... . . . . . . ... ... . . . 0 · · · xij,α · · · x i j     is a local expression of the natural embedding ja,m . 3.2. prolongations of g-structures. let g be a lie subgroup of gl(n), we denote by ga,n the image of t ag by the homomorphism ja,rn, i.e. ga,n = ja,rn(t ag). clearly ga,n is a lie subgroup of gl(n × dim a). 130 p.m.k. wamba, g.f. wankap nono, a. ntyam let (p,m,π) be a g-structure on m, we denote by πa the restriction of the projection ptam : f(t am) → tam to the subbundle t ap = ja,m (tap). then we obtain a ga,n-structure ( t ap,tam,πa ) on the weil bundle tam of m related to a. it is called the a-prolongation of the g-structure p to the weil bundle tam to m. proposition 10. let p (resp. p ′) be a g-structure on m (resp. m ′) and f : m → m ′ be a diffeomorphism. then f is an isomorphism of p on p ′ if and only if taf : tam → tam ′ is an isomorphism of t ap on t ap ′. proof. the diffeomorphism f : m → m ′ is an isomorphism of p on p ′, if and only if f(f)(p) = p ′. by the equality ja,m′ ◦ta(ff) = f ( taf ) ◦ja,m it follows that, if f is an isomorphism of p on p ′, then t ap ′ = ja,m′ ( tap ′ ) = ja,m′ ◦ta(ff) ( tap ) = f ( taf ) ◦ ja,m ( tap ) = f ( taf )( t ap ) . inversely, if taf : tam → tam ′ is an isomorphism of t ap on t ap ′, then ja,m′(t ap ′) = f(taf)(tap) = f(taf) ◦ ja,m (tap) = ja,m′ ◦ta(ff)(tap). therefore, tap ′ = ta(ff)(tap). in particular, p ′ = πa,p ′(t ap ′) = πa,p ′◦ ta(ff)(tap) = f(f)◦πa,p (tap) = f(f)(p). so f is an isomorphism of p on p ′. corollary 1. let f be a diffeomorphism of m into itself, and p be a g-structure on m. then f is an automorphism of p if and only if taf is an automorphism of the a-prolongation t ap . let φ : m → fm be a smooth section, the we define φ̃a = ja,m ◦ta(φ), where ja,m : t a(fm) → f(tam) is the natural embedding from subsection 3.1. it is a smooth section of the frame bundle f(tam) called complete lift of φ to f(tam). remark 10. let ( u,x1, . . . ,xn ) be a local coordinate of m, we introduce the coordinate ( tau,xiα ) of tam. let φ : m → fm be a smooth section such that φ ∣∣ u = φij ( ∂ ∂xi ) ⊗ej, prolongations of g-structures related to weil bundles 131 then φ̃a ∣∣ tau = ( φij )(α−β) ( ∂ ∂xiα ) ⊗ejβ , where { ei } i=1,...,n and { ei,eiα } (i,α)∈{1,...,n}×ba are the dual basis of the canonical basis of rn and ta(rn). definition 4. let (p,m,π) be a g-structure on m. the g-structure p is called integrable (or flat) if for each point x ∈ m, there is a coordinate neighborhood u with local coordinate system ( x1, . . . ,xn ) such that the frame(( ∂ ∂x1 ) y , . . . , ( ∂ ∂xn ) y ) ∈ py for any y ∈ u. proposition 11. let p be a g-structure on a manifold m. then, p is integrable if and only if the a-prolongation t ap of p is integrable. proof. we suppose that p is integrable, then there is a cross section φ : u → p of p over u ⊂ m of fm such that φ = n∑ i=1 ( ∂ ∂xi ) ⊗ei. then φ̃a = ja,m ◦ta(φ) is a cross section of t ap over tau and, φ̃a = ∑ α∈ba ( ∂ ∂xiα ) ⊗eiα so, the a-prolongation t ap of p is integrable. inversely, taking (a1, . . . ,ak) be a basis of na over r. we consider the basis b = (1a,a1, . . . ,ak) as a linear isomorphism a → rk+1 and let παb : a → r be the composition of b with the projection rk+1 → r on α-factor, α = 1, . . . ,k + 1. for a coordinate system ( u,xi ) in m we define the induced coordinate system { xi0,x i α } on tam by:{ xi0 = x i ◦πam , xiα = ( xi )(παb) , α = 1, . . . ,k . using these arguments, the proof is similar as for the case of tangent bundle of higher order establish in [12]. 132 p.m.k. wamba, g.f. wankap nono, a. ntyam 4. prolongations of some classical g-structures 4.1. complex structures. we take j0 : r2n → r2n a linear automorphism such that j0 ◦ j0 = −idrn and denote by g(n,j0) the group of all a ∈ gl(2n) such that a ◦ j0 = j0 ◦ a. we consider {1a,eα, α ∈ ba} be a basis of a over r. we consider this basis as a linear isomorphism ta(r2n) → r2n dim a. the map ta(j0) is a linear automorphism of ta(r2n) such that ta(j0) ◦ta(j0) = −idta(rn). we put, g̃ = ja,r2n ( ta(g(n,j0)) ) . proposition 12. the lie group g̃ is a lie subgroup of g ( n × dim a, ta(j0) ) . proof. let ã ∈ g̃, then there is an element x ∈ ta(g(n,j0)), such that ã = ja,rn(x). we put x = j aϕ, with ϕ : rk → g(n,j0) smooth map. for any jaξ ∈ tarn, we have: ta(j0) ◦ ã ( jaξ ) = ta(j0) ( ja(ϕ∗ ξ) ) = ja(j0 ◦ (ϕ∗ ξ)). as, for any z ∈ rk, j0 ◦ (ϕ∗ ξ)(z) = j0 ◦ϕ(z)(ξ(z)) = ϕ(z) ◦j0(ξ(z)) = ϕ∗ (j0 ◦ ξ)(z) , we deduce that ta(j0) ◦ ã ( jaξ ) = ja(ϕ∗ (j0 ◦ ξ)) = ja,rn ( jaϕ )( ja(j0 ◦ ξ) ) = ja,rn(x) ◦ta(j0) ( jaξ ) . so, ta(j0) ◦ ã ( jaξ ) = ã◦ta(j0) ( jaξ ) , for all jaξ ∈ tarn. remark 11. let m be a smooth manifold of dimension 2n, m has an almost complex structure if and only if m has a g (n,j0)-structure p . applying subsection 2.2, we see that tam has canonically a g̃-structure t ap . by proposition 9, t ap induces canonically a g(n dim a,ta(j0))-structure p̃a. which means that tam has a canonical almost complex structure. theorem 2. the canonical almost complex structure j̃a on tam induced by a g(n dim a,ta(j0))-structure p̃ a is just the complete lift j(c) of the associated almost complex structure j with p . prolongations of g-structures related to weil bundles 133 proof. let φ : m → p be a smooth section, then j(x) = φ(x)◦j0◦φ(x)−1, for any x ∈ m. consider the vector ei,α = ja(zαei), with α ∈ ba and i ∈ {1, . . . , 2n}. the family (ei,α) is a basis of the real vector space ta(rn). if φ ∣∣ u = φ j i ( ∂ ∂xj ) ⊗ei then φ̃a ∣∣ tau = ( φ j i )(α−β) ( ∂ ∂x j α ) ⊗eiβ. in particular φ̃a(ei,α) = ( φ j i )(α−β) ( ∂ ∂x j β ) = (φ(ei)) (α) , so j̃a ◦ φ̃a(ei,α) = j̃a ( (φ(ei)) (α) ) . for any jaξ ∈ tam, we have φ̃a ◦ta(j0)(ei,α) ( jaξ ) = κa,m ◦ ja,tm ( ta(φ) ◦ta(j0) ( ja(zαei) ))( jaξ ) = κa,m ◦ ja,tm ( ja(φ◦ ξ) )( ja(zαj0(ei)) ) = κa,m ( ja ((φ◦ ξ) ∗ (zαj0(ei))) ) . for any z ∈ rk, (φ◦ ξ) ∗ (zαj0(ei))(z) = φ(ξ(z))(zαj0(ei)) = zαφ(ξ(z)) ◦j0(ei) = zαj(ξ(z)) ◦φ(ξ(z))(ei) = j(ξ(z)) ◦φ(zαei)(ξ(z)), we deduce that φ̃a ◦ta(j0)(ei,α) ( jaξ ) = κa,m ◦taj ( χ (α) tm ◦t a(φ(ei)) )( jaξ ) = ( κa,m ◦ta(j) ◦k−1a,m ) ◦ ( κa,m ◦χ (α) tm ◦t a(φ(ei)) )( jaξ ) = j(c) ( (φ(ei)) (α) )( jaξ ) . as φ̃a ◦ ta(j0)(ei,α) = j̃a ◦ φ̃a(ei,α), we deduce that, j̃a ( (φ(ei)) (α) ) = j(c) ( (φ(ei)) (α) ) , for any α ∈ ba. so j̃a is the complete lift of j. 4.2. almost symplectic structure. let f : r2n × r2n → r be a skew-symmetric non degenerate bilinear form on r2n. we denote by g(f) the group of all a ∈ gl(2n) such that f(a(x),a(y)) = f(x,y), for all x,y ∈ r2n. we consider the basis of a over r, {1a,eα, α ∈ ba} as a linear isomorphism ta(r2n) → r2n dim a. we suppose that, a is a weil-frobenius algebra, so 134 p.m.k. wamba, g.f. wankap nono, a. ntyam there is a linear form p : a → r such that the bilinear form q : a×a → r, (a,b) 7→ p (ab) is non degenerate. the map p◦ta(f) : ta(r2n)×ta(r2n) → r is a skew-symmetric non degenerate bilinear form on ta ( r2n ) . we put, f(a) = p◦ta(f) and g̃ = ja,r2n ( ta(g(f)) ) . proposition 13. the lie group g̃ is a lie subgroup of g(f(a)). proof. let u = jaξ ∈ ta(g(f)), then ja,r2n(u) = ũ is the linear automorphism of ta(r2n) defined for any jaϕ ∈ ta(r2n) by: ũ ( jaϕ ) = ja(ξ ∗ϕ) where (ξ ∗ϕ)(z) = ξ(z)(ϕ(z)), for any z ∈ rk. for any jaϕ,jaψ ∈ ta(r2n), we have: f(a) ( ũ ( jaϕ ) , ũ ( jaψ )) = f(a) ( ja(ξ ∗ϕ),ja(ξ ∗ψ) ) = p◦ta(f) ( ja(ξ ∗ϕ),ja(ξ ∗ψ) ) = p ( ja(f(ξ ∗ϕ,ξ ∗ψ)) ) . on the other hand, for any z ∈ rk, f(ξ ∗ϕ,ξ ∗ψ)(z) = f(ξ(z)(ϕ(z)),ξ(z)(ψ(z))) = f(ϕ(z),ψ(z)). therefore, f(a) ( ũ ( jaϕ ) , ũ ( jaψ )) = p◦ta(f) ( jaϕ,jaψ ) = f(a) ( jaϕ,jaψ ) . theorem 3. the almost symplectic form on tam associated with the a-prolongation of a g(f) structure p on a manifold m is the p-prolongation of the almost symplectic form associated with the g-structure p . proof. let φ : m → p be a smooth section, consider the vector ei,α = ja(zαei), with α ∈ ba and i ∈ {1, . . . , 2n}. the family (ei,ei,α) is a basis of the real vector space ta(rn). if φ ∣∣ u = φ j i ( ∂ ∂xj ) ⊗ ei then φ̃a ∣∣ tau =( φ j i )(α−β) ( ∂ ∂x j α ) ⊗eiβ. in particular, φ̃a(ei,α) = ( φ j i )(α−β) ( ∂ ∂x j β ) = (φ(ei)) (α). prolongations of g-structures related to weil bundles 135 we denote by ω the almost symplectic form induced by p and ωa the almost symplectic form induced by t ap . for all i,j ∈ {1, . . . , 2n} and α,β ∈ ba, we have: ωa ( (φ(ei)) (α), (φ(ej)) (β) ) = f(a) (( φ̃a )−1 ( (φ(ei)) (α) ) , ( φ̃a )−1 ( (φ(ej)) (β) )) = p◦ta(f)(ei,α,ej,β) = p◦ta(f) ( ja(zαei),j a(zβej) ) = p ( ja(f(zαei,z βej)) ) = p ( ja(zα+βf(ei,ej)) ) = (ω(φ(ei),φ(ej))) (α+β) = ω(p) ( (φ(ei)) (α), (φ(ej)) (β) ) . it follows that, ωa = ω (p), where ω(p) is the complete p-lift of ω described in [9] and [10]. remark 12. when f : rn × rn → r is a bilinear symmetric non degenerate form and g the lie subgroup generated by all elements of linear group invariant with respect to f, then, the pseudo riemannian metric on tam associated with the a-prolongation of a g-structure p on a manifold m is the p-prolongation of the pseudo riemannian metric associated with the structure p . 4.3. regular foliations induced by a-prolongations of g(v )structures. let v be a vector subspace of rn (dim v = p). we denote by g(v ) the group of all a ∈ gl(n) such that a(v ) = v . we consider the basis {1a,eα, α ∈ ba} of a over r and the linear isomorphism induced ta(rn) → rn dim a. therefore gl(ta(r2n)) is identified to gl(n dim a). proposition 14. the lie group g̃ = ja,rn(t a(g(v ))) is a lie subgroup of g(ta(v )). proof. let x = ja,rn ( jaϕ ) where ϕ : rk → g(v ) is a smooth map. so that, x : ta(rn) → ta(rn) is a linear isomorphism and for any jaξ ∈ ta(rn), x ( jaξ ) = ja(ϕ∗ ξ). for any jaξ ∈ ta(v ), we have x ( jaξ ) = ja(ϕ ∗ ξ), as for any z ∈ rk, (ϕ ∗ ξ)(z) = ϕ(z)(ξ(z)) ∈ v , it follows that x ( jaξ ) ∈ ta(v ). thus, x ( ta(v ) ) ⊂ ta(v ). 136 p.m.k. wamba, g.f. wankap nono, a. ntyam let d be a smooth regular distribution on m of rank p, we denote by xd the set of all local vector fields x such that: for all x ∈ m, x(x) ∈ dx. let us notice that for a completely integrable distribution d, the family xd is a lie subalgebra of the lie algebra of vector fields on m. we denote by d(a) the distribution generated by the family { x(α), 0 ≤ α ≤ h } . as [ x(α),x(β) ] = [x,y ] (α+β) and by the frobenius theorem, it follows that d(a) is a smooth regular and completely integrable distribution on tam. it is called a-complete lift of d from m to tam. in particular d(a) = κa,m ( ta(d) ) ⊂ t ( tam ) . proposition 15. if s ⊂ m is a leaf of regular completely integrable distribution d, then tas is a leaf of the regular distribution d(a). proof. as s is connected, then tas is also connected. in fact, let ξ1,ξ2 ∈ tas, we put πas (ξi) = si, i = 1, 2. we consider x0 : m → t am the smooth section defined by for any x ∈ m by: x0(x) = j a(z 7→ x). in particular πas ◦ x0(si) = si, for i = 1, 2. there is a continuous curve α1 : [0, 1] → tas1m such that α1(0) = ξ1 and α1(1) = x0(s1). by the same way, there is a continuous curve α2 : [0, 1] → tas2m such that α2(0) = x0(s2) and α2(1) = ξ2. let α0 : [0, 1] → s be a continuous curve such that α0(0) = s1 and α0(1) = s2. consider the following curve α : [0, 1] → tas defined by: α(t) =   α1(3t) if 0 ≤ t ≤ 13 x0 ◦α0(3t− 1) if 13 ≤ t ≤ 2 3 , α2(3t− 2) if 23 ≤ t ≤ 1 . the curve α is continuous and α(0) = ξ1, α(1) = ξ2. so, t as is connected. for any ξ ∈ tas, we have, tξ(t as) = tξ (( πam )−1 (s) ) = ( tξπ a m )−1 ( tπa m (ξ)s ) = ( tξπ a m )−1 ( dπa m (ξ) ) = d (a) ξ . thus, tas is a leaf of d(a). prolongations of g-structures related to weil bundles 137 theorem 4. the regular foliation on tam associated with the aprolongation of a g(v )-structure p on a manifold m is the a-complete lift of the regular foliation associated with the structure p . proof. let φ : m → p be a smooth section. if locally φ ∣∣ u = φ j i ( ∂ ∂xj ) ⊗ei then φ̃a ∣∣ tau = ( φ j i )(α−β) ( ∂ ∂x j α ) ⊗eiβ. in particular, φ̃a(ei,α) = ( φ j i )(α−β) ( ∂ ∂x j β ) = (φ(ei)) (α). let d the regular smooth distribution induced by the g(v )-structure p and d̃ the smooth distribution induced by t ap , for any ξ ∈ tam, d̃ξ = φ̃a(ξ) ( tav ) = 〈 φ̃a(ξ)(ei,α), i ∈{1, . . . ,p} , 0 ≤ |α| ≤ h 〉 = 〈 (φ(ei)) (α)(ξ), i ∈{1, . . . ,p} , 0 ≤ |α| ≤ h 〉 . it follows that, d̃ξ = d (a) ξ . references [1] d. bernard, sur la géométrie différentielle des g-structures, ann. inst. fourier (grenoble) 10 (1960), 151 – 270. [2] a. cabras, i. kolář, prolongation of tangent valued forms to weil bundles, arch. math. (brno) 31 (2) (1995), 139 – 145. [3] j. debecki, linear natural operators lifting p-vectors to tensors of type (q, 0) on weil bundles, czechoslovak math. j. 66 (2) (2016), 511 – 525. [4] m. doupovec, m. kureš, some geometric constructions on frobenius weil bundles, differential geom. appl. 35 (2014), 143 – 149. [5] j. gancarzewicz, w. mikulski, z. pogoda, lifts of some tensor fields and connections to product preserving functors, nagoya math. j. 135 (1994), 1 – 41. [6] i. kolář, on the geometry of weil bundles, differential geom. appl. 35 (2014), 136 – 142. [7] i. kolář, covariant approach to natural transformations of weil functors, comment. math. univ. carolin. 27 (4) (1986), 723 – 729. [8] i. kolář, p. michor, j. slovak, “ natural operations in differential geometry ”, springer-verlag, berlin, 1993. [9] p.m. kouotchop wamba, a. ntyam, prolongations of dirac structures related to weil bundles, lobachevskii j. math. 35 (2014), 106 – 121. 138 p.m.k. wamba, g.f. wankap nono, a. ntyam [10] p.m. kouotchop wamba, a. mba, characterization of some natural transformations between the bundle functors ta◦t∗ and t∗◦ta on mfm, imhotep j. afr. math. pures appl. 3 (2018), 21 – 32. [11] m. kures, w. mikulski, lifting of linear vector fields to product preserving gauge bundle functors on vector bundles, lobachevskii j. math. 12 (2003), 51 – 61. [12] a. morimoto, prolongations of g-structure to tangent bundles of higher order, nagoya math. j. 38 (1970), 153 – 179. [13] a. morimoto, lifting of some types of tensor fields and connections to tangent bundles of pr-velocities, nagoya math. j. 40 (1970), 13 – 31. preliminaries lifts of functions and vector fields. lifts of tensor fields of type (1,q). lifts of tensor fields of type (0,s). the natural transformations ja,e: ta(fe) f(tae) the embedding ja,v: ta(gl(v))gl(tav). frame gauge functor on the vector bundles. the natural embedding ja,e: ta(fe)f(tae). prolongations of g-structures to weil bundles the natural embedding ja,m: ta(fm)f(tam). prolongations of g-structures. prolongations of some classical g-structures complex structures. almost symplectic structure. regular foliations induced by a-prolongations of g(v)-structures. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 2 (2022), 153 – 184 doi:10.17398/2605-5686.37.2.153 available online march 25, 2022 extensions, crossed modules and pseudo quadratic lie type superalgebras m. pouye 1, b. kpamegan 2 1 institut de mathématiques et de sciences physiques (imsp), bénin 2 département de mathématiques, fast, uac, bénin received october 13, 2021 presented by consuelo mart́ınez accepted february 17, 2022 abstract: extensions and crossed modules of lie type superalgebras are introduced and studied. we construct homology and cohomology theories of lie-type superalgebras. the notion of left super-invariance for a bilinear form is defined and we consider lie type superalgebras endowed with nondegenerate, supersymmetric and left super-invariant bilinear form. such lie type superalgebras are called pseudo quadratic lie type superalgebras. we show that any pseudo quadratic lie type superalgebra induces a jacobi-jordan superalgebra. by using the method of double extension, we study pseudo quadratic lie type superalgebras and theirs associated jacobi-jordan superalgebras. key words: lie type superalgebras, jacobi-jordan superalgebras, extension, crossed module, homology, cohomology, double extension, pseudo quadratic lie type superalgebras. msc (2020): 17a15, 17a70, 17a60, 20k35, 17a45. introduction recently, in order to investigate commutative non-associative algebras, authors in [5] introduce the so-called jacobi-jordan algebras that are commutative algebras satisfying the jacobi identity. those algebras were first defined in [12] and since then they have been studied in various papers [3, 4, 6, 8] under different name such as jordan algebras of nil rank 3, mock-lie algebras, liejordan algebras or pathological algebras. it turns out that the commutativity and jacobi identity satisfied by the product of an algebra (a,∗) induce two relations x∗(y∗z) = −(x∗y)∗z−y∗(x∗z) and x∗(y∗z) = −(x∗y)∗z−(x∗z)∗y for all x,y,z ∈ a, called respectivelly left lie-type identity and right lie-type identity. this motivated us to introduce and study in [11] a new type of nonassociative (super)-algebra called left or right lie-type superalgebra. a left (resp. right) lie type superalgebra consists of a z2-graded vector space u := u0̄⊕u1̄ endowed with an even bilinear map [ , ] : u ⊗u → u such that [uα,uβ] ⊆ uα+β for all α,β ∈ z2 and [x, [y,z]] = −[[x,y],z] − (−1)|x||y|[y, [x,z]] (resp. issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.37.2.153 https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 154 m. pouye, b. kpamegan [x, [y,z]] = −[[x,y],z] − (−1)|y||z|[[x,z],y]) for all x,y,z ∈u. it is called symmetric lie type superalgebra if it is simultaneously a left and right lie type superalgebra. lie type superalgebras can be seen as generalization of jacobijordan (super)-algebras introduced in [5] which are sub class of the class of jordan algebras that plays an important role in physics (see [10]). in fact, unlike jacobi-jordan (super)-algebras, lie-type superalgebras are not necessary (super)-commutative. for more details about jacobi-jordan algebras (see [5, 9, 3]). in this paper, we introduce and study extension and crossed module of lie type superalgebras. we give a characterization of extension in terms of two bilinear applications and characterize the notion of isomorphism between two extensions in terms of linear applications satisfying some properties. the notion of trivial extension is defined and studied. by following [7] where the authors studied crossed modules of leibniz algebras, we define crossed module for lie type superalgebras that we call lie type crossed module. the notion of normalized, linked and bilateral lie type crossed module are defined and we also characterize the equivalence of two lie type crossed modules. a homology and cohomology theory of lie-type superalgebras is introduced and the first degree cohomology group is given in term of equivalent class of the so-called restricted trivial extensions. in [11], we studied quadratic lie-type superalgebras that are lie-type superalgebras (u, [ , ]) endowed with a nondegenerate, supersymmetic and invariant bilinear form b. we notice that the invariant or associative property of b that is b([x,y],z) = b(x, [y,z]) for all x,y,z ∈ u, plays an important role in the study of quadratic structure of lie-type superalgebras. but the fact that the bracket of lie-type superalgebras is not necessary supercommutative allows us to define a new type of invariant of b by b([x,y],z) = (−1)|x||y|b(y, [x,z]) called left super-invariance that is different from the associative property. another purpose of this paper is the study of the so-called pseudo quadratic lie type superalgebras. a lie type superalgebra is said pseudo quadratic if it is endowed with a nondegenerate, symmetric and left super-invariant bilinear form. we show that any pseudo quadratic lie type superalgebra (u, [ , ]) induces a jacobi-jordan superalgebra (u,∧). by using the double extension extented to lie type superalgebras, we study pseudo quadratic lie type superalgebra (u, [ , ]) and the associated jacobi-jordan superalgebra (u,∧). this paper is organized as follows. the first section is devoted to the definitions and elementary results. in section 2, we study homology and cohomollie type superalgebras 155 ogy of lie type superalgebras. in section 3, we define extension and crossed module of lie type superalgebras and characterize these notions in terms of linear and bilinear applications. we give a characterization of the notion of isomorphism between two lie type crossed modules and the notion of equivalence between extensions of lie type superalgebra. the notion of normalized and bilateral lie type crossed module are defined and studied. in section 4, we define left super-invariance for a bilinear form and by using the notion of double extension, we study pseudo quadratic lie type superalgebras and the induced jacobi-jordan superalgebras. throughout this paper, all vector spaces and algebras considered are defined over an algebraically closed field k of characteristic zero. notations: in this paper we shall keep the same notation as in [2]. 1. preliminaries in this section we give basic definitions and elementary results about lietype superalgebras and jacobi-jordan superalgebras. definition 1.1. let u := u0̄ ⊕u1̄ be a z2-graded vector space endowed with a bilinear map [ , ] : u ⊗ u → u such that [uα,uβ] ⊆ uα+β for all α,β ∈ z2. then (u, [ , ]) is called left lie type superalgebra if [x, [y,z]] = −[[x,y],z]−(−1)|x||y|[y, [x,z]] ∀x ∈u|x|, y ∈u|y|, z ∈u, (1.1) and (u, [ , ]) is called right lie type superalgebra if [x, [y,z]] = −[[x,y],z]−(−1)|y||z|[[x,z],y] ∀x ∈u|x|, y ∈u|y|, z ∈u|z|. (1.2) the superalgebra (u, [ , ]) is called symmetric lie type superalgebra if it is simultaneously a left and right lie type superalgebra. remark 1.1. let (u, [ , ]) be a left lie type superalgebra. define the bilinear map { , } : u⊗u →u by {x,y} = (−1)|x||y|[y,x], then (u,{ ,}) is a right lie type superalgebra. therefore the category of left lie-type superalgebras is isomorphic to the category of right lie-type superalgebras. let (a, ·) be a superalgebra. we define the anti-associator of a by the trilinear application aasso : a⊗a⊗a → a by aasso(x,y,z) := x · (y ·z) + (x·y)·z for all x,y,z ∈ a. the superalgebra (a, ·) is said to be anti-associative if aasso(x,y,z) = 0 for all x,y,z ∈ a. 156 m. pouye, b. kpamegan example 1.1. let (u, ·) be an anti-associative superalgebra. if we define the bilinear map [ , ] : u ⊗u →u by [x,y] = x ·y + (−1)|x||y|y ·x for x ∈u|x| and y ∈u|y|, then (u, [ , ]) is a right lie type superalgebra. a homomorphism f : u →w between two z2-graded vector spaces is said to be homogeneous of degree α ∈ z2 if f(uβ) ⊆ wα+β for all β ∈ z2. given three z2-graded vector spaces u, w and h, a bilinear map g : u ⊗w → h is said to be homogeneous of degree α ∈ z2 if g(uβ,wγ) ⊆ hα+β+γ for all β,γ ∈ z2. the degree of a homogeneous linear or bilinear map f is denoted by | f | and f is said to be an even (resp. odd) map if | f |= 0̄ (resp. | f |= 1̄). for any left lie-type superalgebra (u, [ , ]), the left and the right multiplication l and r defined by lx(y) := [x,y] and rx(y) := (−1)|x||y|[y,x] satisfy the following relations: lemma 1.1. (i) l[x,y] = −lx◦ly−(−1)|x||y|ly◦lx for all x ∈u|x|,y ∈u|y|; (ii) r[x,y] = −lx ◦ry − (−1)|x||y|ry ◦rx for all x ∈u|x|,y ∈u|y|; (iii) r[x,y] = −lx ◦ry − (−1)|x||y|ry ◦lx for all x ∈u|x|,y ∈u|y|; (iv) ry ◦rx = ry ◦lx for all x ∈u|x|,y ∈u|y|. proof. straightforward computation. the left centre zl(u) and the right centre zr(u) are defined by zl(u) = {x ∈ u, [x,u] = 0} and zr(u) = {x ∈ u, [u,x] = 0}. define by ker(u) the subspace generated by elements of the form [x,y] − (−1)|x||y|[y,x] where x ∈u|x| and y ∈u|y|. for any left lie-type superalgebra (u, [ , ]), it holds lemma 1.2. (i) ker(u) ⊆ zl(u); (ii) zl(u) is a two sided ideal and zr(u) is a sub-superalgebra. proof. see [11, lemma 2.6]. the fact that ker(u) ⊆ zl(u) implies that [[x,y],z] = (−1)|x||y|[[y,x],z] for x,y,z ∈u. one can sees that an analogous result of lemma 1.2 holds for right lie-type superalgebras. in fact, if (u, [ , ]) is a right lie-type superalgebra then ker(u) ⊆ zr(u). therefore [x, [y,z]] = (−1)|y||z|[x, [z,y]] ∀x ∈u|x|, y ∈u|y|, z ∈u|z|. (1.3) lie type superalgebras 157 definition 1.2. let u be a left lie-type superalgebra and v a z2-graded vector space. a representation of u over v is a couple of even linear maps (ϕ,λ) where ϕ,λ : u → end(v ) such that ϕ[x,y] = −ϕx ·φy − (−1) |x||y|ϕy ·φx , λ[x,y] = −ϕx ·λy − (−1) |x||y|λy ·λx , λ[x,y] = −ϕx ·λy − (−1) |x||y|λy ·ϕx , for all homogeneous elements x,y ∈ u. if ϕ = λ = 0, the representation is called trivial representation. we denote repuv the set of all representations of u over a given z2-graded vector space v . example 1.2. let u be a left lie-type superalgebra. then according to lemma 1.1, (l,r) ∈ repuu and is called the adjoint representation or the regular representation of u. let (u, [ , ]) be a left (resp. right) lie type superalgebra, v := v0̄ ⊕v1̄ a z2-graded vector space and (ϕ,λ) a representation of u in v. then the even bilinear application ψ : u ⊗u → v is said to be an even bi-cocycle of left (resp. right) lie type superalgebra with respect to (ϕ,λ) if for all x,y,z ∈ u we have ψ(x, [y,z]) + ψ([x,y],z) + (−1)|x||y|ψ(y, [x,z]) + ϕx(ψ(y,z)) + (−1)|x||y|ϕy(ψ(x,z)) + (−1)|z|(|x|+|y|)λz(ψ(x,y)) = 0 (resp. ψ(x, [y,z]) + ψ([x,y],z) + (−1)|y||z|ψ([x,z],y) + ϕx(ψ(y,z)) + (−1)|z|(|x|+|y|)λz(ψ(x,y)) + (−1)|x||y|λy(ψ(x,z)) = 0 ) . let (u, [ , ]) be a left(resp. right) lie type superalgebra, v a z2-graded vector space and ψ : u ⊗u → v an even bilinear map. then, the z2-graded space u := u ⊕v endowed with the product [x + u,y + v]ψ = [x,y] + ψ(x,y) ∀x,y ∈u, u,v ∈v is a left (resp. right) lie type superalgebra if and only if ψ(x, [y,z]) + ψ([x,y],z) + (−1)|x||y|ψ(y, [x,z]) = 0 158 m. pouye, b. kpamegan (resp. ψ(x, [y,z]) + ψ([x,y],z) + (−1)|y||z|ψ([x,z],y) = 0) . moreover, (u, [ , ]ψ) is a symmetric lie type superalgebra if and only if (u, [ , ]) is symmetric and ψ is an even bi-cocycle of u with respect the trivial representation such that ψ(x, [y,z]) = (−1)|x|(|y|+|z|)ψ([y,z],x) ∀x,y,z ∈u . in this case ψ is called an even lie-type bi-cocycle of u on the trivial u-module v. we denote by (zltype(u,v))0̄ the set of even lie-type bi-cocycles of u on the trivial u-module v. lemma 1.3. let u be a left lie-type superalgebra. then u is a right lie-type superalgebra if and only if [x, [y,z]] = (−1)|x|(|y|+|z|)[[y,z],x] ∀x,y,z ∈u . (1.4) proof. see the proof of [11, lemma 3.1]. according to the above lemma, a lie-type superalgebra is symmetric if and only if relation (1.4) holds. definition 1.3. a jacobi-jordan superalgebra is a z2-graded vector space j := j0̄ ⊕j1̄ endowed with an even bilinear map [ , ] : j ⊗j → j such that [jα,jβ] ⊆jα+β for all α,β ∈ z2 and 1. [x,y] = (−1)|x||y|[y,x] for all x ∈j|x|, y ∈j|y|; 2. (−1)|x||z|[x, [y,z]] + (−1)|x||y|[y, [z,x]] + (−1)|y||z|[z, [x,y]] = 0 for all x ∈j|x|, y ∈j|y|, z ∈j|z|. example 1.3. ([1]) the (2n + 1)-dimensional heisenberg jacobi-jordan superalgebra h(2n + 1,k) = (h0̄ ⊕ h1̄, ·) where h0̄ ⊕ h1̄ = {e1, . . . ,en} ⊕ {f1, . . . ,fn,z} and ei ·fi = fi ·ei := z ∀i = 1, . . . ,n. every jacobi-jordan superalgebra is a lie-type superalgebra. a lie-type superalgebra (u, [ , ]) is a jacobi-jordan superalgebra if and only if ker(u) = {0}. lie type superalgebras 159 2. homology and cohomology of lie-type superalgebras in this section we study homology and cohomology of right lie-type superalgebras. definition 2.1. a z2-graded vector space v := v0̄ ⊕ v1̄ is called right u-module if it endowed with an action [ , ] : v ⊗u → v such that [v, [x,y]] = −[[v,x],y]−(−1)|x||y|[[v,y],x] ∀x ∈u|x|, y ∈u|y|, v ∈ v. (2.1) let us consider the canonical surjection ϕ : u → uab := u/[u,u] and v a right u-module. we define cn(u,v ) := v ⊗ (ϕ(u))⊗n for all n ∈ n. then one can easily see that cn(u,v ) is a u-module through the following action [v ⊗x1 ⊗x2 ⊗···⊗xn,x] = (−1)|x| ∑ 16k6n|xk|[v,x] ⊗x1 ⊗···⊗xn for all v ⊗x1 ⊗x2 ⊗···⊗xn ∈ cn(u,v ) and x ∈u|x|. in the sequel for simplicity, we denote by x0⊗x1⊗x2⊗···⊗xn an element of cn(u,v ) with x0 ∈ v . let δ : cn(u,v ) → cn−1(u,v ) be the application defined by δ(x0,x1, . . . ,xn) = n∑ j=1 (−1)|xj| ∑ 0 0 and y,z ∈uab. then we have the following relations: proposition 2.1. (i) δ(x⊗y) = δx⊗y + [x,y]; (ii) [x⊗y,z] = (−1)|y||z|[x,z] ⊗y; (iii) δ[x,y] = −[δx,y]; (iv) δ2 = 0. 160 m. pouye, b. kpamegan proof. for relation (i), let us set xn+1 = y. we have δ(x⊗y) = δ(x0 ⊗x1 ⊗···⊗xn ⊗xn+1) = n+1∑ j=1 (−1)|xj| ∑ 0 0. thus ricci solitons are generalizations of einstein manifolds and they are also called as quasi einstein manifolds by theoretical physicists. in 1923, eisenhart [6] proved that if a positive definite riemannian manifold (m, g) admits a second order parallel symmetric covariant tensor other than a constant multiple of the metric tensor then it is reducible. in 1925, levy [8] obtained the necessary and sufficient conditions for the existence of such tensors. since then, many others investigated the eisenhart problem of finding symmetric and skew-symmetric parallel tensors on various spaces and obtained fruitful results. for instance, by giving a global approach based on the ricci identity. sharma [11] firstly investigated eisenhart problem on non-flat real and complex space forms, in 1989. using eisenhart problem calin and crasmareanu [4], bagewadi and ingalahalli [7, 1], debnath and bhattacharyya [5] have studied the existence of ricci solitons in f-kenmotsu manifolds, α-sasakian, lorentzian α sasakian and trans-sasakian manifolds. in 1989 the author olszak [9] has worked on existence of generalized complex space form. the authors parveena and bagewadi [2, 10] extended the study to some curvature tensors on generalized complex space form. motivated by these ideas, in this paper, we made an attempt to study ricci solitons of generalized complex space form by using eisenhart problem. 2. preliminaries a kähler manifold is an n(even)-dimensional manifold, with a complex structure j and a positive-definite metric g which satisfies the following conditions; j2(x) = −x, g(jx, jy ) = g(x, y ) and (∇xj)(y ) = 0, (2.1) a study on ricci solitons in generalized complex space form 229 where ∇ means covariant derivative according to the levi-civita connection. the formulae [3] r(x, y ) = r(jx, jy ), (2.2) s(x, y ) = s(jx, jy ), (2.3) s(x, jy ) + s(jx, y ) = 0, (2.4) are well known for a kähler manifold. definition 2.1. a kähler manifold with constant holomorphic sectional curvature c is said to be a complex space form and its curvature tensor is given by r(x, y )z = c 4 [ g(y, z)x − g(x, z)y + g(x, jz)jy − g(y, jz)jx + 2g(x, jy )jz ] . the models now are cn, cp n and chn, depending on c = 0, c > 0 or c < 0. definition 2.2. an almost hermition manifold m is called a generalized complex space form m(f1, f2) if its riemannian curvature tensor r satisfies, r(x, y )z = f1{g(y, z)x − g(x, z)y } + f2{g(x, jz)jy − g(y, jz)jx + 2g(x, jy )jz}. (2.5) 3. parallel symmetric second order covariant tensor and ricci soliton in a non-flat generalized complex space form let h be a (0, 2)-tensor which is parallel with respect to ∇ that is ∇h = 0. applying the ricci identity [11] ∇2h(x, y ; z, w) − ∇2h(x, y ; w, z) = 0. (3.1) we obtain the relation [11]: h(r(x, y )z, w) + h(z, r(x, y )w) = 0. (3.2) using equation (2.5) in (3.2) and putting x = w = ei, 1 ≤ i ≤ n after simplification, we get f1{g(y, z)(tr.h) − h(y, z)} + f2{h(jy, jz) − g(y, jz)(tr.hj) + 2h(jz, jy )} − {(n − 1)f1 − 3f2}h(z, y ) = 0, (3.3) 230 m.m. praveena, c.s. bagewadi where h is a (1, 1) tensor metrically equivalent to h. symmetrization and anti-symmetrization of (3.3) yield: [nf1 − 3f2] f1 h(z, y ) − 3f2 f1 h(jy, jz) = (tr.h)g(y, z), (3.4) [(n − 2)f1 − 3f2] f2 h(y, z) + h(jz, jy ) = g(y, jz)(tr.hj). (3.5) replacing y, z by jy, jz respectively in (3.4) and adding the resultant equation from (3.4), provide we obtain: hs(y, z) = β.(tr.h)g(y, z), (3.6) where β = f1 nf1 − 6f2 . replacing y, z by jy, jz respectively in (3.5) and adding the resultant equation from (3.5), provide we obtain: ha(y, z) = f2 [(n − 2)f1 − hf2] (tr.hj)g(y, jz). (3.7) by summing up (3.6) and (3.7) we obtain the expression: h = {β.(tr.h)g + ρ(tr.hj)ω}, (3.8) where ρ = f2 [(n − 2)f1 − hf2] . hence we can state the following. theorem 3.1. a second order parallel tensor in a non-flat generalized complex space form is a linear combination (with constant coefficients) of the underlying kaehlerian metric and kaehlerian 2-form. corollary 3.1. the only symmetric (anti-symmetric) parallel tensor of type (0, 2) in a non-flat generalized complex space form is the kaehlerian metric (kaehlerian 2-form) up to a constant multiple. corollary 3.2. a locally ricci symmetric (∇s = 0) non-flat generalized complex space form is an einstein manifold. a study on ricci solitons in generalized complex space form 231 proof. if h = s in (3.8) then tr.h = r and tr.hj = 0 by virtue of (2.4). equation (3.8) can be written as s(y, z) = βr g(y, z). (3.9) remark 3.1. the following statements for non-flat generalized complex space form are equivalent. 1. einstein 2. locally ricci symmetric 3. ricci semi-symmetric that is r · s = 0 if f1 ̸= 0. proof. the statements (1) → (2) → (3) are trivial. now, we prove the statement (3) → (1) is true. here r · s = 0 means (r(x, y ) · s(u, w)) = 0. which implies s(r(x, y )u, w) + s(u, r(x, y )w) = 0. (3.10) using equations (2.5) in (3.10) and putting y = u = ei, where {ei} is an orthonormal basis of the tangent space at each point of the manifold and taking summation over i (1 ≤ i ≤ n) we get after simplification that f1{ns(x, w) − rg(x, w)} = 0. (3.11) if f1 ̸= 0, then (3.11) reduced to s(x, w) = r n g(x, w). (3.12) therefore, we conclude the following. lemma 3.1. a ricci semi-symmetric non-flat generalized complex space form is an einstein manifold if f1 ̸= 0. 232 m.m. praveena, c.s. bagewadi corollary 3.3. suppose that on a non-flat generalized complex space form, the (0, 2) type field lv g + 2s is parallel where v is a given vector field. then (g, v ) yield a ricci soliton if jv is solenoidal. in particular, if the given non-flat generalized complex space form is ricci semi-symmetric with lv g parallel, we have same conclusion. proof. from theorem (3.1) and corollary (3.2), we have λ = −βr as seen below: (lv g + 2s)(y, z) = [ β tr(lv g + 2s)g(y, z) + ρ.tr((lv g + 2s)j)ω(y, z) ] = [ 2β(div v + r)g(y, z) + ρ[2(div jv )ω(y, z) + 2(tr.sj)ω(y, z) ] , (3.13) by virtue of (2.4) the above equation becomes (lv g + 2s)(y, z) = [ 2β(div v + r)g(y, z) + 2ρ(div jv )ω(y, z) ] . (3.14) by definition (g, v, λ) yields ricci soliton. if div jv = 0 then div v = 0 becouse jv = iv i.e., (lv g + 2s)(y, z) = 2βr g(y, z) = −2λg(y, z). (3.15) therefore λ = −βr. corollary 3.4. let (g, v, λ) be a ricci soliton in a non-flat generalized complex space form. then v is solenoidal if and only if it is shrinking or steady or expanding depending upon the sign of scalar curvature. proof. using equation (3.12) in (1.1) we get (lv g)(y, z) + 2 r n g(y, z) + 2λg(y, z) = 0. (3.16) putting y = z = ei where {ei} is an orthonormal basis of the tangent space at each point of the manifold and taking summation over i (1 ≤ i ≤ n), we get (lv g)(ei, ei) + 2 r n g(ei, ei) + 2λg(ei, ei) = 0. (3.17) the above equation implies div v + r + λn = 0. (3.18) a study on ricci solitons in generalized complex space form 233 if v is solenoidal then div v = 0. therefore the equation (3.18) can be reduced to λ = −r n . references [1] c.s. bagewadi, g. ingalahalli, ricci solitons in lorentzian α-sasakian manifolds, acta mathematica. academiae paedagogicae ny¡regyháziensis 28 (1) (2012), 59 – 68. [2] c.s. bagewadi, m.m. praveena, semi-symmetric conditions on generalized complex space forms, acta math. univ. comenian. (n.s.) 85 (1) (2016), 147 – 154. [3] d.e. blair, “a contact manifolds in riemannian geometry”, lecture notes in mathematics, 509, springer-verlag, berlin-new york, 1976. [4] c. calin, m. crasmareanu, from the eisenhart problem to ricci solitons in f-kenmotsu manifolds, bull. malays. math. sci. soc. (2) 33 (3) (2010), 361 – 368. [5] s. debnath, a. bhattacharyya, second order parallel tensor in transsasakian manifolds and connection with ricci soliton, lobachevskii journal of mathematics 33 (4) (2012), 312 – 316. 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[12] p. topping, “lectures on the ricci flow”, london mathematical society lecture note series, 325, cambridge university press, cambridge, 2006. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 1 (2020), 69 – 97 doi:10.17398/2605-5686.35.1.69 available online april 29, 2020 hom-jordan and hom-alternative bimodules s. attan, h. hounnon, b. kpamegan département de mathématiques, université d’abomey-calavi 01 bp 4521, cotonou 01, bénin syltane2010@yahoo.fr , hi.hounnon@fast.uac.bj , kpamegan bernadin@yahoo.fr received march 21, 2019 and, in revised form, april 8, 2020 presented by consuelo mart́ınez accepted april 15, 2020 abstract: in this paper, hom-jordan and hom-alternative bimodules are introduced. it is shown that jordan and alternative bimodules are twisted via endomorphisms into hom-jordan and homalternative bimodules respectively. some relations between hom-associative bimodules, hom-jordan and hom-alternative bimodules are given. key words: bimodules, alternative algebras, jordan algebras, hom-alternative algebras, homjordan algebras, hom-associative algebras. ams subject class. (2010): 17a30, 17b10, 17c50, 17d05. 1. introduction algebras where the identities defining the structure are twisted by a homomorphism are called hom-algebras. they have been intensively investigated in the literature recently. hom-algebra started from hom-lie algebras introduced and discussed in [6, 10, 11, 12], motivated by quasi-deformations of lie algebras of vector fields, in particular q-deformations of witt and virasoro algebras. hom-associative algebras were introduced in [15] while homalternative and hom-jordan algebras are introduced in [14], [23] as twisted generalizations of alternative and jordan algebra respectively. the reader is referred to [20] for applications of alternative algebras to projective geometry, buildings, and algebraic groups and to [4, 9, 16, 19] for discussions about the important roles of jordan algebras in physics, especially quantum mechanics. the anti-commutator of a hom-alternative algebra gives rise to a homjordan algebra [23]. starting with a hom-alternative algebra (a, ·,α), it is known that the jordan product x∗y = 1 2 (x ·y + y ·x) gives a hom-jordan algebra a+ = (a,∗,α). in other words, hom-alternative algebras are hom-jordan-admissible [23]. issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.35.1.69 mailto:syltane2010@yahoo.fr mailto:hi.hounnon@fast.uac.bj mailto:kpamegan_bernadin@yahoo.fr https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 70 s. attan, h. hounnon, b. kpamegan the notion of bimodule for a class of algebras defined by multilinear identities has been introduced by eilenberg [3]. if h is in the class of associative algebras or in the one of lie algebras then this notion is the familiar one for which we are in possession of well-worked theories. the study of bimodule (or representation) of jordan algebras was initiated by n. jacobson [7]. subsequently the alternative case was considered by schafer [17]. modules over an ordinary algebra has been extended to the ones of homalgebras in many works [2, 18, 21, 22]. the aim of this paper is to introduce hom-alternative bimodules and homjordan bimodules and to discuss about some findings. the paper is organized as follows. in section two, we recall basic notions related to hom-algebras and modules over hom-associative algebras. section three is devoted to the introduction of hom-alternative bimodules . proposition 3.7 shows that from a given hom-alternative bimodule, a sequence of this kind of bimodules can be obtained. theorem 3.8 establishes that, an alternative bimodule gives rise to a bimodule over the corresponding twisted algebra. it is also proved that a direct sum of a hom-alternative algebra and a module over this homalgebra is again a hom-alternative algebra (theorem 3.11). in section four, we introduce hom-jordan modules and attest similar results as in the previous section. furthermore, it is proved that a hom-jordan special left and right module, with an additional condition, has a bimodule structure over this homalgebra (theorem 4.10). finally, proposition 4.12 shows that a bimodule over a hom-associative algebra has a bimodule structure over its plus homalgebra. all vector spaces are assumed to be over a fixed ground field k of characteristic 0. 2. preliminaries we recall some basic notions introduced in [6, 15, 21] related to homalgebras and while dealing of any binary operation we will use juxtaposition in order to reduce the number of braces, i.e., e.g., for “·”, xy · α(z) means (x ·y) ·α(z). also, for the map µ : a⊗2 → a, we will write sometimes µ(a⊗b) as µ(a,b) or ab for a,b ∈ a and if v is another vector space, τ1 : a⊗v → v ⊗a (resp. τ2 : v ⊗a → a⊗v ) denote the twist isomorphism τ1(a⊗v) = v ⊗a (resp. τ2(v ⊗a) = a⊗v). definition 2.1. a hom-module is a pair (m,αm ) consisting of a kmodule m and a linear self-map αm : m → m. a morphism f : (m,αm ) → (n,αn ) of hom-modules is a linear map f : m → n such that f◦αm = αn◦f. hom-jordan and hom-alternative bimodules 71 definition 2.2. ([15, 21]) a hom-algebra is a triple (a,µa,αa) in which (a,αa) is a hom-module, µ : a ⊗2 → a is a linear map. the homalgebra (a,µ,α) is said to be multiplicative if α◦µ = µ◦α⊗2 (multiplicativity). a morphism f : (a,µa,αa) → (b,µb,αb) of hom-algebras is a morphism of the underlying hom-modules such that f ◦µa = µb ◦f⊗2. an important class of hom-algebras that is considered here is the one of hom-alternative algebras. these algebras have been introduced in [14] and more studied in [23]. definition 2.3. let (a,µ,α) be a hom-algebra. (i) the hom-associator of a is the linear map asa : a ⊗3 → a defined as asa = µ ◦ (µ ⊗ α − α ⊗ µ). a multiplicative hom-algebra (a,µ,α) is said to be hom-associative algebra if asa = 0. (ii) a hom-alternative algebra [14] is a multiplicative hom-algebra (a,µ,α) that satisfies asa(x,x,y) = 0 (left hom-alternativity) , (1) asa(x,y,y) = 0 (right hom-alternativity) (2) for all x,y ∈ a. (iii) let (a,µ,α) be a hom-alternative algebra. a hom-subalgebra of (a,µ,α) is a linear subspace h of a, which is closed for the multiplication µ and invariant by α, that is, µ(x,y) ∈ h and α(x) ∈ h for all x,y ∈ h. if furthermore µ(a,b) ∈ h and µ(b,a) ∈ h for all (a,b) ∈ a×h, then h is called a two-sided hom-ideal of a. now, we prove: proposition 2.4. let (a,µ,α) be a hom-alternative algebra and i be a two-sided hom-ideal of (a,µ,α). then (a/i,µ̄, ᾱ) is a hom-alternative algebra where µ̄(x̄, ȳ) = µ(x,y) and ᾱ(x̄) = ¯α(x) for all x̄, ȳ ∈ a/i. proof. first, note that the multiplicativity of µ̄ with respect to ᾱ follows from the one of µ with respect to α. next, pick x̄, ȳ ∈ a/i. then the left hom-alternativity (1) in (a/i,µ̄, ᾱ) is proved as follows asa/i(x̄, x̄, ȳ) = µ̄(µ̄(x̄, x̄), ᾱ(ȳ)) − µ̄(ᾱ(x̄), µ̄(x̄, ȳ) = µ(µ(x,x)α(y)) −µ(α(x),µ(x,y)) = asa(x,x,y)) = 0̄ . similarly, we get (2) and therefore (a/i,µ̄, ᾱ) is a hom-alternative algebra. 72 s. attan, h. hounnon, b. kpamegan as hom-alternative algebras, hom-jordan algebras are fundamental objects of this paper. they appear as cousins of hom-alternative algebras and these two hom-algebras are related as jordan and alternative algebras. definition 2.5. ([23]) (i) a hom-jordan algebra is a multiplicative hom-algebra (a,µ,α) such that µ◦τ = µ (commutativity of µ) and the so-called hom-jordan identity holds asa(µ(x,x, ),α(y),α(x)) = 0,∀ (x,y) ∈ a2 (3) where, τ : a⊗2 → a⊗2, τ(a⊗ b) = b⊗a, is the twist isomorphism. (ii) let (a,µ,α) be a hom-jordan algebra. a hom-subalgebra of (a,µ,α) is a linear subspace h of a, which is closed for the multiplication µ and invariant by α, that is, µ(x,y) ∈ h and α(x) ∈ h for all x,y ∈ h. if furthermore µ(a,b) ∈ h for all (a,b) ∈ a × h, then h is called a two-sided hom-ideal (or simply hom-ideal) of a [5]. similarly as a hom-alternative algebra case, if h is a hom-ideal of a homjordan algebra (a,µ,α), then (a/h,µ̄, ᾱ) is a hom-jordan algebra where µ̄(x̄, ȳ) = µ(x,y) for all x̄, ȳ ∈ a/h and ᾱ : a/h → a/h is naturally induced by α, inherits a hom-jordan algebra structure, which is named quotient homjordan algebra. remark 2.6. in [14] makhlouf defined a hom-jordan algebra as a commutative multiplicative hom-algebra satisfying asa(x 2,y,α(x)) = 0, which becomes the identity (3) if y is replaced by α(y). the proof of the following result can be found in [23] where the product ∗, differs from the one given here by a factor of 1 2 . proposition 2.7. let (a,µ,α) be a hom-alternative algebra. then a+ = (a,∗,α) is a hom-jordan algebra where x∗y = xy + yx for all x,y ∈ a. example 2.8. from the eight-dimensional hom-alternative algebra oα = (o,µα,α) with basis {e0,e1,2 ,e3,e4,e5,e6,e7} [23, example 3.19], constructed from the octonion algebra which is an eight-dimensional alternative algebra, we obtain, the hom-jordan algebra o+α = (o,∗ = µα+µα◦τ,α) where the non zero products are: e0 ∗e0 = 2e0, e0 ∗e1 = e1 ∗e0 = 2e5, e0 ∗e2 = e2 ∗e0 = 2e6, e0 ∗ e3 = e3 ∗ e0 = 2e7, e0 ∗ e4 = e4 ∗ e0 = 2e1, e0 ∗ e5 = e5 ∗ e0 = 2e2, e0 ∗ e6 = e6 ∗ e0 = 2e3, e0 ∗ e7 = e7 ∗ e0 = 2e4, e1 ∗ e1 = e2 ∗ e2 = e3 ∗ e3 = hom-jordan and hom-alternative bimodules 73 e4 ∗ e4 = e5 ∗ e5 = e6 ∗ e6 = e7 ∗ e7 = −2e0 and the twisting map α is given by α(e0) = e0, α(e1) = e5, α(e2) = e6, α(e3) = e7, α(e4) = e1, α(e5) = e2, α(e6) = e3, α(e7) = e4. a. makhlouf proved that the plus algebra of any hom-associative algebra is a hom-jordan algebra as defined in [14]. here, we prove the same result for the hom-jordan algebra as defined in [23] (see also definition 2.5 above). proposition 2.9. let (a, ·,α) be a hom-associative algebra. then a+ = (a,∗,α) is a hom-jordan algebra where x∗y = xy + yx for all x,y ∈ a. proof. the commutativity of ∗ is obvious. we compute the hom-jordan identity as follows: asa+ ( x2,α(x),α(y) ) = (x2 ∗α(y)) ∗α2(x) −α(x2) ∗ (α(y) ∗α(x)) = (x2 ·α(y)) ·α2(x) + (α(y) ·x2) ·α2(x) + α2(x) · (x2 ·α(y)) + α2(x) · (α(y) ·x2) −α(x2) · (α(y) ·α(x)) −α(x2) · (α(x) ·α(y)) − (α(y) ·α(x)) ·α(x2) − (α(x) ·α(y)) ·α(x2) (by a direct computation) = (α(y) ·x2) ·α2(x) + α2(x) · (x2 ·α(y)) −α(x2) · (α(x) ·α(y)) − (α(y) ·α(x)) ·α(x2) (by the hom-associativity) = (α(y) ·x2) ·α2(x) + α2(x) · (x2 ·α(y)) − (α(x) ·α(x)) ·α(x ·y) −α(yx) · (α(x) ·α(x)) (by the multiplicativity) = (α(y) ·x2) ·α2(x) + α2(x) · (x2 ·α(y)) −α2(x) · (α(x) · (x ·y)) − ((yx) ·α(x)) ·α2(x) (by the hom-associativity) = 0 (by the hom-associativity) . then a+ = (a,∗,α) is a hom-jordan algebra. examples 2.10. (i) consider the three-dimensional hom-associative algebra a = (a,µa,αa) over k with basis (e1,e2,e3) defined by µa(e1,e1) = e1, µa(e2,e2) = e2, µa(e3,e3) = e1, µa(e1,e3) = µa(e3,e1) = −e3 and αa(e1) = e1, αa(e3) = −e3 (see [24, theorem 3.12], hom-algebra a′ 3 3). using the product ∗ in proposition 2.9, the triple a+ = (a,∗,αa) is hom-jordan algebra where, e1 ∗e1 = 2e1, e2 ∗e2 = 2e2, e3 ∗e3 = 2e1, e1 ∗e3 = e3 ∗e1 = −2e3. 74 s. attan, h. hounnon, b. kpamegan (ii) from the three-dimensional hom-associative algebra b = (b,µb,αb) over k with basis (e1,e2,e3) defined by µb(e1,e1) = e1, µb(e2,e2) = e1, µb(e3,e3) = e3, µb(e1,e2) = µb(e2,e1) = −e2 and αb(e1) = e1, αb(e2) = −e2 (see [24, theorem 3.12], hom-algebra a′ 3 5). then the triple b+ = (b,∗, αb) is a hom-jordan algebra, where “∗” is the product in proposition 2.9 and e1 ∗e1 = 2e1, e2 ∗e2 = 2e1, e3 ∗e3 = 2e3, e1 ∗e2 = e2 ∗e1 = −2e2. let us consider the following definitions which will be used in next sections. definition 2.11. let (a,µ,αa) be any hom-algebra. (i) a hom-module (v,αv ) is called an a-bimodule if it comes equipped with a left and a right structure maps on v that is morphisms ρl : a⊗v → v , a ⊗ v 7→ a · v and ρr : v ⊗ a → v , v ⊗ a 7→ v · a of hom-modules respectively. (ii) a morphism f : (v,αv ,ρl,ρr) → (w,αw ,ρ′l,ρ ′ r) of a-bimodules is a morphism of the underlying hom-modules such that f ◦ρl = ρ′l ◦ (ida ⊗f) and f ◦ρr = ρ ′ r ◦ (f ⊗ ida) . (iii) let (v,αv ) be an a-bimodule with structure maps ρl and ρr. then the module hom-associator of v is a trilinear map asa,v defined as: asa,v ◦ idv⊗a⊗a = ρr ◦ (ρr ⊗αa) −ρl ◦ (αv ⊗µ) , asa,v ◦ ida⊗v⊗a = ρr ◦ (ρl ⊗αa) −ρl ◦ (αa ⊗ρr) , asa,v ◦ ida⊗a⊗v = ρl ◦ (µ⊗αv ) −ρl ◦ (αa ⊗ρl) . remark 2.12. the module hom-associator given above is a generalization of the one given in [2]. now, let consider the following notion for hom-associative algebras. definition 2.13. let (a,µ,αa) be a hom-associative algebra and (m,αm ) be a hom-module. (i) a left hom-associative a-module structure on m consists of a morphism ρ : a⊗m → m of hom-modules, such that ρ◦ (αa ⊗ρ) = ρ◦ (µ⊗αm ) (4) hom-jordan and hom-alternative bimodules 75 (ii) a right hom-associative a-module structure on m consists of a morphism ρ : m ⊗a → m of hom-modules, such that ρ◦ (αm ⊗µ) = ρ◦ (ρ⊗αa) (5) (iii) a hom-associative a-bimodule structure on m consists of two structure maps ρl : a⊗m → m and ρr : m ⊗a → m such that (m,αm,ρl) is a left a-module, (m,αm,ρr) is a right a-module and that the following hom-associativity (or operator commutativity) condition holds: ρl ◦ (αa ⊗ρr) = ρr ◦ (ρl ⊗αa) (6) remark 2.14. actually, left hom-associative a-module, right hom-associative a-module and hom-associative a-bimodule have been already introduced in [21, 22] where they are called left a-module, right a-module and a-bimodule respectively. the expressions, used in definition 2.13 for these notions, are motivated by the unification of our terminologies. 3. hom-alternative bimodules in this section, we give the definition of hom-alternative (bi)modules. we prove that from a given hom-alternative bimodule, a sequence of this kind of bimodules can be constructed. it is also proved that a direct sum of a homalternative algebra and a bimodule over this hom-algebra is a hom-alternative algebra called a split null extension of the considered hom-algebra. first, we start by the following notion, due to [2], where it is called a module over a left (resp. right) hom-alternative algebra. however, we call it a hom-alternative left (resp. right) module in this paper. definition 3.1. let (a,µ,αa) be a hom-alternative algebra. (i) a left hom-alternative a-module is a hom-module (v,αv ) with a left structure map ρl : a⊗v → v , a⊗v 7→ a ·v such that asa,v (x,y,v) = −asa,v (y,x,v) for all x,y ∈ a and v ∈ v . (ii) a right hom-alternative a-module is a hom-module (v,αv ) with a right structure map ρr : v ⊗a → v , v ⊗a 7→ v ·a such that asa,v (v,x,y) = −asa,v (v,y,x) for all x,y ∈ a and v ∈ v . 76 s. attan, h. hounnon, b. kpamegan now, as a generalization of alternative bimodules [8, 17], one has: definition 3.2. let (a,µ,αa) be a hom-alternative algebra. a homalternative a-bimodule is a hom-module (v,αv ) with a (left) structure map ρl : a⊗v → v , a⊗v 7→ a ·v and a (right) structure map ρr : v ⊗a → v , v ⊗a 7→ v ·a such that the following equalities hold: asa,v (a,v,b) = −asa,v (v,a,b) = asa,v (b,a,v) = −asa,v (a,b,v) (7) for all (a,b,v) ∈ a×2 ×v . remarks 3.3. (i) the relation (7) is equivalent to asa,v (a,v,b) = −asa,v (v,a,b) = asa,v (b,a,v) = −asa,v (b,v,a) or since the field’s characteristic is 0 to asa,v (a,v,b) = −asa,v (v,a,b) = asv (b,a,v) and asa,v (a,a,v) = 0 . (ii) if αa = ida and αv = idv then v is the so-called alternative bimodule for the alternative algebra (a,µ) [8, 17]. examples 3.4. here are some examples of hom-alternative a-bimodules. (i) let (a,µ,αa) be a hom-alternative algebra. then (a,αa) is a homalternative a-bimodule where the structure maps are ρl(a,b) = µ(a,b) and ρr(a,b) = µ(b,a). more generally, if b is a two-sided hom-ideal of (a,µ,αa), then (b,αa) is a hom-alternative a-bimodule where the structure maps are ρl(a,x) = µ(a,x) and ρr(x,b) = µ(x,b) for all x ∈ b and (a,b) ∈ a×2. (ii) if (a,µ) is an alternative algebra and m is an alternative a-bimodule [8] in the usual sense, then (m,idm ) is a hom-alternative a-bimodule where a = (a,µ,ida) is a hom-alternative algebra. (iii) if f : (a,µa,αa) → (b,µb,αb) is a surjective morphism of homalternative algebras, then (b,αb) becomes a hom-alternative a-bimodule via f, i.e, the structure maps are defined as ρl : (a,b) 7→ µb(f(a),b) and ρr : (b,a) 7→ µb(b,f(a)) for all (a,b) ∈ a × b. indeed one can remark that asa,b ◦ (ida ⊗f ⊗ ida) = f ◦asa. in order to give another example of hom-alternative bimodules , let us consider the following hom-jordan and hom-alternative bimodules 77 definition 3.5. an abelian extension of hom-alternative algebras is a short exact sequence of hom-alternative algebras 0 → (v,αv ) i−→ (a,µa,αa) π−−→ (b,µb,αb) → 0 where (v,αv ) is a trivial hom-alternative algebra, i and π are morphisms of hom-algebras. furthermore, if there exists a morphism s : (b,µb,αb) → (a,µa,αa) such that π◦s = idb then the abelian extension is said to be split and s is called a section of π. example 3.6. given an abelian extension as in the previous definition, the hom-module (v,αv ) inherits a structure of a hom-alternative b-bimodule and the actions of the hom-algebra (b,µb,αb) on v are as follows. for any x ∈ b, there exist x̃ ∈ a such that x = π(x̃). let x acts on v ∈ v by x ·v := µa(x̃, i(v)) and v ·x := µa(i(v), x̃). these are well-defined, as another lift x̃′ of x is written x̃′ = x̃+v′ for some v′ ∈ v and thus x·v = µa(x̃, i(v)) = µa(x̃′, i(v)) and v · x = µa(i(v), x̃) = µa(i(v), x̃′) because v is trivial. the actions property follow from the hom-alternativity identity. in case these actions of b on v are trivial, one speaks of a central extension. the following result describes a sequence of hom-alternative bimodules by twisting the structure maps of a given bimodule over this hom-algebra. proposition 3.7. let (a,µ,αa) be a hom-alternative algebra and (v,αv ) be a hom-alternative a-bimodule with the structure maps ρl and ρr. then the maps ρ (n) l = ρl ◦ (α n a ⊗ idv ) ρ(n)r = ρr ◦ (idv ⊗α n a) give the hom-module (v,αv ) the structure of a hom-alternative a-bimodule that we denote by v (n) proof. it is clear that ρ (n) l and ρ (n) r are structure maps on v (n). next, observe that for all x,y ∈ a and v ∈ v , asa,v (n) (x,v,y) = ρ (n) r (ρ (n) l (x,v),αa(y)) −ρ (n) l (αa(x),ρ (n) r (v,y)) = ρr(ρl(α n a(x),v),α n+1 a (y)) −ρl(α n+1 a (x),ρr(v,α n a(y)) = asa,v (α n a(x),v,α n a(y)) 78 s. attan, h. hounnon, b. kpamegan and similarly asa,v (n) (v,x,y) = asa,v (v,α n a(x),α n a(y)) , asa,v (n) (y,x,v) = asa,v (α n a(y),α n a(x),v) , asa,v (n) (x,y,v) = asa,v (α n a(x),α n a(y),v) . therefore, equalities of (7) in v (n) derive from the one in v . we know that alternative algebras can be deformed into hom-alternative algebras via an endomorphism. the following result shows that alternative bimodules can be deformed into hom-alternative bimodules via an endomorphism. this provides a large class of examples of hom-alternative bimodules. theorem 3.8. let (a,µ) be an alternative algebra, v be an alternative a-bimodule with the structure maps ρl and ρr, αa be an endomorphism of the alternative algebra a and αv be a linear self-map of v such that αv ◦ρl = ρl ◦ (αa ⊗αv ) and αv ◦ρr = ρr ◦ (αv ⊗αa). write aαa for the hom-alternative algebra (a,µαa,αa) and vαv for the hom-module (v,αv ). then the maps ρ̃l = αv ◦ρl and ρ̃r = αv ◦ρr give the hom-module vαv the structure of a hom-alternative aαa-bimodule. proof. trivially, ρ̃l and ρ̃r are structure maps on vαv . the proof of (7) for vαv follows directly by the fact that asa,vαv = α 2 v ◦ asa,v and the relation (7) in v . corollary 3.9. let (a,µ) be an alternative algebra, v be an alternative a-bimodule with the structure maps ρl and ρr, αa an endomorphism of the alternative algebra a and αv be a linear self-map of v such that αv ◦ ρl = ρl ◦ (αa ⊗αv ) and αv ◦ρr = ρr ◦ (αv ⊗αa). write aαa for the hom-alternative algebra (a,µαa,αa) and vαv for the hom-module (v,αv ). then the maps ρ̃l (n) = ρl ◦ ( αn+1a ⊗αv ) and ρ̃r (n) = ρr ◦ ( αv ⊗αn+1a ) give the hom-module vα the structure of a hom-alternative aαa-bimodule for each n ∈ n. hom-jordan and hom-alternative bimodules 79 lemma 3.10. let (a,µ,αa) be a hom-alternative algebra and (v,αv ) be a hom-alternative a-bimodule with the structure maps ρl and ρr. then the following relation asa,v (v,a,a) = 0 (8) holds for all a ∈ a and v ∈ v . proof. using (7), for all (a,b) ∈ a×2 and v ∈ v we have −asa,v (v,a,b) = asa,v (a,v,b) and asa,v (v,b,a) = −asa,v (a,b,v). moreover again from (7), we get asa,v (a,v,b) = −asa,v (a,b,v) and then −asa,v (v,a,b) = asa,v (v,b,a). it follows that asa,v (v,a,a) = 0 since the field k is of characteristic 0. the following result shows that a direct sum of a hom-alternative algebra and a bimodule over this hom-algebra, is still a hom-alternative, called the split null extension determined by the given bimodule. theorem 3.11. let (a,µ,αa) be a hom-alternative algebra and (v,αv ) be a hom-alternative a-bimodule with the structure maps ρl and ρr. defining on a⊕v the bilinear map µ̃ : (a⊕v )⊗2 → a⊕v , µ̃(a + m,b + n) := ab + a · n + m ·b and the linear map α̃ : a⊕v → a⊕v , α̃(a + m) := αa(a) + αv (m), then e = (a⊕v,µ̃, α̃) is a hom-alternative algebra. proof. the multiplicativity of α̃ with respect to µ̃ follows from the one of α with respect to µ and the fact that ρl and ρr are morphisms of hom-modules. next ase(a + m,a + m,b + n) = µ̃(µ̃(a + m,a + m), α̃(b + n)) − µ̃(α̃(a + m), µ̃(a + m,b + n)) = µ̃(a2 + a ·m + m ·a,αa(b) + αv (n)) − µ̃(αa(a) + αv (m),ab + a ·n + m · b) = a2αa(b) + a 2 ·αv (n) + (a ·m) ·αa(b) + (m ·a) ·αa(b) −αa(a)(ab) −αa(a) · (a ·n) −αa(a) · (m · b) −αv (m) · (ab) = asa(a,a,b)︸︷︷︸ 0 + asv (a,a,n)︸︷︷︸ 0 + asa,v (a,m,b) + asa,v (m,a,b)︸︷︷︸ 0 (by (1), remarks 3.3 and (7)) = 0 . 80 s. attan, h. hounnon, b. kpamegan similarly, we compute ase(a + m,b + n,b + n) = µ̃(µ̃(a + m,b + n), α̃(b + n)) − µ̃(α̃(a + m), µ̃(b + n,b + n)) = µ̃(ab + a ·n + m · b,αa(b) + αv (n)) − µ̃(αa(a) + αv (m),b2 + b ·n + b · b) = (ab)αa(b) + (ab) ·αv (m) + (a ·n) ·αa(b) + (m · b) ·αa(b) −αa(a)(b2) −αa(a) · (b ·n) −αa(a) · (n · b) −αv (m) · b2 = asa(a,b,b)︸︷︷︸ 0 + asa,v (a,b,n) + asa,v (a,n,b)︸︷︷︸ 0 + asa,v (m,b,b)︸︷︷︸ 0 (by (2), (7) and (8)) = 0 . we then conclude that (a⊕v,µ̃, α̃) is a hom-alternative algebra. remark 3.12. consider the split null extension a⊕v determined by the hom-alternative bimodule (v,αv ) of the hom-alternative algebra (a,µ,αa) in the previous theorem. write elements a + v of a ⊕ v as (a,v). then, there is an injective homomorphism of hom-modules i : v → a⊕v given by i(v) = (0,v) and a surjective homomorphism of hom-modules π : a⊕v → a given by π(a,v) = a. moreover i(v ) is a two-sided hom-ideal of a⊕v such that a ⊕ v/i(v ) ∼= a. on the other hand, there is a morphism of homalgebras σ : a → a⊕v given by σ(a) = (a, 0) which is clearly a section of π. hence, we obtain the abelian split exact sequence of hom-alternative algebras and (v,αv ) is a hom-alternative a-bimodule via π. 4. hom-jordan bimodules in this section, we study hom-jordan bimodules. it is observed that similar results for hom-alternative bimodules hold for hom-jordan bimodules. some of them require an additional condition. furthermore, relations between hom-associative bimodules and hom-jordan bimodules are given on the one hand, and on the other hand, relations between left (resp. right) hom-alternative modules and left(resp. right) special hom-jordan modules are proved. first, we have: definition 4.1. let (a,µ,αa) be a hom-jordan algebra. hom-jordan and hom-alternative bimodules 81 (i) a right hom-jordan a-module is a hom-module (v,αv ) with a right structure map ρr : v ⊗ a → v , v ⊗ a 7→ v · a such that the following conditions hold: αv (v ·a) ·αa(bc) + αv (v · b) ·αa(ca) + αv (v · c) ·αa(ab) = (αv (v) · bc) ·α2a(a) + (αv (v) · ca) ·α 2 a(b) + (αv (v) ·ab) ·α2a(c) , (9) αv (v ·a) ·αa(bc) + αv (v · b) ·αa(ca) + αv (v · c) ·αa(ab) = ((v ·a) ·αa(b)) ·α2a(c) + ((v · c) ·αa(b)) ·α 2 a(a) + α2v (v) · ((ac)αa(b)) (10) for all a,b,c ∈ a and v ∈ v . (ii) a left hom-jordan a-module is a hom-module (v,αv ) with a left structure map ρl : a⊗v → v , a⊗v 7→ a·v such that the following conditions hold: αa(bc) ·αv (a ·v) + αa(ca) ·αv (b ·v) + αa(ab) ·αv (c ·v) = α2a(a) · (bc ·αv (v)) + α 2 a(b) · (ca ·αv (v)) + α2a(c) · (ab ·αv (v)) , (11) αa(bc) ·αv (a ·v) + αa(ca) ·αv (b ·v) + αa(ab) ·αv (c ·v) = α2a(c) · (αa(b) · (a ·v)) + α 2 a(a) · (αa(b) · (c ·v)) + ((ac)αa(b)) ·α2v (v) (12) for all a,b,c ∈ a and v ∈ v . the following result allows to introduce the notion of right special homjordan modules. theorem 4.2. let (a,µ,αa) be a hom-jordan algebra, (v,αv ) be a hom-module and ρr : v ⊗ a → v , a ⊗ v 7→ v · a, be a bilinear map satisfying αv ◦ρr = ρr ◦ (αv ⊗αa) (13) and αv (v) · (ab) = (v ·a) ·αa(b) + (v · b) ·αa(a) (14) for all (a,b) ∈ a×2 and v ∈ v . then (v,α,ρr) is a right hom-jordan amodule called a right special hom-jordan a-module. 82 s. attan, h. hounnon, b. kpamegan proof. it suffices to prove (9) and (10). for all (a,b) ∈ a×2 and v ∈ v , we have: αv (v ·a) ·αa(bc) + αv (v · b) ·αa(ca) + αv (v · c) ·αa(ab) = αv (v ·a) ·αa(b)αa(c) + αv (v · b) ·αa(c)αa(a) + αv (v · c) ·αa(a)αa(b) (multiplicativity) = ((v ·a) ·αa(b)) ·α2(c) + ((v ·a) ·αa(c)) ·α2(b) + ((v · b) ·αa(c)) ·α2(a) + ((v · b) ·αa(a)) ·α2(c) + ((v · c) ·αa(a)) ·α2(b) + ((v · c) ·αa(b)) ·α2(a) (by (14)) = [αv (v) ·ab− (v · b)αa(a)] ·α2(c) + ((v ·a) ·αa(c)) ·α2(b) + [αv (v) · bc− (v · c)αa(b)] ·α2(a) + ((v · b) ·αa(a)) ·α2(c) + [αv (v) · ca− (v ·a) ·αa(c)] ·α2(b) + ((v · c) ·αa(b)) ·α2(a) (again by (14)) = (αv (v) · bc) ·α2a(a) + (αv (v) · ca) ·α 2 a(b) + (αv (v) ·ab) ·α 2 a(c) and thus, we get (9). finally, (10) is proved as follows: αv (v ·a) ·αa(bc) + αv (v · b) ·αa(ca) + αv (v · c) ·αa(ab) = αv (v ·a) ·αa(b)αa(c) + αv (v · b) ·αa(c)αa(a) + αv (v · c) ·αa(a)αa(b) (multiplicativity) = ((v ·a) ·αa(b)) ·α2a(c) + ((v ·a) ·αa(c)) ·α 2 a(b) + ((v · b) ·αa(c)) ·α2a(a) + ((v · b) ·αa(a)) ·α 2 a(c) + ((v · c) ·αa(a)) ·α2a(b) + ((v · c) ·αa(b)) ·α 2 a(a) (by ((14)) = ((v ·a) ·αa(b)) ·α2a(c) + [αv (v) ·ac− ((v · c) ·αa(a)] ·α 2 a(b) + ((v · b) ·αa(c)) ·α2a(a) + ((v · b) ·αa(a)) ·α 2 a(c) + ((v · c) ·αa(a)) ·α2a(b) + ((v · c) ·αa(b)) ·α 2 a(a) (again by (14)) = ((v ·a) ·αa(b)) ·α2a(c) + α 2 v (v) · ((ac)αa(b)) − (αv (v) ·αa(b)) ·αa(ac) + ((v · b) ·αa(c)) ·α2a(a) + ((v · b) ·αa(a)) ·α2a(c) + ((v · c) ·αa(b)) ·α 2 a(a) (again by (14)) hom-jordan and hom-alternative bimodules 83 = ((v ·a) ·αa(b)) ·α2a(c) + α 2 v (v) · ((ac)αa(b)) − (αv (v · b) ·αa(ac) + ((v · b) ·αa(c)) ·α2a(a) + ((v · b) ·αa(a)) ·α 2 a(c) + ((v · c) ·αa(b)) ·α2a(a) (by (13)) = ((v ·a) ·αa(b)) ·α2a(c) + α 2 v (v) · ((ac)αa(b)) − ((v · b) ·αa(a)) ·α 2 a(c) − ((v · b) ·αa(c)) ·α2a(a) + ((v · b) ·αa(c)) ·α 2 a(a) + ((v · b) ·αa(a)) ·α2a(c) + ((v · c) ·αa(b)) ·α 2 a(a) (by (14)) = ((v ·a) ·αa(b)) ·α2a(c) + ((v · c) ·αa(b)) ·α 2 a(a) + α 2 v (v) · ((ac)αa(b)) which is (10). similarly, the following result can be proved. theorem 4.3. let (a,µ,αa) be a hom-jordan algebra, (v,αv ) be a hom-module and ρl : a ⊗ v → v , v ⊗ a 7→ a · v, be a bilinear map satisfying αv ◦ρl = ρl ◦ (αa ⊗αv ) and (ab) ·αv (v) = αa(a) · (b ·v) + αa(b) · (a ·v) (15) for all (a,b) ∈ a×2 and v ∈ v . then (v,α,ρl) is a left hom-jordan a-module called a left special hom-jordan a-module. it is well known that the plus algebra of any hom-alternative algebra is a hom-jordan algebra. the next result shows that any left (resp. right) hom-alternative module a is also a left (resp. right) module over its plus hom-algebra. proposition 4.4. let (a,µ,αa) be a hom-alternative algebra and (v,αv ) be a hom-module. (i) if (v,αv ) is a right hom-alternative a-module with the structure map ρr then (v,αv ) is a right special hom-jordan a +-module with the same structure map ρr. (ii) if (v,αv ) is a left hom-alternative a-module with the structure map ρl then (v,αv ) is a left special hom-jordan a +-module with the same structure map ρl. 84 s. attan, h. hounnon, b. kpamegan proof. it suffices to prove (14) and (15). (i) if (v,αv ) is a right hom-alternative a-module with the structure map ρr, then for all (x,y,v) ∈ a × a × v , asa,v (v,x,y) = −asa,v (v,y,x) by (8), i.e., αv (v) · (xy) + αv (v) · (yx) = (v · x) · αa(y) + (v · y) · αa(x). thus αv (v) · (x ∗ y) = αv (v) · (xy) + αv (v) · (yx) = (v · x) · αa(y) + (v · y) · αa(x). therefore (v,αv ) is a right special hom-jordan a+-module by theorem 4.2. (ii) if (v,αv ) is a left hom-alternative a-module with the structure map ρl, then for all (x,y,v) ∈ a × a × v , asa,v (x,y,v) = −asa,v (y,x,v) by remarks 3.3 and then (xy) ·αv (v) + (yx) ·αv (v) = αa(x) · (y · v) + αa(y) · (x ·v). thus (x∗y) ·αv (v) = (xy) ·αv (v) + (yx) ·αv (v) = αa(x) · (y ·v) + αa(y) · (x ·v). therefore (v,αv ) is a left special hom-jordan a+-module by theorem 4.3. now, we give the definition of a hom-jordan bimodule. definition 4.5. let (a,µ,αa) be a hom-jordan algebra. a hom-jordan a-bimodule is a hom-module (v,αv ) with a left structure map ρl : a⊗v → v , a⊗v 7→ a ·v and a right structure map ρr : v ⊗a → v , v ⊗a 7→ v ·a, such that the following conditions hold: ρr ◦ τ1 = ρl , (16) αv (v ·a) ·αa(bc) + αv (v · b) ·αa(ca) + αv (v · c) ·αa(ab) = (αv (v) · bc) ·α2a(a) + (αv (v) · ca) ·α 2 a(b) + (αv (v) ·ab) ·α2a(c) , (17) αv (v ·a) ·αa(bc) + αv (v · b) ·αa(ca) + αv (v · c) ·αa(ab) = ((v ·a) ·αa(b)) ·α2a(c) + ((v · c) ·αa(b)) ·α 2 a(a) + ((ac)αa(b)) ·α2v (v) , (18) for all a,b,c ∈ a and v ∈ v . in term of the module hom-associator, using the relation (16) and the fact that the structure maps are morphisms, the relations (17) and (18) are respectively (a,b,c) asa,v (αa(a),αv (v),bc) = 0 , (19) asa,v (v ·a,αa(b),αa(c)) + asa,v (v · c,αa(b),αa(a)) + asa,v (ac,αa(b),αv (v)) = 0 . (20) hom-jordan and hom-alternative bimodules 85 remarks 4.6. (i) one can note that (17) and (18) are the same identities as (9) and (10) respectively. (ii) since ρr ◦ τ1 = ρl, nothing is lost in dropping one of the compositions. thus the term hom-jordan module can be used for hom-jordan bimodule. (iii) since the field is of characteristic 0, the identity (19) implies asa,v (αa(a),αv (v),a 2) = 0 . (iv) if αa = ida and αv = idv then v is reduced to the so-called jordan module of the jordan algebra (a,µ) [7, 8]. examples 4.7. here are some examples of hom-jordan bimodules. (i) let (a,µ,αa) be a hom-jordan algebra. then (a,αa) is a homjordan a-bimodule where the structure maps are ρl = ρr = µ. more generally, if b is a hom-ideal of (a,µ,αa), then (b,αa) is a hom-jordan a-bimodule where the structure maps are ρl(a,x) = µ(a,x) = µ(x,a) = ρr(x,a) for all (a,x) ∈ a×b. (ii) if (a,µ) is a jordan algebra and m is a jordan a-bimodule [8] in the usual sense then (m,idm ) is a hom-jordan a-bimodule where a = (a,µ,ida) is a hom-jordan algebra. (iii) if f : (a,µa,αa) → (b,µb,αb) is a surjective morphism of homjordan algebras, then (b,αb) becomes a hom-jordan a-bimodule via f, i.e, the structure maps are defined by ρl : (a,b) 7→ µb(b,f(a)) and ρr : (b,a) 7→ µb(f(a),b) for all (a,b) ∈ a×b. as in the case of hom-alternative algebras, in order to give another example of hom-jordan bimodules, let us consider the following definition 4.8. an abelian extension of hom-jordan algebras is a short exact sequence of hom-jordan algebras 0 → (v,αv ) i−→ (a,µa,αa) π−−→ (b,µb,αb) → 0 where (v,αv ) is a trivial hom-jordan algebra, i and π are morphisms of hom-algebras. furthermore, if there exists a morphism s : (b,µb,αb) → (a,µa,αa) such that π◦s = idb then the abelian extension is said to be split and s is called a section of π. example 4.9. given an abelian extension as in the previous definition, the hom-module (v,αv ) inherits a structure of a hom-jordan b-bimodule 86 s. attan, h. hounnon, b. kpamegan and the actions of the hom-algebra (b,µb,αb) on v are as follows. for any x ∈ b, there exist x̃ ∈ a such that x = π(x̃). let x acts on v ∈ v by x ·v := µa(x̃, i(v)) and v ·x := µa(i(v), x̃). these are well-defined, as another lift x̃′ of x is written x̃′ = x̃+v′ for some v′ ∈ v and thus x·v = µa(x̃, i(v)) = µa(x̃′, i(v)) and v · x = µa(i(v), x̃) = µa(i(v), x̃′) because v is trivial. the actions property follow from the hom-jordan identity. in case these actions of b on v are trivial, one speaks of a central extension. the next result shows that a special left and right hom-jordan module has a hom-jordan bimodule structure under a specific condition. theorem 4.10. let (a,µ,αa) be a hom-jordan algebra and (v,αv ) be both a left and a right special hom-jordan a-module with the structure maps ρ1 and ρ2 respectively such that the hom-associativity (or operator commutativity) condition holds ρ2 ◦ (ρ1 ⊗αa) = ρ1 ◦ (αa ⊗ρ2) . (21) define the bilinear maps ρl : a⊗v → v and ρr : v ⊗a → v by ρl = ρ1 + ρ2 ◦ τ1 and ρr = ρ1 ◦ τ2 + ρ2 . (22) then (v,αv ,ρl,ρr) is a hom-jordan a-bimodule. proof. it is clear that ρl and ρr are structure maps and (16) holds. to prove relations (17) and (18), let put ρl(a⊗v) := a�v, i.e., a�v = a ·v + v ·a for all (a,v) ∈ a×v . we have then ρr(v ⊗a) := v �a = a ·v + v ·a for all (a,v) ∈ a×v . therefore for all (a,b,v) ∈ a×a×v , we have αv (v �a) �αa(bc) + αv (v � b) �αa(ca) + αv (v � c) �αa(ab) = αv (v ·a) ·αa(bc) + αv (a ·v) ·αa(bc) + αa(bc) ·αv (v ·a) + αa(bc) ·αv (a ·v) + αv (v · b) ·αa(ca) + αv (b ·v) ·αa(ca) + αa(ca) ·αv (v · b) + αa(ca) ·αv (b ·v) + αv (v · c) ·αa(ab) + αv (c ·v) ·αa(ab) + αa(ab) ·αv (v · c) + αa(ab) ·αv (c ·v) (by a straightforward computation) = {αv (v ·a) ·αa(bc) + αv (v · b) ·αa(ca) + αv (v · c) ·αa(ab)} + {αa(bc) · (αv (v) ·αa(a)) + αa(ca) · (αv (v) ·αa(b)) + αa(ab) · (αv (v) ·αa(c))} + {αa(bc) ·αv (a ·v) + αa(ca) ·αv (b ·v) hom-jordan and hom-alternative bimodules 87 + αa(ab) ·αv (c ·v)} + {(αa(a) ·αv (v)) ·αa(bc) + (αa(b) ·αv (v)) ·αa(ca) + (αa(c) ·αv (v)) ·αa(ab)} (rearranging terms and noting that ρ1 and ρ2 are morphisms) = {(αv (v) · bc) ·α2a(a) + (αv (v) · ca) ·α 2 a(b) + (αv (v) ·ab) ·α 2 a(c)} + {(bc ·αv (v)) ·α2a(a) + (ca ·αv (v)) ·α 2 a(b) + (ab ·αv (v)) ·α 2 a(c)} + {α2a(a) · (bc ·αv (v)) + α 2 a(b) · (ca ·αv (v)) + α 2 a(c) · (ab ·αv (v))} + {α2a(a) · (αv (v) · bc) + α 2 a(b) · (αv (v) · ca) + α 2 a(c) · (αv (v) ·ab)} (by (9), (11) and (21)) = {(αv (v) � bc) ·α2a(a) + (αv (v) � ca) ·α 2 a(b) + (αv (v) �ab) ·α 2 a(c)} + {α2a(a) · (αv (v) � bc) + α 2 a(b) · (αv (v) � ca) + α 2 a(c) · (αv (v) �ab)} (by the definition of �) = (αv (v) � bc) �α2a(a) + (αv (v) � ca) �α 2 a(b) + (αv (v) �ab) �α 2 a(c) (again by the definition of �). therefore, we get (17). finally, we have: αv (v �a) �αa(bc) + αv (v � b) �αa(ca) + αv (v � c) �αa(ab) = αv (v ·a) ·αa(bc) + αv (a ·v) ·αa(bc) + αa(bc) ·αv (v ·a) + αa(bc) ·αv (a ·v) + αv (v · b) ·αa(ca) + αv (b ·v) ·αa(ca) + αa(ca) ·αv (v · b) + αa(ca) ·αv (b ·v) + αv (v · c) ·αa(ab) + αv (c ·v) ·αa(ab) + αa(ab) ·αv (v · c) + αa(ab) ·αv (c ·v) (by a straightforward computation) = {αv (v ·a) ·αa(bc) + αv (v · b) ·αa(ca) + αv (v · c) ·αa(ab)} + {(αv (a ·v) ·αa(b)αa(c) + (αv (b ·v)) ·αa(c)αa(a) + (αv (c ·v) ·αa(a)αa(b)} + {αa(bc) ·αv (a ·v) + αa(ca) ·αv (b ·v) + αa(ab) ·αv (c ·v)} + {αa(b)αa(c) ·αv (v ·a) + αa(c)αa(a) ·αv (v · b) + αa(a)αa(b) ·αv (v · c)} (rearranging terms and using the multiplicativity of αa) = {((v ·a) ·αa(b)) ·α2a(c)︸︷︷︸ 1 + ((v · c) ·αa(b)) ·α2a(a)︸︷︷︸ 2 + α2v (v) · ((ac)αa(b))︸︷︷︸ 5 } 88 s. attan, h. hounnon, b. kpamegan + {((a ·v) ·αa(b)) ·α2a(c)︸︷︷︸ 1 +((a ·v) ·αa(c)) ·α2a(b) + ((b ·v) ·αa(c)) ·α2a(a) + ((b ·v) ·αa(a)) ·α 2 a(c) + ((c ·v) ·αa(a)) ·α2a(b) + ((c ·v) ·αa(b)) ·α 2 a(a)︸︷︷︸ 2 } + {α2a(c) · (αa(b) · (a ·v))︸︷︷︸ 3 + α2a(a) · (αa(b) · (c ·v))︸︷︷︸ 4 + ((ac)αa(b)) ·α2v (v)︸︷︷︸ 5 } + {α2a(b) · (αa(c) · (v ·a)) + α2a(c) · (αa(b) · (v ·a))︸︷︷︸ 3 +α2a(a) · (αa(c) · (v · b)) + α2a(c) · (αa(a) · (v · b)) + α 2 a(a) · (αa(b) · (v · c))︸︷︷︸ 4 + α2a(b) · (αa(a) · (v · c))} (by (10), (12), (14) and (15)) = ((v �a) ·αa(b)) ·α2a(c) + ((v � c) ·αa(b)) ·α 2 a(a) + α2a(c) · (αa(b) · (v �a)) + α 2 a(a) · (αa(b) · (v � c)) + α2v (v) � ((ac)αa(b)) + ((a ·v) ·αa(c)) ·α 2 a(b) + ((b ·v) ·αa(c)) ·α2a(a) + ((b ·v) ·αa(a)) ·α 2 a(c) + ((c ·v) ·αa(a)) ·α2a(b) + α 2 a(b) · (αa(c) · (v ·a)) + α2a(a) · (αa(c) · (v · b)) + α 2 a(c) · (αa(a) · (v · b)) + α2a(b) · (αa(a) · (v · c)) = ((v �a) ·αa(b)) ·α2a(c) + ((v � c) ·αa(b)) ·α 2 a(a) + α2a(c) · (αa(b) · (v �a)) + α 2 a(a) · (αa(b) · (v � c)) + α2v (v) � ((ac)αa(b)) + (αa(a) · (v · c)) ·α 2 a(b) + (αa(b) · (v · c)) ·α2a(a) + (αa(b) · (v ·a)) ·α 2 a(c) + (αa(c) · (v ·a)) ·α2a(b) + α 2 a(b) · ((c ·v) ·αa(a)) + α2a(a) · ((c ·v) ·αa(b)) + α 2 a(c) · ((a ·v) ·αa(b)) + α2a(b) · ((a ·v) ·αa(c)) (by (21)) hom-jordan and hom-alternative bimodules 89 = ((v �a) ·αa(b)) ·α2a(c) + ((v � c) ·αa(b)) ·α 2 a(a) + α2a(c) · (αa(b) · (v �a)) + α 2 a(a) · (αa(b) · (v � c)) + α2v (v) � ((ac)αa(b)) + α 2 a(a) · ((v · c) ·αa(b))︸︷︷︸ 6 + (αa(b) · (v · c)) ·α2a(a)︸︷︷︸ 7 + (αa(b) · (v ·a)) ·α2a(c)︸︷︷︸ 8 + α2a(c) · ((v ·a) ·αa(b))︸︷︷︸ 9 + (αa(b) · (c ·v)) ·α2a(a)︸︷︷︸ 7 + α2a(a) · ((c ·v) ·αa(b))︸︷︷︸ 6 + α2a(c) · ((a ·v) ·αa(b))︸︷︷︸ 9 + (αa(b) · (a ·v)) ·α2a(c))︸︷︷︸ 8 (again by (21)) = ((v �a) ·αa(b)) ·α2a(c)︸︷︷︸ 10 + ((v � c) ·αa(b)) ·α2a(a)︸︷︷︸ 11 + α2a(c) · (αa(b) · (v �a))︸︷︷︸ 13 + α2a(a) · (αa(b) · (v � c))︸︷︷︸ 12 + α2v (v) � ((ac)αa(b)) + α 2 a(a) · ((v � c) ·αa(b))︸︷︷︸ 12 + (αa(b) · (v � c)) ·α2a(a)︸︷︷︸ 11 + (αa(b) · (v �a)) ·α2a(c)︸︷︷︸ 10 + α2a(c) · ((v �a) ·αa(b))︸︷︷︸ 13 = ((v �a) �αa(b)) ·α2a(c) + ((v � c) �αa(b)) ·α 2 a(a) + α2a(a) · ((v � c) �αa(b)) + α 2 a(c) · ((v �a) �αa(b)) + α2v (v) � ((ac)αa(b)) = ((v �a) �αa(b)) �α2a(c) + ((v � c) �αa(b)) �α 2 a(a) + α2v (v) � ((ac)αa(b)) which is (18). the following result will be used below. it gives a relation between homassociative modules and special hom-jordan modules. 90 s. attan, h. hounnon, b. kpamegan lemma 4.11. let (a,µ,αa) be a hom-associative algebra and (v,αv ) be a hom-module. (i) if (v,αv ) is a right hom-associative a-module with the structure maps ρr then (v,αv ) is a right special hom-jordan a +-module with the same structure map ρr. (ii) if (v,αv ) is a left hom-associative a-module with the structure maps ρl then (v,αv ) is a left special hom-jordan a +-module with the same structure map ρl. proof. it also suffices to prove (14) and (15). (i) if (v,αv ) is a right hom-associative a-module with the structure map ρr then for all (x,y,v) ∈ a×a×v , αv (v)·(a∗b) = αv (v)·(ab)+αv (v)·(ba) = (v ·a)·αa(b) + (v ·b)·αa(a) where the last equality holds by (5). then (v,αv ) is a right special hom-jordan a+-module. (ii) if (v,αv ) is a left hom-associative a-module with the structure map ρl then for all (x,y,v) ∈ a×a×v , (a∗b)·αv (v) = (ab)·αv (v)+(ba)·αv (v) = αa(a)·(b·v) +αa(b)·(a·v) where the last equality holds by (4). then (v,αv ) is a left special hom-jordan a+-module. now, we prove that a hom-associative module gives rise to a hom-jordan module for its plus hom-algebra. proposition 4.12. let (a,µ,αa) be a hom-associative algebra and (v,ρ1,ρ2,αv ) be a hom-associative a-bimodule. then (v,ρl,ρr,αv ) is a hom-jordan a+-bimodule where ρl and ρr are defined as in (22). proof. the proof follows from lemma 4.11 , the hom-associativity condition (6) and theorem 4.10. the following elementary result will be used below. it gives a property of a module hom-associator. lemma 4.13. let (a,µ,αa) be a hom-jordan algebra and (v,αv ) be an hom-jordan a-bimodule with the structure maps ρl and ρr. then αnv ◦asa,v ◦ ida⊗v⊗a = asa,v ◦ (α ⊗n a ⊗α ⊗n v ⊗α ⊗n a ) . (23) hom-jordan and hom-alternative bimodules 91 proof. using twice the fact that ρl and ρr are morphisms of hom-modules, we get αnv ◦asa,v ◦ ida⊗v⊗a = αnv ◦ (ρr ◦ (ρl ⊗αa) −ρl ◦ (αa ⊗ρr)) = αnv ◦ρr ◦ (ρl ⊗αa) −α n v ◦ρl ◦ (αa ⊗ρr) (linearity of α n v ) = ρr ◦ (αnv ◦ρl ⊗α n+1 a ) −ρl ◦ (α n+1 a ⊗α n v ◦ρr) = ρr ◦ (ρl ◦ (αna ⊗α n v ) ⊗α n+1 a ) −ρl ◦ (α n+1 a ⊗ρr ◦ (α n v ⊗α n a)) = (ρr ◦ (ρl ⊗αa) −ρl ◦ (αa ⊗ρr)) ◦ (α⊗na ⊗α ⊗n v ⊗α ⊗n a ) = asa,v ◦ (α⊗na ⊗α ⊗n v ⊗α ⊗n a ). that ends the proof. the next result is similar to the one of proposition 3.7, but an additional condition is needed. proposition 4.14. let (a,µ,αa) be a hom-jordan algebra and (v,αv ) be a hom-jordan a-bimodule with the structure maps ρl and ρr. suppose that there exists n ∈ n such that αnv = idv . then the maps ρ (n) l = ρl ◦ (α n a ⊗ idv ) , (24) ρ(n)r = ρr ◦ (idv ⊗α n a) (25) give the hom-module (v,αv ) the structure of a hom-jordan a-bimodule that we denote by v (n). proof. since the structure map ρl is a morphism of hom-modules, we get: αv ◦ρ (n) l = αv ◦ρl ◦ (α n a ⊗ idv ) (by (24)) = ρl ◦ (αn+1a ⊗αv ) = ρl ◦ (αna ⊗ idv ) ◦ (αa ⊗αv ) = ρ (n) l ◦ (αa ⊗αv ) then, ρ (n) l is a morphism. similarly, we get that ρ (n) r is a morphism and that (16) holds for v (n). next, we compute 92 s. attan, h. hounnon, b. kpamegan (a,b,c) asa,v (n) (αa(a),αv (v),ab) = (a,b,c){ρ (n) r (ρ (n) l (αa(a),αv (v)),αa(bc))−ρ (n) l (α 2 a(a),ρ (n) r (αv (v),bc))} = (a,b,c){ρr(ρ (n) l (αa(a),αv (v)),α n+1 a (bc))−ρl(α n+2 a (a),ρ (n) r (αv (v),bc))} = (a,b,c){ρr(ρl(α n+1 a (a),αv (v)),α n+1 a (bc))−ρl(α n+2 a (a),ρr(αv (v),α n a(bc))} = (a,b,c){ρr(ρl(α n+1 a (a),αv (v)),αa(α n a(bc))) −ρl(αa(αn+1a (a)),ρr(αv (v),α n a(bc))} = (a,b,c) asa,v (α n+1 a (a),αv (v),α n a(bc)) = (a,b,c) asa,v (α n+1 a (a),α n+1 v (v),α n a(bc)) (by the hypothesis αv = α n+1 v ) = αnv ( (a,b,c) asa,v (αa(a),αv (v),bc)) (by (23) and the linearity of α n v ) = 0 (by (19) in v ). then we get (19) for v (n). finally remarking that asa,v (n) (ρ n r (v,a),αa(b),αa(c)) = asa,v (n) (v ·α n a(a),αa(b),αa(c)) = ρnr (ρ n r (v ·α n a(a),αa(b),α 2 a(c)) −ρ n r (αv (v) ·α n+1 a (a),µ(αa(b),α(a(c)) = ρr(ρr(v ·αna(a),α n+1 a (b),α n+2 a (c)) −ρr(αv (v) ·αn+1(a),µ(αn+1a (b),α n+1 a (c)) = αa,v (v ·αna(a),α n+1 a (b),α n+1 a (c)) , and similarly asa,v (n) (ρ n r (v,c),αa(b),αa(a)) = asa,v (v ·α n a(c),α n+1 a (b),α n+1 a (a)) , asa,v (n) (ac,αa(b),αv (v)) = asa,v (α n a(a)α n a(c),α n+1 a (b),αv (v)) (20) is proved for v (n) as it follows: asa,v (n) (ρ n r (v,a),αa(b),αa(c)) + asa,v (n) (ρ n r (v,c),αa(b),αa(a)) + asa,v (n) (ac,αa(b),αv (v)) = αv (v ·αna(a),α n+1 a (b),α n+1 a (c)) + asa,v (v ·α n a(c),α n+1 a (b),α n+1 a (a)) + asa,v (α n a(a)α n a(c),α n+1 a (b),αv (v)) hom-jordan and hom-alternative bimodules 93 = αv (v ·αna(a),αa(α n a(b)),αa(α n a(c))) + asa,v (v ·αna(c),αa(α n a(b)),αa(α n a(a))) + asa,v (α n a(a)α n a(c),αa(α n a(b)),αv (v)) = 0(by (20) in v ). we conclude that v (n) is a hom-jordan a-bimodule. example 4.15. consider the hom-jordan algebra a+ of the examples 2.10 and the subspace v = span(e1,e3) of a. then (v,µv ,αv ) is a homideal of a+ where µv = µa|v and αv = αa|v . it follows that (v,ρl,ρr,αv ) is a hom-jordan a+-bimodule where ρl and ρr are defined as in examples 4.7. we have α2v = idv , then by proposition 4.14, the structure maps ρ (2) l = ρl ◦ (α2a ⊗idv ) and ρ (2) r = ρr ◦ (idv ⊗α2a) give the hom-module (v,αv ) the structure of a hom-jordan a+-bimodule that we denote by v (2). corollary 4.16. let (a,µ,αa) be a hom-jordan algebra and (v,αv ) be a hom-jordan a-bimodule with the structure maps ρl and ρr such that αv is an involution. then (v,αv ) is a hom-jordan a-bimodule with the structure maps ρ (2) l = ρl ◦ (α 2 a ⊗ idv ) and ρ (2) r = ρr ◦ (idv ⊗α2a). example 4.17. consider the hom-jordan algebra b+ of the examples 2.10 and the subspace v = span(e1,e2) of b. then (v,µv ,αv ) is a hom-ideal of b+ where µv = µb|v and αv = αb|v . therefore (v,ρl,ρr,αv ) is a homjordan b+-bimodule where ρl and ρr are defined as in examples 4.7. note that αv is involutive, i.e., α 2 v = idv , then by corollary 4.16, the structure maps ρ (2) l = ρl ◦(α 2 b ⊗idv ) and ρ (2) r = ρr ◦(idv ⊗α2b) give the hom-module (v,αv ) the structure of a hom-jordan b+-bimodule. the following result is similar to theorem 3.8. it says that jordan bimodules can be deformed into hom-jordan bimodules via an endomorphism. theorem 4.18. let (a,µ) be a jordan algebra, v be a jordan a-bimodule with the structure maps ρl and ρr, αa be an endomorphism of the jordan algebra a and αv be a linear self-map of v such that αv ◦ρl = ρl◦(αa⊗αv ) and αv ◦ ρr = ρr ◦ (αv ⊗ αa). write aαa for the hom-jordan algebra (a,µαa,αa) and vαv for the hom-module (v,αv ). then the maps: ρ̃l = αv ◦ρl and ρ̃r = αv ◦ρr give the hom-module vαv the structure of a hom-jordan aαa-bimodule. 94 s. attan, h. hounnon, b. kpamegan proof. it is easy to prove that the relation (16) for vαv holds and both maps ρ̃l, ρ̃r are morphisms. remarking that asa,vαv = α 2 v ◦asa,v (26) we first compute (a,b,c) asa,vαv (αa(a),αv (v),µαa(b,c)) = (a,b,c) α 2 v (asa,v (αa(a),αv (v),αa(bc))) (by (26)) = (a,b,c) α 3 v ((asa,v (a,v,bc)) (by (23)) = α3v ( (a,b,c) (asa,v (a,v,bc)) = 0 (by (19) in v ) and then, we get (19) for vαv . finally, we get asa,vαv (ρ̃r(v,a),αa(b),αa(c)) + asa,vαv (ρ̃r(v,c),αa(b),αa(a)) + asa,vαv (µαa(a,c),αa(b),αv (v)) = α2v (asa,v (ρ̃r(v,a),αa(b),αa(c))) + α 2 v (asa,v (ρ̃r(v,c),αa(b),αa(a))) + α2v (asa,v (µαa(a,c),αa(b),αv (v))) (by (26)) = α2v (asa,v (αv (v ·a),αa(b),αa(c))) + α2v (asa,v (αv (v · c),αa(b),αa(a))) + α2v (asa,v (αa(ac),αa(b),αv (v))) = α3v (asa,v (v ·a,b,c)) + α 3 v (asa,v (v · c,b,a)) + α3v (asa,v (ac,b,v)) (by 23) = α3v (asa,v (v ·a,b,c) + asa,v (v · c,b,a) + asa,v (ac,b,v)) = 0 (by (20) in v ) which is (20) for vαv . therefore the hom-module vαv has a hom-jordan aαa-bimodule structure. corollary 4.19. let (a,µ) be a jordan algebra, v be a jordan abimodule with the structure maps ρl and ρr, αa be an endomorphism of the jordan algebra a and αv be a linear self-map of v such that αv ◦ρl = ρl ◦ (αa ⊗αv ) and αv ◦ρr = ρr ◦ (αv ⊗αa). moreover, suppose that there exists n ∈ n such that αnv = idv . write aαa for the hom-jordan algebra (a,µαa,αa) and vαv for the hom-module hom-jordan and hom-alternative bimodules 95 (v,αv ). then the maps: ρ̃ (n) l = ρl ◦ (α n+1 a ⊗αv ) and ρ̃ (n) r = ρr ◦ (αv ⊗α n+1 a ) (27) give the hom-module vα the structure of a hom-jordan aαa-bimodule for each n ∈ n. proof. the proof follows from proposition 4.14 and theorem 4.18. similarly to hom-alternative algebras, the split null extension, determined by the given bimodule over a hom-jordan algebra, is constructed as follows: theorem 4.20. let (a,µ,αa) be a hom-jordan algebra and (v,αv ) be a hom-jordan a-bimodule with the structure maps ρl and ρr. then (a⊕v,µ̃, α̃) is a hom-jordan algebra where µ̃ : (a⊕v )⊗2 → a⊕v , µ̃(a+m,b+n) := ab+a·n+m·b and α̃ : a⊕v → a⊕v , α̃(a + m) := αa(a) + αv (m) proof. first, the commutativity of µ̃ follows from the one of µ. next, the multiplicativity of α̃ with respect to µ̃ follows from the one of α with respect to µ and the fact that ρl and ρr are morphisms of hom-modules. finally, we prove the hom-jordan identity (3) for e = a⊕v as it follows ase(µ̃(x + m,x + m), α̃(y + n), α̃(x + m)) = µ̃(µ̃(µ̃(x + m,x + m), α̃(y + n)), α̃2(x + m)) − µ̃(α̃(µ̃(x + m,x + m)), µ̃(α̃(y + n), α̃(x + m))) = µ̃(µ̃(x2 + x ·m + m ·x,αa(y) − µ̃(αa(x2) + αv (n)),α2a(x) + α 2 v (m)) + αv (x ·m) + αv (m ·x), µ̃(αa(y) + αv (n),αa(x) + αv (m))) = µ̃(x2αa(y) + x 2 ·αv (n) + (x ·m) ·αa(y) + (m ·x) ·αa(y),α2a(x) + α2v (m)) − µ̃(α 2 a(x 2) + αv (x ·m) + αv (m ·x),αa(y)αa(x) + αa(y) ·αv (m) + αv (n) ·αa(x)) = (x2αa(y))α 2 a(x) + (x 2αa(y)) ·α2v (m) + (x 2 ·αv (n)) ·α2a(x) + ((x ·m) ·αa(y)) ·α2a(x) + ((m ·x) ·αa(y)) ·α 2 a(x)) −αa(x2)(αa(y)αa(x)) −αa(x2) · (αa(y) ·αv (m)) −αa(x2) · (αv (n) ·αa(x)) −αv (x ·m) · (αa(y)αa(x)) −αv (m ·x) · (αa(y)αa(x)) 96 s. attan, h. hounnon, b. kpamegan = asa(x 2,αa(y),αa(x)) + asa,v (x 2,αa(y),αv (m)) + asa,v (x 2,αv (n),αa(x)) + asa,v (x ·m,αa(y),αa(x)) + asa,v (m ·x,αa(y),αa(x)) = asa,v (m ·x,αa(y),αa(x)) + asa,v (m ·x,αa(y),αa(x))︸︷︷︸ 0 + asa,v (x 2,αv (n),αa(x)) + asa,v (x 2,αa(y),αv (m))︸︷︷︸ 0 + asa(x 2,αa(y),αa(x))︸︷︷︸ 0 = 0 , where the first 0 follows from (20), the second from (19) (see remarks 4.6) and the last from the hom-jordan identity (3) in a. we conclude then that (a⊕v,µ̃, α̃) is a hom-jordan algebra. similarly as hom-alternative algebra case, let give the following: remark 4.21. consider the split null extension a⊕v determined by the hom-jordan bimodule (v,αv ) for the hom-jordan algebra (a,µ,αa) in the previous theorem. write elements a + v of a⊕v as (a,v). then there is an injective homomorphism of hom-modules i : v → a⊕v given by i(v) = (0,v) and a surjective homomorphism of hom-modules π : a ⊕ v → a given by π(a,v) = a. moreover, i(v ) is a hom-ideal of a⊕v such that a⊕v/i(v ) ∼= a. on the other hand, there is a morphism of hom-algebras σ : a → a⊕v given by σ(a) = (a, 0) which is clearly a section of π. hence, we obtain the abelian split exact sequence of hom-jordan algebras and (v,αv ) is a hom-jordan bimodule for a via π. references [1] h. ataguema, a. makhlouf, s.d. silvestrov, generalization of nary nambu algebras and beyond, j. math. phys. 50 (8) (2009), 083501, 15 pp. [2] i. bakayoko, b. manga, hom-alternative modules and hom-poisson comodules. arxiv:1411.7957v1 [3] s. eilenberg, extensions of general algebras, ann. soc. polon. math. 21 (1948), 125 – 34. 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[18] y. sheng, representation of hom-lie algebras, algebr. represent. theory 15 (6) (2012), 1081 – 1098. [19] t.a. springer, f.d. veldkamp, “octonions, jordan algebras, and exceptional groups”, springer-verlag, berlin, 2000. [20] j. tits, r.m. weiss, “moufang polygons”, springer-verlag, berlin, 2002. [21] d. yau, hom-algebras and homology. arxiv:0712.3515v1 [22] d. yau, module hom-algebras. arxiv:0812.4695v1 [23] d. yau, hom-maltsev, hom-alternative and hom-jordan algebras, int. electron. j. algebra 11 (2012), 177 – 217. [24] a. zahary, a. makhlouf, structure and classification of hom-associative algebras. arxiv:1906.04969v1[math.ra] arxiv:0712.3515v1 arxiv:0812.4695v1 arxiv:1906.04969v1[math.ra] introduction preliminaries hom-alternative bimodules hom-jordan bimodules e extracta mathematicae vol. 31, núm. 2, 189 – 197 (2016) geometry of foliated manifolds a.ya. narmanov, a.s. sharipov namangan engineering-pedagogical institute, namangan national university of uzbekistan, tashkent, uzbekistan narmanov@yandex.ru , asharipov@inbox.ru presented by manuel de león received march 15, 2016 abstract: in this paper some results of the authors on geometry of foliated manifolds are stated and results on geometry of riemannian (metric) foliations are discussed. key words: foliation, foliated manifold, riemannian submersion, riemannian foliation, gaussian curvature. ams subject class. (2010): 53c12, 57c30. 1. introduction a manifold, with some fixed foliation on it, is called a foliated manifold. theory of foliated manifolds is one of new fields of mathematics. it appeared in the intersection of differential equations, differential geometry and differential topology in the second part of 20th century. in formation and development of the theory of foliations the big contribution was made by famous mathematicians such as c. ehresmann [1], g. reeb [15], a. haefliger [3], r. langevin [8], c. lamoureux [7]. further development of the geometrical theory of foliations is connected with known works of r. hermann [4, 5], p. molino [9], b.l. reinhart [16], ph. tondeur [17]. at present, the theory of foliations (the theory of foliated manifolds) is intensively developing and has wide applications in many fields of science and technique. in the theory of foliations, it is possible to get acquainted with the latest scientific works in work ph. tondeur [18], where the bibliography consisting of more than 2500 works on the theory of foliations is provided. in work [10] applications of the theory of foliations in the qualitative theory of optimal control are discussed. in this paper some results of the authors on geometry of foliated manifolds are stated and results on geometry of riemannian (metric) foliations are discussed. 189 190 a.ya. narmanov, a.s. sharipov at first we will give some necessary definitions and examples. let (m, g) be a smooth riemannian manifold of dimension n, where g is a riemannian metric and 0 < k < n. definition 1. a family f = {lα ⊂ m : α ∈ b} of pathwise connected subsets m is called a k-dimensional smooth foliation if it satisfies the following three conditions: (f1): ∪ α∈b lα = m ; (f2): for all α, β ∈ b if α ̸= β, then lα ∩ lβ = ∅ ; (f3): for any point p ∈ m there is a neighborhood up and a coordinate chart( x1, x2, . . . , xn ) such that if up ∩ lα ̸= ∅ for some α ∈ b, then pathwise connected components of the set up ∩ lα are given by the equations: xk+1 = ck+1, xk+2 = ck+2, . . . , xn = cn, where numbers ck+1, ck+2, . . . , cn are constant on components of pathwise connectedness. the set lα is called a leaf of a foliation f. in the described situation a kdimensional cr-foliation is also called cr-foliation of codimension q = n − k. existence of a foliation f in a manifold m is expressed by a symbol (m, f). conditions (f1), (f2) mean that m consists of mutually disjoint leaves. the condition (f3) means that locally leaves are arranged as the parallel planes. the neighborhood u in the definition is called a foliated neighborhood. the simplest foliations from the point of view of geometry are the foliations generated by submersions, in particular the family of level surfaces of differentiable functions. definition 2. a differentiable mapping f : m → b of maximal rank, where m, b are smooth manifolds of dimension n, m respectively, and n > m, is called a submersion. for submersions the following theorem holds. theorem 1. let f : m → b be a submersion, where m is a smooth manifold of dimension m, n > m. then for each point q ∈ b the set lq = {p ∈ m : f (p) = q} is a manifold of dimension (n − m) and partition of m into connected components of the fibers is k = (n−m)-dimensional foliations. thus, the submersion f : m → b generates a foliation f of dimension on k = (n − m) on the manifold m, leaves of which are connected components of fibers lq = f −1 (q), q ∈ b. geometry of foliated manifolds 191 numerous researches [10-14], [17] are devoted to studying of geometry and topology of foliations generated by submersions. let f be a smooth foliation of dimension k on m. by l(p) denote the leaf of the foliation f passing through a point p, tpl is a tangent space of the leaf l(p) at the point p, h(p) is an orthogonal complement of tpl in tpm, p ∈ m. we get two sub-bundles tf = {tpl : p ∈ m}, h = {h (p) : p ∈ m} of a tangent bundle tm such that tm = tf ⊕ h, where h is an orthogonal complement of tf . in this case each vector field x ∈ v (m) can be represented in the form x = xv + xh, where xv, xh are orthogonal projections of x on tf , h respectively. if x ∈ v (f) (i.e., xh = 0), then x is called a vertical field. if x ∈ v (h) (xν = 0), then x is a horizontal field. definition 3. a submersion f : m → b is riemannian, if the differential of a mapping df preserves the length of horizontal vectors. definition 4. a foliation on a riemannian manifold is called riemannian if every geodesic, orthogonal to a leaf of the foliation f remains orthogonal to all leaves in all its points. for the first time a riemannian foliation was entered in work [16] and was shown that riemannian submersions generate riemannian foliations. this class of foliations plays very important role in the theory of foliations and is substantial from the point of view of geometry. there is a large number of works devoted to geometry of riemannian foliations. a riemannian foliations with singularity were introduced by p. molino [9], and studied in a. narmanov’s works [10], [14] and other authors. 2. previous results an important class of foliations of codimension one are the foliations generated by level surfaces of differentiable functions without critical points. function f : mn → r1 on a riemannian manifold mn, whose length of a gradient vector is constant on each level surface (i.e., for each vertical vector field x it holds x ( |gradf|2 ) = 0), is called metric. for the first time the geometry of foliations generated by surfaces of metric functions is studied in work [17]. 192 a.ya. narmanov, a.s. sharipov the following theorem shows that metric functions are included into a class of riemannian submersions. theorem 2. let f : m → r1 be a metric function. then on r1 there is a riemannian metric such that f : m → r1 is a riemannian submersion. therefore, level surfaces of metric function generate a riemannian foliation. riemannian foliations generated by metric functions are studied in works of a.ya. narmanov, a.m. bayturayev [11], a.ya. narmanov, g.kh. kaipnazarova [12], ph. tondeur [17]. we remind that by definition the gradient vector x = gradf of the function f given on riemannian manifold depends not only on the function f, but also on a riemannian metric. the integral curve of the gradient vector field is called the gradient line of function f. by a.ya. narmanov and g.kh. kaipnazarova in work [12] it is shown that if for each vertical vector field the equality x ( |gradf|2 ) = 0 holds, then each gradient line is the geodesic line of riemannian manifold. in work [12] geometry of foliations is studied generated by level surfaces of metric functions and the whole classification appears in the next form: theorem 3. let f metric function is defined in rn. then the level surface of function makes f surface that has one of these types of n: 1) foliations f consists of parallel hyperplanes; 2) foliations f consists of concentric hyperspheres and a point (that is the center of spheres); 3) foliations f consists of concentric cylinders in the form sn−k−1 × rk and singular foliation rk (that occurs when sphere sn−k−1 shrinks and becomes a point), where k is minimal dimension of critical level surfaces and 1 ≤ k ≤ n − 2. in work [11] the following theorem is proved. theorem 4. let m be a smooth complete and connected riemannian manifold of constant non-negative section curvature, f : m → r1 metric function without critical points. then, level surfaces of function f generate completely geodesic foliation f on m, whose leaves are mutually isometric. geometry of foliated manifolds 193 3. main part before formulating the following theorem about curvature of leaves, we will recall the gaussian curvature of a submanifold. the riemannian metric on the manifold m induces a riemannian metric g̃ on a leaf lp. the canonical injection i : lp → m is an isometric immersion with respect to this metric. connection ∇ induces a connection ∇̃ on lp which coincides with the connection determined by the riemannian metric ∇̃ [6]. let z be a horizontal vector field. for each vertical vector field we will define a vector field s (x, z) = (∇xz)v, where ∇ is the levi-civita connection defined by the riemannian metric g. at the fixed horizontal field we obtain a tensor field of type (1,1) szx = s (x, z) . with the help of this tensor field the bilinear form lz (x, y ) = g (szx, y ) is defined, where g(x, y ) is the scalar product defined by the riemannian metric g. the defined tensor field sz is called the second main tensor, and a form lz (x, y ) is called the second main form with respect to a horizontal field z. the mapping sz : tqf → tqf determined by the formula xq → s (x, z)q is a self-conjugate endomorphism with respect to a scalar product, determined by a riemannian metric g̃. if the vector field z is a field of unit vectors, then eigenvalues of this endomorphism are called the main curvatures of the manifold lp at a point q, and the corresponding eigenvectors are called the main directions. by the main curvatures the gaussian curvature kz = det sz is defined. we will prove that level surfaces of riemannian submersions are surfaces of constant gaussian curvature. theorem 5. let m be a riemannian manifold of constant non-negative curvature, f : m → r1 a riemannian submersion. then each leaf of a foliation f generated by riemannian submersion (connected components of the level surfaces of the function f) is a manifold of constant gaussian curvature. 194 a.ya. narmanov, a.s. sharipov proof. as is known the hessian is given by hf (x, y ) = lz (x, y ) = ⟨∇xz, y ⟩ where z = gradf, ∇the levi-civita connection defined by riemannian metric g. the map x → hf (x) = ∇xz (hesse tensor) is a linear operator and is given by a symmetric matrix a: hf (x) = ∇xz = ax. we denote by χ(λ) the characteristic polynomial of the matrix a with a free term (−1)n det a and define a new polynomial ρ(λ) by the equation λρ(λ) = det a − (−1)nχ(λ) . since χ (a) = 0 we have that aρ (a) = det a · e, where e is the identity matrix. the elements of the matrix ρ (a) are cofactors of the matrix a. this matrix is denoted by hcf. it is well known that the gaussian curvature of the surface is calculated by the formula [2, p. 110] k = det s = 1 |gradf|n+1 ⟨ hcf (gradf) , gradf ⟩ . to prove the theorem it suffices to show that x (k) = 0 for each vertical vector field x at any point q of a leaf lp. by hypothesis of the theorem differential df preserves the length of |gradf|. therefore, we have x ( |gradf|2 ) = 0 and so x ( 1 |gradf|n+1 ) = 0 . therefore we need to show that⟨ ∇xhcfz, z ⟩ + ⟨ hcfz, ∇xz ⟩ = 0 . we know that if x(|gradf|2) = 0 for each vertical vector field x, each gradient line of f is a geodesic line of riemannian manifold [12]. by definition, the gradient line is a geodesic if and only if ∇nn = 0, where n = z|z|. geometry of foliated manifolds 195 we calculate the covariant differential ∇nn = 1 |z| ∇zn = 1 |z| ( 1 |z| ∇zz + z ( 1 |z| ) z ) = 0 and get ∇zz = λz, where λ = −|z|z ( 1 |z| ) . this means that the gradient vector z is the eigenvector of matrix a. let x01, x 0 2, . . . , x 0 n−1, z 0-be mutually orthogonal eigenvectors of a at the point q ∈ lp such that x01, x 0 2, . . . , x 0 n−1 the unit vectors, z 0 the value of the gradient field at a point q. locally, they can be extended to the vector fields x1, x2, . . . , xn−1, z to a neighborhood of (say u) point q so that they formed at each point of an orthogonal basis consisting of eigenvectors. we construct the riemannian normal system of coordinates (x1, x2, . . . , xn) in a neighborhood u via vectors x01, x 0 2, . . . , x 0 n−1, z 0 [2, p. 112]. the components gij of the metric g and the connection components γ k ij in the normal coordinate system satisfies the conditions of [2, p. 132]: gij(q) = δij , γ k ij(q) = 0 . we show that x(λ) = 0 for each vertical field x. from the equality x (λ) = −x (|z|) z ( 1 |z| ) − |z| x ( z ( 1 |z| )) and from the condition x(|z|) = 0 follows equality x ( z ( 1 |z| )) = x (z (ϕ)) = [x, z] (ϕ) − z (x (ϕ)) , where ϕ = 1|z|, [x, z]-lie bracket of vector fields x, z. from the condition of the theorem follows x(ϕ) = 0. in [17] it is shown that x ( |gradf|2 ) = 0 for each of the vertical vector fieldx if and only if [x, z] a vertical field. therefore [x, z] (ϕ) = 0. thus, λ is a constant function on the leaf l. now we denote by λ1, λ2, . . . , λn−1 the eigenvalues of the matrix a corresponding to the eigenvectors x1, x2, . . . , xn−1. then in the basis x1, x2, . . . , xn−1, z matrix a has the form: a =   λ1 0 . . . 0 0 λ2 . . . 0 ... ... ... ... 0 0 . . . λn   196 a.ya. narmanov, a.s. sharipov by hypothesis of the theorem, the vector field ∇xz is vertical field. it follows codazzi equations have the form [6, p. 29] (∇xa)y = (∇y a)x. from this equation we get ∇xiaxj = ∇xj axi , ∇xiaz = ∇zaxi (1) at any point of u for each vector field xi. from first equation of (1) we take following equality xi(λj)xj + λj∇xixj = xj(λi)xi + λi∇xj xi . (2) since ∇xixj = γ k ijxk = 0 at the point q by properties of normal coordinate system, from (2) follows equality xi(λj)xj = xj(λi)xi . (3) by the linear independence x1, x2, · · · , xn−1,we have that xi(λj) = 0 for i ̸= j . from second equation of (1) we take following xi(λ)z + λ∇xiz = z(λi)xi + λi∇zxi . (4) since ∇xiz = ∇zxi = 0 at the point q from the linear independence of vectors xi, z we have that xi(λ) = 0 , z(λi) = 0 for all i . on the other hand ∇zaxi = z(λi)xi + λi∇zxi , ∇xiaz = ∇zaxi , (5) ∇xiz = λixi . from (5) we get that λ2i xi + z(λi)xi = xi(λ)z + λλixi . (6) geometry of foliated manifolds 197 since z(λi) = 0, xi(λ) = 0 from the (6) we get λ2i xi = λλixi . (7) since xi is nonzero vector, from the (7) follows that λ 2 i = λλi. from this equality follows if λ = 0, then λi = 0. if λi ̸= 0 then λi = λ and x(λi) = x(λ) = 0, z(λ) = z(λi) = 0 for all i. thus, in the neighborhood u of the point q non-zero eigenvalues of the matrix a are constant and equal λ. given this fact we compute x(k). we denote by m the number of zero eigenvalues of a. if m = 0, all the eigenvalues are equal to the number λ. in this case, by the definition of the matrix hcf we get that h c fz = λ n−1z and ∇xhcfz = x ( λn−1 ) z + λn−1∇xz . as mentioned above field ∇xgradf is a vertical vector field for each vertical vector field x (the field ax is vertical). from this equalities follows⟨ ∇xhcf(gradf), gradf ⟩ = 0 at the point q. consider the case when m > 0. if m > 1, then hcf = 0. if m = 1 than λi = 0 for some i and axi = ∇xiz = 0. this means that the vector field z is parallel along the integral curve of a vector field xi (along i-coordinate line). if i = n we have λ = λi = 0 for all i and h c f = 0. without loss of generality we assume that i < n. in this case vector hcfz have only one nonzero component bi and h c fz = bi ∂ ∂xi . in this case we get ∇xhcfz = x(bi) ∂ ∂xi + bi∇x ∂ ∂xi . as we know that xi = ∂ ∂xi vertical and ∇x ∂∂xi = 0. thus in the case m = 1 we have ⟨ ∇xhcf(gradf), gradf ⟩ = 0. the theorem 5 is proved. example 1. let m = r3 \{(x, y, z) : x = 0, y = 0}, f(x, y, z) = x2 +y2. level surfaces of this submersion are manifolds of zero gaussian curvature. example 2. let m = r3 \{(0, 0, 0)}, f(x, y, z) = x2 +y2 +z2. level surfaces of this submersion are concentric spheres, gaussian curvature of which is positive. acknowledgements the authors express their sincere gratitude to the anonymous reviewer for a thorough review, which helped to improve the text of the paper. 198 a.ya. narmanov, a.s. sharipov references [1] c. ehresmann, s. weishu, sur les espaces feuilletés: théorème de stabilité, c. r. acad. sci. paris 243 (1956), 344 – 346. 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(2) 69 (1959), 119 – 132. [17] ph. tondeur, “ foliations on riemannian manifolds ”, springer-verlag, new york, 1988. [18] ph. tondeur, www.math.illinois.edu/~tondeur/bib foliations.htm . e extracta mathematicae vol. 32, núm. 2, 255 – 273 (2017) conformal mappings of mixed generalized quasi-einstein manifolds admitting special vector fields s. dey ∗, a. bhattacharyya department of mathematics, jadavpur university kolkata-700032, india santu.mathju@gmail.com bhattachar1968@yahoo.co.in presented by manuel de león received october 28, 2016 abstract: it is known that einstein manifolds form a natural subclass of the class of quasieinstein manifolds and plays an important role in geometry as well as in general theory of relativity. in this work, we investigate conformal mapping of mixed generalized quasieinstein manifolds, considering a conformal mapping between two mixed generalized quasieinstein manifolds vn and v̄n. we also find some properties of this transformation from vn to v̄n and some theorems are proved. considering this mapping, we peruse some properties of these manifolds. later, we also study some special vector fields under these mapping on this manifolds and some theorems about them are proved. key words: mixed generalized quasi-einstein manifolds, φ(ric)-vector field, concircular vector field, codazzi tensor, conformal mapping, conharmonic mapping, σ(ric)-vector field, ν(ric)-vector field. ams subject class. (2010): 53c25, 53c15. 1. introduction the notion of quasi-einstein manifold was introduced by m.c. chaki and r.k. maity [5]. a non-flat riemannian manifold (mn,g), (n ≥ 3) is a quasieinstein manifold if its ricci tensor s satisfies the condition s(x,y ) = ag(x,y ) + bϕ(x)ϕ(y ) and is not identically zero, where a,b are scalars, b ̸= 0 and ϕ is a non-zero 1-form such that g(x,u) = ϕ(x) , for all x ∈ χ(m) , u being a unit vector field. ∗ first author supported by dst/inspire fellowship/2013/1041, government of india. 255 256 s. dey, a. bhattacharyya here a and b are called the associated scalars, ϕ is called the associated 1-form and u is called the generator of the manifold. such an n-dimensional manifold denoted by (qe)n. as a generalization of quasi-einstein manifold in [7], u.c. de and g.c. ghosh defined the generalized quasi-einstein manifold. a non-flat riemannian manifold is called generalized quasi-einstein manifold if its ricci-tensor is non-zero and satisfies the condition s(x,y ) = ag(x,y ) + bϕ(x)ϕ(y ) + cψ(x)ψ(y ) , where a,b and c are non-zero scalars and ϕ, ψ are two 1-forms such that g(x,u) = ϕ(x) and g(x,v ) = ψ(x) , u and v being unit vectors which are orthogonal, i.e., g(u,v ) = 0 . the vector fields u and v are called the generators of the manifold. this type of manifold will be denoted by g(qe)n. the notion of mixed generalized quasi einstein manifold was introduced by a. bhattacharya, t. de and d. debnath in their paper [2]. a non-flat riemannian manifold is called mixed generalized quasi-einstein manifold if its ricci-tensor is non-zero and satisfies the condition s(x,y ) = ag(x,y ) + bϕ(x)ϕ(y ) + cψ(x)ψ(y ) + d[ϕ(x)ψ(y ) + ϕ(y )ψ(x)] , (1.1) where a,b,c and d are non-zero scalars and ϕ, ψ are two 1-forms such that g(x,u) = ϕ(x) and g(x,v ) = ψ(x) , (1.2) u and v being unit vectors which are orthogonal, i.e., g(u,v ) = 0 . the vector fields u and v are called the generators of the manifold. this type of manifold will be denoted by mg(qe)n. putting x = y = ei in (1.1), we get r = na + b + c. (1.3) conformal mappings 257 here r is the scalar curvature of mg(qe)n where {ei}, i = 1,2, . . . ,n is an orthonormal basis of the tangent space at each point of the manifold. quasi-einstein manifolds arose during the study of exact solutions of the einstein field equations as well as during considerations of quasi-umbilical hypersurfaces of semi euclidean spaces. for instance, the robertson-walker spacetimes are quasi-einstein manifolds. so quasi-einstein manifolds have some importance in the general theory of relativity. one of the important concepts of riemannian geometry is conformal mapping. conformal mappings of riemannian manifolds (or semi-riemannian manifolds) have been investigated by many authors. in general relativity, conformal mappings are important since they preserve the causal structure up to time orientation and light-like geodesics up to parametrization [13]. the existence of conformal mappings of riemannian manifolds onto einstein manifolds have been studied by brinkmann [3], mikeš, gavrilchenko, gladysheva [14] and others. also, conformal mappings between two einstein manifolds have been examined by brinkmann. what is more, the problem of finding the invariants under a particular type of mapping is an important and active research topic. in particular, gover and nurowski [9] obtained the polynomial conformal invariants, the vanishing of which is a necessary and sufficient for an n-dimensional suitably generic (pseudo-)riemannian manifold to be conformal to an einstein manifold, and some of the invariants have certain practical significance in physics, such as quantum field theory [4], general relativity [1]. motivated by the above studies the present paper provides conformal mapping on mg(qe)n admitting special vector fields. in the second section, we study conformal mapping of two mixed generalized quasi-einstein manifolds vn and v̄n. we also find some properties of these transformation from vn to v̄n and some theorems are proved. third section deals with conformal mapping on mg(qe)n admitting special vectors fields and in the final section we give an example of mg(qe)n. 2. conformal mapping of two mixed generalized quasi-einstein manifolds in this section, we suppose that vn and v̄n, (n ≥ 3) are two mixed generalized quasi-einstein manifolds with metrics g and ḡ , respectively. definition 1. a conformal mapping is a diffeomorphism of vn onto v̄n such that ḡ = e2σg , (2.1) 258 s. dey, a. bhattacharyya where σ is a function on vn. if σ is constant, then it is called a homothetic mapping. in local coordinates, (2.1) is written as ḡij(x) = e 2σ(x)gij(x) , ḡ ij(x) = e2σ(x)gij(x) , (2.2) besides those equations, we have the christoffel symbols, the components of the curvature tensor, the ricci tensor, and the scalar curvature, respectively γ̄hij = γ h ij + δ h i σj + δ h j σi − σ hgij , r̄hijk = r h ijk + δ h kσij − δ h j σik + g hα(σαkgij − σαjgik + △1σ(δhkgij − δ h j gik) , s̄ij = sij + (n − 2)σij + (△2σ + (n − 2)△1σ)gij , (2.3) r̄ = e−2σ(r + 2(n − 1)△2σ + (n − 1)(n − 2)△1σ) , (2.4) where sij = r α ijα , r = sαβg αβ , σi = ∂σ ∂xi = ∇iσ , σh = σαgαh (2.5) and σij = ∇j∇iσ − ∇iσ∇jσ . (2.6) ∇1σ and ∇2σ are the first and the second beltrami’s symbols which are determined by △1σ = gαβ∇ασ∇βσ , △2σ = gαβ∇β∇ασ , (2.7) where ∇ is the covariant derivative according to the riemannian connection in vn. we denote the objects of space conformally corresponding to vn by a bar, i.e., v̄n. if vn is a mg(qe)n, then, from (1.1), (2.2) and (2.3), we have b̄ϕ̄iϕ̄j + c̄ψ̄iψ̄j + d̄ [ ϕ̄iψ̄j + ϕ̄jψ̄j ] = bϕiϕj + cψiψj + d [ ϕiψj + ϕjψi ] + (n − 2)σij + { △2σ + (n − 2)△1σ + a − āe2σ } gij . (2.8) definition 2. a vector field ξ in a riemannian manifold m is called torse-forming if it satisfies the condition ∇xξ = ρx + λ(x)ξ , conformal mappings 259 where ξ ∈ χ(m), λ(x) is a linear form and ρ is a function, [16]. in the local transcription, this reads ∇iξh = ρδhi + ξ hλi , (2.9) ξh and λi are the components of ξ and ϕ, δ h i is the kronecker symbol. a torse-forming vector field ξ is called recurrent if ρ = 0; concircular if the form λi is a gradient covector, i.e., there is a function ϑ(x) such that λ = dϑ(x); convergent, if it is concircular and ρ = const.exp(ϑ). therefore, recurrent vector fields are characterized by the following equation from (2.9) ∇iξj = λiξj . also, from definition 2., for a concircular vector field ξ, we get ∇iξj = ρigij (2.10) for all x,y ∈ χ(m). a riemannian space with a concircular vector field is called equidistant, [15, 16]. conformal mappings of riemannian spaces (or semi-riemannian spaces) have been studied by many authors, [3, 6, 8, 14]. in this section, we investigate the conformal mappings of mixed generalized quasi-einstein manifolds preserving the associated 1-forms ϕ(x) and ψ(x). theorem 1. if vn admits a conformal mapping preserving the associated 1-forms ϕ(x) and ψ(x) and the associated scalars b and c, then vn is an equidistant manifold. proof. suppose that vn admits a conformal mapping preserving the associated 1-forms ϕ(x) and ψ(x) and the associated scalars b and c. using (2.8), we obtain (n − 2)σij + (β + a − āe2σ)gij = 0 , where β = △2σ + (n − 2)△1σ + a − āe2σ . in this case, we get σij = αgij , (2.11) where α = 1 n − 2 (āe2σ − a − β) is a function. putting ξ = − exp(−σ) and using (2.5), (2.6), (2.10) and (2.11), we get that vn is an equidistant manifold. hence, the proof is complete. 260 s. dey, a. bhattacharyya theorem 2. an equidistant manifold vn admits a conformal mapping preserving the associated 1-forms ϕ(x) and ψ(x) if the associated scalars ā, b̄ and c̄ satisfy both of the conditions d̄ = d, c̄ = c, b̄ = b, ā = e−2σ(a + γ) , where γ = (n − 1) n [ 2△2σ + (n − 2)△1σ ] . proof. suppose that vn is an equidistant manifold. then, there exists a concircular vector field ξ satisfying the condition (2.10), that is, we have ∇jξi = ρgij , (2.12) where ξi = ∇iξ. putting σ = − ln(ξ(x)) and using the condition (2.3), we obtain s̄ij = sij + γgij , where γ = (n − 1) n [ 2△2σ + (n − 2)△1σ ] . considering (1.1) in (2.12) and using (2.2), we get āe2σgij + b̄ϕ̄iϕ̄j + c̄ψ̄iψ̄j + d̄ [ ϕ̄iψ̄j + ϕ̄jψ̄j ] = (a + γ)gij + bϕiϕj + cψiψj + d [ ϕiψj + ϕjψi ] . (2.13) if we take d̄ = d, c̄ = c, b̄ = b and ā = e−2σ(a + γ), then from (2.13) we get ϕ̄iϕ̄j = ϕiϕj , ψ̄iψ̄j = ψiψj , ϕ̄iψ̄j = ϕiψj and ϕ̄jψ̄i = ϕjψi . these completes the proof. the conharmonic transformation is a conformal transformation preserving the harmonicity of a certain function. if the conformal mapping is also conharmonic, then we have [11], ∇iσi + 1 2 (n − 2)σiσi = 0 . (2.14) conformal mappings 261 theorem 3. let vn be a conformal mapping with preservation of the associated 1-forms ϕ(x) and ψ(x) and the associated scalars b and c. a necessary and sufficient condition for this conformal mapping to be conharmonic is that the associated scalar ā be transformed by ā = e−2σa, b̄ = e−2σb, c̄ = e−2σc. proof. we consider a conformal mapping of quasi-einstein manifolds vn and v̄n. then, we have from (1.1) and (2.3), we have b̄ϕ̄iϕ̄j + c̄ψ̄iψ̄j + d̄ [ ϕ̄iψ̄j + ϕ̄jψ̄j ] = bϕiϕj + cψiψj + d [ ϕiψj + ϕjψi ] + (n − 2)σij + { △2σ + (n − 2)△1σ + a − āe2σ } gij . (2.15) multiplying (2.15) by gij and using (1.2), (2.1), (2.6) and (2.7), it can be seen that the following relation is satisfied nā + b̄ + c̄ = e−2σ[na + b + c + 2(n − 1)△2σ + (n − 1)(n − 2)△1σ] . (2.16) if the conformal mapping is also conharmonic, then we have from (2.7) and (2.14) 2△2σ + (n − 2)△1 = 0 . (2.17) considering (2.17) in (2.16), it is found that nā + b̄ + c̄ = nae−2σ + be−2σ + ce−2σ . (2.18) from the equation (2.18), it can be seen that the associated scalars are transformed by ā = e−2σa, b̄ = e−2σb, c̄ = e−2σc. (2.19) conversely, if the associated scalars of the manifolds are transformed by (2.19), then we have from (2.16), 2(n − 1)△2σ + (n − 1)(n − 2)△1σ = 0 and so, we get the relation (2.14). thus, the conformal mapping is also conharmonic. this completes the proof. 262 s. dey, a. bhattacharyya definition 3. a φ(ric)-vector field is a vector field on an n-dimensional riemannian manifold (m,g) and levi-civita connection ∇, which satisfies the condition ∇φ = µric , (2.20) where µ is a constant and ric is the ricci tensor [10]. when (m,g) is an einstein space, the vector field φ is concircular. moreover, when µ = 0, the vector field φ is covariantly constant. in local coordinates, (2.17) can be written as ∇jφi = µsij , where sij denote the components of the ricci tensor and φi = φ αgiα. suppose that vn admits a σ(ric)-vector field. then, we have ∇jσi = µsij , (2.21) where µ is a constant. now, we can state the following theorem: theorem 4. let us consider the conformal mapping (2.1) of a mg(qe)n vn with constant associated scalars being also conharmonic with the σ(ric)vector field. a necessary and sufficient condition for the length of σ to be constant is that the sum of the associated scalars b and c of vn be constant. proof. we consider that the conformal mapping (2.1) of a mg(qe)n vn admitting a σ(ric)-vector field is also conharmonic. in this case, comparing (2.14) and (2.21), we get r = (2 − n) 2µ σiσi , (2.22) where r is the scalar curvature of vn. if vn is of the constant associated scalars, from (1.1) and (2.22), we find b + c = [ (2 − n) 2µ σiσi − na ] . if the length of σ is constant, then σiσj = c1, where c1 is a constant. thus, we can see that b + c is constant. the converse is also true. hence, the proof is complete. in [10], it was shown that riemannian manifolds with a φ(ric)-vector field of constant length have constant scalar curvature. the converse of this theorem is also true. we need the following theorem [12], for later use. conformal mappings 263 theorem 5. let vn be a riemannian manifold with constant scalar curvature. if vn admits a φ(ric)-vector field, then the length of φ is constant. now, we consider a mg(qe)n admitting the generator vector field u as a ϕ(ric)-vector field. then we have from (2.20) ∇jϕi = µsij and ∇jψi = µsij , (2.23) where µ is a constant. then, we give the following theorem: theorem 6. in a mg(qe)n, if the vector fields u and v corresponding to the 1-forms ϕ and ψ are ϕ(ric)-vector field and ψ(ric)-vector field , then u and v are covariantly constant. proof. we consider a mg(qe)n whose generator vector field is a ϕ(ric)vector field. putting (1.1) in (2.23), we obtain ∇jϕi = µsij = µ [ agij + bϕiϕj + cψiψj + d{ϕiψj + ϕjψi} ] . (2.24) multiplying (2.24) by ϕi and using the condition g(u,u) = 1, it can be seen that µsijϕ i = µ(a + b)ϕj + µdψj = 0 . (2.25) now multiplying (2.25) by ϕj, we get µ(a + b) = 0 . (2.26) as µ is non-zero, so from (2.26), we get a = −b. (2.27) similarly putting (1.1) in (2.23), we obtain µsij = ∇jψi = µ [ agij + bϕiϕj + cψiψj + d{ϕiψj + ϕjψi} ] . (2.28) again multiplying (2.28) by ψi, it can be seen that µsijψ i = µ(a + c)ϕj + dµϕj = 0 . (2.29) now multiplying (2.29) by ψj, we get µ(a + c) = 0 . (2.30) 264 s. dey, a. bhattacharyya similarly from (2.30), we obtain a = −c. (2.31) by the aid of (1.1), (2.27) and (2.31), we obtain sij = a[gij − (ϕiϕj + ψiψj)] + d{ϕiψj + ϕjψi} . (2.32) otherwise taking the covariant derivative of the expression sijϕ i and using (2.24), we obtain (∇ksij)ϕi + µsijsik = 0 . (2.33) multiplying (2.33) by gjk, we obtain (∇kski )ϕ i + µsijs ij = 0 , (2.34) where sij = gjksik. it was shown, [10], that riemannian manifolds with a ϕ(ric) vector field of constant length have constant scalar curvature. since the generator u is a unit vector field and it is also a ϕ(ric) vector field, the scalar curvature of the manifold is constant. in this case, using the contracted second bianchi identity and considering that the scalar curvature of the manifold is constant, it is obtained that ∇kski = 1 2 ∇ir = 0 . (2.35) using (2.34) and (2.35) and assuming that µ is a non-zero constant, we obtain sijs ij = 0 . (2.36) by the aid of (2.32) and (2.36) it follows that (n − 2)a2 + 2d2 = 0 . (2.37) since n > 2, from (2.37), it is seen that a and d must be zero, that is, a = c = 0 = d. but, in this case, from (2.32) we get that the ricci tensor vanishes which is a contradiction to the hypothesis. therefore, the constant µ must be zero and so, the generator vector field u is covariantly constant. similalry, if we take ψ(ric)-vector field, then we can show that the generator vector field v is also covariantly constant. this completes the proof. now we prove the following theorem: conformal mappings 265 theorem 7. in a mg(qe)n admits a φ(ric)-vector field and ν(ric)vector field with constant length, then either ϕi, ψi and φi are coplanar or the ricci tensor of the manifold reduces to the following form sij = bϕiϕj + cψiψj + d{ϕiψj + ϕjψi} if ψi and φ i are orthogonal to each other and ψi, ϕi and νi are coplanar or the ricci tensor of the manifold reduces to the following form sij = bϕiϕj + cψiψj + d{ϕiψj + ϕjψi} if ϕi and ν i are orthogonal to each other. proof. we assume that mg(qe)n admits a φ(ric)-vector field and ν(ric)-vector field with constant length. then, we have φiφ i = p(say) and νiν i = q(say) , (2.38) where c is a constant. taking the covariant derivative of the condition (2.38), using the equation (2.23) and considering µ as a non-zero constant (that is φ is proper φ(ric)-vector field), it follows that sikφ i = 0 . (2.39) by the aid of (1.1) and (2.39), we get aφk + b(φ iϕi)ϕk + c(ψiφ i)ψk + d { ϕiψkφ i + ϕkψiφ i } = 0 . (2.40) multiplying (2.40) by ϕk and using (1.2), it is obtained that (a + b)φkϕ k + dψiφ i = 0 . (2.41) if we take ψi and φ i are orthogonal to each other, then from (2.41), we obtain (a + b)φkϕ k = 0 . so either φkϕ k = 0 which gives from (2.40) that aφk = 0 . so, we get a = 0 and so, the ricci tensor of the manifold reduces to the form sij = bϕiϕj + cψiψj + d{ϕiψj + ϕjψi} 266 s. dey, a. bhattacharyya or φkϕ k ̸= 0 which gives from (2.28) that a = −b. (2.42) again taking the covariant derivative of the condition (2.38), using the equation (2.23) and considering µ as a non-zero constant (that is ν is proper ν(ric)-vector field), it follows that sikν i = 0 . (2.43) using the equation (1.1) and (2.43), we get aνk + b(ν iϕi)ϕk + c(ψiν i)ψk + d{ϕiψkνi + ϕkψiνi} = 0 . (2.44) multiplying (2.44) by ψk and using (1.2), it is obtained that (a + c)νkψ k + dϕiν i = 0 . (2.45) if we take ϕi and ν i are orthogonal to each other, then from (2.45), we obtain (a + c)νkψ k = 0 . so either νkψ k = 0 which gives from (2.44) that aνk = 0 . thus, we get a=0. and so, the ricci tensor of the manifold reduces to the form sij = bϕiϕj + cψiψj + d{ϕiψj + ϕjψi} or νkψ k ̸= 0 which gives from (2.45) that a = −c. (2.46) since b ̸= 0 and c ̸= 0 then a ̸= 0 and using the equation (2.40), (2.42) and (2.46), we obtain that φk = (φ iϕi)ϕk + (ψiφ i − dϕiφi)ψk . (2.47) so from (2.47) we say that φk, ϕk and ψk are coplanar. again from (2.44), we obtain νk = (ν iϕi)ϕk + (ψiν i − dϕiνi)ψk , (2.48) i.e., νk, ϕk and ψk are also coplanar. conformal mappings 267 corollary 1. if a mg(qe)n admits φ(ric)-vector field and ν(ric)vector field with constant length which is not orthogonal to the generators, then the associated scalars of the manifold must be constants and the vector fields φ and ν are covariantly constant. proof. as it has been alluded before that a riemannian manifold admitting a φ(ric)-vector field and ν(ric)-vector field with constant length has constant scalar curvature. besides, under the assumptions and from theorem 7., we obtain that the associated scalars of mg(qe)n are connected by a = −b and a = −c, and from (1.3), we obtain r = (n − 2)a. (2.49) since the scalar curvature of the manifold is constant, in this case, from (1.3) and (2.49), we see that the associated scalars of the manifold are constants. for the second part, multiplying the equation (2.47) by φk and using (2.38), it can be ocular that φiϕi is a constant as ψiφ i is also constant. so, (2.47) displays that the generator vector field u is also a ϕ(ric)-vector field. in this case, u must be covariantly constant by theorem 6. again, multiplying (2.48) by νk and using (2.38), it can be seen that ψiν i is a constant as νiϕi is also constant. now due to the coplanarity of φ, u and v , φ is covariantly constant. similarly, due to the coplanarity of ν, u and v , ν is also covariantly constant. hence the proof is completed. 3. conformal mapping of mg(qe)n admitting special vector fields definition 4. a symmetric tensor field t of type (0,2) on a riemannian manifold (m,g) is said to be a codazzi tensor if it satisfies the following condition (∇xt)(y,z) = (∇y t)(x,z) (3.1) for arbitrary vector fields x,y and z. now, we assume that the ricci tensors s′ and s of the mg(qe)n are codazzi tensors with respect to the levi-civita connections r′ and r, respectively. then, from (3.1), we have the following relations ∇̄ks̄ij = ∇̄js̄ik (3.2) and ∇ksij = ∇jsik . (3.3) 268 s. dey, a. bhattacharyya on the other hand, if the ricci tensor of the manifold is a codazzi tensor, then from the second bianchi identity, it can be seen that the scalar curvature is constant. according to our assumptions, the scalar curvatures r′ and r of the quasi-einstein manifolds are constants. so, we state and prove the following theorems. theorem 8. let us consider a conformal mapping ḡ = ge2σ of mg(qe)n whose ricci tensors are codazzi type. if the vector field generated by the 1-form σ is a σ(ric)-vector field, then either this conformal mapping is homothetic or the relation µ = (2 − n)(n − 1)c′ − (na + b + c) 2(n − 1)(na + b + c) is satisfied where c′ is the square of the length of σi = ∂σ ∂xi = ∂iσ and and µ denotes the constant corresponding to the σ(ric)-vector field. proof. suppose that the ricci tensors of vn and v̄n are codazzi tensors and suppose that ḡ = ge2σ is a conformal mapping with a σ(ric)-vector field. by using the second bianchi identity, it can be seen that the scalar curvatures r and r̄ are constants. since r is constant, then the length of σi is constant by theorem 5., (and r ̸= 0 which can be seen from theorem 7 and corollary 2.1) and so we have the condition σiσ i = c′, (3.4) where c′ is a constant. if we assume that the vector field generated by the 1-form σ in the conformal mapping (2.1) is a σ(ric)-vector field, we get ∇jσi = µsij , (3.5) where µ is a constant. using (2.7), (3.4) and (3.5), we have the following relations △2σ = µr, △1σ = c′ (3.6) and so, △1σ and △2σ are constants. using the relations (3.6) in (2.4), we find r̄ = e−2σb , (3.7) where r, r̄ and b = r + 2(n − 1)µr + (n − 1)(n − 2)c are constants. in this case, if r̄ is non-zero then we get from (3.7) that b is non-zero and so, e−2σ conformal mappings 269 is constant. thus, σ is constant. therefore, this mapping is homothetic. if r̄ is zero then b must be zero. so we obtain using (1.3) µ = (2 − n)(n − 1)c′ − (na + b + c) 2(n − 1)(na + b + c) . this completes the proof. next we consider a conformal mapping between two mg(qe)n admitting a concircular vector field σi. theorem 9. let us consider a conformal mapping ḡ = ge2σ of mg(qe)n whose ricci tensors are codazzi type. if σi is a concircular vector field, then either (i) σk is orthogonal to ϕ h k, or (ii) the function ρ is found as ρ = b − [ c n−1 + (n − 2)△1σ ] n + 2 , and (iii) σk is orthogonal to ψ h k, or (iv) the function ρ is found as ρ = c − [ b n−1 + (n − 2)△1σ ] n + 2 , where ϕi, ψi denote the components of the vector field associated 1-form ϕ and ψ, σi = ∂σ ∂xi = ∂iσ, b, c are the associated scalar of vn and ρ denotes the function corresponding to the concircular vector field. proof. let the ricci tensors of vn and v̄n be codazzi tensors and σi be a concircular vector field. in this case, we have from (2.10) ∇jσi = ρgij , (3.8) where ρ is a function. taking the covariant derivative of s̄ij and using (2.4), it can be obtained that (∇̄s̄ij) = ∇sij + (n − 2)∇kσij + ∂k(△2σ + (n − 2)△1σ)gij − 2σksij − σisjk − σisik − 2(△2σ + (n − 2)△1σ)gijσk + σh(sihgjk + shjgik + (n − 2)(σhσhjgik + σhσihgjk − 2σkσij − σiσkj − σjσik). (3.9) 270 s. dey, a. bhattacharyya changing the indices j and k in (3.9) and subtracting the last equation from (3.8) and using (2.6), (3.2), (3.3) and (3.8), it can be seen that 2(n − 1)(ρkgij − ρjgik) + [(n − 2)△1σ + (n + 2)ρ](σjgik − σkgij) + σjsik − σksij + σhshjgik − σhshkgij = 0 . (3.10) multiplying (3.10) by gij, it is obtained that 2(n − 1)2ρk + [(n − 2)(1 − n)△1σ + (n + 2)(1 − n)ρ − r]σk + (2 − n)σhshk = 0 . (3.11) on the other hand, we have from the ricci identity and the equation (3.8) σαr α ijk = ρkgij − ρjgik , (3.12) rαijk denote the components of the curvature tensor. multiplying (3.12) by gij, we obtain σαs α k = (n − 1)ρk . (3.13) substituting ρk obtained from (3.13) in (3.11), it can be obtained that nσhshk + [(n − 2)(1 − n)△1σ + (n + 2)(1 − n)ρ − r]σk = 0 . (3.14) considering (1.1) in (3.14) and using (1.3), we get nσh [ bϕhϕk + cψhψk + d{ϕiψj + ϕjψi} ] + [ (n − 2)(1 − n)△1σ + (n + 2)(1 − n)ρ − b − c ] σk = 0 . (3.15) multiplying (3.15) by ϕk and using (1.2), we obtain[ (n − 1)b − c + (n − 2)(1 − n)△1σ + (n + 2)(1 − n)ρ ] σkϕk + ndσ hψh = 0 . (3.16) multiplying (3.16) by ϕh, we obtain[ (n − 1)b − c + (n − 2)(1 − n)△1σ + (n + 2)(1 − n)ρ ] σkϕhk = 0 . (3.17) again multiplying (3.15) by ψk and using (1.2), we obtain[ (n−1)c−b+(n−2)(1−n)△1σ+(n+2)(1−n)ρ ] σkψk +ndσ hϕh = 0 . (3.18) conformal mappings 271 multiplying (3.18) by ψh, we obtain[ (n − 1)c − b + (n − 2)(1 − n)△1σ + (n + 2)(1 − n)ρ ] σkψhk = 0 . (3.19) from (3.17), we see that either σkϕhk = 0 or (n − 1)b − c + (n − 2)(1 − n)△1σ + (n + 2)(1 − n)ρ = 0 . thus, we obtain that either σk is orthogonal to ϕ h k or the function ρ is found as ρ = b − [ c n−1 + (n − 2)△1σ ] n + 2 . similarly, from (3.19) we obtain either σk is orthogonal to ψ h k or the function ρ is found as ρ = c − [ b n−1 + (n − 2)△1σ ] n + 2 . so the proof is completed. 4. examples let us consider a riemannian metric g on the 4-dimensional real number space m4 by ds2 = gijdx idxj = (1 + 2p) [ (dx1)2 + (dx2)2 + (dx3)2 ] + (dx4)2 (4.1) where i,j = 1,2,3,4, p = e x1 ρ2 and ρ is a non-zero constant and x1,x2,x3,x4 are the standard coordinates of m4. then the only non-vanishing components of the christoffel symbols, the curvature tensor, the ricci tensor and scalar curvature are given by γ122 = − p (1 + 2p) , γ133 = γ 1 44 = −γ 1 11 = −γ 2 12 = −γ 3 13 = −γ 4 14 , r1221 = r1331 = r1441 = p (1 + 2p) , s11 = 3 p (1 + 2p)2 , s22 = s33 = s44 = p (1 + 2p)2 , r = 6 p (1 + 2p)3 ̸= 0 , 272 s. dey, a. bhattacharyya and the components which can be obtained from these by the symmetry properties. therefore m4 is a riemannian manifold (m4,g) of non-vanishing scalar curvature. we shall now show that m4 is a mg(qe)4, i.e., it satisfies (1.1). let us now consider the associated scalars as follows: a = p (1 + 2p)3 , b = − 1 (1 + 2p)4 , c = −(1 + 2p)2 , d = 7 4(1 + 2p) . (4.2) in terms of local coordinate system, let us consider the 1-forms ϕ and ψ as follows: ϕi(x) = {√ p(1 + 2p) for i = 1 , 0 for otherwise , (4.3) and ψi(x) =   2 √ p (1 + 2p)2 for i = 1 , 0 for otherwise , (4.4) at any point x ∈ m4. in terms of local coordinate system, the defining condition (1.1) of a mg(qe)4 can be written as sii = agii + bϕiϕj + cψiψj + 2dϕiψj . (4.5) by virtue of (4.2), (4.3) and (4.4), it can be easily shown that (4.5) holds for i,j = 1,2,3,4. therefore (m4,g) is a mg(qe)4, which is not quasi-einstein. hence we can state the following: let (m4,g) be a riemannian manifold endowed with the metric given in (4.1). then (m4,g) is a mg(qe)4 with non-vanishing scalar curvature which is not quasi-einstein. references [1] k. arslan, r. ezentas, c. murathan, c. özgür, on pseudo riccisymmetric manifolds, balkan j. geom. appl. 6 (2) (2001), 1 – 5. [2] a. bhattacharya, t. de, d. debnath, mixed generalized quasieinstein manifold and some properties, analele stiintifice ale universitatii “ al.i.cuza ” din iasi (s.n) mathematica tomul liii, f.1., (2007). [3] h.w. brinkmann, einstein spaces which are mapped conformally on each other, math. ann. 94 (1) (1925), 119 – 145. [4] u. bruzzo, geometric invariants and quantum field theory, j. geom. phys. 61 (2011), 1157 – 1248. conformal mappings 273 [5] m.c. chaki, r.k. maity, on quasi-einstein manifolds, publ. math. debrecen 57 (2000), 297 – 306. [6] o. chepurna, v. kiosak, j. mikeš, conformal mappings of riemannian spaces which preserve the einstein tensor, aplimat j. appl. math. 3 (1) (2010), 253 – 258. [7] u.c. de, g.c. ghosh, on generalized quasi-einstein manifolds, kyungpook math. j. 44 (2004), 607 – 615. [8] l.p. eisenhart “ riemannian geometry ”, princeton univ. press, princeton, n.j., 1926. [9] a.r. gover, p. nurowski, obstructions to conformally einstein metrics in n dimentions, j. geom. phys. 56 (2006), 450 – 484. [10] i. hinterleitner, v.a. kiosak, ϕ(ric)-vector fields in riemannian spaces, arch. math. (brno) 44 (5) (2008), 385 – 390. [11] y. ishii, on conharmonic transformations, tensor (n.s.) 7 (1957), 73 – 80. [12] b. kirik, f.o. zengin, conformal mappings of quasi-einstein manifolds admitting special vector fields, filomat 29 (3) (2015), 525 – 534. [13] w. kühnel, h.b. rademacher, conformal transformations of pseudoriemannian manifolds, in “ recent developments in pseudo-riemannian geometry ”, esi lect. math. phys., eur. math. soc., zürich, 2008, 261 – 298. [14] j. mikeš, m.l. gavrilchenko, e.i. gladysheva, conformal mappings onto einstein spaces, moscow univ. math. bull. 49 (3) (1994), 10 .– 14. [15] n.s. sinyukov, “ geodesic mappings of riemannian spaces ”, nauka, moscow, 1979. [16] k. yano, concircular geometry, i-iv, proc. imp. acad. tokyo 16 (1940), 195 – 200, 354 – 360, 442 – 448, 505 – 511. e extracta mathematicae vol. 32, núm. 2, 239 – 254 (2017) content semimodules rafieh razavi nazari, shaban ghalandarzadeh faculty of mathematics, k. n. toosi university of technology, tehran, iran rrazavi@mail.kntu.ac.ir ghalandarzadeh@kntu.ac.ir presented by juan antonio navarro received november 27, 2016 abstract: the purpose of this paper is to study content semimodules. we obtain some results on content semimodules similar to the corresponding ones on content modules. we study normally flat content semimodules and multiplication content semimodules. moreover, we characterize content semimodules over discrete valuation semirings and boolean algebras. key words: semiring, content semimodule, multiplication semimodule, normally flat semimodule. ams subject class. (2010): 16y60. introduction semirings and semimodules have many applications in different branches of mathematics (see [7], [8] and [9]). semiring is a generalization of ring and bounded distributive lattice. we recall here some definitions: a semiring is a nonempty set s with two binary operations addition (+) and multiplication (·) such that the following conditions hold: 1) (s, +) is a commutative monoid with identity element 0; 2) (s, .) is a monoid with identity element 1 ̸= 0; 3) 0a = 0 = a0 for all a ∈ s; 4) a(b + c) = ab + ac and (b + c)a = ba + ca for every a, b, c ∈ s. the semiring s is commutative if the monoid (s, .) is commutative. all semirings in this paper are commutative. an ideal i of a semiring s is a nonempty subset of s such that a + b ∈ i and sa ∈ i for all a, b ∈ i and s ∈ s. an ideal i is subtractive if a + b ∈ i and b ∈ i imply that a ∈ i for all a, b ∈ s. a semiring is entire if ab = 0 implies that a = 0 or b = 0. further, an element a of a semiring s is multiplicatively cancellable (abbreviated as mc) if ab = ac implies that b = c. if every nonzero element of s is multiplicatively 239 240 r. razavi nazari, sh. ghalandarzadeh cancellable we say that the semiring s is a semidomain. an element a of a semiring s is multiplicatively idempotent if a2 = a. let i×(s) denote the set of all multiplicatively idempotent elements of s. we say that s is multiplicatively idempotent, if i×(s) = s. let s be a semiring. an s-semimodule is an additive abelian monoid (m, +) with additive identity 0m and a function s × m → m ((s, m) 7→ sm), called scalar multiplication, such that the following conditions hold for all s, s′ ∈ s and all m, m′ ∈ m: 1) (ss′)m = s(s′m); 2) s(m + m′) = sm + sm′; 3) (s + s′)m = sm + s′m; 4) 1m = m; 5) s0m = 0m = 0m; a subset n of an s-semimodule m is a subsemimodule of m if n is closed under addition and scalar multiplication. we say that a subsemimodule n of an s-semimodule m is subtractive if m+m′ ∈ n and m ∈ n imply that m′ ∈ n for all m, m′ ∈ m. let m and m ′ be s-semimodules. then a function α from m to m ′ is an s-homomorphism if α(m + m′) = α(m) + α(m′) for all m, m′ ∈ m and α(sm) = s(α(m)) for all m ∈ m and s ∈ s. the kernel of α is ker(α) = α−1{0}. then ker(α) is a subtractive s-semimodule of m. the set α(m) = {α(m) | m ∈ m} is a subsemimodule of m ′. an s-homomorphism α : m → m ′ is an smonomorphism if αβ = αβ′ implies β = β′ for all s-semimodule k and all s-homomorphisms β, β′ : k → m. if α is an s-monomorphism, then ker(α) = 0. but the converse need not be true. for example, let s be an entire semiring and b ∈ s such that it is not multiplicatively cancellative. thus there exists a ̸= a′ ∈ s such that ab = a′b. define a map ϕ : s → sb by s 7→ sb. then ϕ is an s-homomorphism with ker(ϕ) = 0. but ϕ is not injective, since ϕ(a) = ϕ(a′). an s-homomorphism α : m → m ′ is surjective if α(m) = m ′. let s be a semiring and m an s-semimodule. for any x ∈ m, we define c(x) = ∩{i | i is an ideal of s and x ∈ im}. then c is a function from m to the set of ideals of s and it is called the content function. an ssemimodule m is called a content semimodule if for every x ∈ m, x ∈ c(x)m. in this paper, we study content semimodules and extend some results of [14] to semimodules over semirings. in section 1, we recall some properties of content semimodules 241 content semimodules from [12] and we show that projective semimodules are content semimodules. we study normally flat content semimodules in section 2. in section 3, we characterize content s-semimodules over discrete valuation semirings. in section 4, we investigate some properties of faithful multiplication content semimodules, as a generalization of faithful multiplication modules. in the last section, we prove that if every subsemimodule of a content s-semimodule is a content s-semimodule with restricted content function, then s is a multiplicatively regular semiring. we also characterize content semimodules over boolean algebras. 1. content semimodules the concepts of content modules and content algebras were introduced in [14]. the concept of content semimodules is studied in [12]. let s be a semiring and m an s-semimodule. for any x ∈ m, we define the content of x by, cs,m(x) = ∩ { i | i is an ideal of s and x ∈ im } . therefore cs,m is a function from m to the set of ideals of s which is called the content function. if n is any non-empty subset of m, we define cs,m(n) to be the ideal ∑ x∈n cs,m(x). whenever there is no fear of ambiguity, either or both of the subscripts s and m will be omitted. definition 1. let s be a semiring. an s-semimodule m will be called a content s-semimodule if for every x ∈ m, x ∈ c(x)m. example 2. let s be a multiplicatively idempotent semiring, j an ideal of s and x ∈ j. it is clear that (x) ⊆ c(x) = ∩{i | i ⊆ s, x ∈ ij}. thus (x)j ⊆ c(x)j. but x = x2 ∈ (x)j and hence x ∈ c(x)j. therefore j is a content s-semimodule. now, we recall next results from [12], which will be used repeatedly. theorem 3. let m be an s-semimodule. then the following statements are equivalent: 1) m is a content s-semimodule. 2) for every set of ideals {ii} of s, (∩ii)m = ∩(iim). 3) for every set of finitely generated ideals {ii} of s, (∩ii)m = ∩(iim). 242 r. razavi nazari, sh. ghalandarzadeh 4) there exists a function f from m to the set of ideals of s such that for every x ∈ m and every ideal i of s, x ∈ im if and only if f(x) ⊆ i. moreover, if m is a content s-semimodule and x ∈ m, then c(x) is a finitely generated ideal. theorem 4. let m be a content s-semimodule, and n a subsemimodule of m. then the following statements are equivalent: 1) im ∩ n = in for every ideal i of s. 2) for every x ∈ n, x ∈ cm(x)n. 3) n is a content s-semimodule and cm restricted to n is cn. we know that every free module and every projective module as a direct summand of a free module, are content modules by [14, corollary 1.4]. moreover every free semimodule is a content semimodule by [12, corollary 26]. but not all projective semimodules are direct summands of free semimodules (cf. [3, example 2.3]). we can prove that every projective semimodule is a content semimodule as follows: theorem 5. any projective semimodule is a content semimodule. proof. let s be a semiring and m a projective s-semimodule. then by [18, theorem 3.4.12], there exist {mi}i∈i ⊆ m and {fi}i∈i ⊆ homs(m, s) such that for any x ∈ m, fi(x) = 0 for almost all i ∈ i, and x = ∑ i fi(x)mi. suppose that x ∈ m. then x = ∑n i=1 fi(x)mi and hence x ∈ (f1(x), . . . , fn(x))m. thus c(x) ⊆ (f1(x), . . . , fn(x)). now assume that x ∈ im for some ideal i of s. then there exist m ∈ n, r1, . . . , rm ∈ i and x1, . . . , xm ∈ m such that x = ∑m i=1 rixi. for each 1 ≤ j ≤ n, fj(x) = ∑m i=1 rifj(xi), and hence fj(x) ∈ (r1, . . . , rm). therefore (f1(x), . . . , fn(x)) ⊆ (r1, . . . , rm) ⊆ i. this implies (f1(x), . . . , fn(x)) ⊆ c(x), by definition of content function. therefore (f1(x), . . . , fn(x)) = c(x), and x ∈ (f1(x), . . . , fn(x))m = c(x)m. let m be an s-semimodule, n a subsemimodule of m and i an ideal of s. put (n :m i) = {x | x ∈ m and ix ⊆ n}. then (n :m i) is a subsemimodule of m. theorem 6. let m be a content s-semimodule, and let s ∈ s. then the following statements are equivalent: content semimodules 243 1) s(c(x)) = c(sx) for all x ∈ m. 2) (i :s s)m = (im :m s) for every ideal i of s. proof. (1) ⇒ (2): the proof is similar to [14, theorem 1.5]. (2) ⇒ (1): let x ∈ m. then x ∈ c(x)m since m is a content s-semimodule. thus, sx ∈ sc(x)m. this implies that c(sx) ⊆ sc(x). now by (2), (c(sx) :s s)m = (c(sx)m :m s). on the other hand, since m is a content semimodule, sx ∈ c(sx)m. this implies x ∈ (c(sx)m :m s) = (c(sx) :s s)m. so c(x) ⊆ (c(sx) : s) and hence sc(x) ⊆ c(sx). definition 7. let s be a semidomain. an s-semimodule m is said to be torsionfree if for any 0 ̸= a ∈ s, multiplication by a on m is injective, i.e., if ax = ay for some x, y ∈ m, then x = y. now we give the following theorem for content torsionfree semimodules over a semidomain. theorem 8. let s be a semidomain and m a content torsionfree ssemimodule. then for every s ∈ s and x ∈ m, s(c(x)) = c(sx). proof. since m is a content s-semimodule, x ∈ c(x)m. therefore sx ∈ sc(x)m, which implies c(sx) ⊆ sc(x). now sx ∈ sm, implies that c(sx) ⊆ (s). therefore c(sx) = (s)j, where j = (c(sx) : s). since m is a content semimodule, sx ∈ c(sx)m = sjm. therefore sx = sz, for some element z ∈ jm. then x = z, since m is torsionfree. thus x ∈ jm and hence c(x) ⊆ j. therefore sc(x) ⊆ sj = c(sx). theorem 9. let s be a semiring such that any 0 ̸= s ∈ s is in at most finitely many ideals, and let m be an s-semimodule such that for all ideals i, j of s, (i ∩ j)m = im ∩ jm. then m is a content semimodule if and only if ∩(iim) = 0, whenever {ii} is an infinite set of ideals of s. proof. let m be a content s-semimodule and let {ii} be an infinite set of ideals of s. then by theorem 3, ∩(iim) = (∩ii)m = 0. conversely, let {ii}i∈i be a set of ideals of s. if i is finite, then ∩(iim) = (∩ii)m by assumption. now if i is infinite, then ∩(iim) = (∩ii)m = 0. therefore m is a content s-semimodule. 244 r. razavi nazari, sh. ghalandarzadeh 2. normally flat semimodules and content semimodules in this section, we investigate normally flat content semimodules. the concept of normally flat semimodules was introduced in [2]. let us recall some definitions. let m and n be two s-semimodules. an s-balanced map g : m ×n → g, where g is an abelian monoid, is a bilinear map such that g(ms, n) = g(m, sn) for all m ∈ m, s ∈ s and n ∈ n. a commutative monoid m ⊗s n together with an s-balanced map τ : m × n → m ⊗s n is called a tensor product of m and n over s if for every abelian monoid g with an s-balanced map β : m × n → g, there exists a unique morphism of monoids γ : m ⊗s n → g that γ ◦ τ = β. for more details on tensor product of semimodules see [10], [1] and [15]. definition 10. assume that m is an s-semimodule. we say that a subsemimodule n ≤s m is a normal subsemimodule, and write n ≤ns m, if the embedding n ↪→ m is a normal monomorphism, that is, n = ker(f) for some s-homomorphism f : m → l and some s-semimodule l. note that n ≤ns m if and only if n = n, the normal closure of n, defined by n := {m ∈ m | m + n1 = n2 for some n1, n2 ∈ n}. therefore n ≤ns m if and only if n is a subtractive subsemimodule of m. definition 11. let f and m be s-semimodules. we say that f is normally flat with respect to m (or normally m-flat) if n ⊗s f ≤nn m ⊗s f for every n ≤ns m. we say that f is normally flat, if f is normally m-flat for every s-semimodule m. assume that r is a domain. it is well known that if m is a flat r-module, then m is torsionfree [4, chapter i, §2.5, proposition 3]. we have the following result for normally flat semimodules. theorem 12. let s be a semidomain such that every principal ideal of s is subtractive and let m be a normally flat s-semimodule. then m is a torsionfree s-semimodule. proof. let 0 ̸= a ∈ s. we should show that for all x, y ∈ m, ax = ay implies x = y. define a map f : s → s by f : s 7→ as. if as = as′ for some s, s′ ∈ s, then s = s′ since a is an mc element. therefore f is an injective s-homomorphism. moreover, f(s) = sa is a subtractive subsemimodule of s. since m is normally flat, f̄ : s ⊗s m → s ⊗s m where f̄ : s ⊗ m 7→ as ⊗ m, is content semimodules 245 injective. but s ⊗s m θ∼= m by [15, theorem 7.6]. thus θ ◦ f̄ ◦ θ−1 : m 7→ am is injective. in [14, corollary 1.6], it is proved that a content module is a flat module if and only if for every s ∈ s and x ∈ m, s(c(x)) = c(sx). now from theorem 12, we have the following result. corollary 13. let s be a semidomain such that every principal ideal of s is subtractive and let m be a content normally flat s-semimodule. then for every s ∈ s and x ∈ m, s(c(x)) = c(sx). proof. by theorem 12, m is a torsionfree s-semimodule. thus by theorem 8, for every s ∈ s and x ∈ m, s(c(x)) = c(sx). theorem 14. let s be a semiring and m a content s-semimodule such that for every s ∈ s and every ideal i of s, (i :s s)m = (im :m s). then (i :s j)m = (im :m j) for every pair of ideals i, j of s. proof. since m is a content s-semimodule, by theorem 3, (i : j)m = (∩s∈j(i : s))m = ∩s∈j(i : s)m. but ∩s∈j(i : s)m = ∩s∈j(im : s) = (im : j). corollary 15. assume that s is a semidomain such that every principal ideal of s is subtractive and let m be a content normally flat s-semimodule. then (i :s j)m = (im :m j) for every pair of ideals i, j of s. proof. by theorem 13, for every s ∈ s and x ∈ m, s(c(x)) = c(sx) and by theorem 6, (i :s s)m = (im :m s) for every ideal i of s and s ∈ s. thus by theorem 14, (i :s j)m = (im :m j). 3. content semimodules over discrete valuation semirings discrete valuation semiring was introduced and studied in [13]. similar to [14, proposition 2.1], we will obtain a characterization of content ssemimodules over a discrete valuation semiring. first, we recall some definitions and results from [13]. let (m, +, 0, <) be a totally ordered commutative monoid (abbreviated as tomonoid) with no greatest element and let +∞ be an element such that +∞ /∈ m. put m∞ = m ∪ {+∞}. now set m < +∞ for all m ∈ m and 246 r. razavi nazari, sh. ghalandarzadeh m + (+∞) = (+∞) + m = +∞ for all m ∈ m∞. then m∞ is a tomonoid with the greatest element +∞. definition 16. a map v : s → m∞ is an m-valuation on s if the following properties hold: 1) s is a semiring and m∞ is a tomonoid with the greatest element +∞, which has been obtained from the tomonoid m with no greatest element, 2) v(xy) = v(x) + v(y) for all x, y ∈ s, 3) v(x + y) ≥ min{v(x), v(y)}, whenever x, y ∈ s, 4) v(1) = 0 and v(0) = +∞. if in the above m = z, we will say that v is a discrete valuation on s. definition 17. let s be a semiring. if there exists an m-valuation v on s, then it is obvious that sv = {s ∈ s | v(s) ≥ 0} is a subsemiring of s. in this case we say that “sv is a v -semiring with respect to the triple (s, v, m)”. an element s of a semiring s is a unit if there exists an element s′ of s such that ss′ = 1. we say that s is a semifield if every nonzero element of s is a unit. definition 18. a semiring s is called discrete valuation semiring, if s = kv is a v -semiring with respect to the triple (k, v, z), where k is a semifield and v is surjective. a semiring s is called a local semiring if it has a unique maximal ideal. note that by [13, theorem 3.6] every discrete valuation semiring is a local semiring. theorem 19. let (s, m) be a discrete valuation semiring and let m be an s-semimodule. then m is a content s-semimodule if and only if ∩{mim | i = 1, 2, · · · } = 0. proof. let m be a content s-semimodule. by [13, theorem 3.6], ∩∞i=1m i = 0. thus by theorem 3, ∩∞i=1(m im) = (∩∞i=1m i)m = 0. now let 0 ̸= x ∈ m. since every ideal of s is of the form mi(i ∈ n) [13, lemma 3.3], c(x) = ∩{mi | x ∈ mim}. but ∩{mim | i = 1, 2, · · · } = 0. so there exists a positive integer n such that x ∈ mnm and x /∈ mim for all i > n. therefore c(x) = mn. hence x ∈ c(x)m. content semimodules 247 we know that free semimodules, and more generally, projective semimodules are examples of normally flat semimodules (see [1]), and in section 1 we proved that these semimodules are content semimodules. now, we give an example of a content semimodule which is not normally flat. first, we recall the following definition: let m be an s-semimodule and n a subsemimodule of m. then we can define a congruence relation on m as follows: m ≡n n iff m + a = n + b for some a, b ∈ n. the set of equivalence classes is an s-semimodule and denoted by m/n. the equivalence class of m ∈ m is denoted by m/n. example 20. let s = (n ∪ {+∞}, min, +, +∞, 0). then s is a semidomain. let j = s\{1s} = {−∞} ∪ {1, 2, · · · }. we show that j is a principal ideal of s. if x, y ∈ j and s ∈ s, then x ⊕ y = min{x, y} ∈ j and 0 ̸= s + x = s ⊙ x ∈ j. now let 0s ̸= a ∈ j. then a = 1+ a · · · +1 = 1⊙ a · · · ⊙1 = 1a ∈ (1). therefore j = (1) and j is the unique maximal ideal of s. if i is an ideal of s, then i is a power of j. let i be an ideal of s, 0s ̸= n ∈ i the smallest element in i and 0s ̸= x ̸= n ∈ i. then x ≥ n and hence x − n ∈ s. thus x = x − n + n = (x − n) ⊙ n ∈ (n). therefore i = (n). moreover, n = 1 + · · · + 1 = 1 ⊙ · · · ⊙ 1 = 1n and hence i = (n) = (1n) = (1)n = jn. thus s is a discrete valuation semiring by [13, theorem 3.6]. now let i = (n) be an ideal of s and x, y ∈ s such that x + y, y ∈ i. if y ≥ x, then x + y = min{y, x} = x ∈ i. if x ≥ y, then x − y ∈ s and so x = x−y+y = x−y⊙y ∈ i. therefore i is subtractive. this implies s/i ̸= 0. now consider the s-semimodule m = s/j2. since j2m = 0, ∩{jim | i = 1, 2, · · · } = 0. thus by theorem 19, m is a content s-semimodule. note that s is a semidomain such that every ideal of s is subtractive and m is not torsionfree. thus from theorem 12, m is not a normally flat s-semimodule. 4. multiplication semimodules and content semimodules in this section we study the relation between multiplication semimodules and content semimodules and give some results about multiplication semimodules. it is known that every faithful multiplication module is a content module. here we investigate faithful multiplication content semimodules and extend some results of [6] to semimodules. if n and l are subsemimodules of an s-semimodule m, we set (n : l) = {s ∈ s | sl ⊆ n}. then (n : l) is an ideal of s. 248 r. razavi nazari, sh. ghalandarzadeh definition 21. let s be a semiring and m an s-semimodule. then m is called a multiplication semimodule if for each subsemimodule n of m there exists an ideal i of s such that n = im. in this situation we can prove that n = (n : m)m. cyclic semimodules are examples of multiplication semimodules [19, example 2]. theorem 22. suppose that m is a content s-semimodule and for any subsemimodule n of m and ideal i of s such that n ⊂ im there exists an ideal j of s such that j ⊂ i and n ⊆ jm. then m is a multiplication s-semimodule. proof. the proof is similar to [6, theorem 1.6]. we recall the following results from [16]. let m be an s-semimodule and p a maximal ideal of s. we say that m is p-cyclic if there exist m ∈ m, t ∈ s and q ∈ p such that t + q = 1 and tm ⊆ sm. theorem 23. let m be an s-semimodule. if m is a multiplication semimodule, then for every maximal ideal p of s either m = {m ∈ m | m = qm for some q ∈ p} or m is p-cyclic [16, theorem 1.6]. definition 24. an element m of an s-semimodule m is cancellable if m+m′ = m+m′′ implies that m′ = m′′. the s-semimodule m is cancellative if every element of m is cancellable. a semiring s is yoked if for all a, b ∈ s, there exists an element t of s such that a + t = b or b + t = a. now, we give the following theorem for yoked semirings. theorem 25. let s be a yoked semiring such that every maximal ideal of s is subtractive and let m be a cancellative faithful multiplication ssemimodule. then m is a content s-semimodule. proof. let {iλ}(λ ∈ λ) be any non-empty collection of ideals of s. put i = ∩λ∈λiλ. clearly im ⊆ ∩λ∈λ(iλm). let x ∈ ∩λ∈λ(iλm) and let k = {r ∈ s | rx ∈ im}. if k ̸= s, then there exists a maximal ideal q of s such that k ⊆ q. suppose that m = {m ∈ m | m = pm for some p ∈ q}. then x = px for some p ∈ q. since s is a yoked semiring, there exists t ∈ s such that t + p = 1 or 1 + t = p. suppose that t + p = 1. then px + tx = x. content semimodules 249 since m is a cancellative s-semimodule, tx = 0 and hence t ∈ k ⊆ q which is a contradiction. now suppose that 1 + t = p. then x + tx = px. since m is a cancellative semimodule, tx = 0 and hence t ∈ k ⊆ q. on the other hand, since q is a subtractive ideal, 1 ∈ q which is a contradiction. therefore by theorem 23, m is q-cyclic. hence there exist m ∈ m, t ∈ s and q ∈ q such that t + q = 1 and tm ⊆ sm. then tx ∈ ∩λ∈λ(iλm). thus for each λ ∈ λ, there exists aλ ∈ iλ such that tx = aλm. choose α ∈ λ. then aαm = aλm for each λ ∈ λ. since s is a yoked semiring, there exists rλ ∈ s such that aα + rλ = aλ or aλ + rλ = aα. suppose that aα + rλ = aλ. then aαm + rλm = aλm and hence rλm = 0. thus trλm ⊆ rλ(sm) = 0. since m is a faithful semimodule, trλ = 0. but taα + trλ = taλ and hence taα = taλ. now suppose that aλ + rλ = aα. a similar argument shows that taα = taλ. thus in any case taα ∈ iλ for each λ ∈ λ and hence taα ∈ i. therefore t2x = taαm ∈ im. this implies that t2 ∈ k ⊆ q which is a contradiction. therefore k = s and hence x ∈ im. we call an s-semimodule m multiplicatively cancellative (abbreviated as mc) if for any s, s′ ∈ s and 0 ̸= m ∈ m, sm = s′m implies s = s′ [5]. theorem 26. let m be an mc multiplication s-semimodule. then m is a content s-semimodule. proof. by [16, theorem 2.9], m is a projective s-semimodule and by corollary 5, m is a content s-semimodule. now, by using [16, corollary 2.10], we get the following result. corollary 27. let s be a yoked entire semiring and m a cancellative faithful multiplication s-semimodule. then m is a content s-semimodule. we say that a subsemimodule e of an s-semimodule m is an essential subsemimodule, if for any nonzero subsemimodule n ⊆ m, e ∩ n ̸= 0 [11]. let s be a semiring and m a faithful multiplication content s-semimodule. then similar to [6, theorem 2.13] we can prove that a subsemimodule n of m is essential if and only if there exists an essential ideal e of s such that n = em. assume that m is an s-semimodule. we define the socle of m, denoted by soc(m), to be soc(m) = ∩ {n | n ⊆e m} (see also [11]). now if m is a faithful multiplication content s-semimodule, then similar to [6, corollary 2.14], we conclude that soc(m) = soc(s)m. 250 r. razavi nazari, sh. ghalandarzadeh an s-semimodule m is called finitely cogenerated if for every set a of subsemimodules of m, ∩ a = 0 if and only if ∩ f = 0 for some finite set f ⊆ a [11]. the semiring s is called finitely cogenerated if it is finitely cogenerated as an s-semimodule. let s be a semiring and m a faithful multiplication content s-semimodule. then with a similar proof for [6, corollary 1.8], we can show that, m is finitely cogenerated if and only if s is finitely cogenerated. assume that m is an s-semimodule. now we give some properties of the ideal c(m). theorem 28. (see [14, corollary 1.6]) let m be a content s-semimodule. then c(m) = s iff mm ̸= m for every maximal ideal m of s. proof. (⇒) let c(m) = s and m a maximal ideal of s such that mm = m. if x ∈ m = mm, then c(x) ⊆ m. hence c(m) ⊆ m which is a contradiction. (⇐) let m be a maximal ideal of s and mm ̸= m. then there exists x ∈ m\mm. thus c(x) * m since x ∈ c(x)m. therefore c(m) * m. since for all maximal ideal m of s, c(m) * m, we have c(m) = s. let s be a semiring and m an s-semimodule. put a = {i ⊆ s|m = im} and τ(m) = ∩i∈ai. then τ(m) is an ideal of s. theorem 29. let m be a content s-semimodule. then c(m) = τ(m). proof. if x ∈ m, then x ∈ c(x)m ⊆ c(m)m. therefore c(m)m = m and hence τ(m) ⊆ c(m). now let i be an ideal of s such that m = im. then for each x ∈ m = im, c(x) ⊆ i and hence c(m) ⊆ i. therefore c(m) ⊆ τ(m). theorem 30. let m be a faithful multiplication content s-semimodule and i = τ(m). then: 1) m ∈ im for each m ∈ m; 2) i2 = i; 3) ann(i) = 0. proof. the proof is similar to [6, lemma 3.2]. content semimodules 251 5. regular semirings and content semimodules an element a of a semiring s is multiplicatively regular if there exists an element x of s such that axa = a. a semiring s is multiplicatively regular if every element of s is multiplicatively regular. bounded distributive lattices, and in particular, boolean algebras are multiplicatively regular semirings. example 31. let s be a semifield and a a nonempty set. suppose that f ∈ sa. define a map g : a → s by g(a) = f(a)−1 if f(a) ̸= 0, and g(a) = 0 if f(a) = 0. then f = fgf. therefore sa is a multiplicatively regular semiring. theorem 32. let s be a multiplicatively regular semiring. then every ideal of s is generated by idempotents. proof. let i be an ideal of s and x ∈ i. then x = x2s for some s ∈ s. thus xs ∈ i×(s) and (x) = (xs). therefore i = ∑ x∈i sx is generated by idempotents. in [14], it is shown that a ring r is regular if and only if every submodule of a content r-module is a content module with restricted content function. we can extend this result to multiplicatively regular semirings as follows: theorem 33. assume that s is a semiring. if every subsemimodule of a content s-semimodule is a content s-semimodule with restricted content function, then s is a multiplicatively regular semiring. proof. by [17, proposition 1], it is sufficient to show that every principal ideal of s is generated by an idempotent. suppose that a ∈ s. then by theorem 4, (a)s ∩ (a) = (a2). thus there exists r ∈ s such that a = ra2. hence ar ∈ i×(s) and (a) = (ar). let s be a semiring. an element a of s is complemented if there exists an element c of s such that ac = 0 and a + c = 1. let comp(s) denote the set of all complemented elements of s. note that comp(s) ⊆ i×(s). since if a ∈ comp(s), then a = a1 = a(a + c) = a2 + ac = a2. theorem 34. let s be a semiring such that comp(s) = i×(s). let i = (e, f) be an ideal of s such that e, f ∈ i×(s). then i = (g) for some g ∈ i×(s). 252 r. razavi nazari, sh. ghalandarzadeh proof. since e, f ∈ comp(s), there exist elements x, y ∈ i×(s) such that x + e = 1, y + f = 1, xe = 0 and yf = 0. then 1 = xy + ye + xf + fe. put g = ye+xf +fe ∈ i. then 1 = xy+g and g2 = g. moreover e = exy+eg = eg and f = fxy + fg = fg. therefore i = (e, f) ⊆ (g) ⊆ i. hence i = (g). example 35. let s be a semiring such that i×(s) = {0, 1}. let a be a nonempty set and f ∈ i×(sa). then for each a ∈ a, f(a) ∈ i×(s) = {0, 1}. define a map g : a → s by g(a) = 1 if f(a) = 0, and g(a) = 0 if f(a) = 1. then f + g = 1sa, and fg = 0. thus f ∈ comp(sa). therefore sa is a semiring such that comp(sa) = i×(sa). theorem 36. let s be a semiring. if every finitely generated ideal in s is generated by an idempotent, then every subsemimodule of a content s-semimodule is a content s-semimodule with restricted content function. proof. let m be a content s-semimodule and n ⊆ m. by theorem 4, we should show that for every x ∈ n, x ∈ cm(x)n. let x ∈ n. then x ∈ cm(x)m. since m is a content s-semimodule, cm(x) is a finitely generated ideal. hence there exists an element e ∈ i×(s) such that cm(x) = (e). thus x ∈ (e)m and hence there exists m ∈ m such that x = em = e2m = ex. therefore x ∈ cm(x)n. here we study content semimodules over boolean algebras (see [14, section 4]). note that, by theorem 34, every finitely generated ideal of a boolean algebra is generated by an idempotent. lemma 37. let s be a semiring and m an s-semimodule. if every finitely generated ideal of s is generated by an idempotent then for all ideals i, j ⊆ s, (i ∩ j)m = im ∩ jm. proof. it is clear that (i ∩j)m ⊆ im ∩jm. suppose that x ∈ im ∩jm. then x = ∑m i=1 rimi = ∑n j=1 sjm ′ j , where mi, m ′ j ∈ m, ri ∈ i and sj ∈ j for all 1 ≤ i ≤ m and 1 ≤ j ≤ n. put i′ = (r1, . . . , rm), j ′ = (s1, . . . , sn). then there exist e, u ∈ i×(s) such that i′ = (e) ⊆ i and j′ = (u) ⊆ j. then for each i, 1 ≤ i ≤ m, there exists ui ∈ s such that ri = eui. moreover, for each j, 1 ≤ j ≤ n, there exists tj ∈ s such that sj = utj. hence x = ∑m i=1 rimi = ∑m i=1 euimi = e ∑m i=1 euimi = ex. similarly x = ∑n j=1 sjm ′ j = ∑m j=1 utjm ′ j = u ∑m j=1 utjm ′ j = ux. thus x = ex = eux. therefore x ∈ (i ∩ j)m. content semimodules 253 lemma 38. let s be a boolean algebra and m an s-semimodule. then for all s ∈ s, (0 :s s)m = (0 :m s). proof. clearly, (0 :s s)m ⊆ (0 :m s). let x ∈ m such that sx = 0. since s is a boolean algebra, there exists an element t ∈ s such that t + s = 1 and ts = 0. thus x = tx ∈ (0 :s s)m. therefore (0 :s s)m = (0 :m s). lemma 39. let s be a boolean algebra, m an s-semimodule and x ∈ m. then c(x) = ann(ann(x)). proof. let i be a finitely generated ideal of s such that x ∈ im. then anni ⊆ ann(x) and hence ann(ann(x)) ⊆ ann(ann(i)). but ann(ann(i)) = i and so ann(ann(x)) ⊆ i. by theorem 3, ann(ann(x)) ⊆ c(x). conversely, let s ∈ s such that sx = 0. then x ∈ (0 :m s) = (0 :s s)m by theorem 38. thus c(x) ⊆ (0 :s s) and hence c(x) ⊆ ∩ s∈ann(x)(0 :s s) = ann(ann(x)). in the following theorem we characterize content s-semimodules over boolean algebras. theorem 40. let s be a boolean algebra and m an s-semimodule. then m is a content s-semimodule if and only if for all x ∈ m, ann(x) is a finitely generated ideal. proof. suppose that m is a content s-semimodule and x ∈ m. then c(x) = ann(ann(x)) is a finitely generated ideal. thus there exists e ∈ s such that ann(ann(x)) = (e). moreover there exists u ∈ s such that ue = 0 and u + e = 1. we show that ann(x) = (u). since m is a content s-semimodule, x ∈ c(x)m = (e)m. thus there exists z ∈ m such that x = ez. hence ux = uez = 0. therefore u ∈ ann(x) and hence (u) ⊆ ann(x). for the reverse inclusion, let r ∈ ann(x). then r = ur + er = ur. thus r ∈ (u) and hence ann(x) ⊆ (u). now suppose that x ∈ m and ann(x) is a finitely generated ideal. note that by lemma 37, (i∩j)m = im ∩jm for all ideals i, j ⊆ s. let ann(x) = (s1, . . . , sn). then x ∈ ∩n i=1(0 :m si) = ∩n i=1(0 :s si)m = ( ∩n i=1(0 :s si))m = (ann(ann(x)))m. thus by theorem 39, x ∈ c(x)m and hence m is a content s-semimodule. 254 r. razavi nazari, sh. ghalandarzadeh references [1] j.y. abuhlail, uniformly flat semimodules, arxiv: 1201.0591, 2012. [2] j.y. abuhlail, some remarks on tensor products and flatness of semimodules, semigroup forum 88 (3) (2014), 732 – 738. [3] h.m.j. al-thani, characterizations of projective and k-projective semimodules,int. j. math. math. sci. 32 (7) (2002), 439 – 448. [4] n. bourbaki, “commutative algebra”, hermann/addison-wesley, 1972. [5] r. ebrahimi atani, s. ebrahimi atani, on subsemimodules of semimodules, bul. acad. ştiinţe repub. mold. mat. 2 (63) (2010), 20 – 30. [6] z.a. el-bast, p.f. smith, multiplication modules, comm. algebra 16 (4) (1988), 755 – 779. [7] k. glazek, “a guide to the literature on semirings and their applications in mathematics and information sciences”, kluwer academic publishers, dordrecht, 2002. [8] j.s. golan, “semirings and their applications”, kluwer academic publishers, dordrecht, 1999. [9] u. hebisch, h.j. weinert, “semirings-algebraic theory and applications in computer science”, world scientific publishing co., singapore, 1998. 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[18] m. shabir, “some characterizations and sheaf representations of regular and weakly regular monoids and semirings”, phd thesis, quaid-i-azam university, pakistan, 1995. [19] g. yesilot, k. orel, u. tekir, on prime subsemimodules of semimodules, int. j. algebra 4 (1) (2010), 53 – 60. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 2 (2020), 229 – 252 doi:10.17398/2605-5686.35.2.229 available online september 16, 2020 multifractal formalism of an inhomogeneous multinomial measure with various parameters a. samti analysis, probability & fractals laboratory lr18es17 university of monastir, faculty of sciences of monastir department of mathematics, 5019-monastir, tunisia amal samti@yahoo.fr received may 18, 2020 presented by mostafa mbekhta accepted july 7, 2020 abstract: in this paper, we study the refined multifractal formalism in a product symbolic space and we estimate the spectrum of a class of inhomogeneous multinomial measures constructed on the product symbolic space. key words: hausdorff dimension, packing dimension, fractal, multifractal. ams subject class. (2010): 28a80, 28a78, 28a12, 11k55. 1. introduction the multifractal formalism of a measure µ aims to establish a relationship between the dimension of level set of the local hölder exponent of µ to the legendre transform of what is called the ”free energy” function. a problem initially raised and studied for physical motivations [8, 9, 11, 12, 10]. it will be convenient to give a brief description of the multifractal formalism. let x be a metric space. the local hölder exponent αµ(x) at the point x ∈ x is defined to be αµ(x) = lim r→0 log µ(b(x,r)) log r where b(x,r) stands for the ball of radius r centered at x. the measure µ is said to satisfy the multifractal formalism at α ≥ 0, if the hausdorff dimension (dim) and the packing dimension (dim) of the level set e(α) which is defined by e(α) = {x ∈ supp(µ) : αµ(x) = α} , are equal respectively to the value of the legendre transform at α of a scale function τµ associated to the measure µ, i.e., dim e(α) = dim e(α) = τ∗µ(α), issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.35.2.229 mailto:amal_samti@yahoo.fr https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 230 a. samti where f∗(x) = inf y (xy + f(y)) is the legendre transform of a function f and supp(µ) is the topological support of µ. the upper bound for dim e(α) (respectively dim e(α)) is obtained by a standard covering argument as besicovisch’s covering theorem and vitali’s lemma [13]. however, the lower bound is usually much harder to prove, it is related to the existence of an auxiliary measure such as a gibbs measure [13] or a frostman measure [3] which is supported by the set to be analyzed. for this reason, f. ben nasr et al. [4] improved the olsen’s result in describing a class of measures satisfying the multifractal formalism and proposed a new sufficient condition that gives the lower bound. in such a situation, they concluded that bµ(q) = bµ(q), where bµ and bµ are olsen’s functions. besides, they constructed inhomogeneous bernoulli products, such measures whose both multifractal dimension functions bµ and bµ agree at one or two points only. which implies a valid refined multifractal formalism no more than two points. in [5], ben nasr and peyrière constructed an example of a “bad” measure on the interval {0, 1}n for which the olsen’s functions bµ and bµ differ and the hausdorff dimensions of the sets e(α) are given by the legendre transform of bµ, and their packing dimensions by the legendre transform of bµ, i.e., bµ(q) < bµ(q) for all q ∈{0, 1} and dim e(α) = b∗µ(α) and dim e(α) = b ∗ µ(α), for some α ≥ 0. shen [14] and wu et al. [17, 18, 19] revisited this example such that the functions bµ and bµ can be real analytic. motivated by these examples n. attia and b. selmi [1, 2] introduced and studied a new multifractal formalism based on the hewitt-stromberg measures and showed that this formalism is completely parallel to olsen’s multifractal formalism based on the hausdorff and packing measures. in the present work, let 2 ≤ r1 < r2 be two integers, we consider a class of measures defined on a product symbolic space a1 × a2 endowed with the distance product where ai = {0, . . . ,ri − 1} for i = 1, 2, and constructed on the rectangles that flatten as their diameters tend to zero. however, these rectangles do not allow the calculation of the hausdorff dimension, hence the difficulty of the problem. the aim of this paper is to study the validity of the refined multifractal formalism of this class of measures. the paper is organized as follows. in section 2, we give some notations and definitions which will be useful. in the third section we consider a sequence of finite partitions of a product symbolic space made of rectangles and we show through an example that the almost squares allow the calculation of the multifractal formalism of an inhomogeneous measure 231 hausdorff and packing dimensions. in section 4, we consider a variant of the refined multifractal formalism as already introduced by ben nasr and peyrière [5] which we adapt it to almost squares and estimate the dimensions of the level sets e(α). finally, we apply our results to a class of inhomogeneous measures defined on the product symbolic space. 2. notations and definitions in this section, we will recall the hausdorff and packing measures and their dimensions. let (x,d) be a separable metric space. the diameter of a non-empty set e ⊆ x is given by diam e = sup{d(x,y) : x,y ∈ e} , with the convention that diam(∅) = 0. we define the closed ball with center x ∈ x and radius r > 0 as b(x,r) = {y ∈ x : d(x,y) ≤ r} . a finite or countable collection of subsets {ui}i of x is called a δ-cover of e ⊆ x, if for each i we have diam ui ≤ δ and e ⊂ ⋃ i ui. suppose that e is a subset of x and s is a non-negative number. for any δ > 0 we define hsδ(e) = inf {∑ i diam(ui) s : {ui}i is a δ-cover of e } . as δ decreases, the class of δ-covers of e is reduced. therefore, this infimum increases and approaches a limit as δ ↘ 0. thus we define hs(e) = lim δ→0 hsδ(e). we term hs(e) the s-dimensional hausdorff measure of e. then we define the hausdorff dimension of e as dim(e) = sup{s ≥ 0 : hs(e) = ∞} = inf {s ≥ 0 : hs(e) = 0} . remark 1. notice that the covering of e with centered balls in e allow the calculation of the hausdorff dimension of e, for more details see [7]. 232 a. samti we will now define the packing measure. first, let define a δ-packing of e ⊂ x to be a finite or countable collection of disjoint balls {b(xi,ri)}i of diameter at most δ and with centers in e. for s ≥ 0 and δ > 0, let psδ(e) = sup {∑ i (2ri) s : {b(xi,ri)}i is a δ-packing of e } . from this the s-dimensional pre-packing measure ps of e is defined by ps(e) = lim δ→0 psδ(e). finally, we define the s-dimensional packing measure ps(e) of e by ps(e) = inf {∑ i ps(ei) : e ⊂ ∞⋃ i=1 ei } . the packing dimension of e, denoted by dim(e), is defined in the same way as hausdorff dimension, that means dim(e) = sup{s ≥ 0 : ps(e) = ∞} = inf {s ≥ 0 : ps(e) = 0} . for more details about the hausdorff, packing measures and their dimensions see [15, 16, 7]. 3. calculation of the hausdorff and packing dimensions on the product symbolic space on different basis for practical reasons, we shall need basic notions about the set of words on an alphabet. let 2 ≤ r1 < r2 be two integers. for i ∈ {1, 2}, given ai = {0, . . . ,ri − 1} a finite alphabet. for all n ∈ n∗, each element in ani is denoted by a string of n letters or digits in ai that we call a word; by convention a0i is reduced to the empty word ∅. let a ∗ i = ⋃ n≥0 a n i be the set of finite words built over ai and ai = an ∗ i the symbolic space over ai. the set a∗i ∪ ai is endowed with the concatenation operation: if ω ∈ a ∗ i and ω′ ∈ a∗i ∪ ai, we denote by ω.ω ′ the word obtained by juxtaposition of the two words ω and ω′. for each finite word ω ∈ a∗i , [ω] is the cylinder ω ·ai = {ω ·ω ′ : ω′ ∈ ai}. furthermore, if ω = ω1 · · ·ωk · · · ∈ ai and n ∈ n then ω|n stands for the prefix ω1 · · ·ωn of ω for n ≥ 1 and the empty word otherwise. each set ai multifractal formalism of an inhomogeneous measure 233 is endowed with the ultrametric distance di : (z,z ′) ∈ a2i 7−→ r −|z∧z′| i , where z∧z′ is defined to be the longest prefix common to both z and z′ and |z| the length of a word z ∈ a∗i ∪ ai. then the product symbolic space a1 × a2 is endowed with the ultrametric distance. d((x,y), (x′,y′)) = max(d1(x,x ′),d2(y,y ′)). in the next, if ω ∈ ak1 and ω ′ ∈ ak ′ 2 , we call r(ω,ω ′) the rectangle obtained as the product of the cylinders [ω] and [ω′]. we denote by ∣∣r(ω,ω′)∣∣ m = sup ( 1 rk1 , 1 rk ′ 2 ) , and ∣∣r(ω,ω′)∣∣ m = inf ( 1 rk1 , 1 rk ′ 2 ) . we say that a sequence {ξn}n≥1 of finite partitions of a1 × a2 made of rectangles satisfies condition (1) if lim n→∞ sup r∈ξn diam(r) = 0 and ξn+1 is a refinement of ξn. (1) in all over this work, we will consider a sequence {ξn}n≥1 of finite partitions of a1 × a2 made of rectangles verifying (1) and we put ξ = ⋃ n≥1 ξn. if r belongs to ξn+1, we define by p(r) the element of ξn that contains it. let e be a nonempty subset of a1 × a2 and s a strictly positive real number. for all ε > 0, a finite or countable collection of rectangles {rj}j is called an ε-covering of e if diam(rj) ≤ ε for all j and e ⊂ ⋃ j rj. let hsξ,ε(e) = inf  ∑ j diam(rj) s : rj ∈ ξ,{rj}j is an ε-covering of e   and hsξ(e) = lim ε→0 hsξ,ε(e). finally, the dimension dimξ(e) is given by dimξ(e) = inf { s > 0 : hsξ(e) = 0 } = sup { s > 0 : hsξ(e) = ∞ } . 234 a. samti here, we define an ε-packing of e ⊂ a1 × a2 to be a finite or countable collection of disjoint rectangles {rj}j of diameter not exceeding ε and with rj ∩e 6= ∅. for ε > 0, we define psξ,ε(e) = sup  ∑ j diam(rj) s : rj ∈ ξ,{rj}j is an ε-packing of e   . then psξ,ε(e) decreases as ε increases, so we may take the limit psξ(e) = lim ε→0 psξ,ε(e). unfortunately, psξ(e) is not an outer measure, to overcome this difficulty we define psξ (e) = inf  ∑ j psξ(ej) : e ⊆ ⋃ j ej   . the definition of packing dimension parallels that of hausdorff dimension. so, let dimξ(e) defined such that dimξ(e) = inf { s > 0 : psξ (e) = 0 } = sup { s > 0 : psξ (e) = ∞ } . in the following proposition we will give some conditions on a family ξ of rectangles of the symbolic space a1×a2 such that for every part e of a1×a2, we have dim(e) = dimξ(e) and dim(e) = dimξ(e). proposition 3.1. suppose that (i) lim n→∞ sup r∈ξn log |r |m / log |r |m = 1, (ii) lim n→∞ sup r∈ξn log |r |m / log |p(r)|m = 1. then for any part e of a1 ×a2, we have dimξ(e) = dim(e), (2) dimξ(e) = dim(e). (3) multifractal formalism of an inhomogeneous measure 235 proof. in order to prove the equality (2), we start by proving that dimξ(e) ≤ dim(e). let t > dim(e) and η > 0 such that t 1+η > dim(e). it follows from assumption (i) that there exists an integer n0 such that for all n ≥ n0 and for all r ∈ ξn, we have |r|1+ηm ≤ |r|m . take {ej}j a cover of e and choose {rk}k an ε-packing of ej with ε ≤ inf r∈ξn0 diam(r). for j ∈ n, fix xk ∈ rk ∩ ej, we denote by bk = b(xk, |rk|m). it is clear that {bk}k is an ε-packing of ej. as ε ≤ inf r∈ξn0 diam(r) we get for all integer k, |rk| 1+η m ≤ |rk|m (4) and ∑ k diam(rk) t ≤ ∑ k diam(bk) t 1+η . then, ptξ,ε(ej) ≤p t 1+η ε (ej) and as ε goes to 0, yields ptξ(ej) ≤p t 1+η (ej). therefore, ptξ(e) ≤p t 1+η (e) < +∞ consequently, dimξ(e) < t, for all t > dim(e), which implies that dimξ(e) ≤ dim(e). in order to obtain the other inequality, fix t > dimξ(e) and η > 0 such that t 1+η > dimξ(e). using assumption (ii) there exists an integer n0 such that for all n ≥ n0 and for all r ∈ ξn, we have |p(r)|1+ηm ≤ |r |m . (5) 236 a. samti let {ej}j be a cover of e and {bk = b(xk,rk)}k an ε-packing of ej with ε ≤ inf r∈ξn0 diam(r). if rk is a rectangle such that rk ⊂ b(xk,rk) and p(rk) * b(xk,rk), (6) then {rk}k is an ε-packing of ej. since ε ≤ inf r∈ξn0 diam(r), we have for all k ∈ n, |p(rk)| 1+η m ≤ |rk|m . (7) taking into account relations (6) and (7), we have∑ k diam(bk) t ≤ ∑ k diam(p(rk)) t ≤ ∑ k diam(rk) t 1+η . so, ptε(ej) ≤p t 1+η ξ,ε (ej). as ε goes to zero, pt(ej) ≤p t 1+η ξ (ej). then, we obtain pt(e) ≤p t 1+η ξ (e) < +∞. hence, dim(e) ≤ dimξ(e) which achieves the proof of equality (2). now, we will be interested in proving the equality (3). it is easy to see that ht(e) ≤htξ(e) then dim(e) ≤ dimξ(e). let’s prove that dimξ(e) ≤ dim(e). fix t > dim(e) and η > 0 such that t 1+η + (2−2(1 +η)3) > dim(e). let ε be a positive number such that ε ≤ inf r∈ξn0 diam(r). pick an ε-covering {rj}j of e and set bj = b(xj, |rj|m ) such that rj ⊆ bj. for all j ∈ n, there exists a family of disjoint rectangles {rjk}k∈lj such that ⋃ k∈lj rjk ⊂ bj , p(rjk) * bj and bj ⊆ ⋃ k∈lj p(rjk). multifractal formalism of an inhomogeneous measure 237 in a first step, we will calculate the number of p(rjk) that cover bj. we denote by λ the lebesgue measure on a1 × a2. using relations (4) and (5), we have λ(p(rjk)) (1+η)2 ≤ λ(rjk) and diam(bj) 2(1+η)3 ≤ λ(rjk). (8) let s and s′ be two positive integers such that r −(s+1) 1 < |rjk|m ≤ r −s 1 and r −(s′+1) 2 < |rjk|m ≤ r −s′ 2 . we have ∑ k∈lj λ(rjk) ≤ λ(bj) ≤ r−s1 r −s′ 2 ≤ (r1r2) diam(bj) 2. (9) it follows from inequalities (8) and (9) that∑ k∈lj diam(bj) 2(1+η)3 ≤ r1r2 diam(bj)2. hence, card(lj) ≤ r1r2 diam(bj)2−2(1+η) 3 . in a second step, we have |p(rjk)| 1+η m ≤ |rjk|m ≤ diam(bj) and ∑ k∈lj |p(rjk)| t m ≤ ∑ k∈lj diam(bj) t 1+η . so, ∑ j diam(rjk) t ≤ ∑ j ∑ k∈lj |p(rjk)|tm ≤ ∑ j (r1r2) diam(bj) 2−2(1+η)3 diam(bj) t 1+η and htξ,ε(e) ≤ (r1r2)h t 1+η +(2−2(1+η)3) ε (e). 238 a. samti letting ε tend to 0 implies htξ(e) ≤ (r1r2)h t 1+η +2−(1+η)3 (e). finally, we obtain dimξ(e) ≤ t. and the result yields. next, we set a generalization of the billingsley theorem [6] in our case. for this purpose, we introduce the following notations. if e is a non empty subset of a1 × a2 and x = (x1,x2) ∈ e, let ξ = ⋃ n≥1 ξn be a family of rectangles satisfying assumptions (i) and (ii) of proposition 3.1 and rn(x) be the rectangle of ξn containing x. in the sequel, we define by p(a1×a2) the set of borel probability measures on a1 ×a2. for all µ ∈p(a1 ×a2) and ε > 0, and e ∈ a1 ×a2, we define µ]ε(e) = inf  ∑ j µ(rj) : rj ∈ ξ,{rj}j an ε-covering of e   , µ](e) = lim ε→0 µ]ε(e) and ess sup x∈e,µ] a(x) = inf { t ∈ r : µ]({x ∈ e : a(x) > t}) = 0 } . proposition 3.2. let e be a subset of a1 × a2 and µ ∈ p(a1 × a2), we have (a) dimξ(e) ≤ sup x∈e lim inf n→∞ log µ(rn(x)) log(diam(rn(x))) ; (b) dimξ(e) ≤ sup x∈e lim sup n→∞ log µ(rn(x)) log(diam(rn(x))) . if µ](e) > 0, then we have (c) dimξ(e) ≥ ess sup x∈e,µ] lim inf n→∞ log µ(rn(x)) log(diam(rn(x))) ; (d) dimξ(e) ≥ ess sup x∈e,µ] lim sup n→∞ log µ(rn(x)) log(diam(rn(x))) . multifractal formalism of an inhomogeneous measure 239 proof. let us prove assumption (a). take δ> sup x∈e lim inf n→∞ log µ(rn(x)) log(diam(rn(x))) , then for all x ∈ e, there exists k ≥ n such that µ(rk(x)) ≥ diam(rk(x))δ. let ε be a positive number, there exists {rj}j a family of pairwise disjoint rectangles such that e ⊂ ⋃ j rj with µ(rj) ≥ diam(rj)δ and diam(rj) ≤ ε. we have ∑ j diam(rj) δ ≤ ∑ j µ(rj) < ∞. therefore, hδξ,ε(e) < ∞. finally, when ε → 0, we get dimξ(e) ≤ δ and the result easily follows. to prove the assumption (b), take δ > sup x∈e lim sup n→∞ log µ(rn(x)) log(diam(rn(x))) . for all x ∈ e, there exists n ∈ n such that, for all k ≥ n one has µ(rk(x)) ≥ diam(rk(x))δ. consider the set e(n) = { x ∈ e : for each k ≥ n, µ(rk(x)) ≥ diam(rk(x))δ } . let {ek}k be a cover of e and {rj}j be an ε-packing of e(n) ∩ ek with ε < infr∈ξn0 diam(r). one has∑ j diam(rj) δ ≤ ∑ j µ(rj) < ∞. from which pδξ,ε(e(n) ∩ ek) < ∞. then we get p δ ξ(e(n) ∩ ek) < ∞ when ε → 0. since e = ⋃ n e(n), we obtain dimξ(e) ≤ δ. hence (b). let us prove assumption (c). take δ < ess sup x∈e,µ] lim inf n→∞ log µ(rn(x)) log(diam(rn(x))) and set eδ = { x ∈ e : lim inf n→∞ log µ(rn(x)) log(diam(rn(x))) > δ } . 240 a. samti let en = { x ∈ eδ : for each k ≥ n, µ(rk(x)) ≤ diam(rk(x))δ } . it is clear that eδ = ⋃ n en. as we have µ ](eδ) > 0, there exists n ∈ n such that µ](en) > 0. then, for any ε-covering {rj}j of en, one has µ]ε(en) ≤ ∑ j µ(rj) ≤ ∑ j diam(rj) δ. therefore, µ]ε(en) ≤h δ ξ,ε(en). so, 0 < µ](en) ≤hδξ(en), which implies dimξ(e) ≥ dimξ(eδ) ≥ dimξ(en) ≥ δ and assumption (c) yields. in order to prove assumption (d), let δ < ess sup x∈e,µ] lim sup n→∞ log µ(rn(x)) log(diam(rn(x))) , and put eδ = { x ∈ e : lim sup n→∞ log(µ(rn(x))) log(diam(rn(x))) > δ } . we have µ](eδ) > 0, so there exists a subset f of eδ such that µ ](f) > 0. if x ∈ f , then for all n ∈ n there exists k ≥ n such that µ(rk(x)) ≤ diam(rk(x))δ (10) let ε > 0 and {rj}j an ε-packing of f satisfying (10). so, µ]ε(f) ≤ ∑ j µ(rj) ≤ ∑ j diam(rj) δ. then µ]ε(f) ≤p δ ξ,ε(f). this implies 0 < µ](f) ≤pδξ(f). multifractal formalism of an inhomogeneous measure 241 hence, if f = ⋃ j fj, one has 0 < µ](f) < ∑ j µ](fj) ≤ ∑ j pδξ(fj). thus, pδξ (f) > 0. therefore, dimξ(eδ) ≥ δ, from which the result follows and we achieve the proof of proposition 3.2. as a consequence of proposition 3.2, we obtain the following corollary. we adopt the following convention log 0 log ρ = +∞, for each ρ > 0. corollary 1. let γ ∈ r. if µ is a probability borel measure on a1 ×a2 such that µ(e) > 0, we consider a family ξ of rectangles verifying the assumptions of proposition 3.1 and e ⊂ { x ∈ a1 ×a2 : lim n→∞ log µ(rn(x)) log(diam(rn(x))) = γ } , we have dimξ(e) = dimξ(e) = γ. next, we will be interested in adding an example of application of corollary 1. example. let {ξn}n≥1 be a sequence of finite partitions of a1 ×a2 made of rectangles in the form [ω]×[ω′], for all (ω,ω′) ∈ aq(n)1 ×a n 2 and ξ = ⋃ n≥1 ξn, where the integer q(n) is defined such that, for n ∈ n∗ n log(r2) log(r1) ≤ q(n) < n log(r2) log(r1) + 1. it is clear that the family ξ satisfies the assumptions of proposition 3.1. for α ≥ 0, we consider the set eα = { x ∈ a1 ×a2 : lim n→∞ n ω,ω′ n n (x) = αω,ω′ for all (ω,ω ′) ∈ a1 ×a2 } 242 a. samti where for (ω,ω′) ∈ a1 ×a2, n ω,ω′ n (x) stands for the number of appearances of the couple (ω,ω′) in the product word x|n×y|n and α = (αω,ω′)(ω,ω′)∈a1×a2 is a family of positive numbers such that∑ (ω,ω′)∈a1×a2 αω,ω′ = 1. we propose to calculate the hausdorff dimension of the set eα. for this purpose, we consider the bernoulli measure µ in a1 ×a2 defined by µ([ω1 · · ·ωn] × [ω′1 · · ·ω ′ n]) = n∏ k=1 αωk,ω′k for each n ∈ n∗. we have µ([ω1 · · ·ωq(n)] × [ω ′ 1 · · ·ω ′ n]) = n∏ k=1 αωk,ω′k q(n)∏ k=n+1 λωk with λωk = ∑ ω′ k αωk,ω′k . it is clear that eα ⊂ { x ∈ a1 ×a2 : lim n→∞ log µ(rn(x)) log(diam(rn(x))) = γ } , where γ = − ∑ ω,ω′ αω,ω′ log αω,ω′ log r2 + ( 1 log r2 − 1 log r1 )∑ ω λω log λω. so, according to the strong law of large numbers we have µ(eα) = 1. by using corollary 1 we have, dimξ(eα) = dimξ(eα) = γ, which implies from proposition 3.1 that dim(eα) = dim(eα) = γ. thus, any borel set of eα with dimension inferior to γ is of measure µ-zero. multifractal formalism of an inhomogeneous measure 243 4. a variant of the refined multifractal formalism in the product space a1 ×a2 4.1. problematic. in this section, we will consider a sequence {ξn}n≥1 of finite partitions of a1 ×a2 made of rectangles satisfying condition (1) and we put ξ = ⋃ n≥1 ξn. in the following, we consider a borel probability measure µ on a1 × a2 and one defines its support supp(µ) to be the complement of the set⋃ {r ∈ ξ : µ(r) = 0} . then, we intend to underestimate the dimensions of the fractal sets eµ(γ) for some values of γ, where eµ(γ) = { x ∈ supp(µ) : lim n→∞ log µ(rn(x)) log(diam(rn(x))) = γ } . notice that the natural coverings of these iso-hölder sets are made of rectangles which become thinner and thinner as their diameter tends to zero which doesn’t allow the calculation of the hausdorff and packing dimensions. for this purpose, we will consider a variant of the refined multifractal formalism as already introduced by f. ben nasr and j. peyrière [5], adapted to rectangles. let us consider an auxiliary borel probability measure ν on a1 ×a2. if e is a nonempty subset of a1 ×a2 then for q,t ∈ r and ε > 0, we introduce the following quantities: hq,tµ,ν,ε(e) = inf {∑ j µ(rj) q diam(rj) tν(rj) : rj ∈ ξ,{rj}j an ε-covering of e } , hq,tµ,ν(e) = lim ε→0 hq,tµ,ν,ε(e), and p q,t µ,ν,ε(e) = sup {∑ j µ(rj) q diam(rj) tν(rj) : rj ∈ ξ, {rj}j an ε-packing of e } , p q,t µ,ν(e) = lim ε→0 p q,t µ,ν,ε(e). 244 a. samti the function p q,t µ,ν is called the packing pre-measure. in order to deal with an outer measure, one defines pq,tµ,ν(e) = inf  ∑ j p q,t µ,ν(ej) : e ⊂ ⋃ j ej   . let ϕ be the following function ϕ(q) = inf { t ∈ r : pq,tµ,ν(supp(µ)) = 0 } . (11) 4.2. main results. let µ be a borel probability measure on a1 × a2. for α,β ∈ r, one sets eµ(α,β) = eµ(α) ∩eµ(β), where eµ(α) = { x ∈ supp(µ) : lim inf n→∞ log µ(rn(x)) log(diam(rn(x))) ≥ α } and eµ(β) = { x ∈ supp(µ) : lim sup n→∞ log µ(rn(x)) log(diam(rn(x))) ≤ β } . theorem 4.1. assume that ϕ(0) = 0 and ν](supp(µ)) > 0. then one has dimξ eµ ( −ϕ′r(0),−ϕ ′ l(0) ) ≥ inf { lim inf n→∞ log ν(rn(x)) log(diam(rn(x))) : x ∈ eµ(−ϕ′r(0),−ϕ ′ l(0)) } and dimξ eµ ( −ϕ′r(0),−ϕ ′ l(0) ) ≥ inf { lim sup n→∞ log ν(rn(x)) log(diam(rn(x))) : x ∈ eµ(−ϕ′r(0),−ϕ ′ l(0)) } , where ϕ′r,ϕ ′ l are respectively the left-hand and right-hand derivatives of ϕ. multifractal formalism of an inhomogeneous measure 245 remark 2. the same result holds with ψ(q) = inf { t ∈ r : pq,tµ,ν(supp(µ)) = 0 } . the proof of theorem 4.1 is an immediate consequence of the following proposition. proposition 4.1. assume that ϕ(0) = 0 and ν](supp(µ)) > 0. then one has ν](eµ(−ϕ′r(0),−ϕ ′ l(0)) c) = 0. proof. take δ > −ϕ′l(0), there exist two positive reals t and δ ′ such that δ > δ′ > −ϕ′l(0) and δ ′t > ϕ(−t) which implies p−t,δ ′t µ,ν (supp(µ)) = 0. so, there exists a partition {ej}j of supp(µ) such that∑ j p −t,δ′t µ,ν (ej) ≤ 1. it results that p −t,δt µ,ν (ej) = 0 for all j. now, consider the set eδ = { x ∈ supp(µ) : lim sup n→∞ log µ(rn(x)) log(diam(rn(x))) > δ } . if x ∈ eδ, for all n ∈ n there exists k ≥ n such that µ(rk(x)) ≤ diam(rk(x))δ. let e be a subset of eδ and set fj = e ∩ej. for 0 < ε ≤ inf r∈ξn diam(r) and for all j, one can find an ε-packing {rjk}k of fj such that µ(rjk) ≤ diam(rjk) δ. so, we have ν]ε(fj) ≤ ∑ j ν(rj) ≤ ∑ j ∑ k ν(rjk) ≤ ∑ j ∑ k µ(rjk) −t diam(rjk) δtν(rjk) ≤ ∑ j p −t,δt µ,ν,ε (fj) = 0. 246 a. samti then ν](eδ) = 0. we conclude that ν] ({ x ∈ supp(µ) : lim sup n→∞ log µ(rn(x)) log(diam(rn(x))) > −ϕ′l(0) }) = 0. in the same way, one proves that ν] ({ x ∈ supp(µ) : lim inf n→∞ log µ(rn(x)) log(diam(rn(x))) < −ϕ′r(0) }) = 0. proof of theorem 4.1. assume that ϕ(0) = 0 and ν](supp(µ)) > 0. then we have according to proposition 4.1 ν](eµ(−ϕ′r(0),−ϕ ′ l(0))) > 0. so, it is easy to see from proposition 3.2 that dimξeµ(−ϕ′r(0),−ϕ ′ l(0)) ≥ ess sup x∈eµ(−ϕ′r(0),−ϕ′l(0)),ν ] lim inf n→∞ log ν(rn(x)) log(diam(rn(x))) , and dimξ eµ(−ϕ′r(0),−ϕ ′ l(0)) ≥ ess sup x∈eµ(−ϕ′r(0),−ϕ′l(0)),ν ] lim sup n→∞ log ν(rn(x)) log(diam(rn(x))) . however, as a property of ess sup, we know that if ν](eµ(−ϕ′r(0),−ϕ′l(0))) > 0, then inf x∈eµ(−ϕ′r(0),−ϕ′l(0)) { lim inf n→0 log ν(r(x)) log(diam(rn(x))) } ≤ ess sup x∈eµ(−ϕ′r(0),−ϕ′l(0)),ν ] lim inf n→∞ log ν(r(x)) log(diam(rn(x))) and the proof of the theorem follows. multifractal formalism of an inhomogeneous measure 247 5. an example in this section we give a large class of measures satisfying the result of theorem 4.1. let {ξn}n≥1 be the sequence of finite partitions of a1 × a2 made of rectangles of the form [ω] × [ω′], for all (ω,ω′) ∈ aq(n)1 × a n 2 and ξ = ⋃ n≥1 ξn, where the integer q(n) is defined such that, for n ∈ n∗ n log(r2) log(r1) ≤ q(n) < n log(r2) log(r1) + 1. for (i,j) ∈ a1×a2, take (pi,j)i,j and (qi,j)i,j two sequences of non negative numbers such that∑ i,j pi,j = ∑ i,j qi,j = 1 and λi = ∑ j pi,j = ∑ j qi,j. let (tn)n≥1 be a sequence of integers defined by t1 = 1 , tn < tn+1 and lim n→∞ tn tn+1 = 0. consider the family of parameters αik,jk αik,jk = { pik,jk if t2n−1 ≤ k < t2n, qik,jk if t2n ≤ k < t2n+1. we define the measure µ on a1 ×a2 as follows µ ( [i1 · · ·in] × [j1 · · ·jn] ) = n∏ k=1 αik,jk. it is easy to see that µ ( [i1 · · ·iq(n)] × [j1 · · ·jn] ) = µ ( [i1 · · ·in] × [j1 · · ·jn] ) ·λin+1 · · ·λiq(n). in the sequel we will impose those monotony hypotheses p0,0 < p0,1 < · · · < p0,r2−1 < p1,0 < · · · < p1,r2−1 < · · · · · · < pr1−1,0 < · · · < pr1−1,r2−1, q0,0 < q0,1 < · · · < q0,r2−1 < q1,0 < · · · < q1,r2−1 < · · · · · · < qr1−1,0 < · · · < qr1−1,r2−1, p0,0 < q0,0 and pr1−1,r2−1 > qr1−1,r2−1, 248 a. samti which prove the existence of a real x0 such that t(x0) = w(x0), where t(x) = ∑ i,j pxi,j∑ i,j pxi,j logr2 pi,j + ( 1 log r1 − 1 log r2 )∑ i,j pxi,j∑ i,j pxi,j log λi and w(x) = ∑ i,j qxi,j∑ i,j qxi,j logr2 qi,j + ( 1 log r1 − 1 log r2 )∑ i,j qxi,j∑ i,j qxi,j log λi. for this real x0, we denote by p̃i,j = px0i,j∑ i,j px0i,j and q̃i,j = qx0i,j∑ i,j qx0i,j . our aim is to estimate the dimensions of the sets eµ(γ) for certain values of γ. to be done, we consider an auxiliary measure ν on a1 × a2 defined as µ with the parameters p̃i,j and q̃i,j instead of pi,j and qi,j by ν([i1 · · ·in] × [j1 · · ·jn]) = n∏ k=1 α̃ik,jk where α̃ik,jk = { p̃ik,jk if t2n−1 ≤ k < t2n, q̃ik,jk if t2n ≤ k < t2n+1. let λ̃i = ∑ j p̃i,j = ∑ j q̃i,j. then, we have the following result. theorem 5.1. for every γ ∈ ( − logr2 ( qr1−1,r2−1λ log(r2) log(r1) −1 r1−1 ) ,− logr2 ( q0,0λ log(r2) log(r1) −1 0 )) we have dim eµ(γ) ≥ min{h(p̃),h(q̃)} and dim eµ(γ) ≥ max{h(p̃),h(q̃)} , multifractal formalism of an inhomogeneous measure 249 where h(p̃) = − ∑ i,j p̃i,j logr2 p̃i,j + ( 1 log r2 − 1 log r1 )∑ i λ̃i log λ̃i and h(q̃) = − ∑ i,j q̃i,j logr2 q̃i,j + ( 1 log r2 − 1 log r1 )∑ i λ̃i log λ̃i. in order to prove this theorem we will calculate the function ϕ defined in equation (11). for that, we need to use the following lemma. lemma 5.1. for t ∈ r, one has ϕ(t) = lim sup n→∞ 1 n log r2 log ∑ rn∩supp(µ)6=∅ µ(rn) tν(rn). proof. for t ∈ r, we denote by φ(t) = lim sup n→∞ 1 n log r2 log ∑ rn∩supp(µ)6=∅ µ(rn) tν(rn). we will prove that ϕ(t) = φ(t). let’s begin by proving that ϕ(t) ≤ φ(t). for α > 0 satisfying φ(t) ≤ α, there exists n0 ∈ n such that for all n ≥ n0, 1 n log r2 log ∑ rn∩supp(µ)6=∅ µ(rn) tν(rn) ≤ α. so, ∑ rn∩supp(µ) 6=∅ µ(rn) tν(rn)r −nα 2 ≤ 1, for each n ≥ n0. then p t,α µ,ν(supp(µ)) ≤ 1, and α ≥ ϕ(t), which gives that φ(t) ≥ ϕ(t). 250 a. samti next, we prove that ϕ(t) ≥ φ(t). let α > ϕ(t), then p t,α µ,ν(supp(µ)) = 0. for ε > 0, there exists an ε-packing {rn}n of supp(µ) such that∑ rn∩supp(µ)6=∅ µ(rn) tν(rn)r −nα 2 ≤ 1. thus 1 n log r2 log ∑ rn∩supp(µ)6=∅ µ(rn) tν(rn) ≤ α. so, lim sup n→∞ 1 n log r2 log ∑ rn∩supp(µ)6=∅ µ(rn) tν(rn) ≤ α and φ(t) ≤ α, which prove lemma 5.1. now, we are able to prove theorem 5.1. it is easy to see that ϕ(t) = sup  logr2 ∑ i,j pti,jp̃i,j, logr2 ∑ i,j qti,jq̃i,j   + ( 1 log r1 − 1 log r2 ) log ∑ i λtiλ̃i and ϕ(0) = 0. by the way, using the definitions of the sequences (p̃i,j) and (q̃i,j) and a simple computation of the derivative of ϕ at 0 we obtain ϕ′(0) = ∑ i,j p̃i,j logr2 pi,j + ( 1 log r1 − 1 log r2 )∑ i λ̃i log λi. let γ = −ϕ′(0), it is clear that γ ∈ ( − logr2 ( qr1−1,r2−1λ log(r2) log(r1) −1 r1−1 ) ,− logr2 ( q0,0λ log(r2) log(r1) −1 0 )) . multifractal formalism of an inhomogeneous measure 251 besides, using the strong law of large numbers we can see that lim inf n→∞ logr2 ν(rn(x)) −n = min{h(p̃),h(q̃)} and lim sup n→∞ logr2 ν(rn(x)) −n = max{h(p̃),h(q̃)} , for ν-almost every x. then, it follows from theorem 4.1 and proposition 3.2 that dim eµ(γ) ≥ min{h(p̃),h(q̃)} and dim eµ(γ) ≥ max{h(p̃),h(q̃)} which achieve 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[19] j. xiao, m. wu, the multifractal dimension functions of homogeneous moran measure, fractals 16 (2008), 175 – 185. introduction notations and definitions calculation of the hausdorff and packing dimensions on the product symbolic space on different basis a variant of the refined multifractal formalism in the product space a1a2 problematic. main results. an example e extracta mathematicae vol. 31, núm. 1, 11 – 23 (2016) on a ρn-dilation of operator in hilbert spaces † a. salhi, h. zerouali pb 1014, departement of mathematics, siences faculty, mohamed v university in rabat, rabat, morocco radi237@gmail.com, zerouali@fsr.ac.ma presented by mostafa mbekhta received march 21, 2016 abstract: in this paper we define the class of ρn−dilations for operators on hilbert spaces. we give various properties of this new class extending several known results ρ−contractions. some applications are also given. key words: ρn-dilation, ρ-dilation. ams subject class. (2010): 47a20. 1. introduction sz-nagy and foias introduced in [8], the subclass cρ of the algebra l(h) of all bounded operators on a given complex hilbert space h. more precisely, for each fixed ρ > 0, an operator t ∈ cρ if there exists a hilbert space k containing h as a subspace and a unitary transformation u on k such that; tn = ρprun|h for all n ∈ n ∗. (1) where pr : k → h is the orthogonal projection on h. the unitary operator u is then called a unitary ρ-dilation of t, and the operator t is a ρ-contraction. recall that t is power bounded if ∥t n∥ ≤ m for some fixed m and every nonegative integer n. from equation (1), it follows that every ρ-contraction is power bounded since ∥tn∥ ≤ ρ for all n ∈ n∗. computing the spectral radius of t, it comes that the spectrum of the operator t satisfies σ(t) ⊂ d, where d = d(0, 1) is the open unit disc of the set of complex numbers c. operators in the class cρ enjoy several nice properties, we list below the most known, we refer to [7] for proofs and further information. †this work is partially supported by hassan ii academy of siences and the cnrst project urac 03. 11 12 a. salhi, h. zerouali (1) the function ρ 7→ cρ is nondecreasing, that is cρ cρ′ if ρ < ρ′. we will denote by c∞ = ∪ ρ>0 cρ. (2) c1 coincides whith the class of contractions (see [6]) and c2 is the class of operators t having a numerical radius less or equal to 1 (see [1]). the numerical radius is given by the expression, w(t) = sup{|⟨th; h⟩| : ∥h∥ = 1}. (3) if t ∈ cρ so is t n. it is however not true in general that the product of two operators in cρ is in cρ. also it is not always true that ξt belongs to cρ when t ∈ cρ for |ξ| ̸= 1. (4) for any m a t−invariant subspace, the restriction of t to the subspace m is in the class cρ whenever t is. (5) any operator t such that σ(t) ⊂ d belongs to c∞. numerous papers where devoted the the study of differents aspects of cρ; we refer to [2, 4, 5] for more information. the next theorem provides a useful characterization of the class cρ in term of some positivity conditions, theorem 1.1. let t be a bounded operator on the hilbert space h and ρ be a nonnegative real. the following are equivalent (1) the operator t belongs to the class cρ ; (2) for all h ∈ h; z ∈ d(0; 1) ( 2 ρ − 1)∥zth∥2 + (2 − 2 ρ )re (zth, h) ≤ ∥h∥2; (2) (3) for all h ∈ h; z ∈ d(0; 1) (ρ − 2)∥h∥2 + 2re ((i − zt)−1h, h) ≥ 0. (3) 2. unitary ρn-dilation we extend the notion of ρ-contractions to a more general setting. more precisely, let (ρn)n∈n be a sequence of nonnegative numbers. we will say that the operator t on a complex hilbert space h belongs to the class cρn if, there on a ρn-dilation of operator in hilbert spaces 13 exists a hilbert space k containing h as a subspace and a unitary operator u such that tn = ρnpru n |h for all n ∈ n ∗. (4) we say in this case that the unitary operator u is a ρn-dilation for the operator t and the operator t will be called a ρn-contraction. remark 2.1. . (1) for any bounded operator t, the operator t∥t∥ is a contraction and hence admits a unitary dilation. we deduce that, t ∈ cρn for ρn = ∥t∥ n for all n ∈ n. we notice at this level that, without additional restrictive assumptions on the sequence (ρn)n∈n, there is no hope to construct a reasonable ρn-dilation theory. our goal will be to extend the most usefull properties of ρ−contraction to this more general setting. (2) from equation 4, for t ∈ cρn with u a ρn-dilation, we obtain ∥tn∥ ≤ ∥ρnprun|h∥ ≤ ρn. therefore the condition limn→∞ (ρn) 1 n ≤ 1 will ensure that σ(t) ∈ d(0; 1). (3) in contrast with the class cρ, the class c(ρn) is not stable by powers. however, if t ∈ cρn and k ≥ 1 is a given integer, we obtain tk ∈ cρkn. this latter fact can be seen as a trivial extension of the case ρn = ρ0 for every n. in the remaining part of this paper, we will assume that (ρn)n∈n is a sequence of nonnegative numbers satisfying lim n→∞ (ρn) 1 n ≤ 1. (5) we associate with the sequence (ρn)n∈n, the following function, ρ(z) = ∑ n≥0 zn ρn . it is easy to see that condition limn→∞ (ρn) 1 n ≤ 1 implies that ρ ∈ h(d). here h(d) is the set of holomorphic functions on the open unit disc d. also, the valued-operators function ρ(zt) = ∑ n≥0 zntn ρn 14 a. salhi, h. zerouali is well defined and converges in norm for every |z| < 1. we give next a necessary and sufficient condition to the membership to the class cρn; theorem 2.2. let t be an operator on a hilbert space h and (ρn)n∈n is a sequence of nonnegative numbers. the operator t has a ρn-dilation if and only if (1 − 2 ρ0 )∥h∥2 + 2re ⟨ρ(zt)(h); h⟩ ≥ 0 for all h ∈ h; z ∈ d(0; 1). (6) we recall first the next well known lemma from [7, theorem 7.1] that will be needed in the proof of the previous theorem. lemma 2.3. let h be a hilbert space, g be a multiplicative group and ψ be an operator valued function s ∈ g 7→ ψ(s) ∈ l(h) such that  ψ(e) = i, e is the identity element of g ψ(s−1) = ψ(s)∗∑ s∈g ∑ t∈g(ψ(t −1s)h(s); h(t)) ≥ 0 for finitely non-zero function h(s) from g. then, there exists a hilbert space k containing h as a subspace and a unitary representation u of g, such that ψ(s) = pr(u(s)) (s ∈ g) and k = ∨ s∈g u(s)h proof of theorem 2.2. let t be a bounded operator in the class cρn and u be the unitary ρn-dilation of t, given by the expression 4. we have clearly, i + 2 ∑ n≥1 znun converges to (i + zu)(i − zu)−1 for all complex numbers z such that |z| < 1. and pr(i + 2 ∑ n≥1 znun) = i + 2 ∑ n≥1 zn ρn tn. on a ρn-dilation of operator in hilbert spaces 15 by writing, i + 2 ∑ n≥1 zn ρn tn = (1 − 2 ρ0 )i + 2 ∑ n≥0 zn ρn tn = (1 − 2 ρ0 )i + 2ρ(zt), we get pr((i + zu)(i − zu)−1) = (1 − 2 ρ0 )i + 2ρ(zt). on the other hand, ⟨(i + zu)k; (i − zu)k⟩ = ∥k∥2 + ⟨zuk; k⟩ − ⟨k; zuk⟩ − ∥zuk∥2 it follows that for every k ∈ k, we have re ⟨(i + zu)k; (i − zu)k⟩ = ∥k∥2 − ∥zuk∥2 = ∥k∥2 − |z|2∥k∥2 = ∥k∥2(1 − |z|2) ≥ 0 since |z| < 1. now if we take h = (i − zu)k we will find, re ⟨(i + zu)(i − zu)−1h; h⟩ = re ⟨pr(i + zu)(i − zu)−1h; h⟩ = re ⟨(1 − 2 ρ0 )h + 2ρ(zt)(h); h⟩, and hence for every h ∈ h, we obtain re ⟨(1 − 2 ρ0 )h + 2ρ(zt)(h); h⟩ ≥ 0 or equivalently, (1 − 2 ρ0 )∥h∥2 + 2re ⟨ρ(zt)(h); h⟩ ≥ 0 for every h ∈ h and all complex number z such that |z| < 1. conversely, let us show that condition (6) implies that the operator t belongs to the class cρn. to this aim, assume that (6) is satisfied and take 0 ≤ r < 1 and 0 ≤ ϕ < 2π. we introduce the next operator valued function q(r; ϕ) = i + ∑ n≥1 rn ρn (einϕtn + e−inϕt∗n). 16 a. salhi, h. zerouali then q(r; ϕ) converges in the norm operator for every r and ϕ. moreover, from the inequality 6, we have ⟨q(r; ϕ)l; l⟩ ≥ 0 for every l ∈ h. therefore j = 1 2π ∫ 2π 0 ⟨q(r; ϕ)h(ϕ); h(ϕ)⟩dϕ ≥ 0 for every h(ϕ) = ∑+∞ −∞ hne −inϕ where (hn)n∈z is a sequence with only finite number of nonzero elements in h. we have j =: +∞∑ −∞ ∥hn∥2+ ∑ m ∑ n>m rn−m ρn−m ⟨tn−mhn; hm⟩+ ∑ m ∑ n 0. it is immediate that ψ(ρn)(n) is nonnegative on the additive group z of integers. using lemma 2.3, there exists a unitary operator u on a hilbert space k containing h as a subspace and such that ψ(ρn)(n) = pru(n) for all n ∈ z. therefore for all n ∈ n∗ tn = ρnpru n |h. the proof is completed. remark 2.4. in the case where (ρn)n∈n is a constant sequence, that is ρn = ρ for all n ∈ n with ρ > 0, we obtain ρ(z) = 1 1 − z and hence, the inequality 6 becomes (1 − 2 ρ )∥h∥2 + 2re ⟨(i − zt)−1h; h⟩ ≥ 0 for all h ∈ h and z ∈ d. we substitute h by l = (i − zt)−1h to retrieve relation 3 and by theorem (2.1) we obtain t is a ρ-contraction. on a ρn-dilation of operator in hilbert spaces 17 the next two corollaries are immediate consequences of equation (6) and are related to analogous results of ρ−contraction. corollary 2.5. let t ∈ cρn and m a t−invariant subspace. then t|m ∈ cρn. proof. it suffices to see that equation 6 is close to restrictions. corollary 2.6. let t be in the class cρn and r ≥ 1 be a real number, then t is in the class crρn. proof. the inequality 6 is equivalent to (ρ0 − 2)∥h∥2 + 2ρ0re ⟨ρ(zt)(h); h⟩ ≥ 0. pluging rρn instead of ρn, we get (rρ0 − 2)∥h∥2 + 2rρ0re ⟨ 1 r ρ(zt)(h); h⟩ ≥ 0, and thus (1 − 2 rρ0 )∥h∥2 + 2re ⟨ 1 r ρ(zt)(h); h⟩ ≥ 0. therefore t ∈ c(rρn). we also have, proposition 2.7. let t be a bounded operator on a hilbert space h. then for every α > 2, there exists γ(α) > 0 such that the operator t belongs to c(ρn), where ρn is a sequence given by ρn = γ(α).∥t n∥(1 + nα). proof. let γ > 0 and ρα(z) = ∑ n≥1 zn γ.∥t n∥(1+nα) for all |z| ≤ 1. then ρα(zt) = ∑ n≥1 zntn γ.∥t n∥(1 + nα) for all |z| ≤ 1. for every vector h in h, we set a(z) = ⟨ρα(zt)h; h⟩ 18 a. salhi, h. zerouali |a(z)| = ∣∣ ∑ n≥0 ⟨ zn γ.∥t n∥(1 + nα) t nh; h⟩ ∣∣ ≤ ∑ n≥0 | ⟨t nh; h⟩ γ.∥t n∥(1 + nα) zn|. setting an = ⟨t nh;h⟩ γ.∥t n∥(1+nα), we have |an| ≤ ∥t n∥∥h∥2 γ.∥t n∥(1 + nα) = ∥h∥2 γ.(1 + nα) < ∞. we conclude that a(z) is holomorphic in the unit disc and continuous on the boundary. since the maximun is attained of the circle |z| = 1, we obtain |a(z)| = | ∑ n≥0 anz n| ≤ ∑ n≥0 |an||z|n = ∑ n≥1 |an| ≤ ∑ n≥0 ∥h∥2 γ.(1 + nα) now, since ∑ n≥0 1 1+(n)α is a convergent sequence (α > 2), then choosing γ = 2 ∑ n≥0 1 1+nα will leads to |a(z)| ≤ 1 2 ∥h∥2, and then ∥h∥2 + 2re ⟨ρα(zt)h; h⟩ ≥ 0 for all h ∈ h and z ∈ d. finally , inequality (6) is satisfied and the operator t belongs to the class c(ρn). 3. the bergmann shift we devote this section to the membership of the bergmann shift to the class c(ρn) for some suitable sequence ρn. let h be a hilbert space and (ei)i∈n∗ be an orthonormal basis of h. recall that for a given sequence (ωn)n∈n of non negative numbers; the weighted shift sω associated with ωn is defined on the on a ρn-dilation of operator in hilbert spaces 19 basis by sω(en) = ωnen+1. a detailed study on weighted shifts can be found in the survey [9]. on the other hand; the membership of weighted shifts to the class cρ is investigated in [3]. the bergman shift is the weighted shift defined on the basis by the expression ten = wnen+1, where wn = n + 1 n for all integer n ∈ n∗. it is easy to see that, • ∥t∥ = sup(wn)n∈n∗ = 2. • the weight (wn)n∈n∗ is decreasing and then ∥tn∥ = n∏ i=1 wi = n + 1. in particular t is not power bounded and hence does not belong to the class cρ for any ρ > 0. we have proposition 3.1. let t be the bergmann shift and ρn be the sequence given by ρn = n α for some α > 0. then for every α > 2, there exists γ(α) such that t ∈ cγ(α)ρn. proof. let γ > 0 and ρα(z) = ∑ n≥1 zn γnα for all |z| ≤ 1 in that ρ(zt) = ∑ n≥1 zntn γnα for all |z| ≤ 1. we set, s = ρ(zt) and let x be a vector in h. therefore s(x) = ρ(zt)(x) = ∑ i≥1 zntnx γnα . writing x = ∑ i≥1 xiei, we get s(x) = ∑ i≥1 ⟨s(x); ei⟩ei = ∑ i≥1 ( ∑ j≥1 xj⟨sej; ei⟩)ei, 20 a. salhi, h. zerouali and ⟨sej; ei⟩ = ⟨ ∑ n≥1 zntn γnα (ej); ei⟩ = ∑ n≥1 zn γnα ⟨tn(ej); ei⟩ = ∑ n≥1 zn γnα ⟨( j+n−1∏ p=j wp)ej+n; ei⟩. it follows that ⟨sej; ei⟩ = zi−j γ (i − j)α i−1∏ p=j wp, and then s(x) = ∑ i≥2 ( i−1∑ j=1 ( i−1∏ p=j wp)xj zi−j γ(i − j)α )ei. for the bergman shift, we have ∏i−1 p=j wp = i j and thus ρ(zt)(x) = ∑ i≥2 ( i−1∑ j=1 i γj(i − j)α xjz i−j)ei. finally, we conclude that the inequality (6) is equivalent to ∑ i≥1 |xi|2 + 2re ( ∑ i≥2 i−1∑ j=1 i γj(i − j)α xixjz i−j) ≥ 0. if we consider the function a(z) = 2 ∑ i≥2 i−1∑ j=1 i γj(i − j)α xixjz i−j and we write n = i − j, we will obtain, a(z) = 2 ∑ i≥2 i−1∑ n=1 i γ(i − n)γnα xixi−nz n = 2 ∑ n≥1 ∑ i≥n+1 i γ(i − n)nα xixi−nz n. we denote by (â(n))n = (an)n∈n∗ the sequence of coefficients of a, an = 1 2 ∑ i≥n+1 i γnα(i − n) xixi−n on a ρn-dilation of operator in hilbert spaces 21 since i n(i−n) = 1 n + 1 i−n ≤ 2 for every i ≥ n + 1, we obtain |an| = ∣∣∣1 2 ∑ i≥n+1 i γnα(i − n) xixi−n ∣∣∣ ≤ 1 γnα−1 ∑ i≥n+1 |xixi−n| ≤ 1 γnα−1 ∑ i≥n+1 |xixi−n|; and by the cauchy-schwartz inequality, it follows, |an| ≤ 1 γnα−1 ∥x∥2 ≤ ∞. we deduce that a(z) is holomorphic in the open unit disc and continuous on the closed unit disc. as the maximum is attained on the circle |z| = 1, we have |a(z)| = ∣∣∣1 2 ∑ n≥1 ( ∑ i≥n+1 i 1 γnα (i − n) xixi−n)z n ∣∣∣ ≤ ∑ n≥1 |an||z|n = ∑ n≥1 |an|. now, since ∑ n≥1 1 (n)α−1 is a convergente sequence (α ≥ 2), choosing γ =∑ n≥1 1 (n)α−1 would lead us to |a(z)| ≤ ∥x∥2 = ∑ i≥1 |xi|2. we derive that, |re (a(z))| ≤ |a(z)| ≤ ∥x∥2 = ∑ i≥1 |xi|2, and hence ∣∣∣1 2 re ( ∑ i≥2 i−1∑ j=1 i jγ(i − j)α xixjz i−j) ∣∣∣ ≤ ∑ i≥1 |xi|2. therefore for all x ∈ h and a complex z such that |z| ≤ 1 we have ∑ i≥1 |xi|2 + 2re ( ∑ i≥2 i−1∑ j=1 i jγ(i − j)α xixjz i−j) ≥ 0. we conclude that the weighted shift {wn} is a ρn-contraction with ρn = γn α. 22 a. salhi, h. zerouali remark 3.2. we claim that for every α ≥ 1 the bergmann shift belongs to a class c∞,nα. a proof is not available for this claim; however it is motivated by the incomplete computations below. let us set, for exemple, ρn = 4.n for all integer n ≥ 1, let h be a hilbert space and (ei)i∈n∗ be a an orthonormal basis for the hilbert space h. consider the bergmann shift defined on the basis by ten = n+1 n en+1 for all n ∈ n∗. then as in the proof of the previous proposition, we show that inequality (6) is equivalent to the next ∑ i≥1 |xi|2 + re ( ∑ i≥2 i−1∑ j=1 i 2j(i − j) xixjz i−j) ≥ 0. (7) we write ∑ i≥1 |xi|2+re ( ∑ i≥2 i−1∑ j=1 i 2j(i − j) xixjz i−j) ≥ ∑ i≥1 |xi|2− ∑ i≥2 i−1∑ j=1 i 2j(i − j) |xi||xj|, and ∑ i≥1 |xi|2 + ∑ i≥2 i−1∑ j=1 i 2j(i − j) |xi||xj| = ∑ i,j≥1 ai,j|xi||xj|, with { ai;i = 1 for all i ≥ 1 ai;j = i 4j|(i−j)| for all j ̸= i then to show inequality (7), it suffices to prove that the infinite symmetric matrix with the real entries m = [ai;j] is nonnegative. to this aim, we compute the determinant of the first n × n-corner, to check if it is nonnegative. an attempt on classical softwares allow to show this fact for n ≤ 150. it is hence reasonable to conjecture that the bergman shift belongs to c∞,n. references [1] c.a. berger, a strange dilation theorem, notices amer. math. soc. 12 (1965), 590. [2] g. cassier, h. zerouali, operator matrices in class cρ, linear algebra and its applications 420 (1-2) (2007), 361 – 376. [3] g. eckstein, a. racz, weighted shifts of class cρ, acta sci. math. (szeged) 35 (1973), 187 – 194. on a ρn-dilation of operator in hilbert spaces 23 [4] g. eckstein, sur les opérateurs de la classe cρ, acta sci. math.(szeged) 33 (3-4) (1972), 349 – 352. [5] h. mahzouli, vecteurs cycliques, opérateurs de toeplitz généralisés et régularité des algèbres de banach, thèse, université claude bernard, lyon 1, 2005. [6] b. sz.-nagy, sur les contractions de l’éspace de hilbert, acta sci. math. (szeged) 15 (1-1) (1953), 87 – 92. [7] b. sz.-nagy, c. foias, “harmonic analysis of operators on hilbert space”, north-holland, amsterdam, 1970. [8] b. sz nagy, c. foias, on certain class of power bounded operators in hilbert space, acta. sci. math. (szeged) 27 (1-2) (1996), 17 – 25. [9] a.l. shields, weighted shift operators and analytic function theory, in “topics in operator theory”, mathematical survey and monograph 13, amer. math. soc., providence, ri, 1974, 49 – 128. e extracta mathematicae vol. 32, núm. 1, 105 – 123 (2017) ostrowski type fractional integral inequalities for generalized (g, s, m, φ)-preinvex functions artion kashuri, rozana liko department of mathematics, faculty of technical science, university “ismail qemali”, vlora, albania artionkashuri@gmail.com rozanaliko86@gmail.com presented by horst martini received september 9, 2016 abstract: in the present paper, a new class of generalized (g, s, m, φ)-preinvex function is introduced and some new integral inequalities for the left hand side of gauss-jacobi type quadrature formula involving generalized (g, s, m, φ)-preinvex functions are given. moreover, some generalizations of ostrowski type inequalities for generalized (g, s, m, φ)-preinvex functions via riemann-liouville fractional integrals are established. at the end, some applications to special means are given. key words: ostrowski type inequality, hölder’s inequality, power mean inequality, riemannliouville fractional integral, s-convex function in the second sense, m-invex, p-function. ams subject class. (2010): 26a51, 26a33, 26d07, 26d10, 26d15. 1. introduction and preliminaries the following notation is used throughout this paper. we use i to denote an interval on the real line r = (−∞, +∞) and i◦ to denote the interior of i. for any subset k ⊆ rn, k◦ is used to denote the interior of k. rn is used to denote a generic n-dimensional vector space. the nonnegative real numbers are denoted by r◦ = [0, +∞). the set of integrable functions on the interval [a, b] is denoted by l1[a, b]. the following result is known in the literature as the ostrowski inequality (see [11]) and the references cited therein, which gives an upper bound for the approximation of the integral average 1 b−a ∫ b a f(t)dt by the value f(x) at point x ∈ [a, b]. theorem 1.1. let f : i −→ r, where i ⊆ r is an interval, be a mapping differentiable in the interior i◦ of i, and let a, b ∈ i◦ with a < b. if |f ′(x)| ≤ m for all x ∈ [a, b], then 105 106 a. kashuri, r. liko ∣∣∣∣∣f(x) − 1b − a ∫ b a f(t)dt ∣∣∣∣∣ ≤ m(b − a) [ 1 4 + ( x − a+b 2 )2 (b − a)2 ] , ∀x ∈ [a, b]. (1.1) for other recent results concerning ostrowski type inequalities (see [11]) and the references cited therein, also (see [12]) and the references cited therein. fractional calculus (see [10]) and the references cited therein, was introduced at the end of the nineteenth century by liouville and riemann, the subject of which has become a rapidly growing area and has found applications in diverse fields ranging from physical sciences and engineering to biological sciences and economics. definition 1.2. let f ∈ l1[a, b]. the riemann-liouville integrals jαa+f and jαb−f of order α > 0 with a ≥ 0 are defined by jαa+f(x) = 1 γ(α) ∫ x a (x − t)α−1f(t)dt, x > a and jαb−f(x) = 1 γ(α) ∫ b x (t − x)α−1f(t)dt, b > x, where γ(α) = ∫ +∞ 0 e−uuα−1du. here j0a+f(x) = j 0 b−f(x) = f(x). in the case of α = 1, the fractional integral reduces to the classical integral. due to the wide application of fractional integrals, some authors extended to study fractional ostrowski type inequalities for functions of different classes (see [10]) and the references cited therein. now, let us recall some definitions of various convex functions. definition 1.3. (see [2]) a nonnegative function f : i ⊆ r −→ r◦ is said to be p-function or p-convex, if f(tx + (1 − t)y) ≤ f(x) + f(y), ∀x, y ∈ i, t ∈ [0, 1]. definition 1.4. (see [3]) a function f : r◦ −→ r is said to be s-convex in the second sense, if f(λx + (1 − λ)y) ≤ λsf(x) + (1 − λ)sf(y) (1.2) for all x, y ∈ r◦, λ ∈ [0, 1] and s ∈ (0, 1]. ostrowski type fractional integral inequalities 107 it is clear that a 1-convex function must be convex on r◦ as usual. the s-convex functions in the second sense have been investigated in (see [3]). definition 1.5. (see [4]) a set k ⊆ rn is said to be invex with respect to the mapping η : k × k −→ rn, if x + tη(y, x) ∈ k for every x, y ∈ k and t ∈ [0, 1]. notice that every convex set is invex with respect to the mapping η(y, x) = y −x, but the converse is not necessarily true. for more details please see (see [4], [5]) and the references therein. definition 1.6. (see [6]) the function f defined on the invex set k ⊆ rn is said to be preinvex with respect η, if for every x, y ∈ k and t ∈ [0, 1], we have that f (x + tη(y, x)) ≤ (1 − t)f(x) + tf(y). the concept of preinvexity is more general than convexity since every convex function is preinvex with respect to the mapping η(y, x) = y − x, but the converse is not true. the gauss-jacobi type quadrature formula has the following∫ b a (x − a)p(b − x)qf(x)dx = +∞∑ k=0 bm,kf(γk) + r ⋆ m|f|, (1.3) for certain bm,k, γk and rest r ⋆ m|f| (see [7]). recently, liu (see [8]) obtained several integral inequalities for the left hand side of (1.3) under the definition 1.3 of p-function. also in (see [9]), özdemir et al. established several integral inequalities concerning the lefthand side of (1.3) via some kinds of convexity. motivated by these results, in section 2, the notion of generalized (g, s, m, φ)-preinvex function is introduced and some new integral inequalities for the left hand side of (1.3) involving generalized (g, s, m, φ)-preinvex functions are given. in section 3, some generalizations of ostrowski type inequalities for generalized (g, s, m, φ)-preinvex functions via fractional integrals are given. in section 4, some applications to special means are given. 2. new integral inequalities for generalized (g, s, m, φ)-preinvex functions definition 2.1. (see [1]) a set k ⊆ rn is said to be m-invex with respect to the mapping η : k × k × (0, 1] −→ rn for some fixed m ∈ (0, 1], if 108 a. kashuri, r. liko mx + tη(y, x, m) ∈ k holds for each x, y ∈ k and any t ∈ [0, 1]. remark 2.2. in definition 2.1, under certain conditions, the mapping η(y, x, m) could reduce to η(y, x). we next give new definition, to be referred as generalized (g, s, m, φ)preinvex function. definition 2.3. let k ⊆ rn be an open m-invex set with respect to η : k × k × (0, 1] −→ rn, g : [0, 1] −→ [0, 1] be a differentiable function and φ : i −→ r is a continuous increasing function. for f : k −→ r and any fixed s, m ∈ (0, 1], if f ( mφ(x)+g(t)η(φ(y), φ(x), m) ) ≤ m(1−g(t))sf(φ(x))+gs(t)f(φ(y)) (2.1) is valid for all x, y ∈ k, t ∈ [0, 1], then we say that f is a generalized (g, s, m, φ)-preinvex function with respect to η. remark 2.4. in definition 2.3, it is worthwhile to note that the class of generalized (g, s, m, φ)-preinvex function is a generalization of the class of sconvex in the second sense function given in definition 1.4. also, for g(t) = λ, λ ∈ [0, 1] and φ(x) = x, ∀x ∈ k, we get the notion of generalized (s, m)preinvex function (see [1]). example 2.5. let f(x) = −|x|, g(t) = t, φ(x) = x, s = 1 and η(y, x, m) =   y − mx, if x ≥ 0, y ≥ 0; y − mx, if x ≤ 0, y ≤ 0; mx − y, if x ≥ 0, y ≤ 0; mx − y, if x ≤ 0, y ≥ 0. then f(x) is a generalized (t, 1, m, x)-preinvex function of with respect to η : r × r × (0, 1] −→ r and any fixed m ∈ (0, 1]. however, it is obvious that f(x) = −|x| is not a convex function on r. in this section, in order to prove our main results regarding some new integral inequalities involving generalized (g, s, m, φ)-preinvex functions, we need the following new lemma: lemma 2.6. let φ : i −→ r be a continuous increasing function and g : [0, 1] −→ [0, 1] is a differentiable function. assume that f : k = [mφ(a), mφ(a) + η(φ(b), φ(a), m)] −→ r ostrowski type fractional integral inequalities 109 is a continuous function on the interval of real numbers k◦ with respect to η : k × k × (0, 1] −→ r, for φ(a), φ(b) ∈ k, a < b and mφ(a) < mφ(a) + η(φ(b), φ(a), m). then for any fixed m ∈ (0, 1] and p, q > 0, we have∫ mφ(a)+η(φ(b),φ(a),m) mφ(a) ( x − mφ(a) )p( mφ(a) + η ( φ(b), φ(a), m ) − x )q f(x)dx = η ( φ(b), φ(a), m )p+q+1 ∫ 1 0 gp(t) ( 1 − g(t) )q × f ( mφ(a) + g(t)η ( φ(b), φ(a), m )) d[g(t)]. proof. it is easy to observe that∫ mφ(a)+η(φ(b),φ(a),m) mφ(a) ( x − mφ(a) )p( mφ(a) + η ( φ(b), φ(a), m ) − x )q f(x)dx = η ( φ(b), φ(a), m )∫ 1 0 ( mφ(a) + g(t)η ( φ(b), φ(a), m ) − mφ(a) )p × ( mφ(a) + η ( φ(b), φ(a), m ) − mφ(a) − g(t)η ( φ(b), φ(a), m ))q × f ( mφ(a) + g(t)η ( φ(b), φ(a), m )) d[g(t)] = η ( φ(b), φ(a), m )p+q+1 ∫ 1 0 gp(t) ( 1 − g(t) )q × f ( mφ(a) + g(t)η ( φ(b), φ(a), m )) d[g(t)]. the following definition will be used in the sequel. definition 2.7. the euler beta function is defined for x, y > 0 as β(x, y) = ∫ 1 0 tx−1(1 − t)y−1dt = γ(x)γ(y) γ(x + y) . theorem 2.8. let φ : i −→ r be a continuous increasing function and g : [0, 1] −→ [0, 1] is a differentiable function. assume that f : k = [mφ(a), mφ(a) + η(φ(b), φ(a), m)] −→ r is a continuous function on the interval of real numbers k◦ with φ(a), φ(b) ∈ k, a < b with mφ(a) < mφ(a) + η(φ(b), φ(a), m). let k>1. if |f| k k−1 is a generalized (g, s, m, φ)-preinvex function on an open m-invex set k with respect to η : k × k × (0, 1] −→ r for 110 a. kashuri, r. liko any fixed s, m ∈ (0, 1], then for any fixed p, q > 0,∫ mφ(a)+η(φ(b),φ(a),m) mφ(a) ( x − mφ(a) )p( mφ(a) + η ( φ(b), φ(a), m ) − x )q f(x)dx ≤ ∣∣η(φ(b), φ(a), m)∣∣p+q+1 (s + 1) k−1 k b 1 k ( g(t); k, p, q ) × [ m (( 1 − g(0) )s+1 − (1 − g(1))s+1)∣∣f(φ(a))∣∣ kk−1 + ( gs+1(1) − gs+1(0) )∣∣f(φ(b))∣∣ kk−1 ]k−1 k , where b ( g(t); k, p, q ) = ∫ 1 0 gkp(t) ( 1 − g(t) )kq d[g(t)]. proof. since |f| k k−1 is a generalized (g, s, m, φ)-preinvex function on k, combining with lemma 2.6 and hölder inequality for all t ∈ [0, 1] and for any fixed s, m ∈ (0, 1], we get∫ mφ(a)+η(φ(b),φ(a),m) mφ(a) ( x − mφ(a) )p( mφ(a) + η ( φ(b), φ(a), m ) − x )q f(x)dx ≤ ∣∣η(φ(b), φ(a), m)∣∣p+q+1 [∫ 1 0 gkp(t) ( 1 − g(t) )kq d[g(t)] ]1 k × [∫ 1 0 ∣∣∣f(mφ(a) + g(t)η(φ(b), φ(a), m))∣∣∣ kk−1 d[g(t)] ]k−1 k ≤ ∣∣η(φ(b), φ(a), m)∣∣p+q+1b 1k (g(t); k, p, q) × [∫ 1 0 ( m(1 − g(t) )s∣∣f(φ(a))∣∣ kk−1 + gs(t)∣∣f(φ(b))∣∣ kk−1)d[g(t)] ]k−1 k = ∣∣η(φ(b), φ(a), m)∣∣p+q+1 (s + 1) k−1 k b 1 k ( g(t); k, p, q ) × [ m (( 1 − g(0) )s+1 − (1 − g(1))s+1)∣∣f(φ(a))∣∣ kk−1 + ( gs+1(1) − gs+1(0) )∣∣f(φ(b))∣∣ kk−1]k−1k . ostrowski type fractional integral inequalities 111 corollary 2.9. under the conditions of theorem 2.8 for g(t) = t, we get ∫ mφ(a)+η(φ(b),φ(a),m) mφ(a) ( x − mφ(a) )p( mφ(a) + η ( φ(b), φ(a), m ) − x )q f(x)dx ≤ |η ( φ(b), φ(a), m ) |p+q+1 (s + 1) k−1 k [ β(kp + 1, kq + 1) ]1 k × ( m ∣∣f(φ(a))∣∣ kk−1 + ∣∣f(φ(b))∣∣ kk−1)k−1k . theorem 2.10. let φ : i −→ r be a continuous increasing function and g : [0, 1] −→ [0, 1] is a differentiable function. assume that f : k = [mφ(a), mφ(a) + η(φ(b), φ(a), m)] −→ r is a continuous function on the interval of real numbers k◦ with φ(a), φ(b) ∈ k, a < b with mφ(a) < mφ(a)+η(φ(b), φ(a), m). let l ≥ 1. if |f|l is a generalized (g, s, m, φ)-preinvex function on an open m-invex set k with respect to η : k × k × (0, 1] −→ r for any fixed s, m ∈ (0, 1], then for any fixed p, q > 0, ∫ mφ(a)+η(φ(b),φ(a),m) mφ(a) ( x − mφ(a) )p( mφ(a) + η ( φ(b), φ(a), m ) − x )q f(x)dx ≤ ∣∣η(φ(b), φ(a), m)∣∣p+q+1b l−1l (g(t); p, q) × [ m ∣∣f(φ(a))∣∣lb(g(t); p, q + s) + ∣∣f(φ(b))∣∣lb(g(t); p + s, q)]1l , where b(g(t); p, q) = ∫ 1 0 gp(t)(1 − g(t))qd[g(t)]. proof. since |f|l is a generalized (s, m, φ)-preinvex function on k, combining with lemma 2.6 and the well-known power mean inequality for all t ∈ [0, 1] and for any fixed s, m ∈ (0, 1], we get ∫ mφ(a)+η(φ(b),φ(a),m) mφ(a) ( x − mφ(a) )p( mφ(a) + η ( φ(b), φ(a), m ) − x )q f(x)dx = η ( φ(b), φ(a), m )p+q+1 ∫ 1 0 [ gp(t) ( 1 − g(t) )q]l−1l [ gp(t) ( 1 − g(t) )q]1l × f ( mφ(a) + g(t)η ( φ(b), φ(a), m )) d[g(t)] 112 a. kashuri, r. liko ≤ ∣∣η(φ(b), φ(a), m)∣∣p+q+1 [∫ 1 0 gp(t) ( 1 − g(t) )q d[g(t)] ]l−1 l × [∫ 1 0 gp(t) ( 1 − g(t) )q∣∣∣f(mφ(a) + g(t)η(φ(b), φ(a), m))∣∣∣ld[g(t)] ]1 l ≤ ∣∣η(φ(b), φ(a), m)∣∣p+q+1b l−1l (g(t); p, q) × [∫ 1 0 gp(t) ( 1 − g(t) )q( m ( 1 − g(t) )s∣∣f(φ(a))∣∣l + gs(t)∣∣f(φ(b))∣∣l)d[g(t)] ]1 l = ∣∣η(φ(b), φ(a), m)∣∣p+q+1b l−1l (g(t); p, q) × [ m ∣∣f(φ(a))∣∣lb(g(t); p, q + s) + ∣∣f(φ(b))∣∣lb(g(t); p + s, q)]1l . corollary 2.11. under the conditions of theorem 2.10 for g(t) = t, we get∫ mφ(a)+η(φ(b),φ(a),m) mφ(a) ( x − mφ(a) )p( mφ(a) + η ( φ(b), φ(a), m ) − x )q f(x)dx ≤ ∣∣η(φ(b), φ(a), m)∣∣p+q+1[β(p + 1, q + 1)]l−1l × [ m ∣∣f(φ(a))∣∣lβ (p + 1, q + s + 1) + ∣∣f(φ(b))∣∣lβ (p + s + 1, q + 1)]1l . 3. ostrowski type fractional integral inequalities for generalized (g, s, m, φ)-preinvex functions in this section, in order to prove our main results regarding some generalizations of ostrowski type inequalities for generalized (g, s, m, φ)-preinvex functions via fractional integrals, we need the following new lemma: lemma 3.1. let φ : i −→ r be a continuous increasing function and g : [0, 1] −→ [0, 1] is a differentiable function. suppose k ⊆ r be an open m-invex subset with respect to η : k ×k ×(0, 1] −→ r for any fixed m ∈ (0, 1] and let φ(a), φ(b) ∈ k, a < b with mφ(a) < mφ(a)+η(φ(b), φ(a), m). assume ostrowski type fractional integral inequalities 113 that f : k −→ r is a differentiable function on k◦ and f ′ is integrable on [mφ(a), mφ(a) + η(φ(b), φ(a), m)]. then for α > 0, we have η ( φ(x), φ(a), m )α η ( φ(b), φ(a), m ) [gα(1)f(mφ(a) + g(1)η(φ(x), φ(a), m)) − gα(0)f ( mφ(a) + g(0)η ( φ(x), φ(a), m ))] − η ( φ(x), φ(b), m )α η ( φ(b), φ(a), m ) [gα(1)f(mφ(b) + g(1)η(φ(x), φ(b), m)) − gα(0)f ( mφ(b) + g(0)η ( φ(x), φ(b), m ))] (3.1) − α η ( φ(b), φ(a), m )[∫ mφ(a)+g(1)η(φ(x),φ(a),m) mφ(a)+g(0)η(φ(x),φ(a),m) ( t − mφ(a) )α−1 f(t)dt − ∫ mφ(b)+g(1)η(φ(x),φ(b),m) mφ(b)+g(0)η(φ(x),φ(b),m) ( t − mφ(b) )α−1 f(t)dt ] = η ( φ(x), φ(a), m )α+1 η ( φ(b), φ(a), m ) ∫ 1 0 gα(t)f ′ ( mφ(a) + g(t)η ( φ(x), φ(a), m )) d[g(t)] − η ( φ(x), φ(b), m )α+1 η ( φ(b), φ(a), m ) ∫ 1 0 gα(t)f ′ ( mφ(b) + g(t)η ( φ(x), φ(b), m )) d[g(t)]. proof. a simple proof of the equality can be done by performing an integration by parts in the integrals from the right side and changing the variable. the details are left to the interested reader. remark 3.2. clearly, if we choose m = 1, g(t) = t, η(φ(y), φ(x), m) = φ(y) − mφ(x) and φ(x) = x, ∀x, y ∈ k in lemma 3.1, we get lemma 1 in [11]. let denote sf,g,η,φ(x; α, m, a, b) = η ( φ(x), φ(a), m )α+1 η ( φ(b), φ(a), m ) ∫ 1 0 gα(t)f ′ ( mφ(a) + g(t)η ( φ(x), φ(a), m )) d[g(t)] − η ( φ(x), φ(b), m )α+1 η ( φ(b), φ(a), m ) ∫ 1 0 gα(t)f ′ ( mφ(b) + g(t)η ( φ(x), φ(b), m )) d[g(t)]. by using lemma 3.1, one can extend to the following results. 114 a. kashuri, r. liko theorem 3.3. let φ : i −→ r be a continuous increasing function and g : [0, 1] −→ [0, 1] is a differentiable function. suppose a ⊆ r be an open m-invex subset with respect to η : a × a × (0, 1] −→ r for some fixed s, m ∈ (0, 1] and let φ(a), φ(b) ∈ a, a < b with mφ(a) < mφ(a) + η(φ(b), φ(a), m). assume that f : a −→ r is a differentiable function on a◦. if |f ′|q is a generalized (g, s, m, φ)-preinvex function on [mφ(a), mφ(a)+η(φ(b), φ(a), m)], q > 1, p−1 + q−1 = 1, then for α > 0, we have |sf,g,η,φ(x; α, m, a, b)| (3.2) ≤ 1 (s + 1)1/q ( gpα+1(1) − gpα+1(0) pα + 1 )1 p 1∣∣η(φ(b), φ(a), m)∣∣ × {∣∣η(φ(x), φ(a), m)∣∣α+1[m((1 − g(0))s+1 − (1 − g(1))s+1)∣∣f ′(φ(a))∣∣q + ( gs+1(1) − gs+1(0) )∣∣f ′(φ(x))∣∣q]1q + ∣∣η(φ(x), φ(b), m)∣∣α+1[m((1 − g(0))s+1 − (1 − g(1))s+1) |f ′(φ(b))|q + ( gs+1(1) − gs+1(0) )∣∣f ′(φ(x))∣∣q]1q } . proof. suppose that q > 1. using lemma 3.1, generalized (g, s, m, φ)preinvexity of |f ′|q, hölder inequality and taking the modulus, we have |sf,g,η,φ(x; α, m, a, b)| ≤ ∣∣η(φ(x), φ(a), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ ∫ 1 0 gα(t) ∣∣∣f ′(mφ(a) + g(t)η(φ(x), φ(a), m))∣∣∣d[g(t)] + ∣∣η(φ(x), φ(b), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ ∫ 1 0 gα(t) ∣∣∣f ′(mφ(b) + g(t)η(φ(x), φ(b), m))∣∣∣d[g(t)] ≤ ∣∣η(φ(x), φ(a), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ (∫ 1 0 gpα(t)d[g(t)] )1 p × (∫ 1 0 ∣∣∣f ′(mφ(a) + g(t)η(φ(x), φ(a), m))∣∣∣qd[g(t)])1q ostrowski type fractional integral inequalities 115 + ∣∣η(φ(x), φ(b), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ (∫ 1 0 gpα(t)d[g(t)] )1 p × (∫ 1 0 ∣∣∣f ′(mφ(b) + g(t)η(φ(x), φ(b), m))∣∣∣qd[g(t)])1q ≤ ∣∣η(φ(x), φ(a), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ (∫ 1 0 gpα(t)d[g(t)] )1 p × [∫ 1 0 ( m ( 1 − g(t) )s∣∣f ′(φ(a))∣∣q + gs(t)∣∣f ′(φ(x))∣∣q)d[g(t)] ]1 q + ∣∣η(φ(x), φ(b), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ (∫ 1 0 gpα(t)d[g(t)] )1 p × [∫ 1 0 ( m ( 1 − g(t) )s∣∣f ′(φ(b))∣∣q + gs(t)∣∣f ′(φ(x))∣∣q)d[g(t)] ]1 q = 1 (s + 1)1/q ( gpα+1(1) − gpα+1(0) pα + 1 )1 p 1∣∣η(φ(b), φ(a), m)∣∣ × {∣∣η(φ(x), φ(a), m)∣∣α+1[m((1 − g(0))s+1 − (1 − g(1))s+1) |f ′(φ(a))|q + ( gs+1(1) − gs+1(0) )∣∣f ′(φ(x))∣∣q]1q + ∣∣η(φ(x), φ(b), m)∣∣α+1[m((1 − g(0))s+1 − (1 − g(1))s+1) |f ′(φ(b))|q + ( gs+1(1) − gs+1(0) )∣∣f ′(φ(x))∣∣q]1q } . corollary 3.4. under the conditions of theorem 3.3 for g(t) = t and |f ′| ≤ k, we get 116 a. kashuri, r. liko 1∣∣η(φ(b), φ(a), m)∣∣ ∣∣∣∣∣η(φ(x), φ(a), m)αf ( mφ(a) + η ( φ(x), φ(a), m )) − η ( φ(x), φ(b), m )α f ( mφ(b) + η ( φ(x), φ(b), m )) − γ(α + 1) [ jα( mφ(a)+η(φ(x),φ(a),m) ) − f ( mφ(a) ) − jα( mφ(b)+η(φ(x),φ(b),m) ) − f ( mφ(b) )]∣∣∣∣∣ ≤ k (pα + 1)1/p ( m + 1 s + 1 )1 q [∣∣η(φ(x), φ(a), m)∣∣α+1 + ∣∣η(φ(x), φ(b), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ ] . theorem 3.5. let φ : i −→ r be a continuous increasing function and g : [0, 1] −→ [0, 1] is a differentiable function. suppose a ⊆ r be an open minvex subset with respect to η : a×a×(0, 1] −→ r for some fixed s, m ∈ (0, 1] and let φ(a), φ(b) ∈ a, a < b with mφ(a) < mφ(a)+η(φ(b), φ(a), m). assume that f : a −→ r is a differentiable function on a◦. if |f ′|q is a generalized (g, s, m, φ)-preinvex function on [mφ(a), mφ(a)+η(φ(b), φ(a), m)], q ≥ 1, then for α > 0, we have ∣∣sf,g,η,φ(x; α, m, a, b)∣∣ ≤ (gα+1(1) − gα+1(0) α + 1 )1− 1 q 1∣∣η(φ(b), φ(a), m)∣∣ (3.3) × { |η ( φ(x), φ(a), m ) |α+1 [ m ∣∣f ′(φ(a))∣∣qb(g(t); α, s) + ( gα+s+1(1) − gα+s+1(0) α + s + 1 )∣∣f ′(φ(x))∣∣q ]1 q + ∣∣η(φ(x), φ(b), m)∣∣α+1 [ m ∣∣f ′(φ(b))∣∣qb(g(t); α, s) + ( gα+s+1(1) − gα+s+1(0) α + s + 1 )∣∣f ′(φ(x))∣∣q ]1 q } , where b(g(t); α, s) = ∫ 1 0 gα(t)(1 − g(t))sd[g(t)]. ostrowski type fractional integral inequalities 117 proof. suppose that q ≥ 1. using lemma 3.1, generalized (g, s, m, φ)preinvexity of |f ′|q, the well-known power mean inequality and taking the modulus, we have |sf,g,η,φ(x; α, m, a, b)| ≤ ∣∣η(φ(x), φ(a), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ ∫ 1 0 gα(t) ∣∣∣f ′(mφ(a) + g(t)η(φ(x), φ(a), m))∣∣∣d[g(t)] + ∣∣η(φ(x), φ(b), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ ∫ 1 0 gα(t) ∣∣∣f ′(mφ(b) + g(t)η(φ(x), φ(b), m))∣∣∣d[g(t)] ≤ ∣∣η(φ(x), φ(a), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ (∫ 1 0 gα(t)d[g(t)] )1− 1 q × (∫ 1 0 gα(t) ∣∣∣f ′(mφ(a) + g(t)η(φ(x), φ(a), m))∣∣∣qd[g(t)])1q + ∣∣η(φ(x), φ(b), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ (∫ 1 0 gα(t)d[g(t)] )1− 1 q × (∫ 1 0 gα(t) ∣∣∣f ′(mφ(b) + g(t)η(φ(x), φ(b), m))∣∣∣qd[g(t)])1q ≤ ∣∣η(φ(x), φ(a), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ (∫ 1 0 gα(t)d[g(t)] )1− 1 q × [∫ 1 0 gα(t) ( m ( 1 − g(t) )s∣∣f ′(φ(a))∣∣q + gs(t)∣∣f ′(φ(x))∣∣q ) d[g(t)] ]1 q + ∣∣η(φ(x), φ(b), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ (∫ 1 0 gα(t)d[g(t)] )1− 1 q × [∫ 1 0 gα(t) ( m ( 1 − g(t) )s∣∣f ′(φ(b))∣∣q + gs(t)∣∣f ′(φ(x))∣∣q ) d[g(t)] ]1 q 118 a. kashuri, r. liko = ( gα+1(1) − gα+1(0) α + 1 )1− 1 q 1∣∣η(φ(b), φ(a), m)∣∣ × {∣∣η(φ(x), φ(a), m)∣∣α+1 [ m ∣∣f ′(φ(a))∣∣qb(g(t); α, s) + ( gα+s+1(1) − gα+s+1(0) α + s + 1 )∣∣f ′(φ(x))∣∣q ]1 q + ∣∣η(φ(x), φ(b), m)∣∣α+1 [ m ∣∣f ′(φ(b))∣∣qb(g(t); α, s) + ( gα+s+1(1) − gα+s+1(0) α + s + 1 )∣∣f ′(φ(x))∣∣q ]1 q } . corollary 3.6. under the conditions of theorem 3.5 for g(t) = t and |f ′| ≤ k, we get 1∣∣η(φ(b), φ(a), m)∣∣ ∣∣∣∣∣η(φ(x), φ(a), m)αf ( mφ(a) + η ( φ(x), φ(a), m )) − η ( φ(x), φ(b), m )α f ( mφ(b) + η ( φ(x), φ(b), m )) − γ(α + 1) [ jα(mφ(a)+η(φ(x),φ(a),m))−f ( mφ(a) ) − jα(mφ(b)+η(φ(x),φ(b),m))−f ( mφ(b) )]∣∣∣∣∣ ≤ k (1 + α) 1− 1 q ( mβ(α + 1, s + 1) + 1 α + s + 1 )1 q × [∣∣η(φ(x), φ(a), m)∣∣α+1 + ∣∣η(φ(x), φ(b), m)∣∣α+1∣∣η(φ(b), φ(a), m)∣∣ ] . remark 3.7. for a particular choices of a differentiable function g : [0, 1] −→ [0, 1], for example: e−t, ln(t + 1), sin ( πt 2 ) , cos ( πt 2 ) , etc., by our theorems mentioned in this paper we can get some special kinds of ostrowski type fractional inequalities. ostrowski type fractional integral inequalities 119 4. applications to special means in the following we give certain generalizations of some notions for a positive valued function of a positive variable. definition 4.1. (see [13]) a function m : r2+ −→ r+, is called a mean function if it has the following properties: 1. homogeneity: m(ax, ay) = am(x, y), for all a > 0, 2. symmetry: m(x, y) = m(y, x), 3. reflexivity: m(x, x) = x, 4. monotonicity: if x ≤ x′ and y ≤ y′, then m(x, y) ≤ m(x′, y′), 5. internality: min{x, y} ≤ m(x, y) ≤ max{x, y}. we consider some means for arbitrary positive real numbers α, β (α ̸= β). 1. the arithmetic mean: a := a(α, β) = α + β 2 2. the geometric mean: g := g(α, β) = √ αβ 3. the harmonic mean: h := h(α, β) = 2 1 α + 1 β 4. the power mean: pr := pr(α, β) = ( αr + βr 2 )1 r , r ≥ 1. 5. the identric mean: i := i(α, β) = { 1 e ( ββ αα ) , α ̸= β; α, α = β. 120 a. kashuri, r. liko 6. the logarithmic mean: l := l(α, β) = β − α ln |β| − ln |α| ; |α| ≠ |β|, αβ ̸= 0. 7. the generalized log-mean: lp := lp(α, β) = [ βp+1 − αp+1 (p + 1)(β − α) ]1 p ; p ∈ r \ {−1, 0}, α ̸= β. 8. the weighted p-power mean: mp ( α1, α2, · · · αn u1, u2, · · · un ) = ( n∑ i=1 αiu p i )1 p where 0 ≤ αi ≤ 1, ui > 0 (i = 1, 2, . . . , n) with ∑n i=1 αi = 1. it is well known that lp is monotonic nondecreasing over p ∈ r with l−1 := l and l0 := i. in particular, we have the following inequality h ≤ g ≤ l ≤ i ≤ a. now, let a and b be positive real numbers such that a < b. consider the function m := m(φ(a), φ(b)) : [φ(a), φ(a) + η(φ(b), φ(a))] × [φ(a), φ(a) + η(φ(b), φ(a))] −→ r+, which is one of the above mentioned means, φ : i −→ r be a continuous increasing function and g : [0, 1] −→ [0, 1] is a differentiable function. therefore one can obtain various inequalities using the results of section 3 for these means as follows: replace η(φ(y), φ(x), m) with η(φ(y), φ(x)) and setting η(φ(a), φ(b)) = m(φ(a), φ(b)) for m = 1 in (3.2) and (3.3), one can obtain the following ostrowski type fractional integral inequalities 121 interesting inequalities involving means: 1∣∣m(φ(a), φ(b))∣∣ ∣∣∣∣∣m(φ(a), φ(x))α [ gα(1)f ( φ(a) + g(1)m ( φ(a), φ(x) )) − gα(0)f ( φ(a) + g(0)m ( φ(a), φ(x) ))] − m ( φ(b), φ(x) )α[ gα(1)f ( φ(b) + g(1)m ( φ(b), φ(x) )) − gα(0)f ( φ(b) + g(0)m ( φ(b), φ(x) ))] − α [∫ φ(a)+g(1)m(φ(a),φ(x)) φ(a)+g(0)m ( φ(a),φ(x) ) (t − φ(a))α−1f(t)dt − ∫ φ(b)+g(1)m(φ(b),φ(x)) φ(b)+g(0)m ( φ(b),φ(x) ) (t − φ(b))α−1f(t)dt ]∣∣∣∣∣ ≤ 1 (s + 1)1/q ( gpα+1(1) − gpα+1(0) pα + 1 )1 p 1 m ( φ(a), φ(b) ) (4.1) × { m ( φ(a), φ(x) )α+1[(( 1 − g(0) )s+1 − (1 − g(1))s+1)∣∣f ′(φ(a))∣∣q + ( gs+1(1) − gs+1(0) )∣∣f ′(φ(x))∣∣q]1q + m ( φ(b), φ(x) )α+1[(( 1 − g(0) )s+1 − (1 − g(1))s+1)∣∣f ′(φ(b))∣∣q + ( gs+1(1) − gs+1(0) )∣∣f ′(φ(x))∣∣q]1q } , 1 m ( φ(a), φ(b) )∣∣∣∣∣m(φ(a), φ(x))α [ gα(1)f ( φ(a) + g(1)m ( φ(a), φ(x) )) − gα(0)f ( φ(a) + g(0)m ( φ(a), φ(x) ))] − m ( φ(b), φ(x) )α[ gα(1)f ( φ(b) + g(1)m ( φ(b), φ(x) )) − gα(0)f ( φ(b) + g(0)m ( φ(b), φ(x) ))] − α [∫ φ(a)+g(1)m(φ(a),φ(x)) φ(a)+g(0)m ( φ(a),φ(x) ) (t − φ(a))α−1f(t)dt 122 a. kashuri, r. liko − ∫ φ(b)+g(1)m(φ(b),φ(x)) φ(b)+g(0)m ( φ(b),φ(x) ) (t − φ(b))α−1f(t)dt ]∣∣∣∣∣ ≤ ( gα+1(1) − gα+1(0) α + 1 )1− 1 q 1 m ( φ(a), φ(b) ) (4.2) × { m ( φ(a), φ(x) )α+1[∣∣f ′(φ(a))∣∣qb(g(t); α, s) + ( gα+s+1(1) − gα+s+1(0) α + s + 1 )∣∣f ′(φ(x))∣∣q ]1 q + m ( φ(b), φ(x) )α+1[∣∣f ′(φ(b))∣∣qb(g(t); α, s) + ( gα+s+1(1) − gα+s+1(0) α + s + 1 )∣∣f ′(φ(x))∣∣q ]1 q } . letting m(φ(a), φ(b)) = a, g, h, pr, i, l, lp, mp in (4.1) and (4.2), we get the inequalities involving means for a particular choice of a differentiable generalized (g, s, 1, φ)-preinvex functions f. the details are left to the interested reader. references [1] t. s. du, j. g. liao, y. j. li, properties and integral inequalities of hadamard-simpson type for the generalized (s, m)-preinvex functions, j. nonlinear sci. appl. 9 (5) (2016), 3112 – 3126. 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[13] p. s. bullen, “handbook of means and their inequalities”, kluwer academic publishers group, dordrecht, 2003. e extracta mathematicae vol. 31, núm. 1, 1 – 10 (2016) on small combination of slices in banach spaces sudeshna basu, t. s. s. r. k. rao department of mathematics, george washington university, washington dc 20052, usa sbasu@gwu.edu, sudeshna66@gmail.com stat-math division, indian statistical institute, r. v. college, p. o. bangalore 560059, india tss@isibang.ac.in, srin@fulbrightmail.org presented by pier l. papini received august 13, 2015 abstract: the notion of small combination of slices (scs) in the unit ball of a banach space was first introduced in [4] and subsequently analyzed in detail in [12] and [13]. in this work, we introduce the notion of bscsp, which can be seen as a generalization of dentability in terms of scs. we study certain stability results for the w∗-bscsp leading to a discussion on bscsp in the context of ideals of banach spaces. we prove that the w∗-bscsp can be lifted from a m-ideal to the whole banach space. we also prove similar results for strict ideals and u-subspaces of a banach space. we note that the space c(k, x)∗ has w∗-bscsp when k is dispersed and x∗ has the w∗-bscsp. key words: small combination of slices, m-ideals, strict ideals, u-subspaces. ams subject class. (2010): 46b20, 46b28. 1. introduction let x be a real banach space and x∗ its dual. we will denote by bx, sx and bx(x, r) the closed unit ball, the unit sphere and the closed ball of radius r > 0 and center x. we refer to the monograph [2] for notions of convexity theory that we will be using here. definition 1. (i) we say a ⊆ bx∗ is a norming set for x if ∥x∥ = sup{|x∗(x)| : x∗ ∈ a}, for all x ∈ x. a closed subspace f ⊆ x∗ is a norming subspace if bf is a norming set for x. (ii) let f ∈ x∗, α > 0 and c ⊆ x. then the set s(c, f, α) = {x ∈ c : f(x) > sup f(c) − α} is called the open slice determined by f and α. we assume without loss of generality that ∥f∥ = 1. one can analogously define w∗ slices in x∗ 1 2 s. basu, t.s.s.r.k. rao (iii) a point x ̸= 0 in a convex set k ⊆ x is called a scs (small combination of slices) point of k, if for every ε > 0, there exist slices si of k, and a convex combination s = ∑n i=1 λisi such that x ∈ s and diam(s) < ε. one can analogously define w∗-scs point in x∗. we introduce the following definition analogous to that of a unit ball being dentable, see [2]. definition 2. a banach space is said to have ball-small combination of slices property (bscsp) if the unit ball has small combination of slices of arbitrarily small diameter. analogously we can define w∗-bscsp in a dual space. remark 3. (i) it is clear that if bx has a scs point, then it has bscsp. (ii) strongly regular spaces studied in [4] and [13] were referred to as small combination of slices property (scsp) in [12]. scs points were first introduced in [4] as a “slice generalization” of the notion pc (i.e. points for which the identity mapping on the unit ball, from weak topology to norm topology is continuous). it was proved in [4] that x is strongly regular (respectively, x∗ is w∗-strongly regular) if and only if every non empty bounded convex set k in x (respectively k in x∗) is contained in the norm closure (respectively, w∗-closure) of scs(k) (respectively w∗scs(k)), i.e. the scs points (w∗-scs points) of k. later, it was proved in [13] that banach space has radon nikodym property (rnp) if and only if it is strongly regular and has the krein-milman property (kmp). subsequently, the concept of scs points was used in [12] to investigate the structure of non dentable closed bounded convex sets in banach spaces. in this work, we study certain stability results for w∗−bscsp leading to a discussion on bscsp in the context of ideals of banach spaces, see [5] and [12]. we use various techniques from the geometric theory of banach spaces to achieve this. the spaces that we will be considering have been well studied in the literature. a large class of function spaces like the bloch spaces, lorentz and orlicz spaces, spaces of vector-valued functions and spaces of compact operators are examples of the spaces we will be considering: for details, see [6]. we provide some descriptions of w∗-scs points in banach spaces in different contexts. we need the following definition. on small combination of slices in banach spaces 3 definition 4. let x be a banach space. (i) a linear projection p on x is called an m-projection if ∥x∥ = max{∥px∥, ∥x − px∥}, for all x ∈ x; a linear projection p on x is called an l-projection if ∥x∥ = ∥px∥ + ∥x − px∥ for all x ∈ x. (ii) a subspace m ⊆ x is called an m-summand if it is the range of an mprojection. a closed subspace m ⊆ x is called an l-summand if it is the range of an l-projection. (iii) a subspace m ⊆ x is called an m-ideal if m⊥ is the kernel of an lprojection in x∗ we recall from [6, chapter i] that when m ⊂ x is an m-ideal, elements of m∗ have unique norm-preserving extension to x∗ and one has the identification, x∗ = m∗ ⊕1 m⊥. several examples from among function spaces and spaces of operators that satisfy these geometric properties can be found in the monograph [6], see also [8]. first, we prove that for an l-summand m ⊂ x, if a scs point of bx has a non-zero component m ∈ m, then m is a scs point of bm. for an mideal m ⊂ x, this yields: any w∗-scs point of bx∗, if its restriction to m, say m∗, has the same norm, then m∗ it is a w∗-scs point of bm∗ . we prove a similar result for a u-subspace of a banach space of x. we prove a converse statement for a strict ideal y ⊂ x (see section 2 for the definition) i.e., we prove that a w∗-scs point of a strict ideal continues to be so in the bigger space. we also prove corresponding results for the bscsp. 2. stability results we will use the standard notation of ⊕1, ⊕∞ to denote the ℓ1 and ℓ∞-direct sum of two or more banach spaces. proposition 5. suppose x, y , z are banach spaces such that z = x ⊕1 y ; suppose z0 = (x0, y0) ∈ bz is a scs point of bz with both the components non-zero, then x0 and y0 are scs points of bx and by respectively. 4 s. basu, t.s.s.r.k. rao proof. since z0 is a scs point of bz, we have for any ε > 0, z0 = ∑n i=1 λizi, where zi ∈ si and for z∗i = (x ∗ i , y ∗ i ) with 1 = ∥z ∗ i ∥ = max{∥x ∗ i ∥, ∥y ∗ i ∥}, si = {z ∈ bz/z∗i (z) > 1 − εi} and diam( ∑n i=1 λisi) < ε, si = {z ∈ bz/z∗i (x, y) > 1 − εi} = {z ∈ bz/x ∗ i (x) + y ∗ i (y) > 1 − εi}. since zi = (xi, yi) ∈ si, then x∗i (xi) + y ∗ i (yi) > 1 − εi. case 1 : ∥z∗i ∥ = ∥x ∗ i ∥ = 1. then, x∗i (xi) + y ∗ i (yi) > 1 − εi = ∥x ∗ i ∥ − εi, =⇒ x∗i (xi) > ∥x ∗ i ∥ − εi − y ∗ i (yi), =⇒ 1 ≥ x∗i (xi) > ∥x ∗ i ∥ − βi, where βi = εi + y ∗ i (yi), =⇒ εi + y∗i (yi) > 0. so we have, xi ∈ six ={x ∈bx/x∗i (x)>1−βi}. then (xi, yi)∈ six×{yi}⊆si. case 2: ∥z∗i ∥ = ∥y ∗ i ∥ = 1. we may assume that 0 < ∥x ∗ i ∥ < 1, and let δi = ∥y∗i ∥ − ∥x ∗ i ∥. then, x∗i (xi) + y ∗ i (yi) > 1 − εi = ∥y ∗ i ∥ − εi = ∥x ∗ i ∥ + δi − εi =⇒ x∗i (xi) > ∥x ∗ i ∥ + δi − εi − y ∗ i (yi), =⇒ ∥x∗i ∥ ≥ x ∗ i (xi) > ∥x ∗ i ∥ − ri, where ri = δi − εi − y ∗ i (yi) > 0, =⇒ xi ∈ six = {x ∈ bx/x∗i (x) > 1 − ri}. then (xi, yi) ∈ six × {yi} ⊆ si. let x0 = ∑n i=1 λixi and y0 = ∑n i=1 λiyi. now x0 ∈ ∑n i=1 λisix. also, n∑ i=1 λi[six × yi] ⊆ n∑ i=1 λisi, =⇒ n∑ i=1 λi[six] × {y0} ⊆ n∑ i=1 λi[six × yi] ⊆ n∑ i=1 λisi, =⇒ diam ( n∑ i=1 λisix ) < ε, =⇒ x0 is a scs point of bx. similarly it follows that y0 is a scs point of by . on small combination of slices in banach spaces 5 arguments similar to the ones given above in the context of a ℓ∞-sum yield the following corollary. corollary 6. suppose x, y , z are banach spaces such that z = x ⊕∞ y , suppose z∗ = (x∗, y∗) ∈ bz∗ is a w∗-scs point of bz∗ with both the components non-zero, then x∗ and y∗ are w∗-scs points of bx∗ and by ∗ respectively. remark 7. since in the sequence space ℓ∞ any weakly open set has norm diameter 2, by taking x = c0 and y = ℓ 1, z = x ⊕∞ y , any w∗-scs point of bz∗ has its second component 0. we thank the referee for this observation. definition 8. we recall that a closed subspace y of a banach space x is called a u-subspace if for y∗ ∈ y ∗ there exists a unique norm preserving extension of y∗ in x∗. we continue to denote the unique extension also by y∗. see the discussion on [6, page 44] and the references in that monograph for several examples of u-subspaces from among classical function spaces and spaces of operators. before the next result we also need a definition from [5]. see also [11] for more information and several examples from spaces of operators and tensor product spaces. definition 9. a closed subspace y of a banach space x is said to be an ideal of x if there is a linear projection p : x∗ → x∗ of norm one such that ker(p) = y ⊥. for x∗ ∈ x∗ since p(x∗) − x∗ = 0 on y , as ∥p∥ = 1, we see that p(x∗) is a norm-preserving extension of x∗|y . theorem 10. suppose y is an ideal which is also a u-subspace of x. if y∗ ∈ sy ∗ is a w∗-scs point of bx∗, then y∗ is a w∗-scs point of by ∗. proof. let y∗0 ∈ sy ∗ be a w ∗-scs point of bx∗ , hence for any ε > 0 there exist w∗ slices si, 0 ≤ λi ≤ 1, i = 1, 2, . . . , n, si = {x∗ ∈ bx∗/x∗(xi) > 1−αi} and diam( ∑n i=1 λisi) < ε and y ∗ 0 = ∑ λix ∗ 0i. since y ∗ 0 ∈ sy ∗ and y is a usubspace, y∗0 has unique norm preserving extension in x ∗. let p : x∗ −→ x∗ be the canonical projection. then ∥p(y∗0)∥ = ∥y ∗ 0∥ = 1, also, 1 = ∥y∗0∥ = ∥∥∥∥ n∑ i=1 λix ∗ 0i ∥∥∥∥ ≤ n∑ i=1 λi∥p(x ∗ 0i)∥ ≤ 1. 6 s. basu, t.s.s.r.k. rao this implies ∥p(x∗0i)∥ = ∥x0i ∗∥ = 1 for all i = 1, . . . , n. thus by hypothesis, p(x∗0i) and the restriction of x ∗ 0i to y are denoted by y ∗ 0i. now y ∗ 0i ∈ si, then y∗0i(xi) > 1 − αi. also, since y is an ideal, there exists an operator t : span{xi} −→ y such that ∥t(xi)∥ ≤ (1 + ε)∥xi∥ = 1 + ε. let yi = t(xi). hence, y∗0i(xi) > 1 − αi =⇒ y ∗ 0i(yi − yi + xi) > 1 − αi, =⇒ y∗0i(yi) + y ∗ 0i(xi − yi) > 1 − αi, =⇒ y∗0i(yi) > 1 − αi − y ∗ 0i(xi − yi). case 1: ∥yi∥ =1. so we have 1 > y∗0i(yi) > 1 − αi−y ∗ 0i(xi − yi) = 1 − βi, =⇒ y∗0i ∈ siy = {y ∗ ∈ by ∗/y∗(yi) > 1 − βi}. case 2: ∥yi∥ < 1. let ∥yi∥ = 1 − δi. then ∥yi∥ > y∗0i(yi) > ∥yi∥ + δi − βi = ∥yi∥ − (βi − δi) = ∥yi∥ − γi, γi > 0, =⇒ y∗0i ∈ siy = {y ∗ ∈ by ∗/y∗(yi) > ∥yi∥ − γi}. case 3: ∥yi∥ = 1 + δi. then 1 + δi > y ∗ 0i(yi) > 1 − βi = 1 + δi − (βi + δi), =⇒ y∗0i ∈ siy = {y ∗ ∈ by ∗/y∗(yi) > ∥yi∥ − (βi + δi}. hence y∗0 = n∑ i=1 λiy ∗ 0i ∈ n∑ i=1 λisiy ⊆ n∑ i=1 λisi. hence diam ( n∑ i=1 λisiy ) < diam ( n∑ i=1 λisi ) < ε. thus y∗0 is w ∗-scs point of by ∗. let m ⊆ x be an m-ideal. it follows from the results in [6, chapter i] that any x∗ ∈ x∗, if ∥m∗∥ = ∥x∗|m∥ = ∥x∗∥, then x∗ is the unique norm preserving extension of m∗. for notational convenience we denote both the functionals by m∗. clearly any m-ideal is also an ideal. thus the following corollary answers a natural question in this context for w∗-scs points of the unit sphere. we omit its easy proof. on small combination of slices in banach spaces 7 corollary 11. suppose m ⊆ x is a m-ideal in x. if m∗ ∈ sx∗ is w∗-scs point of bx∗, then m ∗ ∈ sm∗ is a w∗-scs point of bm∗. remark 12. the referee has kindly pointed out an independent proof to show that for z = x ⊕1 y , z has the bscsp if and only if x or y has the bscsp. arguments similar to the ones given during the proof of proposition 5 can be used to show that for z = x ⊕∞ y , if x∗ or y ∗ has the w∗-bscsp then so does z∗. in the case of an m-ideal m ⊂ x, for the sake of completeness we give a detailed proof of the following result. proposition 13. let m ⊆ x be a m-ideal, then if m∗ has the w∗bscsp then x∗ has the w∗-bscsp. proof. suppose m∗ has the w∗-bscsp, then for any ε > 0 there exists slices sim and 0 ≤ λi ≤ 1, i = 1, 2, . . . , n, sim = {m∗ ∈ bm∗/m∗(mi) > 1 − αi} and diam( ∑n i=1 λisim) < ε. since m is an mideal, for any x ∗ ∈ x∗ we have the unique decomposition, x∗ = m∗ + m⊥, where m∗ ∈ m∗ and m⊥ ∈ m⊥. suppose we have 0 < µi < αi. then six = {x∗ ∈ bx∗/x∗(mi) > 1 − µi} = {x∗ ∈ bx∗/m∗(mi) + m⊥(mi) > 1 − µi}, ⊆ sim × µibm⊥, =⇒ n∑ i=1 λisix ⊆ n∑ i=1 λisim × µibm⊥. choose βi = min(µi, ε). then s′ix = {x ∗ ∈ bx∗/x∗(mi) > 1 − βi} ⊆ six × βibm⊥, =⇒ n∑ i=1 λis ′ ix ⊆ ( n∑ i=1 λisim × βibm⊥ ) =⇒ n∑ i=1 λis ′ ix ⊆ ( n∑ i=1 λisim × βibm⊥ ) . thus diam( ∑n i=1 λis ′ ix) ≤ diam( ∑n i=1 λisim) + 2ε < ε + 2ε = 3ε. also, since ∥mi∥ = 1, there exists m∗i ∈ bm∗ such that m ∗ i (mi) > 1−βi. hence m ∗ i ∈ s ′ ix. similarly, ∑n i=1 λim ∗ i ∈ ∑n i=1 λis ′ ix =⇒ ∑n i=1 λis ′ i ̸= ∅. 8 s. basu, t.s.s.r.k. rao since any summand in a ℓ∞-direct sum is in particular an m-ideal of the sum, the following corollary is easy to prove. corollary 14. suppose x = ⊕ℓ∞xi. if x∗i has the w ∗-bscsp for some i, then x∗ has the w∗-bscsp. the above arguments extend easily to vector-valued continuous functions. we recall that for a compact hausdorff space k, c(k, x) denotes the space of continuous x-valued functions on k, equipped with the supremum norm. we recall from [9] that dispersed compact hausdorff spaces have isolated points. corollary 15. suppose k is a compact hausdorff space with an isolated point. if x∗ has the w∗-bscsp, then c(k, x)∗ has the w∗-bscsp. proof. suppose x∗ has the w∗-bscsp. for an isolated point k0 ∈ k, the map f → χk0f is an m-projection in c(k, x) whose range is isometric to x. hence we see that c(k, x)∗ has the w∗-bscsp. we recall that an ideal y is said to be a strict ideal if for a projection p : x∗ → x∗ with ∥p∥ = 1, ker(p) = y ⊥, one also has bp(x∗) is w∗-dense in bx∗ or in other words bp(x∗) is a norming set for x. in the case of an ideal also one has that y ∗ embeds (though there may not be uniqueness of norm-preserving extensions) as p(x∗). thus we continue to write x∗ = y ∗ ⊕y ⊥. in what follows we use a result from [11], that identifies strict ideals as those for which y ⊂ x ⊂ y ∗∗ under the canonical embedding of y in y ∗∗. proposition 16. suppose y is a strict ideal of x. if y∗ ∈ by ∗ is a w∗-scs point of by ∗, then y ∗ is a w∗-scs point of bx∗. proof. since y∗ ∈ by ∗ is a w∗-scs point of by ∗, for any ε > 0 there exists w∗ slices si and 0 ≤ λi ≤ 1, i = 1, 2, . . . , n, si = {y∗ ∈ by ∗/y∗(yi) > 1 − αi} and diam( ∑n i=1 λisi) < ε. since y is a strict ideal in x, we have bx∗ = by ∗ w∗ , hence we have the following: s′i = {x ∗ ∈ bx∗/x∗(xi) > 1 − αi} = {x∗ ∈ by ∗ w∗ /x∗(xi) > 1 − αi}, =⇒ diam ( n∑ i=1 λis ′ i ) ⊆ diam ( n∑ i=1 λisi ) < ε, =⇒ diam ( n∑ i=1 λis ′ i ) < ε. on small combination of slices in banach spaces 9 hence y∗ is a w∗-scs point of by ∗. arguing similarly it follows that: proposition 17. suppose y is a strict ideal of x. if y ∗ has w∗-bscsp then x∗ has w∗-bscsp. remark 18. a prime example of a strict ideal is a banach space x under its canonical embedding in x∗∗. it is known that any w∗-denting point of bx∗∗ is a point of x. now let x∗∗ ∈ bx∗∗ be a w∗-scs point. the referee has kindly pointed out that since bx is weak ∗ dense in bx∗∗, for any ϵ > 0, there is a convex combination ∑n i=1 λixi of vectors in x so that ∥x ∗∗ − ∑n i=1 λixi∥ ≤ ϵ. hence x∗∗ ∈ x. we conclude the paper with a set of remarks and questions. see also the recent paper [1] for other possible geometric connections. let us consider the following densities of w∗-scs points of bx∗. (i) all points of sx∗ are w ∗-scs points of bx∗. (ii) the w∗-scs points of bx∗ are dense in sx∗. (iii) bx∗ is contained in the closure of w ∗-scs points of bx∗. (iv) bx∗ is the closed convex hull of w ∗-scs points of bx∗. (v) x∗ is the closed linear span of w∗-scs points of bx∗. questions: (i) how can each of these properties be realized as a ball separation property considered in [3]? (ii) what stability results will hold for these properties? acknowledgements this work was done when the first author was visiting indian statistical institute bangalore center. she would like to express her deep gratitude to professor t. s. s. r. k. rao and everyone at isi bangalore center, for the warm hospitality provided during her stay. the second author currently is a fulbright-nehru academic and professional excellence fellow, at the department of mathematical sciences, university of memphis. the authors would also like to thank professor pradipta bandyopadhyay for some discussions they had with him. the authors thank the referee for pointing out inaccuracies in earlier versions and other comments for reorganizing and improving the presentation. 10 s. basu, t.s.s.r.k. rao references [1] m. d. acosta, a. kamińska, m. masty lo, the daugavet property in rearrangement invariant spaces, trans. amer. math. soc. 367 (6) (2015), 4061 – 4078. [2] r. d. bourgin, “geometric aspects of convex sets with the radon-nikodym property”, lecture notes in mathematics 993, springer-verlag, berlin, 1983. [3] d. chen, b. l. lin, ball topology on banach spaces, houston j. math. 22 (2) (1996), 821 – 833. [4] n. ghoussoub, g. godefroy, b. maurey, w. schachermayer, “some topological and geometrical structures in banach spaces”, mem. amer. math. soc. 70 (378), 1987. [5] g. godefroy, n. j. kalton, p. d. saphar, unconditional ideals in banach spaces, studia math. 104 (1) (1993), 13 – 59. [6] p. harmand, d. werner, w. werner, “m-ideals in banach spaces and banach algebras”, lecture notes in mathematics 1547, springer-verlag, berlin, 1993. [7] z. hu, b. l. lin, rnp and cpcp in lebesgue-bochner function spaces, illinois j. math. 37 (2) (1993), 329 – 347. [8] ü. kahre, l. kirikal, e. oja, on m-ideals of compact operators in lorentz sequence spaces, j. math. anal. appl. 259 (2) (2001), 439 – 452. [9] h. e. lacey, “the isometric theory of classical banach spaces”, die grundlehren der mathematischen wissenschaften 208, springer-verlag, new york-heidelberg, 1974. [10] b. l. lin, p. k. lin, s. l. troyanski, characterization of denting points, proc. amer. math. soc. 102 (3) (1988), 526 – 528. [11] t. s. s. r. k. rao, on ideals in banach spaces, rocky mountain j. math. 31 (2) (2001), 595 – 609. [12] h. p. rosenthal, on the structure of non-dentable closed bounded convex sets, adv. in math. 70 (1) (1988), 1 – 58. [13] w. schachermayer, the radon nikodym property and the krein-milman property are equivalent for strongly regular sets, trans. amer. math. soc. 303 (2) (1987), 673 – 687. e extracta mathematicae vol. 31, núm. 1, 25 – 36 (2016) adjoints of generalized composition operators with linear fractional symbol aliakbar salaryan, hamid vaezi department of mathematics, faculty of mathematical sciences, university of tabriz, p.o. box: 51666, tabriz, iran, a_goshabulaghi@tabrizu.ac.ir department of mathematics, faculty of mathematical sciences, university of tabriz, p.o. box: 51666,tabriz, iran, hvaezi@tabrizu.ac.ir presented by alfonso montes received march 30, 2015 abstract: given a positive integer n and φ : u → u, an analytic self-map of the open unit disc in the complex plane, the generalized composition operator c (n)φ is defined by c (n) φ f = f (n) ◦ φ for f belonging to some hilbert space of analytic functions on u. in this paper, we investigate some properties of generalized composition operators on the weighted hardy spaces. then we obtain adjoints of generalized composition operators with linear fractional symbol acting on the hardy, bergman and dirichlet spaces. key words: generalized composition operator, adjoint, weighted hardy space. ams subject class. (2010): 47b33, 47b38, 47a05. 1. introduction let u denote the open unit disc of the complex plane. for each sequence β = {βn} of positive numbers, the weighted hardy space h 2(β) consists of analytic functions f (z) = ∑∞ n=0 anz n on u for which the norm ∥f ∥β = ( ∞∑ n=0 |an|2β2n )1 2 is finite. notice that the above norm is induced by the following inner product⟨ ∞∑ n=0 anz n, ∞∑ n=0 bnz n ⟩ β = ∞∑ n=0 anbnβ 2 n, and that the monomials zn form a complete orthogonal system for h 2(β). consequently, the polynomials are dense in h 2(β). observe that particular instances of the sequence β = {βn} yield well known hilbert spaces of analytic 25 26 a. salaryan, h. vaezi functions. indeed, βn = 1 corresponds to the hardy space h 2(u). if β0 = 1, βn = n1/2 for n ≥ 1, the norm obtained is the one in the dirichlet space d and if βn = (n + 1)−1/2, we get the bergman space a2(u). the inner product of two functions f and g in mentioned spaces may also be computed by integration. for the hardy space, h 2(u), ⟨f, g⟩h 2(u) = ∫ 2π 0 f (eiθ)g(eiθ) dθ 2π , where f and g are defined a.e. on ∂u via radial limits (see [4]). in case of the bergman space, ⟨f, g⟩a2(u) = ∫ u f (z)g(z)da(z), where da is the normalized area measure on u and for the dirichlet space, the inner product is given by ⟨f, g⟩d = f (0)g(0) + ∫ u f ′(z)g′(z)da(z). if u is analytic on the open unit disc u and φ is an analytic map of the unit disc into itself, the weighted composition operator on h 2(β) with symbols u and φ is the operator (wu,φf )(z) = u(z)f (φ(z)) for f in h 2(β). when u(z) ≡ 1 we call the operator a composition operator and denote it by cφ. the multiplication operator mu corresponds to the case φ(z) = z and is given by muf (z) = u(z)f (z). for general information in this context one can see the excellent monographs [3], [13] and [15]. in recent years the concept of composition and weighted composition operator has been generalized in the literate. the generalized weighted composition operator dnφ,u for nonnegative integer n, which introduced by zhu [16] (see also [14]), is defined by (dnφ,uf )(z) = u(z)f (n)(φ(z)). we denote the generalized composition operator by c(n)φ . motivation to study this type of operators apart from their own importance is that they appear in the adjoint of composition operators on the space of analytic functions with derivative in the hardy space [12]. one of the most fundamental questions related to composition and weighted composition operators and their generalizations is how to obtain a reasonable representation for their adjoints. the problem of computing the adjoint of a composition operator induced by linear fractional symbol on the hardy space was solved by cowen [2]. hurst [8] using an analogous argument obtained the solution in the weighted bergman space a2α(u). mentioned adjoints of generalized composition operators 27 demonstrations was by composition of another composition operator and two toeplitz operators. in 2003, gallardo and montes [5] computed the adjoint of a composition operator with linear fractional symbol acting on the dirichlet space by a different method from those used by cowen and hurst. hammond, moorhouse and robbins [7] solved the case for rationally induced composition operators on the hardy space, h 2(u). bourdon and shapiro [1] obtained the hammond-moorhouse-robbins formula in a straightforward algebraic fashion. for more investigation we refer the interested reader to [10] and [11]. goshabulaghi and vaezi in [6] gave the adjoint formula for rationally induced composition operators on the dirichlet and bergman spaces. in this paper we investigate some properties of generalized composition operators on the weighted hardy spaces. then we obtain the adjoint of linear fractionally induced generalized composition operators acting on the hardy, bergman and dirichlet spaces. 2. generalized composition operators on h 2(β) every weighted hardy space h 2(β) contains a family of reproducing kernels {kw : w ∈ u}; that is, ⟨f, kw⟩ = f (w) for every f ∈ h 2(β). this property extends to higher derivatives of elements of h 2(β). indeed, for any w ∈ u there exists kw,n ∈ h 2(β) such that for any f ∈ h 2(β), f (n)(w) = ⟨f, kw,n⟩β . we call kw,n, the generalized reproducing kernel function. theorem 2.1. [3, theorem 2.16] the generalized reproducing kernel function kw,n for the weighted hardy space h 2(β) is given by kw,n(z) = ∞∑ m=n m(m − 1) · · · (m − (n − 1)) β2m w̄ m−n zm. simple computations based on theorem 2.1 gives the following proposition. proposition 2.2. the generalized reproducing kernel function kw,n at the point z ∈ u on the hardy, bergman and dirichlet spces is given by n!zn (1 − w̄z)n+1 , (n + 1)!zn (1 − w̄z)n+2 , (n − 1)!zn (1 − w̄z)n , respectively. as a consequence of littlewood’s subordination principle every composition operator on the hardy space is bounded. by [9, proposition 3.4] this fact 28 a. salaryan, h. vaezi also holds for every composition operator on the bergman space. the case for generalized composition operators is rather different. as we will see, if ∥φ∥∞ < 1, then c (n) φ even belongs to the class of hilbert-schmidt operators on the aforementioned spaces. in spite of this, there exist examples of φ in which ∥φ∥∞ = 1 and c (n) φ is unbounded. proposition 2.3. let h 2(β) satisfies limm→∞ βm−n βm ̸= 0 and φ(z) = az with |a| = 1. then c(n)φ : h 2(β) → h 2(β) is unbounded. proof. for m ≥ n, define fm(z) = 1m(m−1)···(m−(n−1))βm z m. then ∥fm∥β = 1 m(m−1)···(m−(n−1)) and so {fm} ∞ m=n converges to zero on h 2(β). but (c(n)φ fm)(z) = 1βm a m−nzm−n and accordingly, ∥c(n)φ fm∥ = βm−n βm and hence c (n) φ fm does not converges to zero as we expect. consequently, c (n) φ is unbounded on h 2(β). by proposition 2.3 for φ(z) = az with |a| = 1, c(n)φ is unbounded on the hardy, bergman and dirichlet spaces. theorem 2.4. let φ be an analytic self map of u such that for any positive integer k ≥ n, ∥φk∥β ≤ ck for some constant 0 < c < 1. then c (n) φ is a hilbert-schmidt operator on h 2(β). proof. let {em}∞m=0 be defined by em(z) = 1 βm zm. then {em}∞m=0 forms an orthonormal basis for h 2(β) and for each m ≥ n, e(n)m (z) = m(m − 1) · · · (m − (n − 1)) βm zm−n. accordingly, ∞∑ m=0 ∥c(n)φ em∥ 2 β = ∞∑ m=0 ∥e(n)m ◦ φ∥ 2 β = ∞∑ m=n m2(m − 1)2 · · · (m − (n − 1))2 β2m ∥φm−n∥2β ≤ ∞∑ m=n m2(m − 1)2 · · · (m − (n − 1))2 β2m c2(m−n) = ∥kc,n∥2β < ∞, which leads to c(n)φ being hilbert-schmidt on h 2(β). adjoints of generalized composition operators 29 in view of proposition 2.3, for φ(z) = az with |a| = 1, c(n)φ is unbounded on the hardy and bergman spaces. if φ is an analytic self map of u and ∥φ∥∞ < 1, the situation is very different than when ∥φ∥∞ = 1, as the following corollary shows. corollary 2.5. let φ be an analytic self map of u and ∥φ∥∞ < 1. then c (n) φ is hilbert-schmidt on the hardy and bergman spaces. theorem 2.6. let φ be an analytic self map of u such that ∥φ∥∞ < 1 . then c(n)φ is hilbert-schmidt on the dirichlet space d. proof. let {em}∞m=0 be defined by em(z) = 1√ m zm. then {em}∞m=0 forms an orthonormal basis for d and for each m ≥ n + 1, e(n)m (z) = m(m − 1) · · · (m − (n − 1)) √ m zm−n. it is clear that φm−n ∈ d. moreover, for each m ≥ n + 1 we have ∥φ′φm−n−1∥2a2 ≤ ∥φ ′∥2a2 ∥φ∥ 2(m−n−1) ∞ ≤ ∥φ∥ 2 d∥φ∥ 2(m−n−1) ∞ , and hence ∥φm−n∥2d = |φ(0)| 2(m−n) + (m − n)2∥φ′φm−n−1∥2a2 ≤ |φ(0)|2(m−n) + (m − n)2∥φ∥2d∥φ∥ 2(m−n−1) ∞ . accordingly, using the root test we see that ∞∑ m=n+1 ∥c(n)φ em∥ 2 d = ∞∑ m=n+1 ∥e(n)m ◦ φ∥ 2 d = ∞∑ m=n+1 m2(m − 1)2 · · · (m − (n − 1))2 m ∥φm−n∥2d ≤ ∞∑ m=n+1 m(m − 1)2 · · · (m − (n − 1))2|φ(0)|2(m−n) + ∞∑ m=n+1 m(m − 1)2 · · · (m − n)2∥φ∥2d∥φ∥ 2(m−n−1) ∞ < ∞, which leads to c(n)φ being hilbert-schmidt on d. 30 a. salaryan, h. vaezi 3. adjoints of generalized composition operator with linear fractional symbol corollary 2.5 and theorem 2.6 of section 2 guaranties that a wide class of generalized composition operators are bounded on the hardy, bergman and dirichlet spaces. specially, there always exist bounded linear fractionally induced generalized composition operators on the mentioned spaces and therefore computing the adjoint of generalized composition operators with linear fractional symbol makes sense. now let φ(z) = az+b cz+d be a linear fractional self map of u. it was shown by cowen [3] that corresponding to φ, σ(z) = āz−c̄−b̄z+d̄ is a self map of u. in the sequel, for n ≥ 1, we compute the adjoint of linear fractionally induced generalized composition operator c(n)φ on the hardy, bergman and dirichlet spaces. theorem 3.1. let c(n)φ be a bounded generalized composition operator on h 2(u) with linear fractional symbol. then for each f ∈ h 2(u) and z ∈ u, c(n) ∗ φ f (z) = f (0)g0(z) + d n σ,umvf (z), where u(z) = z n (−b̄z+d̄)n+1 , v(z) = (d̄z+c̄)n+1 z and g0(z) = − zn (−b̄z + d̄)n+1 ( (d̄z + c̄)n+1 z )(n) (σ(z)) + n!d̄ n+1 zn (−b̄z + d̄)n+1 . proof. a simple computation shows that 1 − φ(w)z = (−b̄z + d̄) ( 1 − σ(z)w̄ c̄w̄ + d̄ ) . (3.1) moreover for every g ∈ h 2(u), proposition 2.2 implies that g(n)(w) = ⟨g, kw,n⟩ = ∫ 2π 0 n!g(eiθ)e−inθ (1 − we−iθ)n+1 dθ 2π . hence for f ∈ h 2(u) with f (0) = 0 and z ̸= 0 we have adjoints of generalized composition operators 31 c(n) ∗ φ f (z) = ⟨c (n)∗ φ f, kz ⟩ = ⟨f, c (n) φ kz ⟩ = ⟨f, k (n) z ◦ φ⟩ = ⟨f, n!z̄ n (kz ◦ φ)n+1⟩ = n!zn ∫ 2π 0 f (eiθ) (1 − φ(eiθ)z)n+1 dθ 2π = n!zn ∫ 2π 0 f (eiθ) (−b̄z + d̄)n+1 ( 1−σ(z)e−iθ c̄e−iθ +d̄ )n+1 dθ2π = n!zn (−b̄z + d̄)n+1 ∫ 2π 0 (c̄e−iθ + d̄)n+1f (eiθ) (1 − σ(z)e−iθ)n+1 dθ 2π = zn (−b̄z + d̄)n+1 ∫ 2π 0 n! (d̄e iθ +c̄)n+1 eiθ f (eiθ)e−inθ (1 − σ(z)e−iθ)n+1 dθ 2π = zn (−b̄z + d̄)n+1 ⟨h, kσ(z),n⟩ = zn (−b̄z + d̄)n+1 h(n)(σ(z)), (3.2) where h(z) = (d̄z+c̄) n+1 z f (z) for z ̸= 0 and h(0) = c̄n+1f ′(0). notice that letting g1(z) = (d̄z + c̄)n+1 we have g1 ∈ h ∞(u) and hence h ∈ h 2(u). now let f ∈ h 2(u) be arbitrary. then for z ̸= 0, (c(n) ∗ φ f (0))(z) = ⟨c (n)∗ φ f (0), kz ⟩ = ⟨f (0), c (n) φ kz ⟩ = ⟨f (0), k (n) z ◦ φ⟩ = ⟨f (0), n!z̄ n (kz ◦ φ)n+1⟩ = n!f (0)zn(kz ◦ φ)(0) n+1 = n!f (0)zn(kφ(0)(z)) n+1 = n!f (0)zn (1 − φ(0)z)n+1 = n!d̄ n+1 f (0)zn (−b̄z + d̄)n+1 . (3.3) on the other hand, by (3.2) (c(n) ∗ φ (f − f (0)))(z) = zn (−b̄z + d̄)n+1 h (n) 0 (σ(z)), (3.4) where h0(z) = (d̄z+c̄)n+1 z (f − f (0)))(z). combining equalities (3.3) and (3.4) 32 a. salaryan, h. vaezi results c(n) ∗ φ f (z) = f (0)   n!d̄n+1 zn (−b̄z + d̄)n+1 − zn (−b̄z + d̄)n+1 ( (d̄z + c̄)n+1 z )(n) (σ(z))   + zn (−b̄z + d̄)n+1 ( (d̄z + c̄)n+1 z f (z) )(n) (σ(z)) = f (0)g0(z) + dnσ,umvf (z). now, by analyticity the statement of theorem holds for arbitrary z ∈ u. theorem 3.2. let c(n)φ and cn+1σ be bounded generalized composition operators on a2(u) with linear fractional symbol. then for each f ∈ a2(u) and z ∈ u, c(n) ∗ φ f (z) = f (0)g0(z) + d n+1 σ,u mvqf (z), where g0(z) = zn (−b̄z + d̄)n+2  (n + 1)!d̄n+2 − ( (d̄z + c̄)n+2 z )(n+1) (σ(z))   , u(z) = z n (−b̄z+d̄)n+2 , v(z) = (d̄z+c̄)n+2 z2 and qf = f is the antiderivative of f with f (0) = 0. proof. for arbitrary g ∈ a2(u), proposition 2.2 implies that g(n)(w) = ⟨g, kw,n⟩ = ∫ u (n + 1)!g(t)t̄ n (1 − t̄w)n+2 da(t). furthermore kz (t) = 1(1−z̄t)2 and hence k (n) z (t) = (n+1)!z̄ n (1−z̄t)n+2 . therefore (k(n)z ◦ φ)(t) = (n + 1)!z̄ n (1 − z̄φ(t))n+2 . let g(w) = (cw+d) n+2 (1−σ(z)w)n+2 . then g ∈ h 2(u) and by lemma 2 of [10] for any f contained in the dirichlet space, and hence for any polynomial f , ⟨f ′, g⟩a2 = ⟨f, ιg⟩h 2 , (3.5) adjoints of generalized composition operators 33 where ι is the identity map ι(z) = z. now, using (3.1) and (3.5), for any polynomial f with f (0) = 0 we have c(n) ∗ φ f (z) = ⟨c (n)∗ φ f, kz ⟩ = ⟨f, c (n) φ kz ⟩ = ⟨f, k (n) z ◦ φ⟩ = ⟨f, (n + 1)!z̄ n (kz ◦ φ)n+2⟩ = ∫ u (n + 1)!f (w)zn (1 − φ(w)z)n+2 da(w) = (n + 1)!zn ∫ u f (w) (−b̄z + d̄)n+2 ( 1−σ(z)w̄ c̄w̄+d̄ )n+2 da(w) = (n + 1)!zn (−b̄z + d̄)n+2 ∫ u (c̄w̄ + d̄)n+2f (w) (1 − σ(z)w̄)n+2 da(w) = (n + 1)!zn (−b̄z + d̄)n+2 ⟨f, g⟩a2 = (n + 1)!zn (−b̄z + d̄)n+2 ⟨f, ιg⟩h 2 = (n + 1)!zn (−b̄z + d̄)n+2 ∫ 2π 0 f (eiθ)e−iθ(c̄e−iθ + d̄)n+2 (1 − σ(z)e−iθ)n+2 dθ 2π = (n + 1)!zn (−b̄z + d̄)n+2 ∫ 2π 0 f (eiθ) (d̄e iθ +c̄)n+2 e2iθ e−i(n+1)θ (1 − σ(z)e−iθ)n+2 dθ 2π = zn (−b̄z + d̄)n+2 ( (d̄z + c̄)n+2f (z) z2 )(n+1) (σ(z)). (3.6) the last equality follows from generalized reproducing kernel property of h 2(u). let a20(u) = {f ∈ a 2(u) : f (0) = 0} and define the operator t0 : a20(u) → a 2 0(u) by t f (z) = f (z) z2 for z ̸= 0 and t f (0) = 12 f ′(0). then t0 is bounded on a20(u). therefor, by continuity (3.6) holds for any f ∈ a 2(u) with f (0) = 0. moreover for arbitrary f ∈ a2(u), (c(n) ∗ φ f (0))(z) = ⟨f (0), k (n) z ◦ φ⟩ = f (0)(k (n) z ◦ φ)(0) = f (0)(n + 1)!zn (1 − φ(0)z)n+2 = f (0)(n + 1)!d̄ n+2 zn (−b̄z + d̄)n+2 . (3.7) it is clear that the antiderivative of f − f (0) at z is f (z) − f (0)z, hence by 34 a. salaryan, h. vaezi (3.6), (c(n) ∗ φ (f − f (0)))(z) = zn (−b̄z + d̄)n+2 ( (d̄z + c̄)n+2 z2 (f (z) − f (0)z) )(n+1) (σ(z)). (3.8) combine (3.7) and (3.8) and obtain c(n) ∗ φ f (z) = f (0)zn (−b̄z + d̄)n+2 ( (n + 1)!d̄ n+2 − ( (d̄z + c̄)n+2 z )(n+1)(σ(z)) ) + zn (−b̄z + d̄)n+2 ( (d̄z + c̄)n+2 z2 f (z) )(n+1) (σ(z)) = f (0)g0(z) + dn+1σ,u mvqf (z). now, by analyticity the statement of theorem holds for arbitrary z ∈ u. theorem 3.3. let c(n)φ be a bounded generalized composition operator on d with linear fractional symbol. then for f ∈ d and z ∈ u, c(n) ∗ φ f (z) = f (0)g0(z) + d n σ,umvf (z), where g0(z) = (n−1)!d̄ n zn (−b̄z+d̄)n , u(z) = (ād̄−b̄c̄)zn+1 (−b̄z+d̄)n+1 and v(z) = (d̄z + c̄) n−1. proof. on the dirichlet space d, kz (t) = 1 + ln 11−z̄t and hence k (n) z (t) = (n−1)!z̄ n (1−z̄t)n . therefore using (3.1) we have (k(n)z ◦ φ)(w) = (n − 1)!z̄ n (1 − z̄φ(w))n = (n − 1)!z̄ n (−bz̄ + d)n ( cw + d 1 − σ(z)w )n , which implies (k(n)z ◦ φ) ′(w) = n!(ad − bc)z̄ n+1 (−bz̄ + d)n+1 . (cw + d)n−1 (1 − σ(z)w)n+1 . (3.9) adjoints of generalized composition operators 35 now, abusing of notation, generalized reproducing kernel property of h 2(u) implies that ⟨f (w), w(cw + d)n−1 (1 − σ(z)w)n+1 ⟩h 2 = ∫ 2π 0 f (eiθ)e−iθ(c̄e−iθ + d̄)n−1 (1 − σ(z)e−iθ)n+1 dθ 2π = ∫ 2π 0 f (eiθ)(d̄eiθ + c̄)n−1e−inθ (1 − σ(z)e−iθ)n+1 dθ 2π = ( (d̄z + c̄)n−1f (z) )(n) (σ(z)) n! , since (k(n)z ◦ φ)′ ∈ h 2(u), using (3.9) we see that c(n) ∗ φ f (z) = ⟨c (n)∗ φ f, kz ⟩d = ⟨f, c (n) φ kz ⟩d = ⟨f, k (n) z ◦ φ⟩d = f (0)k(n)z (φ(0)) + ⟨f ′, (k(n)z ◦ φ) ′⟩a2 = f (0)k(n)z (φ(0)) + ⟨f (w), w(k(n)z ◦ φ) ′(w)⟩h 2 = f (0) (n − 1)!d̄ n zn (−b̄z + d̄)n + n!(ād̄ − b̄c̄)zn+1 (−b̄z + d̄)n+1 ⟨f (w), w(cw + d)n−1 (1 − σ(z)w)n+1 ⟩h 2 = f (0) (n − 1)!d̄ n zn (−b̄z + d̄)n + n!(ād̄ − b̄c̄)zn+1 (−b̄z + d̄)n+1 . ( (d̄z + c̄)n−1f (z) )(n) (σ(z)) n! = f (0) (n − 1)!d̄ n zn (−b̄z + d̄)n + (ād̄ − b̄c̄)zn+1 (−b̄z + d̄)n+1 ( (d̄z + c̄)n−1f (z) )(n) (σ(z)) = f (0)g0(z) + dnσ,umvf (z). finally, by analyticity the statement of theorem holds for arbitrary z ∈ u. references [1] p.s. bourdon, j.h. shapiro, adjoints of rationally induced composition operators, j. funct. anal. 255 (8) (2008), 1995 – 2012. [2] c.c. cowen, linear fractional composition operators on h 2, integral equations operator theory 11 (2) (1988), 151 – 160. [3] c.c. cowen, b.d. maccluer, “composition operators on spaces of analytic functions", crc press, boca raton, fl, 1995. 36 a. salaryan, h. vaezi [4] p.l. duren, “theory of h p spaces", academic press, new york, 1970. [5] e. gallardo-gutiérrez, a. montes-rodríguez, adjoints of linear fractional composition operators on the dirichlet space, math. ann. 327 (1) (2003), 117 – 134. 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[13] j.h. shapiro, “composition operators and classical function theory", universitext: tracts in mathematics, springer-verlag, new york, 1993. [14] a.k. sharma, generalized composition operators between hardy and weighted bergman spaces, acta sci. math. (szeged), 78 (1,2) (2012), 187 – 211. [15] r.k. singh, j.s. manhas, “composition operators on function spaces", north-holland mathematics studies, 179. north-holland publishing co., amsterdam, 1993. [16] x. zhu, products of differentiation, composition and multiplication from bergman type spaces to bers type spaces, integral transforms spec. func. 18 (3,4) (2007), 223 – 231. e extracta mathematicae vol. 31, núm. 1, 37 – 46 (2016) local spectral theory for operators r and s satisfying rsr = r2 pietro aiena, manuel gonzález dipartimento di metodi e modelli matematici, facoltà di ingegneria, università di palermo (italia), paiena@unipa.it departamento de matemáticas, facultad de ciencias, universidad de cantabria, e-39071 santander (spain), manuel.gonzalez@unican.es received february 5, 2016 abstract: we study some local spectral properties for bounded operators r, s, rs and sr in the case that r and s satisfy the operator equation rsr = r2. among other results, we prove that s, r, sr and rs share dunford’s property (c) when rsr = r2 and srs = s2. key words: local spectral subspace, dunford’s property (c), operator equation. ams subject class. (2010): 47a10, 47a11, 47a53, 47a55. 1. introduction and preliminaries the equivalence of dunford’s property (c) for products rs and sr of operators r ∈ l(y, x) and s ∈ l(x, y ), x and y banach spaces, has been studied in [2]. as noted in [13] the proof of theorem 2.5 in [2] contains a gap which was filled up in [13, theorem 2.7]. in [2] it was also studied property (c) for operators r, s ∈ l(x) which satisfy the operator equations rsr = r2 and srs = s2. (1) a similar gap exists in the proof of theorem 3.3 in [2], which states the equivalence of property (c) for r, s, rs and sr, when r, s satisfy (1). in this paper we give a correct proof of this result and we prove further results concerning the local spectral theory of r, s, rs and sr, in particular we show several results concerning the quasi-nilpotent parts and the analytic cores of these operators. it should be noted that these results are established in a more general framework, assuming that only one of the operator equations in (1) holds. supported in part by micinn (spain), grant mtm2013-45643. 37 38 p. aiena, m. gonzález we shall denote by x a complex infinite dimensional banach space. given a bounded linear operator t ∈ l(x), the local resolvent set of t at a point x ∈ x is defined as the union of all open subsets u of c such that there exists an analytic function f : u → x satisfying (λi − t)f(λ) = x for all λ ∈ u . (2) the local spectrum σt (x) of t at x is the set defined by σt (x) := c \ ρt (x). obviously, σt (x) ⊆ σ(t), where σ(t) denotes the spectrum of t . the following result shows that σt (tx) and σt (x) may differ only at 0. it was proved in [7] for operators satisfying the svep. lemma 1.1. for every t ∈ l(x) and x ∈ x we have σt (tx) ⊆ σt (x) ⊆ σt (tx) ∪ {0}. (3) moreover, if t is injective then σt (tx) = σt (x) for all x ∈ x. proof. take s = t and r = i in [6, proposition 3.1 ]. for every subset f of c, the local spectral subspace of t at f is the set xt (f) := {x ∈ x : σt (x) ⊆ f}. it is easily seen from the definition that xt (f) is a linear subspace t-invariant of x. furthermore, for every closed f ⊆ c we have (λi − t)xt (f) = xt (f) for all λ ∈ c \ f. (4) see [9, proposition 1.2.16]. an operator t ∈ l(x) is said to have the single valued extension property at λo ∈ c (abbreviated svep at λo), if for every open disc dλo centered at λo the only analytic function f : dλo → x which satisfies the equation (λi − t)f(λ) = 0 (5) is the function f ≡ 0. an operator t ∈ l(x) is said to have the svep if t has the svep at every point λ ∈ c. clearly, the svep is inherited by the restrictions to invariant subspaces. a variant of xt (f) which is more useful for operators without svep is the glocal spectral subspace xt (f). for an operator t ∈ l(x) and a closed local spectral theory for operators r and s 39 subset f of c, we define xt (f) as the set of all x ∈ x for which there exists an analytic function f : c \ f → x which satisfies (λi − t)f(λ) = x for all λ ∈ c \ f. clearly xt (f) ⊆ xt (f) for every closed f ⊆ c. moreover t has svep if and only if xt (f) = xt (f) for all closed subsets f ⊆ c. see [9, proposition 3.3.2]. note that xt (f) and xt (f) are not closed in general. given a closed subspace z of x and t ∈ l(x), we denote by t |z the restriction of t to z. lemma 1.2. [2, lemmas 2.3 and 2.4] let f be a closed subset of c and t ∈ l(x). (1) if 0 ∈ f and tx ∈ xt (f) then x ∈ xt (f). (2) suppose t has svep, z := xt (f) is closed, and a := t |xt (f). then xt (k) = za(k) for all closed k ⊆ f. lemma 1.3. suppose that t has svep and f is a closed subset of c such that 0 /∈ f. if xt (f ∪ {0}) is closed then xt (f) is closed. proof. set z := xt (f ∪ {0}) and s := t |z. by [9, proposition 1.2.20] we have σ(s) ⊆ f ∪ {0}. we suppose first that 0 /∈ σ(s). then σ(s) ⊆ f, hence z = zs(f). by lemma 1.2 we have zs(f) = xt (f), so xt (f) is closed. for the case 0 ∈ σ(s), we set f0 := σ(s) ∩ f. then σ(s) = f0 ∪ {0}. since 0 ∈ σ(s), by lemma 1.2 we have z = zs(f0) ⊕ zs({0}) and zs(f0) = zs(σ(s) ∩ f) = zs(f) = xt (f), hence xt (f) is closed. 2. operator equation rsr = r2 operators s, r ∈ l(x) satisfying the operator equations rsr = r2 and srs = s2 were studied first in [12], and more recently in [10], [11], [8], and other papers. an easy example of operators for which these equations hold is 40 p. aiena, m. gonzález given in the case that r = pq and s = qp, where p, q ∈ l(x) are idempotents. a remarkable result of vidav [12, theorem 2] shows that if r, s are self-ajoint operators on a hilbert space then the equations (1) hold if and only if there exists an (uniquely determined) idempotent p such that r = pp ∗ and s = p ∗p , where p ∗ is the adjoint of p . the operators r, s, sr and rs for which the equations (1) hold share many spectral properties ([10], [11]), and local spectral properties as decomposability, property (β) and svep ([8]). in this section we consider the permanence of property (c), property (q) in this context. it is easily seen that if 0 /∈ σ(r) ∩ σ(s) then r = s = i, so this case is trivial. thus we shall assume that 0 ∈ σ(r) ∩ σ(s). evidently, the operator equation rsr = r2 implies (sr)2 = sr2 and (rs)2 = r2s. lemma 2.1. suppose that r, s ∈ l(x) satisfy rsr = r2. then for every x ∈ x we have σr(rx) ⊆ σsr(x) and σsr(srx) ⊆ σr(x). (6) proof. for the first inclusion, suppose that λ0 /∈ σsr(x). then there exists an open neighborhood u0 of λ0 and an analytic function f : u0 → x such that (λi − sr)f(λ) = x for all λ ∈ u0. from this it follows that rx = r(λi − sr)f(λ) = (λr − rsr)f(λ) = (λr − r2)f(λ) = (λi − r)(rf)(λ), for all λ ∈ u0. since rf : u0 → x is analytic we get λ0 /∈ σr(rx). for the second inclusion, let λ0 /∈ σr(x). then there exists an open neighborhood u0 of λ0 and an analytic function f : u0 → x such that (λi − r)f(λ) = x for all λ ∈ u0. consequently, srx = sr(λi − r)f(λ) = (λsr − sr2)f(λ) = (λsr − (sr)2)f(λ) = (λi − sr)(srf)(λ), for all λ ∈ u0, and since (sr)f is analytic we obtain λ0 /∈ σsr(srx). local spectral theory for operators r and s 41 theorem 2.2. let s, r ∈ l(x) satisfy rsr = r2, and let f be a closed subset of c with 0 ∈ f. then xr(f) is closed if and only if so is xsr(f). proof. suppose that xr(f) is closed and let (xn) be a sequence of xsr(f) which converges to x ∈ x. we need to show that x ∈ xsr(f). for every n ∈ n we have σsr(xn) ⊆ f and hence, by lemma 2.1, we have σr(rxn) ⊆ f, i.e. rxn ∈ xr(f). since 0 ∈ f, by lemma 1.2 we have xn ∈ xr(f), and since xr(f) is closed, x ∈ xr(f), i.e. σr(x) ⊆ f. now from lemma 2.1 we derive σsr(srx) ⊆ f, and this implies srx ∈ xsr(f). again by lemma 1.2, we obtain x ∈ xsr(f), thus xsr(f) is closed. conversely, suppose that xsr(f) is closed and let (xn) be a sequence of xr(f) which converges to x ∈ x. then σr(xn) ⊆ f for every n ∈ n, hence σsr(srxn) ⊆ f, i.e. srxn ∈ xsr(f) by lemma 2.1. but 0 ∈ f, so, by lemma 1.2, xn ∈ xsr(f). since xsr(f) is closed, x ∈ xsr(f), hence σsr(x) ⊆ f. now from lemma 2.1 we obtain σr(rx) ⊆ f, i.e. rx ∈ xr(f), and the condition 0 ∈ f implies x ∈ xr(f). the following result is inspired by [8, theorem 2.1]. lemma 2.3. let s, r ∈ l(x) be such that rsr = r2 and one of the operators r, sr, rs has svep. then all of them have svep. additionally, if srs = s2 and one of r, s, sr, rs has svep then all of them have svep. proof. by [6, proposition 2.1], sr has svep if and only if rs has svep. so it is enough to prove that r has svep at λ0 if an only if so has rs. suppose that r has svep at λ0 and let f : u0 → x be an analytic function on an open neighborhood u0 of λ0 for which (λi − rs)f(λ) ≡ 0 on u0. then rsf(λ) = λf(λ) and 0 = rs(λi − rs)f(λ) = (λrs − (rs)2)f(λ) = (λrs − (r2s)f(λ) = (λi − r)rsf(λ). the svep of r at λ0 implies that rsf(λ) = λf(λ) = 0 for all λ ∈ u0. hence f ≡ 0 on u0, and we conclude that rs has svep at λ0. conversely, suppose that rs has svep at λ0 and let f : u0 → x be an 42 p. aiena, m. gonzález analytic function on an open neighborhood u0 of λ0 such that (λi−r)f(λ) ≡ 0 on u0. then r 2f(λ) = λrf(λ) = λ2f(λ) for all λ ∈ u0. moreover, 0 = rs(λi − r)f(λ) = λrsf(λ) − r2f(λ) = λrsf(λ) − λ2f(λ) = (λi − rs)(−λf(λ)), and since rs has svep at λ0 we have λf(λ) ≡ 0, hence f(λ) ≡ 0, so r has svep at λ0. the second assertion is clear, if srs = s2, just interchanging r and s in the argument above, the svep fo s holds if and only if sr, or equivalently rs, has svep. we now consider the result of theorem 2.2 when 0 /∈ f. theorem 2.4. let f be a closed subset of c such that 0 /∈ f. suppose that r, s ∈ l(x) satisfy rsr = r2 and r has svep. then we have (1) if xr(f ∪ {0}) is closed then xsr(f) is closed. (2) if xsr(f ∪ {0}) is closed then xr(f) is closed. proof. (1) let us denote f1 := f ∪ {0}. the set f1 is closed, and by assumption xr(f1) is closed. since 0 ∈ f1 then xsr(f1) is closed, by theorem 2.2. moreover, the svep for r is equivalent to the svep for sr by lemma 2.3. then xsr(f) is closed by lemma 1.3. (2) the argument is similar: if xsr(f ∪ {0}) is closed then xr(f ∪ {0}) by theorem 2.2, and since r has svep, xr(f) is closed by lemma 1.3. definition 2.5. an operator t ∈ l(x) is said to have dunford’s property (c) (abbreviated property (c)) if xt (f) is closed for every closed set f ⊆ c. it should be noted that dunford property (c) implies svep. theorem 2.6. suppose that s, r ∈ l(x) satisfy rsr = r2, and any one of the operators r, sr, rs, has property (c).then all of them have property (c). if, additionally, srs = s2 and one of r, s, rs, sr has property (c), then all of them have property (c). proof. since property (c) implies svep, all the operators have svep by lemma 2.3. moreover the equivalence of property (c) for sr and rs has local spectral theory for operators r and s 43 been proved in [2] (see also [13]). so it is enough to prove that r has property (c) if an only if so has rs. suppose that r has property (c) and let f be a closed set. if 0 ∈ f then xsr(f) is closed, by theorem 2.2, while in the case where 0 /∈ f we have that xr(f ∪ {0}) is closed, and hence, by part (1) of theorem 2.4, the svep for r ensures that also in this case xsr(f) is closed. therefore, sr has property (c). conversely, suppose that sr has property (c). for every closed subset f containing 0, xr(f) is closed by theorem 2.2. if 0 /∈ f then xsr(f ∪ {0}) is closed, hence xr(f) is closed by part (2) of theorem 2.4 and we conclude that r has property (c). if additionally, srs = s2 then, by interchanging s with r, the same argument above proves the second assertion, so the proof is complete. next we consider the case when f is a singleton set, say f := {λ}. the glocal spectral subspace xt ({λ}) coincides with the quasi-nilpotent part h0(λi − t) of λi − t defined by h0(λi − t) := {x ∈ x : lim sup n→∞ ∥(λi − t)nx∥1/n = 0}. see [1, theorem 2.20]. in general h0(λi − t) is not closed, but it coincides with the kernel of a power of λi − t in some cases [3, theorem 2.2]. definition 2.7. an operator t ∈ l(x) is said to have the property (q) if h0(λi − t) is closed for every λ ∈ c. it is known that if h0(λi − t) is closed then t has svep at λ ([4]), thus, property (c) ⇒ property (q) ⇒ svep. therefore, for operators t having property (q) we have h0(λi−t)=xt ({λ}). in [13, corollary 3.8] it was observed that if r ∈ l(y, x) and s ∈ l(x, y ) are both injective then rs has property (q) precisely when sr has property (q). recall that t ∈ l(x) is said to be upper semi-fredholm, t ∈ φ+(x), if t(x) is closed and the kernel ker t is finite-dimensional, and t is said to be lower semi-fredholm, t ∈ φ−(x), if the range t(x) has finite codimension. theorem 2.8. let r, s ∈ l(x) satisfying rsr = r2, and r, s ∈ φ+(x) or r, s ∈ φ−(x). then r has property (q) if and only if so has sr. 44 p. aiena, m. gonzález proof. suppose that r, s ∈ φ+(x) and r has property (q). then r has svep and, by lemma 2.3, also sr has svep. consequently, the local and glocal spectral subspaces relative to the a closed set coincide for r and sr. by assumption h0(λi − r) = xr({λ}) is closed for every λ ∈ c, and h0(sr) = xsr({0}) is closed by theorem 2.2. let 0 ̸= λ ∈ c. by [9, proposition 3.3.1, part (f)] xr({λ} ∪ {0}) = xr({λ}) + xr({0}) = h0(λi − r) + h0(r). since r ∈ φ+(x) the svep at 0 implies that h0(r) is finite-dimensional, see [1, theorem 3.18 ], so xr({λ} ∪ {0}) is closed. then part (1) of theorem 2.4 implies that h0(λi − sr) = xsr({λ}) is closed, hence sr has property (q). conversely, suppose that sr has property (q). if λ = 0 then h0(sr) = xrs({0}) is closed by assumption, and h0(r) = xr({0}) is closed by theorem 2.2. in the case λ ̸= 0 we have xsr({λ} ∪ {0}) = xsr({λ}) + xsr({0}) = h0(λi − sr) + h0(sr). since sr has svep and sr ∈ φ+(x), h0(sr) is finite dimensional by [1, theorem 3.18]. so xsr({λ} ∪ {0}) is closed. by part (2) of theorem 2.4, xr({λ}) = h0(λi − r) is closed. therefore r has property (q). the proof in the case where r, s ∈ φ−(x) is analogous. corollary 2.9. let s, r ∈ l(x) satisfy the operator equations (1). if one of the operators r, s, rs and sr is bounded below and has property (q), then all of them have property (q). proof. note that all the operators r, s, rs, and sr are injective when one of them is injective [8, lemma 2.3], and the same is true for being upper semi-fredholm [8, theorem 2.5]. hence, if one of the operators is bounded below, then all of them are bounded below. by theorem 2.8 property (q) for r and for sr are equivalent. so the same is true for s and rs, and also for rs and sr since r and s are injective. the analytical core k(t) of t ∈ l(x) is defined [1, definition 1.20] as the set of all λ ∈ c for which there exists a constant δ > 0 and a sequence (un) in x such that x = u0, and tun+1 = un and ∥un∥ ≤ δn∥x∥ for each n ∈ n. the following characterization can be found in [1, theorem 2.18]: k(t) = xt (c \ {0}) = {x ∈ x : 0 /∈ σt (x)}. local spectral theory for operators r and s 45 the analytical core of t is an invariant subspace and, in general, is not closed. theorem 2.10. suppose that r, s ∈ l(x) satisfy rsr = r2. (1) if 0 ̸= λ ∈ c, then k(λi −r) is closed if and only k(λi −sr) is closed, or equivalently k(λi − rs) is closed. (2) if r is injective, then k(r) is closed if and only k(sr) is closed, or equivalently k(rs) is closed. proof. (1) suppose λ ̸= 0 and k(λi − r) closed. let (xn) be a sequence of k(λi − sr) which converges to x ∈ x. then λ /∈ σsr(xn) and hence, by lemma 2.1, λ /∈ σr(rxn), thus rxn ∈ k(λi − r). since rxn → rx and k(λi − r) is closed, it then follows that rx ∈ k(λi − r), i.e., λ /∈ σr(rx). since λ ̸= 0, by lemma 1.1 we have λ /∈ σr(x), hence λ /∈ σsr(srx) again by lemma 2.1. by lemma 1.1 this implies λ /∈ σsr(x). therefore x ∈ k(λi − sr), and consequently, k(λi − sr) is closed. conversely, suppose that λ ̸= 0 and k(λi − sr) is closed. let (xn) be a sequence of k(λi − r) which converges to x ∈ x. then λ /∈ σr(xn) and, by lemma 2.1, we have λ /∈ σsr(srxn). by lemma 1.1 then we have λ /∈ σsr(xn), so xn ∈ k(λi − sr), and hence x ∈ k(λi − sr), since the last set is closed. this implies that λ /∈ σsr(x), and hence λ /∈ σr(rx), again by lemma 2.1. by lemma 1.1 we have λ /∈ σr(x), so x ∈ k(λi − r). therefore, k(λi − r) is closed. the equivalence k(λi − sr) is closed if and only if k(λi − rs) is closed was proved in [13, corollary 3.3]. (2) the proof is analogous to that of part (1) applying lemma 1.1. corollary 2.11. suppose rsr = r2, srs = s2 and λ ̸= 0. then the following statements are equivalent: (1) k(λi − r) is closed; (2) k(λi − sr) is closed; (3) k(λi − rs) is closed; (4) k(λi − s) is closed. when r is injective, the equivalence also holds for λ = 0. proof. the equivalence of (3) and (4) follows from theorem 2.10, interchanging r and s. since, as noted in the proof of corollary 2.9, the injectivity of r is equivalent to the injectivity of s, the equivalence of (1) and (4) also holds for λ = 0. 46 p. aiena, m. gonzález references [1] p. aiena, “fredholm and local spectral theory, with application to multipliers”, kluwer acad. publishers, dordrecht, 2004. [2] p. aiena, m. gonzález, on the dunford property (c) for bounded linear operators rs and sr, integral equations operator theory 70 (4) (2011), 561 – 568. 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[10] c. schmoeger, on the operator equations aba = a2 and bab = b2, publ. inst. math, (beograd) (n.s.) 78(92) (2005), 127 – 133. [11] c. schmoeger, common spectral properties of linear operators a and b such that aba = a2 and bab = b2, publ. inst. math. (beograd) (n.s.) 79(93) (2006), 109 – 114. [12] i. vidav, on idempotent operators in a hilbert space, publ. inst. math. (beograd) (n.s.) 4(18), (1964), 157 – 163. [13] q. zeng, h. zhong, common properties of bounded linear operators ac and ba: local spectral theory, j. math. anal. appl. 414 (2) (2014), 553 – 560. e extracta mathematicae vol. 31, núm. 1, 109 – 117 (2016) a note on rational approximation with respect to metrizable compactifications of the plane m. fragoulopoulou, v. nestoridis department of mathematics, university of athens panepistimiopolis, athens 157 84, greece fragoulop@math.uoa.gr, vnestor@math.uoa.gr presented by manuel maestre received february 10, 2015 abstract: in the present note we examine possible extensions of runge, mergelyan and arakelian theorems, when the uniform approximation is meant with respect to the metric ϱ of a metrizable compactification (s, ϱ) of the complex plane c. key words: compactification, arakelian’s theorem, mergelyan’s theorem, runge’s theorem, uniform approximation in the complex domain. ams subject class. (2010): 30e10. 1. introduction it is well known that the class of uniform limits of polynomials in d = {z ∈ c : |z| ≤ 1} coincides with the disc algebra a(d). a function f : d → c belongs to a(d) if and only if it is continuous on d and holomorphic in the open unit disc d. it is less known (see [3, 7]) what is the corresponding class when the uniform convergence is not meant with respect to the usual euclidean metric on c, but it is meant with respect to the chordal metric χ on c∪{∞}. the class of χ–uniform limits of polynomials on d is denoted by ã(d) and contains a(d). a function f : d → c∪{∞} belongs to ã(d) if and only if f ≡ ∞, or it is continuous on d, f(d) ⊂ c and f|d is holomorphic. the function f(z) = 1 1−z , z ∈ d, belongs to ã(d), but not to a(d); thus, it cannot be uniformly approximated on d, by polynomials with respect to the usual euclidean metric on c, but it can be uniformly approximated by polynomials with respect to the chordal metric χ. more generally, if k ⊂ c is a compact set with connected complement, then according to mergelyan’s theorem [10] polynomials are dense in a(k) with respect to the usual euclidean metric on c. we recall that a function f : k → c belongs to a(k) if and only if it is continuous on k and holomorphic in the interior k◦ of k. 109 110 m. fragoulopoulou, v. nestoridis an open problem is to characterize the class ã(k) of χ–uniform limits of polynomials on k. conjecture. ([1, 6]) let k ⊂ c be a compact set with connected complement kc. a function f : k → c ∪ {∞} belongs to ã(k) if and only if it is continuous on k and for each component v of k◦, either f(v ) ⊂ c and f|v is holomorphic, or f|v ≡ ∞. extensions of this result have been obtained in [5] when kc has a finite number of components and k is bounded by a finite set of disjoint jordan curves. in this case, the χ–uniform approximation is achieved using rational functions with poles out of k instead of polynomials. furthermore, extensions of runge’s theorem are also proved in [5]. finally a first result has been obtained in [5] concerning an extension of the approximation theorem of arakelian ([2]). instead of considering the one point compactification c∪{∞} of the complex plane c, we can consider an arbitrary metrizable compactification (s, ϱ) of c and investigate the analogues of all previous results. this is the content of the present paper. 2. preliminaries we say that (s, ϱ) is a metrizable compactification of the plane c, if ϱ is a metric on s, s is compact, s ⊃ c and c is an open dense subset of s. obviously, s\c is a closed subset of s. we say that the points in s\c are the points at infinity. let (s, ϱ) be a metrizable compactification of c with metric ϱ. many such compactifications can be found in [1]. the one point compactification c∪{∞} with the chordal metric χ is a distinct one of them. we note that in this case, the continuous function π : s → c ∪ {∞}, such that π(c) = c, for every c ∈ c and π(x) = ∞, for every x ∈ s\c, is useful. another metrizable compactification is the one defined in [8] and constructed as follows: consider the map ϕ : c −→ d = {λ ∈ c : |λ < 1} z 7−→ z 1 + |z| , which is a homeomorphism. a compactification of the image d of ϕ is d, the closure of d, with the usual metric. this leads to the following compactificarational approximation of the plane 111 tion of c (2.1) s1 := c ∪ { ∞eiϑ : 0 ≤ ϑ ≤ 2π } , with metric d given by d(z, w) = ∣∣∣∣ z1 + |z| − w1 + |w| ∣∣∣∣ if z, w ∈ c , d ( z, ∞eiϑ ) = ∣∣∣∣ z1 + |z| − eiϑ ∣∣∣∣ if z ∈ c, ϑ ∈ r ,(2.2) d ( ∞eiϑ, ∞eiφ ) = ∣∣∣eiϑ − eiφ∣∣∣ if ϑ, φ ∈ r . in what follows, with a compactification (s, ϱ) of c, we shall always mean a metrizable compactification. an important question for a given compactification of c is, whether for c ∈ c and x ∈ s\c, the addition c + x is well defined. in other words, having two convergent sequences {zn}, {wn} in c, such that zn → c and wn → x does the sequence {zn + wn} have a limit in s? if the answer is positive for any such sequences {zn}, {wn} in c, then the limit y ∈ s of the sequence {zn + wn} is uniquely determined and we write c + x = y = x + c. we are interested in compactifications (s, ϱ), where c + x is well defined for any c ∈ c and x ∈ s (it suffices to take x ∈ s\c). in this case, the map c × s → s, (c, x) 7→ c + x, is automatically continuous. indeed, let x ∈ s\c, y ∈ c and w = x + y ∈ s\c. let {zn} in s and {yn} in c, such that zn → x and yn → y. if all but finitely many zn belong to c, then by our assumption zn + yn → x + y. suppose that infinitely many zn belong to s\c. without loss of generality we may assume that all zn belong to s\c and by compactness we can assume that zn + yn → l ̸= w = x + y. let d = ϱ(l, w) > 0. then there exists n0 ∈ n, such that ϱ(zn + yn, l) < d 2 for all n ≥ n0 . fix n ≥ n0. since, zn + yn is well defined, there exists z′n ∈ c, such that ϱ(zn, z ′ n) < 1 n and ϱ(zn + yn, z ′ n + yn) < 1 n . it follows that ϱ(z′n, x) ≤ ϱ(z ′ n, zn) + ϱ(zn, x) < 1 n + ϱ(zn, x) → 0 . 112 m. fragoulopoulou, v. nestoridis hence, z′n → x, yn → y and z′n, yn ∈ c. by our assumption, it follows that z′n + yn → x + y = w. but ϱ(z′n + yn, l) ≤ ϱ(z ′ n + yn, zn + yn) + ϱ(zn + yn, l) ≤ 1 n + ϱ(zn + yn, l) < 1 n + d 2 → d 2 . thus, for all n large enough we have ϱ(z′n + yn, l) ≤ 3d 4 < d = ϱ(l, w) . it follows that ϱ(z′n + yn, w) ≥ d 4 , for all n large enough. therefore, we cannot have z′n + yn → w. consequently, one concludes that the addition map is continuous at every (x, y) with x ∈ s\c and y ∈ c. obviously, it is also continuous at every (x, y) with x and y in c. thus, addition is continuous on s × c. furthermore, the following holds: let k ⊂ c be compact. obviously, the map k × s → s, (c, x) 7→ c + x, is uniformly continuous. remark 1. the preceding certainly holds for the compactification (s1, d) (see (2.1)), since c + ∞eiϑ = ∞eiϑ for all c ∈ c and ϑ ∈ r , and we have continuity. remark 2. if we identify r with the interval (−1, 1), up to a homeomorphism, then c ∼= r2 is identified with the square (−1, 1)×(−1, 1). an obvious compactification of c is then the closed square with the usual metric. the points at infinity are those on the boundary of the square, for instance, those points on the side {1} × [−1, 1]. if x ∈ {1} × (−1, 1) and c ∈ c, then c + x is a point in the same side; if im c ̸= 0, then c + x ̸= x. if x = (1, 1) and c ∈ c, then x + c = x. if im c > 0, then c + x lies higher than x in the side {1} × (−1, 1). in this example, the addition is well defined and continuous, but the points at infinity are not stabilized as in remark 1. question. is there a metrizable compactification of c such that the addition c + x is not well defined for some c ∈ c and x ∈ s\c ? rational approximation of the plane 113 the answer is “yes”. an example comes from the previous square in remark 2, if we identify all the points of {1} × [−1 2 , 1 2 ] and make them just one point. 3. runge and mergelyan type theorems in this section using a compactification of c satisfying all properties discussed in the preliminaries, we obtain the following theorem, that extends [5, theorem 3.3]. theorem 3.1. let ω ⊂ c be a bounded domain, whose boundary consists of a finite set of pairwise disjoint jordan curves. let k = ω and a a set containing one point from each component of (c ∪ {∞})\k. let (s, ϱ) be a compactification of c, such that the addition + : c × s → s is well defined. let f : k → s be a continuous function, such that f(ω) ⊂ c and f �ω is holomorphic. let ε > 0. then, there exists a rational function r with poles only in a and such that ϱ(f(z), r(z)) < ε, for all z ∈ k. proof. if ω is a disk, the proof has been given in [1]. if ω is the interior of a jordan curve, the proof is given again in [1], but also in [6]. in the general case, we imitate the proof of [5, theorem 3.3]. namely, we consider the laurent decomposition of f, given by f = f0 +f1 +· · ·+fn (see [4]). the function f0 is defined on a simply connected domain, bounded by a jordan curve, and it can be uniformly approximated by a polynomial or a rational function r0 with pole in the unbounded component. similarly, f1 is approximated by a rational function r1 with pole in a and so on. thus, the function r0 + r1 + · · · + rn approximates, with respect to ϱ, the function f = f0 + f1 + · · · + fn. this is due to the fact that at every point z all the fi’s, i = 1, 2, · · · , n, except maybe one, take values in c and the one, maybe has as a value, an infinity point in s\c. in this way, the addition map c × s → s, (c, x) 7→ c + x, is well defined and uniformly continuous on compact sets and so we are done. another runge–type theorem is the following, where we do not need any assumption for the compactification s, or the addition map + : c × s → s. theorem 3.2. let ω ⊂ c be open, f : ω → c be holomorphic and (s, ϱ) a compactification of c. let a be a set containing one point from each component of (c ∪ ∞)\ω. let ε > 0 and l ⊂ ω compact. then, there 114 m. fragoulopoulou, v. nestoridis exists a rational function r with poles in a, such that ϱ(f(z), r(z)) < ε for all z ∈ l. proof. clearly the subset f(l) of c is compact. then, from the classical theorem of runge, there exist rational functions {rn}, with poles only in a, converging uniformly to f on l, with respect to the euclidean metric | · |. hence, there is a positive integer n0 and a compact k, such that f(l) ⊂ k ⊂ c and rn(l) ⊂ k for all n ≥ n0 . but on k the metrics | · | and ϱ are uniformly equivalent. therefore, rn → f uniformly on l, with respect to ϱ. to conclude the proof, it suffices to put r = rn, for n large enough. theorem 3.2 easily yields the following corollary 3.3. under the assumptions of theorem 3.2 there exists a sequence {rn} of rational functions with poles in a, such that rn → f, ϱ–uniformly, on each compact subset of ω. remark. according to corollary 3.3, some of the ϱ–uniform limits, on compacta, of rational functions with poles in a, are the holomorphic functions f : ω → c. those are limits of the finite type. the other limits of sequences {rn} as above may be functions f : ω → s\c of infinite type, continuous (but maybe not all of them, as the example (s1, d) shows; cf. [8]). question. is a characterization possible for such limits f : ω → s1\c ? an imitation of the arguments in [8, p. 1007] gives that f must be of the form f(z) = ∞eiϑ(z), z ∈ ω, where ϑ is a multivalued harmonic function. the following extends [5, section 5]. theorem 3.4. let ω ⊂ c be open and f a meromorphic function on ω. let b denote the set of poles of f. let (s, ϱ) be a compactification of c, such that the addition + : c × s → s is well defined. let ε > 0 and k ⊂ ω be a compact set. then, there is a rational function g, such that ϱ(f(z), g(z)) < ε, for every z ∈ k\b. proof. since b ∩k is a finite set, the function f decomposes to f = h+w, where h is a rational function with poles in b ∩k and w is holomorphic on an open set containing k. by runge’s theorem there exists a rational function r rational approximation of the plane 115 with poles off k, such that |w(z)−r(z)| < ε′ on k. since w(k) is a compact subset of c and the addition + : c × s → s is well defined, a suitable choice of ε′ gives ϱ ( [h(z) + w(z)] , [h(z) + r(z)] ) < ε on k\b . we set g = h + r and the result follows. 4. arakelian sets a closed set f ⊂ c is said a set of approximation if every function f : f → c continuous on f and holomorphic in f ◦ can be approximated by entire functions, uniformly on the whole f. this is equivalent to the fact that f is an arakelian set (see [2]), that is (c ∪ {∞})\f is connected and locally connected (at ∞). we can now ask about an extension of the arakelian theorem in the context of metrizable compactifcations. a result in this direction is the following proposition 4.1. let f ⊂ c be a closed arakelian set with empty interior, i.e., f ◦ = ∅. we consider the compactification (s1, d) of c (see (2.1) and (2.2)) and let f : f → s1 be a continuous function. let ε > 0. then, there is an entire function g such that d(f(z), g(z)) < ε, for every z ∈ f. proof. according to (1.1), the compactification s1 is homeomorphic to d = {z ∈ c : |z| ≤ 1}. for each 0 < r < 1 let us define ϕr : d −→ {z ∈ c : |z| ≤ r} ⊂ d z 7−→   z , if |z| ≤ r ,rz |z| , if r ≤ |z| ≤ 1 . in other words, the whole line segment [ reiϑ, eiϑ ] is mapped at the end point reiϑ. the function ϕr is continuous and induces a continuous function ϕ̃r : s1 → s1. it suffices to take ϕ̃r := t −1 ◦ ϕr ◦ t , where t : s1 → {w ∈ c : |w| ≤ 1} is defined as follows t(z) := z 1 + |z| for z ∈ c ⊂ s1 , t ( ∞eiϑ ) := eiϑ for ϑ ∈ r . 116 m. fragoulopoulou, v. nestoridis if ε > 0 is given, then there exists rε < 1, such that for rε ≤ r < 1 and z ∈ s1, we have d ( z, ϕ̃r(z) ) < ε 2 . let now f be as in the statement of the proposition 4.1. then, d ( f(z), ( ϕ̃r ◦ f ) (z) ) < ε 2 for all z ∈ f . moreover, the function ϕ̃r ◦ f : f → c is continuous. since f is a closed arakelian set, with empty interior, and ( ϕ̃r ◦ f ) (f) ⊂ k, is included in a compact subset k of c, there exists g entire, such that∣∣∣(ϕ̃r ◦ f)(z) − g(z)∣∣∣ < ε′ for all z ∈ f . since ( ϕ̃r ◦ f ) (f) is contained in a compact subset k of c, for a suitable choice of ε′, it follows that d (( ϕ̃r ◦ f ) (z), g(z) ) < ε 2 for all z ∈ f . the triangle inequality completes the proof. an analogue of proposition 4.1 for the one point compactification c ∪ {∞} of c has been established in [5]. references [1] i. androulidakis, v. nestoridis, extension of the disc algebra and of mergelyan’s theorem, c.r. math. acad. sci. paris 349 (13–14) (2011), 745 – 748. [2] n.u. arakelian, uniform approximation on closed sets by entire functions, izv. akad. nauk sssr ser. mat. 28 (1964), 1187 – 1206 (russian). [3] l. brown, p.m. gauthier, w. hengartner, continuous boundary behaviour for functions defined in the open unit disc, nagoya math. j. 57 (1975), 49 – 58. [4] g. costakis, v. nestoridis, i. papadoperakis, universal laurent series, proc. edinb. math. soc. (2) 48 (3) (2005), 571 – 583. [5] m. fragoulopoulou, v. nestoridis, i. papadoperakis, some results on spherical approximation, bull. lond. math. soc. 45 (6) (2013), 1171 – 1180. [6] v. nestoridis, compactifications of the plane and extensions of the disc algebra, in “ complex analysis and potential theory ” , crm proc. lecture notes, 55, amer. math. soc., providence, ri, 2012, 61 – 75. [7] v. nestoridis, an extension of the disc algebra, i, bull. lond. math. soc. 44 (4) (2012), 775 – 788. rational approximation of the plane 117 [8] v. nestoridis, n. papadatos, an extension of the disc algebra, ii, complex var. elliptic equ. 59 (7) (2014), 1003 – 1015. [9] v. nestoridis, i. papadoperakis, a remark on two extensions of the disc algebra and mergelian’s theorem, preprint 2011, arxiv: 1104.0833. [10] w. rudin, “ real and complex analysis ”, mcgraw-hill book co., new yorktoronto, ont.-london, 1966. e extracta mathematicae vol. 31, núm. 2, 235 – 250 (2016) on ls-category of a family of rational elliptic spaces ii khalid boutahir, youssef rami département de mathématiques et informatique, faculté des sciences, université my ismail, b. p. 11 201 zitoune, meknès, morocco khalid.boutahir@edu.umi.ac.ma y.rami@fs-umi.ac.ma presented by antonio m. cegarra received june 24, 2016 abstract : let x be a finite type simply connected rationally elliptic cw-complex with sullivan minimal model (λv, d) and let k ≥ 2 the biggest integer such that d = ∑ i≥k di with di(v ) ⊆ λiv . if (λv, dk) is moreover elliptic then cat(λv, d) = cat(λv, dk) = dim(v even)(k − 2) + dim(v odd). our work aims to give an almost explicit formula of lscategory of such spaces in the case when k ≥ 3 and when (λv, dk) is not necessarily elliptic. key words: elliptic spaces, lusternik-schnirelman category, toomer invariant. ams subject class. (2010): 55p62, 55m30. 1. introduction the lusternik-schirelmann category (c.f. [7]), cat(x), of a topological space x is the least integer n such that x can be covered by n + 1 open subsets of x, each contractible in x (or infinity if no such n exists). it is an homotopy invariant (c.f. [3]). for x a simply connected cw complex, the rational l-s category, cat0(x), introduced by félix and halperin in [2] is given by cat0(x) = cat(xq) ≤ cat(x). in this paper, we assume that x is a simply connected topological space whose rational homology is finite dimensional in each degree. such space has a sullivan minimal model (λv,d), i.e. a commutative differential graded algebra coding both its rational homology and homotopy (cf. §2). by [1, definition 5.22] the rational toomer invariant of x, or equivalently of its sullivan minimal model, denoted by e0(λv,d), is the largest integer s for which there is a non trivial cohomology class in h∗(λv,d) represented by a cocycle in λ≥sv , this coincides in fact with the toomer invariant of the fundamental class of (λv,d). as usual, λsv denotes the elements in λv of “wordlength” s. for more details [1], [3] and [14] are standard references. in [4] y. felix, s. halperin and j. m. lemaire showed that for poincaré duality spaces, the rational l-s category coincides with the rational toomer 235 236 k. boutahir, y. rami invariant e0(x), and in [9] a. murillo gave an expression of the fundamental class of (λv,d) in the case where (λv,d) is a pure model (cf. §2). let then (λv,d) be a sullivan minimal model. the differential d is decomposable, that is, d = ∑ i≥k di, with di(v ) ⊆ λ iv and k ≥ 2. recall first that in [8] the authors gave the explicit formula cat(λv,d) = dim v odd + (k − 2) dim v even in the case when (λv,dk) is also elliptic. the aim of this paper is to consider another class of elliptic spaces whose sullivan minimal model (λv,d) is such that (λv,dk) is not necessarily elliptic. to do this we filter this model by fp = λ≥(k−1)pv = ∞⊕ i=(k−1)p λiv. (1) this gives us the main tool in this work, that is the following convergent spectral sequence (cf. §3): hp,q(λv,δ) ⇒ hp+q(λv,d). (2) notice first that, if dim(v ) < ∞ and (λv,δ) has finite dimensional cohomology, then (λv,d) is elliptic. this gives a new family of rationally elliptic spaces. in the first step, we shall treat the case under the hypothesis assuming that hn (λv,δ) is one dimensional, being n the formal dimension of (λv,d) (cf. [5]). for this, we will combine the method used in [8] and a spectral sequence argument using (2). we then focus on the case where dim hn (λv,δ) ≥ 2. our first result reads: theorem 1. if (λv,d) is elliptic, (λv,dk) is not elliptic and h n (λv,δ) = q.α is one dimensional, then cat0(x) = cat(λv,d) = sup{s ≥ 0, α = [ω0] with ω0 ∈ λ≥sv}. let us explain in what follow, the algorithm that gives the first inequality, cat(λv,d) ≥ sup{s ≥ 0, α = [ω0] with ω0 ∈ λ≥sv} := r. i) initially we fix a representative ω0 ∈ λ≥rv of the fundamental class α with r being the largest s such that ω0 ∈ λ≥sv. on ls-category of a family of rational elliptic spaces ii 237 ii) a straightforward calculation gives successively: ω0 = ω 0 0 + ω 1 0 + · · · + ω l 0 with ωi0 = (ω i,0 0 ,ω i,1 0 , . . . ,ω i,k−2 0 )∈λ (k−1)(p+i)v ⊕ λ(k−1)(p+i)+1v ⊕ ··· ⊕ λ(k−1)(p+i)+k−2v. using δ(ω0) = 0 we obtain dω0 = a 0 2 + a 0 3 + · · · + a 0 t+l with a0i = (a 0,0 i ,a 0,1 i , . . . ,a 0,k−2 i )∈λ (k−1)(p+i)v ⊕ λ(k−1)(p+i)+1v ⊕ ··· ⊕ λ(k−1)(p+i)+k−2v. iii) we take t the largest integer satisfying the inequality: t ≤ 1 2(k − 1) ( n − 2(k − 1)(p + l) − 2k + 5 ) . since d2 = 0, it follows that a02 = δ(b2) for some b2 ∈ k−2⊕ j=0 λ(k−1)(p+2)−(k−1)+jv. iv) we continue with ω1 = ω0 − b2. v) by the imposition iii), the algorithm leads to a representative ωt+l−1 ∈ λ≥rv of the fundamental class of (λv,d) and then e0(λv,d) ≥ r. now, dim(v ) < ∞ imply dim hn (λv,δ) < ∞. notice also that the filtration (1) induces on cohomology a graduation such that hn (λv,δ) = ⊕p+q=nhp,q(λv,δ). there is then a basis {α1, ...,αm} of hn (λv,δ) with αj ∈ hpj,qj (λv,δ), (1 ≤ j ≤ m). denote by ω0j ∈ λ≥rjv a representative of the generating class αj with rj being the largest sj such that ω0j ∈ λ≥sjv. here pj and qj are filtration degrees and rj ∈{pj(k−1), . . . ,pj(k−1)+(k−2)}. the second step in our program is given as follow: theorem 2. if (λv,d) is elliptic and dim hn (λv,δ) = m with basis {α1, . . . ,αm}, then, there exists a unique pj, such that cat0(x) = sup{s ≥ 0, αj = [ω0j] with ω0j ∈ λ≥sv} := rj. 238 k. boutahir, y. rami remark 1. the previous theorem gives us also an algorithm to determine ls-category of any elliptic sullivan minimal model (λv,d). knowing the largest integer k ≥ 2 such that d = ∑ i≥k di with di(v ) ⊆ λ iv and the formal dimension n (this one is given in terms of degrees of any basis elements of v ), one has to check for a basis {α1, . . . ,αm} of hn (λv,δ) (which is finite dimensional since dim(v ) < ∞). the np-hard character of the problem into question, as it is proven by l. lechuga and a. murillo (cf [12]), sits in the determination of the unique j ∈{1, . . . ,m} for which a represent cocycle ω0j of αj survives to reach the e∞ term in the spectral sequence (2). 2. basic facts we recall here some basic facts and notation we shall need. a simply connected space x is called rationally elliptic if dim h∗(x,q) < ∞ and dim(x) ⊗q < ∞. a commutative graded algebra h is said to have formal dimension n if hp = 0 for all p > n , and hn 6= 0. an element 0 6= ω ∈ hn is called a fundamental class. a sullivan algebra ([3]) is a free commutative differential graded algebra (cdga for short) (λv , d) (where λv = exterior(v odd) ⊗ symmetric(v even)) generated by the graded k-vector space v = ⊕i=∞ i=0 v i which has a well ordered basis {xα} such that dxα ∈ λv<α. such algebra is said minimal if deg(xα) < deg(xβ) implies α < β. if v 0 = v 1 = 0 this is equivalent to saying that d(v ) ⊆ ⊕i=∞ i=2 λ iv . a sullivan model ([3]) for a commutative differential graded algebra (a,d) is a quasi-isomorphism (morphism inducing isomorphism in cohomology) (λv,d) −→ (a,d) with source, a sullivan algebra. if h0(a) = k, h1(a) = 0 and dim(hi(a,d)) < ∞ for all i ≥ 0, then, [6, th.7.1], this minimal model exists. if x is a topological space any minimal model of the polynomial differential forms on x, apl(x), is said a sullivan minimal model of x. (λv,d) (or x) is said elliptic, if both v and h∗(λv,d) are finite dimensional graded vector spaces (see for example [3]). a sullivan minimal model (λv,d) is said to be pure if d(v even) = 0 and d(v odd) ⊂ λv even. for such one, a. murillo [9] gave an expression of a cocycle representing the fundamental class of h(λv,d) in the case where (λv,d) is elliptic. we recall this expression here: assume dim v < ∞, choose homogeneous basis {x1, . . . ,xn}, {y1, . . . ,ym} on ls-category of a family of rational elliptic spaces ii 239 of v even and v odd respectively, and write dyj = a 1 jx1 + a 2 jx2 + · · · + a n−1 j xn−1 + a n j xn, j = 1, 2, . . . ,m, where each aij is a polynomial in the variables xi,xi+1, . . . ,xn, and consider the matrix, a =   a11 a 2 1 an1 a12 a 2 2 an2 a1m a 2 m a n m   . for any 1 ≤ j1 < · · · < jn ≤ m, denote by pj1...jn the determinant of the matrix of order n formed by the columns i1, i2, . . . , in of a:  a1j1 anj1 a1jn a n jn   . then (see [9]) if dim h∗(λv,d) < ∞, the element ω ∈ λv , ω = ∑ 1≤j1<···2k−2 dk′ω0 on ls-category of a family of rational elliptic spaces ii 243 that is: dω0 = δ(ω0) + l∑ i=0 ( d2k−2ω i,1 0 + (d2k−2 + d2k−3)ω i,2 0 + · · · + (d2k−2 + . . . + dk+1)ω i,k−2 0 ) + ∑ k′>2k−2 dk′ω0. as δ(ω0) = 0, we can rewrite: dω0 = a 0 2 +a 0 3 +· · ·+a 0 t+l with a 0 i = (a 0,0 i , . . . ,a 0,k−2 i ) ∈ k−2⊕ j=0 λ(k−1)(p+i)+jv. note also that t is a fixed integer. indeed, the degree of a0t+l is greater than or equal to 2 ( (k − 1)(p + t + l) + k − 2 ) and it coincides with n + 1, n being the formal dimension of (λv,d). then n + 1 ≥ 2 ( (k − 1)(p + t + l) + k − 2 ) . hence t ≤ 1 2(k − 1) ( n − 2(k − 1)(p + l) + 5 − 2k ) . in what follows, we take t the largest integer satisfying this inequality. now, we have: d2ω0 = da 0 2 + da 0 3 + · · · + da 0 t+l = d(a 0,0 2 ,a 0,1 2 , . . . ,a 0,k−2 2 ) + d(a 0,0 3 ,a 0,1 3 , . . . ,a 0,k−2 3 ) + · · · + d(a 0,0 t+l,a 0,1 t+l, . . . ,a 0,k−2 t+l ), with d(a 0,0 2 ,a 0,1 2 , . . . ,a 0,k−2 2 ) = dk(a 0,0 2 ,a 0,1 2 , . . . ,a 0,k−2 2 ) + dk+1(a 0,0 2 ,a 0,1 2 , . . . ,a 0,k−2 2 ) + · · · = ( dka 0,0 2 , ∑ i′+i′′=1 dk+i′a 0,i′′ 2 , . . . , ∑ i′+i′′=k−2 dk+i′a 0,i′′ 2 ) + ( d2k−1a 0,0 2 + d2k−2a 0,1 2 + · · · , . . . ) + · · · 244 k. boutahir, y. rami d(a 0,0 3 ,a 0,1 3 , . . . ,a 0,k−2 3 ) = dk(a 0,0 3 ,a 0,1 3 , . . . ,a 0,k−2 3 ) + dk+1(a 0,0 3 ,a 0,1 3 , . . . ,a 0,k−2 3 ) + · · · = ( dka 0,0 3 , ∑ i′+i′′=1 dk+i′a 0,i′′ 3 , . . . , ∑ i′+i′′=k−2 dk+i′a 0,i′′ 3 ) + ( d2k−1a 0,0 3 + d2k−2a 0,1 3 + · · · , . . . ) + · · · · · · it follows that: d2ω0 = ( dka 0,0 2 , ∑ i′+i′′=1 dk+i′a 0,i′′ 2 , . . . , ∑ i′+i′′=k−2 dk+i′a 0,i′′ 2 ) + ( d2k−1a 0,0 2 + d2k−2a 0,1 2 + · · · , . . . ) + · · · + ( d2k−1a 0,0 3 + d2k−2a 0,1 3 + · · · , . . . ) + · · · since d2ω0 = 0, we have( dka 0,0 2 , ∑ i′+i′′=1 dk+i′a 0,i′′ 2 , . . . , ∑ i′+i′′=k−2 dk+i′a 0,i′′ 2 ) = δ(a02) = 0 with a02 = (a 0,0 2 , . . . ,a 0,k−2 2 ) ∈ ⊕k−2 j=0 λ (k−1)(p+2)+jv . consequently a02 is a δ-cocycle. claim 1. a02 is an δ-coboundary. proof. recall first that the general rth-term of the spectral sequence (6) is given by the formula: ep,qr = z p,q r /z p+1,q−1 r−1 + b p,q r−1, where zp,qr = { x ∈ [fp(λv )]p+q | dx ∈ [fp+r(λv )]p+q+1 } and bp,qr = d ( [fp−r(λv )]p+q−1 ) ∩fp(λv ) = d ( z p−r+1,q+r−2 r−1 ) . on ls-category of a family of rational elliptic spaces ii 245 recall also that the differential dr : e p,q r → e p+r,q−r+1 r in e ∗,∗ r is induced from the differential d of (λv,d) by the formula dr([v]r) = [dv]r, v being any representative in z p,q r of the class [v]r in e p,q r . we still assume that dim hn (λv,δ) = 1 and adopt notations of §4.1.1. notice then ω0 ∈ z p,q 2 and it represents a non-zero class [ω0]2 in e p,q 2 . otherwise ω0 = ω ′ 0 + d(ω ′′ 0 ), where ω ′ 0 ∈ z p+1,q−1 1 and ω ′′ 0 ∈ b p,q 1 , so that α = [ω0] = [ω ′ 0 − (d− δ)(ω ′′ 0 )]. but ω ′ 0 − (d− δ)(ω ′′ 0 ) ∈ λ ≥r+1 is a contradiction to the definition of ω0. now, using the isomorphism e ∗,∗ 2 ∼= h∗,∗(λv,δ), we deduce that, [ω0]2 ∈ e p,q 2 (being the only generating element) must survive to e p,q 3 , otherwise, the spectral sequence fails to converge. whence d2([ω0]2) = [a 0 2]2 = 0 in e p+2,q−1 2 , i.e., a 0 2 ∈ z p+3,q−2 1 + b p+2,q−1 1 . however a02 ∈ ⊕k−2 j=0 λ (k−1)(p+2)+jv , so a02 ∈ b p+2,q−1 1 , that is a 0 2 = d(x), x ∈⊕k−2 j=0 λ (k−1)(p+1)+jv . by wordlength argument, we have necessary a02 = δ(x), which finishes the proof of claim 1. notice that this is the first obstruction to [ω0] to represent a non zero class in the term e ∗,∗ 3 of (6). the others will appear progressively as one advances in the algorithm. let then b2 ∈ ⊕k−2 j=0 λ (k−1)(p+2)−(k−1)+jv such that a02 = δ(b2) and put ω1 = ω0 − b2. reconsider the previous calculation for it: dω1 = dω0 −db2 = (a02 + a 0 3 + · · · + a 0 t+l) − (dkb2 + d4b2 + · · ·), with dkb2 = dk(b 0 2,b 1 2, . . . ,b k−2 2 ) = (dkb 0 2,dkb 1 2, . . . ,dkb k−2 2 ) ∈ k−2⊕ j=0 λ(k−1)(p+2)+jv, dk+1b2 = dk+1(b 0 2,b 1 2, . . . ,b k−2 2 ) = (dk+1b 0 2,dk+1b 1 2, . . . ,dk+1b k−2 2 ) ∈ k−2⊕ j=0 λ(k−1)(p+2)+j+1v, · · · 246 k. boutahir, y. rami this implies that dω1 = a 0 2 + a 0 3 + · · · + a 0 t+l − ( dkb 0 2, ∑ i′+i′′=1 dk+i′b i′′ 2 , . . . , ∑ i′+i′′=k−2 dk+i′b i′′ 2 ) − ( d2k−1b 0 2 + · · · , . . . ) = a02 −δ(b2) + a 0 3 − ( d2k−1b 0 2 + · · · , . . . ) + · · · = a03 − ( d2k−1b 0 2 + · · · , . . . ) + · · · , and then: dω1 = a 1 3 + a 1 4 + · · · + a 1 t+l, with a 1 i ∈ k−2⊕ j=0 λ(k−1)(p+i)+jv. so, d2ω1 = da 1 3 + da 1 4 + · · · + da 1 t+l = ( dka 1,0 3 , ∑ i′+i′′=1 dk+i′a 1,i′′ 3 , . . . , ∑ i′+i′′=k−2 dk+i′a 1,i′′ 3 ) + ( d2k−1a 1,0 3 + · · · , . . . ) + · · · since d2ω1 = 0, by wordlength reasons,( dka 1,0 3 , ∑ i′+i′′=1 dk+i′a 1,i′′ 3 , . . . , ∑ i′+i′′=k−2 dk+i′a 1,i′′ 3 ) = δ(a13) = 0. we claim that a13 = δ(b3) and consider ω2 = ω1 − b3. we continue this process defining inductively ωj = ωj−1−bj+1, j ≤ t+l−2 such that: dωj = a j j+2 + a j j+3 + · · · + a j t+l, with a j i ∈ k−2⊕ h=0 λ(k−1)(p+i)+hv and a j j+2 a δ-cocycle. on ls-category of a family of rational elliptic spaces ii 247 claim 2. a j j+2 is a δ-coboundary, i.e., there is bj+2 ∈ k−2⊕ j=0 λ(k−1)(p+j+2)−(k−1)+jv such that δ(bj+2) = a j j+2; 1 ≤ j ≤ t + l− 2. proof. we proceed in the same manner as for the first claim. indeed, we have clearly for any 1 ≤ j ≤ t+l−2, ωj = ωj−1−bj+1 = ω0−b2−b3−···−bj+1 ∈ z p,q j+2 and it represents a non zero class [ωj]j+2 in e p,q j+2 which is also one dimensional. whence as in claim 1, we conclude that, a j j+2 is a δ-coboundary for all 1 ≤ j ≤ t + l− 2. consider ωt+l−1 = ωt+l−2 − bt+l, where δ(bt+l) = at+l−2t+l . notice that |dωt+l−1| = |dωt+l−2| = n +1, but by the hypothesis on t, we have d(ωt+l−2) = at+l−2t+l and then |d(ωt+l−2 −bt+l)| = |at+l−2t+l −δ(bt+l) − (d−δ)bt+l| = |− (d−δ)bt+l| > n + 1. it follows that dωt+l−1 = 0, that is ωt+l−1 is a d-cocycle. but it can’t be a d-coboundary. indeed suppose that ωt+l−1 = (ω 0 0 + ω 1 0 + · · · + ω l 0) − (b2 + b3 + · · · + bt+l), were a d-coboundary, by wordlength reasons, ω00 would be a δ-coboundary, i.e., there is x ∈ ⊕k−2 j=0 λ (k−1)p−(k−1)+jv such that δ(x) = ω00. then ω0 = δ(x) + ω 1 0 + · · · + ω l 0. since δ(ω0) = 0, we would have δ(ω 1 0 +· · ·+ω l 0) = 0 and then [ω0] = [ω 1 0 +· · ·+ ωl0]. but ω 1 0 + · · ·+ ω l 0 ∈ λ >rv , contradicts the property of ω0. consequently ωt+l−1 represents the fundamental class of (λv,d). finally, since ωt+l−1 ∈ λ≥rv we have e0(λv,d) ≥ r. 4.1.2. for the second inequality. denote s = e0(λv,d) and let ω ∈ λ≥sv be a cocycle representing the generating class α of hn (λv,d). write ω = ω0 + ω1 + · · · + ωt, ωi ∈ λs+iv . we deduce that: dω = ( dkω0 + ∑ i+i′=1 dk+iωi′ + · · · + ∑ i+i′=k−2 dk+iωi′ ) + dkωk−1 + d2k−1ω0 + · · · = δ(ω0,ω1, . . . ,ωk−2) + · · · 248 k. boutahir, y. rami since dω = 0, by wordlength reasons, it follows that δ(ω0,ω1, . . . ,ωk−2) = 0. if (ω0,ω1, . . . ,ωk−2), were a δ-boundary, i.e., (ω0,ω1, . . . ,ωk−2) = δ(x), then ω −dx = (ω0,ω1, . . . ,ωk−2) −δ(x) + (ωk−1 + · · · + ωt) − (d−δ)(x) = (ωk−1 + · · · + ωt) − (d−δ)(x), so, ω − dx ∈ λ≥s+k−1v , which contradicts the fact s = e0(λv,d). hence (ω0,ω1, . . . ,ωk−2) represents the generating class of h n (λv,δ). but (ω0,ω1, . . . ,ωk−2) ∈ λ≥sv implies that s ≤ r. consequently, e0(λv,d) ≤ r. thus, we conclude that e0(λv,d) = r. 4.2. proof of theorem 2. it suffices to remark that since (λv,d) is elliptic, it has poincaré duality property and then dim hn (λv,d) = 1. the convergence of (6) implies that dim e ∗,∗ ∞ = 1. hence there is a unique (p,q) such that p+q = n and e ∗,∗ ∞ = e p,q ∞ . consequently only one of the generating classes α1, . . . ,αm had to survive to e∞. let αj this representative class and (pj,qj) its pair of degrees. example 1. let d = ∑ i≥3 di and (λv,d) be the model defined by v even = < x2,x , 2 >, v odd =< y5,y7,y , 7 > , dx2 = dx , 2 = 0, dy5 = x 3 2, dy7 = x ,4 2 and dy , 7 = x 2 2x ,2 2 , in which subscripts denote degrees. for k ≥ 3, l ≥ 0, we have xk2x ,l 2 = x k−3 2 x 3 2x ,l 2 = d ( y5x k−3 2 x ,l 2 ) . for k ≥ 4, l ≥ 0, x ,k 2 x l 2 = x l 2x ,k−4 2 x ,4 2 = d ( xl2x ,k−4 2 y7 ) . clearly we have dim h(λv,d) < ∞ and dim h(λv,d3) = ∞. using a. murillo’s algorithm (cf. §2) the matrix determining the fundamental class is: a =   x 2 2 0 0 x ,3 2 x2x ,2 2 0   , on ls-category of a family of rational elliptic spaces ii 249 so, ω = −x22x ,3 2 y , 7 + x2x ,5 2 y5 ∈ λ ≥6v is a generator of this fundamental cohomology class. it follows that e0(λv,d) = 6 6= m + n(k − 2). example 2. let d = ∑ i≥3 di and (λv,d) be the model defined by v even = < x2,x , 2 >, v odd =< y5,y9,y , 9 > , dx2 = dx , 2 = 0, dy5 = x 3 2, dy9 = x ,5 2 and dy , 9 = x 3 2x ,2 2 . clearly we have dim h(λv,d) < ∞ and dim h(λv,d3) = ∞. using a. murillo’s algorithm (cf. §2) the matrix determining the fundamental class is: a =   x 2 2 0 0 x ,4 2 x22x ,2 2 0   , so, ω = −x22x ,4 2 y , 9 + x 2 2x ,6 2 y5 ∈ λ ≥7v is a generator of this fundamental cohomology class. it follows that e0(λv,d) = 7 6= m + n(k − 2). 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[14] d. sullivan, infinitesimal computations in topology, inst. hautes études sci. publ. math. 47 (1978), 269 – 331. e extracta mathematicae vol. 32, núm. 1, 83 – 103 (2017) resolutions of cohomology algebras and other struggles with integer coefficients a. jaleel, a. percy villa college, maldives ahsan.jaleel@villacollege.edu.mv ciao, federation university australia, australia andrew.percy@federation.edu.au presented by antonio m. cegarra received august 26, 2016 abstract: there is a well known homotopy π-algebra resolution of a space by wedges of spheres. an attempt to construct the eckmann-hilton dual gives a nice resolution for fp coefficients which can then be used in a spectral sequence. for z coefficients the dual construction has several compounding problems illustrating that integral cohomology becomes relatively problematic when we try to include primary operations. key words: primary cohomology operations, integer coefficients, free resolution, derived functors. ams subject class. (2010): 55-02, 55s05. 1. introduction the cohomology groups of a topological space x are well defined over coefficients in any abelian group, however, there is a richer, natural structure that includes the primary cohomology operations. it is within this extra structure that cohomology over integer coefficients becomes problematic when compared to cohomology over coefficients in a finite field or the rationals. over the integers we may think of the primary structure as the integral version of the steenrod algebra, however, the viewpoint taken here is to consider these cohomology algebras as the eckmann-hilton dual of the well known π-algebras of homotopy theory [20, 34, 2, 13]. to observe the eckmann-hilton duality we will consider the reduced spectral cohomology over the eilenberg-mac lane spectrum of a ring. this ordinary, reduced theory is defined as the homotopy classes of maps into eilenbergmac lane spaces (e-m spaces), h̃n(x; r) = [x, k(r, n)]. the ‘naturality’ of the operations increases the precision of calculations by allowing a map between spaces to induce a morphism between cohomology algebras rather than simply cohomology groups (see [33] for a survey of 83 84 a. jaleel, a. percy applications of additional structure). primary operations are operations that are globally natural and are induced by a universal arrow between products of e-m spaces, known as generalised e-m spaces (gems). for coefficients in a field, the künneth formula shows that cup product and composition are the only primary operations and the cartan formula tells us how these distribute. for integer coefficients, there is no known analogue of the cartan formula. moreover, there are binary operations with no known formulation for the universal arrows representing them (see example 3.5). consequently, for integer coefficients we have only a partial formulation for the primary operation structure on cohomology groups. with no explicit formulation for the primary structure, we might take inspiration from the eckmann-hilton dual in which the primary homotopy structure can be encoded in a category of operations π. functors from π to pointed sets are called π-algebras and the image of a π-algebra are homotopy groups with the primary homotopy operations acting on them. we can encode the primary cohomology operations in the category h(z), of products of integral eilenberg-mac lane spaces and homotopy classes of maps. we will call the eckmann-hilton dual of a π-algebra an h(z)-algebra, which will be a functor from h(z) to the category of pointed sets. generalizing we can define an h(r)-algebra to be a functor from the category h(r), of products of eilenberg-mac lane spaces over a ring r and homotopy classes of maps, to the category of pointed sets. for r = fp it is well known that an h(fp)-algebra is an unstable algebra over the mod p steenrod algebra [8, 4]. taking further inspiration from the theory of π-algebras, we might attempt to study the relation between the h(r)-algebra of x and x itself, using a free cosimplicial resolution and a spectral sequence. in homotopy theory, stover [34] constructs a free simplicial resolution x• which is homotopy equivalent to a wedge of spheres in each simplicial dimension. taking the p-th homotopy of this simplicial space gives a simplicial group and the homotopy groups of that simplicial group fits into the e2 page of the bousfield-friedlander spectral sequence [16]. the sequence commutes with primary operations and by design of the stover resolution, the sequence collapses on the third page, to the π-algebra of x. a simplicial resolution of an h(r)-algebra is a simplicial h(r)-algebra which is weakly equivalent to the constant simplicial h(r)-algebra. a free simplicial resolution additionally has a free h(r)-algebra in each simplicial dimension. a free cosimplicial resolution of a space is a cosimplicial space, x•, with h∗(x•) a free simplicial resolution of the h(r)-algebra of x. the resolutions of cohomology algebras 85 model category structure, defining the weak equivalences on simplicial h(r)algebras, is given by blanc and peschke for h(r) containing finite products of e-m spaces [8]. although a dual stover construction will work fairly well for coefficients in a finite field, there is a difficulty for integer coefficients. unlike the finite fields, z is not algebraically compact, consequently, maps out of an infinite product of e-m spaces over z do not factor through a finite sub-product. in order that the cosimplicial resolution be free, h(z) needs to contain infinite products and we need to allow infinitary primary operations. fortunately, we are still able to define a model category structure for h(z)-algebras but the free resolutions are larger than for field coefficients [26]. we would like to employ a second quadrant cohomology spectral sequence e −p,q 2 = πph q(x•; r) ⇒ hq−p(x; r) which commutes with primary cohomology operations. the spectral sequence exists [18, 21, 5] although convergence is not guaranteed. the spectral sequence would actually converge to h∗(tot(x•); r), where tot(x•) is the total space associated to the cosimplicial space x• [17]. then, to meet our purposes, we also require that tot(x•) has the same cohomology algebra as x. this requires tot(x•) to have the same r-cohomology type as the r-completion of x, which, according to bousfield [15], will occur if h∗(x•; r) is acyclic over all r-module coefficients. for integer coefficients, this requires an expansion of the category h(z) to include all gems, that is, (countable) products of eilenberg-mac lane spaces, k(m, n), for arbitrary z-modules m, whereby h(z)-algebras will no longer be well defined [29]. we expect integer cohomology to contain more topological information than cohomology over a quotient ring of z. when we look at primary operations the additional information contained over integral coefficients is not well understood. in fact, the research has focused, with great success, on cohomology over finite fields and seems to have abandoned the integers since kochman’s work [27]. perhaps the research has not been abandoned, it’s just that no results have been achieved to report. one caveat for eckmann-hilton duality should be “integers for homotopy, finite fields for cohomology”. after some notation in section 2, section 3 looks at the problem of defining the universal arrows for primary integral operations. section 4 develops the encoding of operations in a category and defines h(r)-algebras. section 5 gives the construction of free resolutions by simplicial h(r)-algebras and cosimplicial spaces. the need to consider infinitary integral operations is also explained in this section. section 6 looks at the spectral sequence for calculat86 a. jaleel, a. percy ing h(r)-algebras and the obstacles occurring for integer coefficients. section 7 is a brief conclusion. 2. notation we assume that all statements and results are for some fixed but arbitrary ring r. we will denote e-m spaces by kn = k(r, n). hence the reduced cohomology groups of a space with coefficients in r are given by h̃n(x) = [x, kn]. for the various constructions given we will then discuss the obstacles for r = z compared to r = fp. for convenience we will consider the sub-category, t∗, of pointed topological spaces which are simply-connected, cw-complexes. for spaces x, y ∈ t∗, we write [x, y ] for the homotopy classes of maps from x to y . precomposition by a map f : x → y is denoted f∗(g) = gf, for g : y → z, and post-composition denoted by f∗. the category of pointed sets is denoted set ∗ and graded pointed sets by grset ∗. for a product ∏n i=1 xi, let pri denote the canonical projection onto the factor xi, and for maps xi : y → x let {x1, x2, . . . , xn} : y → ∏n i=1 xi denote the canonical product map. let sc denote the category of simplicial objects over a category c and cc that of cosimplicial objects. given functors u, v : c →c, an object x ∈c and natural transformations µ : u → v, composition is defined as uµ : uu → uv, (uµ)x := u(µx) and µu : uu → vu, (µu)x := µux. 3. primary operations and universal arrows in this section we give a brief overview of primary cohomology operations and explain where the integral theory differs from cohomology over fp. definition 3.1. a cohomology operation of type (g, n, a, m) is a family of functions θx : h n(x; g) → hm(x; a), one for each space x, satisfying the naturality condition f∗θy = θxf ∗ for any map f : x → y . operations exist for any abelian group coefficients, g and a, whether finitely generated or not and any integers n ≤ m. the set of all (g, n, a, m) operations is denoted θ(g, n, a, m) and it follows from the naturality condition that θ(g, n, a, m) ∼= hm(k(g, n); a) so that the cohomology classes of e-m spaces are often called primary cohomology operations [33, 22]. resolutions of cohomology algebras 87 the cohomology operations are generally functions on the underlying sets of the cohomology groups, however, they can be semi-additive or additive. additive operations induce homomorphisms on cohomology groups and, for any spectral cohomology theory, there is a range called the stable range in which operations form families of additive operations. for g = a = z/2, the stable range are called steenrod squares, sqi : hn(x; z/2) −→ hn+1(x; z/2) , 0 ≤ i ≤ n , which, under composition form the mod-2 steenrod algebra. similarly, if g = a = z/p for an odd prime p, there are stable operations called the reduced powers, p i : hn(x; z/p) −→ hn+2i(p−1)(x; z/p) , 0 ≤ i ≤ n 2 , which, under composition form the mod-p steenrod algebra. if g = a = z/p for any prime p there are the bockstein operations, β : hn(x; z/p) → hn+1(x; z/p) and if g = z/pk, for p prime and k ≥ 1, and a = z/pk+1, there are the pontrjagin p-th powers, βp : h2n(x; z/pk) → h2np(x; z/pk+1). remark 3.2. for the case where g is a finitely generated abelian group and a = z/p for any prime, from the results of cartan [19] all cohomology operations are generated (by compositioins) from the steenrod squares, the reduced powers, the bockstein operations and the pontrjagin p-th powers [33]. this can equivalently be shown using the serre spectral sequence [30]. however, if a = z, the process is much more difficult because of mixed torsion occuring on the e2 page, and no complete calculation has been given for the operations hm(k(z, n); z) (see remark 3.4). we should also note that if g = a = q, rational cohomology is an algebra. hence, there are only the q-vector space structure for n = m and a power operation x 7→ x2 in even degrees, satisfying definition 3.1 [28]. now, if g = a = r, for r a commutative ring, there exists the binary operation of cup product ∪x : hn(x; r)×hm(x; r) −→ hn+m(x; r) , satisfying the naturality condition, f∗∪y = ∪x(f∗×f∗) for any map f : x → y (see diagram (1)), which gives the graded cohomology groups a graded 88 a. jaleel, a. percy ring structure and the mapf induces a graded ring homomorphism. this introduces the concept of an n-ary operation and we now use fp for the finite field rather than the cyclic group z/p, so cup products will exist. definition 3.3. an n-ary operation θ : h̃m1(x) × ··· × h̃mn(x) → h̃q(x), n ∈ n, is primary if, given any spaces x and y and any map f : x → y , the following naturality diagram commutes h̃m1(x)×···× h̃mn(x) θ // h̃q(x) h̃m1(y )×···× h̃mn(y ) f∗×···×f∗ oo θ // h̃q(y ) f∗ oo . (1) there may be other operations (such as secondary or higher) that satisfy diagram (1) for some x, y and mi, but only primary operations satisfy this diagram universally. since all spaces are simply connected there is no action of π1(x) on the cohomology groups of x. it follows directly from (1) that the homotopy class θ(pr1, . . . , prn) : k m1 ×···×kmn −→ kq is a universal arrow and the operation θ(x1, x2, . . . , xn) is given by the composition x {x1,...,xn}−−−−−−−−→ km1 ×···×kmn θ(pr1,...,prn)−−−−−−−−−→ kq the proof [31] is eckmann-hilton dual to that for homotopy operations [36]. composition with a representative element θ ∈ h̃m(kn) are the only possible unary operations (for a fixed coefficient ring r). the universal arrow is θ and the operation θ∗ : h̃ n(x) → h̃m(x). the group addition is given by identifying kn with the loop space ωkn+1, up to homotopy. then addition + : kn×kn → kn is given by concatenation of loops. other binary operations are given by the künneth formula applied to a product of two eilenberg-mac lane spaces. iteration of the künneth formula shows all finitary operations are generated by the binary and unary operations. the künneth formula is derived from a short exact sequence which splits nonnaturally giving h̃n (kr ×ks) ∼= ⊕ i+j=n h̃i(kr)⊗h̃j(ks) ⊕ i+j=n+1 tor ( h̃i(kr), h̃j(ks) ) (2) resolutions of cohomology algebras 89 elements α ⊗ β ∈ h̃i(kr) ⊗ h̃j(ks) are included into h̃n (kr ×ks) by the “external cup product” given by α∗(prr)∪β∗(prs) where ∪ is the standard cup product (see [23, p. 210 and p. 278]). if we denote the universal arrow for cup product by h∪ then the universal arrow for the binary operation α⊗β is h∪∗({α∗(prr), β∗(prs)}). to give all binary operations and hence, by iteration, all n-ary operations, it remains to describe elements of the summands⊕ i+j=n+1 tor ( h̃i(kr), h̃j(ks) ) . here it should be noted that integer coefficients differ from coefficients in fp. for coefficients in fp there are no tor terms [23, theorem 3.16] so that all primary operations are combinations of compositions and cup products as for α ⊗ β above. in fact, as a refinement to remark 3.2, by explicitely listing generators of the groups hn(k(z/p, m); z/q), cartan showed that the only operations over fp are freely generated by the stable reduced powers and cup product [19]. in [22, section 12] eilenberg and mac lane defined a certain type of secondary operation, called cross-cap products, and showed these were in bijective correspondence with the tor groups of the homology künneth formula. the construction dualises from chain to cochain complexes and we can define a cross-cap product, α×̄β on elements α ∈ h̃i(kr) and β ∈ h̃j(ks) of cohomology groups with torsion (for details see [31]). cross-cap products are generalized by secondary massey products [35]. these cross-caps correspond to elements of ⊕ i+j=n+1 tor(h̃i(kr), h̃j(ks)) by considering these tor groups as cokernels of the external cup products, hence cosets h̃i+j−1(kr ×ks) / ⊕ i+j=n+1 [ h̃i(kr)⊗ h̃j−1(ks) + h̃i−1(kr)⊗ h̃j(ks) ] . this means the cross-cap product α×̄β is not uniquely defined and consequently cannot be identified with a unique universal arrow. moreover, since the künneth short exact sequence splits non-naturally, there is no natural way to identify a representative arrow in each equivalence class corresponding to the cross-cap products. that these secondary operations correspond to 90 a. jaleel, a. percy cosets of h̃n (kr ×ks) in equation (2) is the closest description we have of an unknown type of primary operation for integer coefficients. remark 3.4. appendix 8 contains a table of primary, unary cohomology operations over z coefficients and a short discussion of differences with operations over fp coefficients. in particular, integer operations do not generate freely and compositions can be unstable and multiple. the calculations shown in table 1 can be used to demonstrate the existence of the unknown primary operations in example 3.5. the binary operations (for z coefficients) h̃14 ( k3 ×k3 ) ∼= h̃3(k3)⊗ h̃11(k3)⊕tor (h̃6(k3), h̃9(k3)) ∼= z/3⊕z/2 (3) are generated by two types of universal arrows. the first is an external cup product universal arrow of order 3 and the second, of unknown formulation and order 2, corresponding to the generator of z/2. 4. h(r)-algebras since being able to list all integral cohomology operations and relations between them is not possible, we may wish to take another approach in which this information is encoded in a category and does not need to be known explicitly. this was the point of view, adopted in the 1990’s, for the study of primary homotopy operations and their relations acting on homotopy groups to give a π-algebra [20, 34]. let h(r) be the category of finite products of eilenberg-mac lane spaces over r, including the trivial product ∗, with pointed homotopy classes of maps. for any x ∈ t∗ consider the functor of homotopy classes of pointed maps [x, ] : h(r) → set ∗. the image of these functors (with all copies of h̃n(x), n ≥ 0, identified as one isomorphism class) give reduced cohomology groups and the morphisms of h(r) induce primary operations on those groups. such functors, as well as their image in set ∗ will be called topological h(r)-algebras. figure 1 indicates the identification of functor with graded cohomology algebra. resolutions of cohomology algebras 91 h(r) set ∗ k4 k1 × k2 k3 h̃4(x) h̃1(x) h̃2(x) ... [x, ]h+ + ∗ 0 h̃3(x) h∪ ∪ α α∗ figure 1: topological h(r)-algebras similarly to π-algebras we can abstract the notion of topological h(r)algebras by definition 4.1. an h(r)-algebra is a functor z : h(r) → set ∗ sending the point to the singleton set, 0, and with the property that the map z (∏n i=1 k ni ) → ∏n i=1 z(k ni) is a natural isomorphism of abelian groups. the category of h(r)-algebras will be denoted set ∗h(r). there are several definitions of algebraic structures describing h(r)-algebras. they could be considered a variety of algebras [29], universal graded algebras [11], models of the algebraic theory h(r) [12] or models of the finite product sketch h(r) [8]. mac lane states that set ∗h(r) contains all limits and colimits and that there exists a free functor, left adjoint to the underlying (forgetful) functor to set ∗ [29]. remark 4.2. borceux [12] shows that the cohomology h(r)-algebras of the objects of h(r) are free h(r)-algebras. in addition, blanc and stover show that simplicial objects over any category of universal graded algebras, such as set ∗h(r), has a closed simplicial model category structure [11]. blanc and peschke give a resolution model category structure on simplicial models of a finite product sketch arising from an adjunction of free and forgetful functors. therefore we have a definition of weak equivalence between simplicial h(r)-algebras. remark 4.3. in addition, the model category structure is enriched, forming a simplicial model category with tensor and cotensor products [8]. 92 a. jaleel, a. percy 5. standard constructions and resolutions a standard construction of simplicial and cosimplicial functors from a comonad and a monad respectively, is given by huber in [24]. definition 5.1. a comonad, ⟨t, µ, η⟩, in a category c consists of an endofunctor t : c →c and natural transformations, the counit µ : t → 1 and comultiplication η : t → t 2 such that the following diagrams commute [29]: t η −−−−→ t 2 η y ytη t 2 −−−−→ ηt t 3 , t t t∥∥∥ yη ∥∥∥ 1t µt ←−−−− t 2 −−−−→ tµ t1 . any comonad generates a simplicial functor v• = (vn, d i n, s i n), n ≥−1 with a sequence of functors vn : c →c, face maps din : vn → vn−1 and degeneracy maps sin : vn → vn+1, 0 ≤ i ≤ n. this is achieved by letting t 0 = 1 and t n+1 = tt n then letting vn = t n+1, din = t iµt n−i and sin = t iηt n−i. one may also consider the simplicial functors v• = (vn, d i n, s i n), n ≥ 0 with an augmentation µ : v0 → 1 with µd00 = µd 1 0. the construction of a cosimplicial functor from a monad is categorically dual to the construction above [24]. a monad ⟨t, ϵ, δ⟩ with t : c → c, ϵ : 1 → t (the unit) and δ : t 2 → t (the multiplication), is used to build a cosimplicial functor v • = (v n, dni , s n i ), n ≥−1 as a sequence of functors v n : c →c, coface maps dni : v n−1 → v n and codegeneracy maps sni : v n+1 → v n, 0 ≤ i ≤ n. this is achieved by letting t 0 = 1 and t n+1 = tt n then letting v n = t n+1, dni = t iϵt n−i and sni = t iδt n−i. one may also consider the cosimplicial functors f • = (v n, dni , s n i ), n ≥ 0 with an augmentation ϵ : 1 → v 0 with d00ϵ = d 0 1ϵ. 5.1. simplicial h(r)-algebras. every adjunction between categories gives rise to both a comonad and a monad [29]. let f : grset ∗ →set ∗h(r) and u be the free and underlying functors for h(r)-algebras with unit of adjunction α : 1 → uf and counit of adjunction β : fu → 1. a comonad, ⟨t, ϵ, δ⟩, is formed by letting the endofunctor be t = fu, the counit ϵ = β and the comultiplication δ = fαu. from this comonad we would like to construct a simplicial h(r)-algebra, t•z, for any given h(r)-algebra, z, and would hope that t•z can be used as a free algebraic resolution of any z in the sense of homotopical algebra. this resolutions of cohomology algebras 93 requires a free object in each simplicial dimension and a homotopy equivalence with the constant simplicial h(r)-algebra over z. the underlying graded set of the h(r)-algebra, z, may contain sets of infinite cardinality. consequently, the first simplicial dimension of the resolution, that is, tz, will be the h(r)-algebra of an infinite product of e-m spaces. by remark 4.2, this h(r)-algebra will be free if it is an object of h(r). for r = fp coefficients we have that h∗(p ; fp) ∼= lim← α h∗(pα; fp) (4) where pα are finite subproducts of the infinite product p [25, proposition 2.1]. this means we can replace tz with the h(fp)-algebra of a finite subproduct and h(fp) need only contain finite products of e-m spaces. in addition, by remark 3.2, all finitary operations in h(fp) are iterations of the künneth formula on the steenrod operations, reduced powers and bocksteins, so h(fp) is well understood. because z is not algebraically compact property (4) does not hold and maps out of infinite products of e-m spaces over z do not factor through a finite subproduct and we must extend our definition of primary operations to cover “infinary” operations and hence let h(z)∞ contain arbitrary products of integral e-m spaces. this then brings up the question of whether the model category structure on set ∗h(r) for h(r) containing finite products is still valid for z coefficients. sketch categories containing infinite products have many of the properties of finite sketch categories under certain restrictions. a good overview is given in [1]. it turns out that even when a sketch category contains infinite products the category of models will be locally presentable [12, 5.6.8]. as such, the category of h(z)∞-algebras has all limits and colimits [12, 5.5.8]. then the model category structure of [8] can be extended to simplicial h(z)∞-algebras. according to this model category structure the homotopy groups πn(t•z), n ≥ 0, of a simplicial h(r)-algebra are the n-th homotopy groups of the underlying simplicial graded group. it can then be shown [26] that πq (t•z) ∼= { 0 if q ̸= 0 z if q = 0 so that t•z has the property of a free simplicial resolution of z. 94 a. jaleel, a. percy 5.2. resolution by cosimplicial space. we would also like to construct a simplicial resolution of a topological h(r)-algebra as the h(r)algebra of a cosimplicial space. this will allow the use of spectral sequences in the calculation of topological h(r)-algebras. that is, for a given space x we would like to construct an augmented cosimplicial space, x• ← x such that the augmented simplicial group h̃q(x•) → h̃q(x) satisfies πph q(x•) = { hq(x) if p = 0 0 otherwise (5) for all q, with the homomorphism π0h̃ q(x•) ∼= h̃q(x) induced by the augmentation. we would also require h̃∗(x•) to have a free h(r)-algebra in each simplicial dimension which, by remark 4.2, requires x• to have an object of h(r) in each cosimplicial dimension. figure 2 shows the simplicial h(r)-algebra, h̃∗(x•) where each column in set ∗ is an h(r)-algebra and each row a simplicial (abelian) group. h k4 k1 × k2 k3 [x•, ]h+ + ∗ h∪ ∪ α α ∗ x• cosimplicial space x0 x1 x2 s 0 d 0 s 1 s 0 s 1 s 2 d 0 d 1 h̃1(x0) h̃2(x0) 0 h̃3(x0) h̃4(x0) d0 s0 s1 ... h̃1(x1) h̃2(x1) ... 0 h̃3(x1) h̃4(x1) ∪ + α∗ d0 s0 s1 d1 s2 set ∗ figure 2: simplicial h(r)-algebra of a cosimplicial space a cosimplical space can indeed be constructed, homotopic to a product of e-m spaces in each cosimplicial dimension, dual to stover’s pushout construction on wedges of spheres and discs [34]. let pkn be the path space over kn and e be the evaluation map sending a based path to it’s end point. if a map f : x → kn is null-homotopic there are maps g : x → pkn such that f ∼= e∗(g) [3, proposition 1.4.9]. dually to stover’s construction, we define an endofunctor t : t∗ → t∗ such that, for a resolutions of cohomology algebras 95 space x, t(x) is the pullback of diagram (6). ϕ is the projection onto the subfactors kn e∗(g) such that f = e∗(g), followed by indentifying the indices e∗(g) and g : tx // �� ∏ n∈n ∏ g:x→pkn pkng ∏ e ��∏ n∈n ∏ f:x→kn knf ϕ // ∏ n∈n ∏ g:x→pkn kng (6) effectively tx has a factor of kn for each map f : x → kn. if f = e∗(g) is a null-homotopic map then the points of knf are identified with the end point of the paths in pkng . contracting any path-space leaves the set of loops on kn. since ωkn ∼= kn−1 it follows that tx is homotopy equivalent to a product of eilenberg-mac lane spaces and hence will have a free h(r)algebra. remark 5.2. the space tx will be less connected than x because of the loop spaces in the product. consequently, at some cosimplicial dimension the product space will cease to be simply connected. the unit for the monad is the natural transformation µ : 1 → t defined on any space x as the unique map given by the universal property of a pullback: x {g} ,,yyyyy yyyyyy yyyyyy yyyyyy yyyyyy yyyyyy yyyyyy {f} ��? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ?? ? µx ((qq qqq qqq qqq qqq qqq qq t(x) // �� ∏ n∈n ∏ g:x→pkn pkng ∏ e ��∏ n∈n ∏ f:x→kn knf ϕ // ∏ n∈n ∏ g:x→pkn kng the multiplication is a natural transformation η : t 2 → t defined for any x ∈t∗ as follows: 96 a. jaleel, a. percy t 2(x) φ2 ,,yyyyy yyyyyy yyyyyy yyyyyy yyyyyy yyyyyy yyyyyy φ1 a aa aa aa aa aa aa aa aa aa aa aa aa ηx ((qq qqq qqq qqq qqq qqq qq t(x) // �� ∏ n∈n ∏ g:x→pkn pkng ∏ e ��∏ n∈n ∏ f:x→kn knf ϕ // ∏ n∈n ∏ g:x→pkn kng the map φ1 is defined by projecting the factors k n prf onto the factor knf and similarly, φ2 is defined by projecting the factors pk n prg onto the factor pkng . from the monad, ⟨t, µ, η⟩ we can then form a cosimplicial functor by the standard construction and apply this to a space x to give a cosimplicial space x•. for the simplicial h(r)-algebra, h̃∗(x•), dually to stover [34], we can show that defining πph q(x•) as the p-th homotopy group of the underlying simplicial group h̃q(x•) we have formed a free resolution [26]. that is, each simplicial dimension is a free h(r)-algebra and equations (5) hold. remark 5.3. it is in the proof that the simplicial groups are acyclic that the path spaces must be included in the pullback construction t in diagram (6), to provide a null-homotopy for dn+1i prf, 1 ≤ i ≤ n + 1, so that for every f in the n-th moore chain complex prf is in the (n + 1)-th moore complex with dn+10 prf = f. 6. calculating with the cosimplicial resolution if we had a convergent spectral sequence for a cosimplical space e −p,q 2 = πph q(x•) ⇒ hq−p(tot(x•)) , then we would be able to calculate the cohomology of more complicated spaces, such as map(y, x). this would be achieved by identifying the e2 page as derived functors of map(y, ) applied dimensionwise to a free cosimplicial resolution x• [7]. bökstedt and ottosen [5] were similarly motivated to construct spectral sequences for calculating the cohomology of fp-completions of the string space resolutions of cohomology algebras 97 (λx)ht of a simply-connected space, x of finite type. using a free cosimplicial resolution rx• → x by simplicial fp-modules [17], they dualize the construction of bousfield’s homology spectral sequence [14]. using (λ )ht applied dimensionwise to rx• gives a spectral sequence e −p,q 2 = ( πph ∗((λrx•)ht; fp )q ⇒ h∗(tot(((λrx•)ht); fp) and associated convergence criteria. for a resolution by fp-modules, tot(rx•) is the fp-completion of a space x [17], so the spectral sequence converges to the fp-cohomology algebra h∗((λx)ht); fp). to explain the e2 terms, (πph ∗((λrx•)ht; fp)q are the collection of fpmodules πph q((λrxq)ht; fp) with the mod p steenrod operations acting columnwise. bökstedt and ottosen [5] used the known endofunctor l, on the category of unstable algebras over the steenrod algebra, to show that e −p,q 2 is the derived functor hp(h ∗(x); l)q where the homology of h∗(x) with coeficients in the functor l is defined by hp(h ∗(x), l) = πpl(rx •). when dualizing bousfield’s homology spectral sequence for use with integer coefficients we must remember that we would want convergence to the cohomology h(z)-algebra of the space x or its z-completion. to do so, it is important that the cosimplicial space x• be fibrant so tot(x•) is the inverse limit of a tower of fibrations from which a convergent spectral sequence could be defined [17, §6]. fibrancy is ensured by the group object property of gem’s [17, x §4.9]. however, unfortunately for integer coefficients tot(x•) need not have the same cohomology as x itself. in order to have the same cohomology, tot(x•) must be homotopy equivalent to the r-completion of x and this is known to occur only when h̃∗(x•) is an acyclic simplicial group for cohomology in any r-module coefficients [15, §7]. we have seen in section 5 that by constructing x• with gem’s over z, h̃∗(x•) is acyclic over z coefficients. however, by remark 5.3, h̃∗(x•) is not acyclic for cohomology in other z-modules, for instance z/2, because the pullback construction (6) supplies null homotopies only for z coefficients. on the other hand, for fp coefficients the construction (6) will give h̃∗(x•) acyclic for cohomology in any fp-module since any fp-module is a direct sum of copies of fp. hence, for fp coefficients tot(x•) is the fp-completion of x. to ensure h̃∗(x•) is acyclic over all z-modules we could expand construction (6) to be a pullback over indexed products of e-m spaces and path spaces 98 a. jaleel, a. percy over all z-module coefficients. this requires the construction to contain arbitrary products over a proper class of e-m spaces. as such, the construction does not exist in t∗, since topological spaces must in particular be sets, not proper classes. furthermore, by remark 4.2, the gem in each cosimplicial dimension must be an object in h(z). functors from the resulting sketch category h(z) may not have a well defined image in set ∗ [29]. remark 6.1. it is possible, however, to restrict the gem in each cosimplicial dimension, and hence the category h(z), to products indexed by a set of bounded (infinite) cardinality, λ, and still achieve acyclicity over all z-modules. the construction of blanc and sen [9] is complex and requires the topological h(z)-algebra h̃∗(x; z), that is, the cardinality λ depends on the space x. primarily the construction relies on the enrichment of h(z) of remark 4.3, which also enriches the resultant h(z)-algebras. it turns out that this enrichment can identify sufficient information, whilst putting an upper bound on λ, to ensure tot(x•) is the z-completion of x [9, corollary 4.28]. finally then, it becomes a question of whether the spectral sequence e −p,q 2 = πph q(x•) ⇒ hq−p(tot(x•)) can be shown to converge. a general result does not exist in the literature. what can be said is that wherever the well known bousfield-kan resolution rx• → x by r-modules [17] can be used in a spectral sequence, the restricted resolution by gems of remark 6.1 can be used. one advantage of resolutions by gems is recognizing the e2 page of spectral sequences as derived functors. for coefficients in a general r, the yoneda lemma gives an embedding of the free representable h(r)-algebras into h(r)op [31]. applying a functor dimensionwise to a cosimplical resolution by gems is equivalent to a simplicial functor from h(r)-algebras and then taking (co)homotopy gives the e2 page of spectral sequences as a derived functor. 7. conclusion in their introduction to part i of [17], bousfield and kan justify the importance of completions and localizations as being able to decompose a homotopy type into fp type and, together with coherence information over the rationals, be able to reconstruct it. they point out that many problems can be solved with fp information alone. in this review of properties of integral cohomology resolutions of cohomology algebras 99 we have seen that often more can be obtained with fp cohomology, since integral cohomology is both overly complex (section 3) and unsuited to many of the powerful tools available for calculation (section 5). these problems with the primary integral operations do not seem to get a mention in homology texts which usually switch from z to z/2 coefficients, without explanation, when introducing the higher structure of operations. since eckmann-hilton duality is not a categorical duality it should not be expected to hold universally. the construction of a free simplicial resolution of a space using wedges of spheres and discs has been of great benefit to homotopy theory and algebraic topology generally however, the dual resolution is more complex and has not yet delivered the same benefits when working with integer coefficients. it is true that, in comparison, the cosimplicial resolution by simplicial rmodules of bousfield and kan [17, §4] is smaller and remains simply connected for a simply connected space (see remark 5.2), even for z coefficients. however, the resolution by products of gems contains the additional information of higher order operations [9], which can be used to identify enough information to ensure tot(x•) is the r-completion of x, for any commutative ring r. the additional information has been sufficient to reconstruct x, up to r-completion from h̃∗(x; r), but, once again, this important result has only been achieved for r = fp or q [9, 10]. resolutions by products of e-m spaces over fp coefficients have been used to solve the realisation problem of which unstable coalgebras over the steenrod algebra can be realised as the cohomology h(fp)-algebras of a space, h̃∗(x; fp) [6]. this work is also dual to work done in homotopy using stover’s resolution but does not extend to z as it requires (co)homology to consist of vector spaces rather than modules. the attempt to eckmann-hilton dualise stover’s seminal construction has been achieved with considerable effort for integer coefficients. however, for fp coefficients, construction (6) is quite straightforward and new results, such as the realisation problem, are commencing to be attained. this survey should suffice to convince the reader that any extension of results for fp coefficients to z coefficients are likely to be difficult to achieve. 100 a. jaleel, a. percy 8. appendix: unary integral cohomology operations table 1 shows results of the (leray)-serre spectral sequence calculation of the groups h̃m(kn) , 2 ≤ m ≤ 14 , 2 ≤ n ≤ 7 (extended from [32] to include m = 14). angle brackets ⟨x⟩ indicate that the class x generates the summand. the cup product structure is given by the s.s. and written as juxtaposition so b2 = bb = b ∪ b. the generators of other summands have been tested for decomposability into compositions or crosscap products using dimensional arguments or known relations. the generator ................................................................................... n m 2 3 4 5 6 7 14 z⟨a7⟩ 0 z/2⟨g2⟩ z/3⟨n⟩ ⊕ z/5⟨o⟩ 0 z/2⟨t2⟩ ⊕ z/2⟨u⟩ 13 0 z/2⟨bd⟩ z/5⟨j⟩ ⊕ z/3⟨fh⟩ z/2⟨kω−1g⟩ z/2⟨s⟩ 0 12 z⟨a6⟩ z/2⟨b4⟩ ⊕ z/5⟨e⟩ z⟨f3⟩ z/2⟨m⟩ z⟨p2⟩ 11 0 z/3⟨bc⟩ z/2⟨fg⟩ ⊕ z/2⟨?⟩ 0 z/2⟨q⟩ ⊕ z/3⟨r⟩ 10 z⟨a5⟩ z/2⟨d⟩ 0 z/2⟨k2⟩ ⊕ z/3⟨l⟩ 9 0 z/2⟨b3⟩ z/3⟨h⟩ 0 8 z⟨a4⟩ z/3⟨c⟩ z⟨f2⟩ 7 0 0 z/2⟨g⟩ ⟨t⟩ 6 z⟨a3⟩ z/2⟨b2⟩ ⟨p⟩ 5 0 0 ⟨k⟩ 4 z⟨a2⟩ ⟨f⟩ 3 0 ⟨b⟩ 2 z⟨a⟩ table 1: the groups h̃m(k(z, n); z) resolutions of cohomology algebras 101 of one summand of h̃11(k4) remains undetermined as either indecomposable or f2 ◦ω−4g, hence is given as ⟨?⟩. the stable range, where ⟨ωx⟩ = h̃m(kn) ∼= h̃m+1(kn+1) = ⟨x⟩ , n ≤ m ≤ 2n−1 , is left blank, although the fundamental class is denoted by a new letter rather than as ω−ia, where ω−1 is the transgression. there are some notable differences to cohomology groups, h̃m(k(fp, n); fp) , of eilengerg-mac lane spaces over fp. we see that there are non-stable compositions for integer coefficients, moreover, h11(k6) shows that there can be more than one stable composition of a given degree, possibly of different order. h̃14(k3) ∼= 0 would be a surprising result if we had coefficients in a field since in those coefficients cup product generates freely [19] whereas with the integers we have b2c = 0. some relation is involved here forcing a cup product of non-trivial generators to be trivial. we may also expect elements created by compositions ω−10g◦(bc), ω−3r◦(b3) and ω−3q◦(b3) within h̃14(k3) ∼= 0. there is also the external cup product and “unknown” primary operation of example (3.5) which could act by universal example on two copies of the fundamental class. acknowledgements the authors would like to thank david blanc for generously giving advise during their visits. we would also like to thank the referee for helpful comments to improve the paper. references [1] j. adámek, h-e. porst, algebraic theories of quasivarieties, j. algebra 208 (2) (1998), 379 – 398. 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[36] g.w. whitehead, “ elements of homotopy theory ”, graduate texts in mathematics, 61, springer-verlag, new york-berlin, 1978. e extracta mathematicae vol. 31, núm. 2, 169 – 188 (2016) more indecomposable polyhedra krzysztof przes lawski, david yost wydzia l matematyki, informatyki i ekonometrii, uniwersytet zielonogórski, ul. prof. z. szafrana 4a, 65 − 516 zielona góra, poland and wydzia l matematyki, informatyki i architektury krajobrazu, katolicki uniwersytet lubelski, ul. konstantynów 1 h, 20 − 708 lublin, poland k.przeslawski@wmie.uz.zgora.pl centre for informatics and applied optimization, faculty of science and technology, federation university, po box 663, ballarat, vic. 3353, australia d.yost@federation.edu.au received june 12, 2016 abstract : we apply combinatorial methods to a geometric problem: the classification of polytopes, in terms of minkowski decomposability. various properties of skeletons of polytopes are exhibited, each sufficient to guarantee indecomposability of a significant class of polytopes. we illustrate further the power of these techniques, compared with the traditional method of examining triangular faces, with several applications. in any dimension d 6= 2, we show that of all the polytopes with d2 + 1 2 d or fewer edges, only one is decomposable. in 3 dimensions, we complete the classification, in terms of decomposability, of the 260 combinatorial types of polyhedra with 15 or fewer edges. key words: polytope, decomposable. ams subject class. (2010): 52b10, 52b11, 52b05. dedicated to the memory of carlos beńıtez what happens if you have two line segments in the plane, oriented in different directions, and you calculate all the sums of all pairs of elements, one from each segment? of course you end up with a rectangle, or at least a parallelogram. do the same again with a triangle and a line segment in three dimensions: this time, you get a prism. thus the prism and the parallelogram are decomposable; they can be expressed as the (minkowski) sum of two dissimilar convex bodies. (recall that that two polytopes are similar if one can be obtained from the other by a dilation and a translation.) on the other hand, any triangle, tetrahedron or octahedron is indecomposable. we refer to [7] for a general introduction to the theory of polytopes, as well as for specific results. determining the decomposability of a polytope 169 170 k. przes lawski, d. yost can be reduced to a computational problem in linear algebra [12, 17]. that is, given the co-ordinates of its vertices, all we have to do is calculate the rank of a rather large matrix. however that is not the approach to be taken here. the edges and vertices of any polytope obviously constitute a graph, sometimes known as its skeleton. in the case of a polyhedron, this will be isomorphic to a planar graph. all the geometric conclusions of this paper will be established by considering the properties of this graph. section 1 develops a number of sufficient conditions for indecomposability (or decomposability). our results have wider applicability than earlier results in this area, which generally relied on the existence large families of triangular faces. section 2 applies them to complete the classification of 3-dimensional polyhedra with up to 15 edges. section 3 applies them to completely classify, as indecomposable or decomposable, all d-dimensional polytopes with up to d2 + 1 2 d edges. we also show that there is no d-dimensional polytope at all with 2d vertices and d2 + 1 edges, for d 6= 3. 1. geometric graphs and indecomposability we will not give a thorough history of this topic, but it is important to recall some preliminary information. we depend heavily on the concept of a geometric graph, which was pioneered by kallay [9]. he defined a geometric graph as any graph g whose vertex set v is a subset of a finite-dimensional real vector space x, and whose edge set e is a subset of the line segments joining members of v . (of course, x will be isomorphic to rd for some d, but we prefer this basis-free formulation.) it is largely a formality whether we consider an edge to be an unordered pair or a line segment. it is significant that such a graph need not be the edge graph of any polytope. he then extended the notion of decomposability to such graphs in the following manner. for convenience, let us say that a function f : v → x is a decomposing function for the graph (v,e) if it has the property that f(v)−f(w) is a scalar multiple of v − w for each edge [v,w] ∈ e. (this is slightly different from kallay’s local similarity; he insisted on strictly positive multiples.) a geometric graph g = (v,e) is then called decomposable if there is a decomposing function which is neither constant, nor the restriction of a homothety on x. if the only non-constant decomposing functions are homotheties then g is called indecomposable. significantly, kallay showed [9, theorem 1] that a polytope is indecomposmore indecomposable polyhedra 171 able if and only if its edge graph is indecomposable in this sense. exploiting an idea of mcmullen [11] and kallay [9, theorem 1b], we showed in [14, theorem 8] that it is even sufficient just to have an indecomposable subgraph which contains at least one vertex from every facet (maximal face). a strategy for proving indecomposability of a polytope is thus to prove that certain simple geometric graphs are indecomposable, and by building up to show that the entire skeleton of our polytope is indecomposable. (it also would be interesting to formulate somehow a notion of primitivity for such graphs.) building on the concept introduced in [18, p. 139], let us say that a geometric graph g = (v,e) is a simple extension of a geometric graph g0 = (v0,e0) if g has one more vertex and two more edges than g0. more precisely, we mean that there is a unique v ∈ v \v0, and distinct vertices u and w in v0, such that e = e0∪{[u,v]}∪{[v,w]}. observe that the existence of these two edges means that the value of any decomposing function at v is determined by its values at u and w. no assumption is made about whether [u,w] is an edge of either graph. our first result is a special case of the next one, but it is so useful and so easy to prove that it is worth stating separately. proposition 1. suppose that g0,g1, . . . ,gn are geometric graphs, that gi+1 is a simple extension of gi for each i, and that g0 is indecomposable. then gn is also indecomposable. proof. it is clearly sufficient to prove this when n = 1, and this follows from the observation in the preceding paragraph. let us illustrate how this can be applied in the simplest cases, polyhedra for which “sufficiently many” faces are triangles [15, §3]. any edge is obviously indecomposable, and then proposition 1 easily implies that any triangle is indecomposable. furthermore if an indecomposable geometric graph shares an edge with a triangle, then their union is easily proved to be indecomposable. it follows that the union of a chain of triangles, as defined in [15, p. 92], is an indecomposable graph. this makes it clear that a polyhedron must be indecomposable if every face is a triangle. if every face but one is a triangle, it remains true that the triangular faces can be ordered into a chain, whose union is the entire skeleton of the polyhedron; again indecomposability is assured. the same holds if all faces but two are triangular, and the nontriangular faces do not share an edge. if all faces but two are triangular, but the non-triangular faces do share an edge, then the triangular faces can still be ordered into a chain, whose union will contain every vertex of the polyhedron 172 k. przes lawski, d. yost a d c b ef a c d e b figure 1: two indecomposable examples without many triangles and every edge bar one. so, indecomposability is assured, whenever there are two or fewer non-triangular faces. this conclusion no longer holds if we have three non-triangular faces, since the triangular prism is decomposable. on the other hand, there are also many indecomposable polyhedra with precisely three non-triangular faces. for comparison, let us mention that that a polyhedron with only three, or fewer, triangular faces is automatically decomposable [17, §6]. to show how powerful proposition 1 is, we note that it guarantees indecomposability of any polyhedron whose graph is either of those shown in figure 1. in neither example is there a chain of triangles touching every face. for the first example, begin with the edge ab, which is indecomposable, then successively add the vertices c,d,e and f. each additional vertex is adjacent to two of the preceding ones, so the resulting geometric graph is indecomposable. since it touches every face, the polyhedron is indecomposable. the second example is even quicker; beginning with the edge ab, it is enough to add the vertices c,d then e. a similar argument also gives a particularly easy proof of the indecomposability of the example in [9, §6]. further applications are given in [3]. kallay [9, theorem 8] showed that if two indecomposable graphs have two common vertices, then their union is indecomposable. a prime example for this result is the 199th polyhedron in the catalogue [4], which will be discussed more indecomposable polyhedra 173 figure 2: bd199 is indecomposable again in the next section. in figure 1, it should be clear that the six triangular faces can be partitioned in two groups, each of which constitutes a chain of 3 triangles. it is clear that the resulting two indecomposable geometric graphs have two vertices (but no edge) in common, and that their union contains every vertex. our next result is a generalization of both proposition 1 and [9, theorem 8]. our proof is no different from kallay’s but, as we shall soon see, our formulation is somewhat more powerful. it is clear from the definition that adding an edge but no vertex to an already indecomposable graph preserves its indecomposability. the point of part (i) is that, with a little care, we can throw away some edges and still preserve indecomposability. part (ii) says that if one edge of an indecomposable graph is replaced by another indecomposable graph, then the new graph is indecomposable. theorem 2. (i) suppose that g1 = (v1,e1) and g2 = (v2,e2) are two geometric graphs in the same vector space, and that v12 = v1 ∩v2 contains at least two distinct vertices. let e12 be the collection of those edges of g1, both of whose vertices lie in v12. let g = (v,e) be another geometric graph with vertex set v = v1 ∪v2 and whose edge set e contains (e1 \e12) ∪e2. if both g1 and g2 are indecomposable, then so is g. (ii) suppose that g1 = (v1,e1) and g2 = (v2,e2) are two indecomposable 174 k. przes lawski, d. yost geometric graphs, that v1 ∩ v2 contains at least two distinct vertices u and w. define a new geometric graph g = (v,e) with vertices v = v1 ∪v2 and edges e = (e1 \{[u,w]}) ∪e2. then g is also indecomposable. proof. (i) let f : v → x be a decomposing function, where x is the ambient vector space. since g2 is indecomposable, f|v2 must be the restriction of a similarity, i.e. there are a scalar α and a vector x such that f(v) = αv +x for all v ∈ v2. in particular, f(u) −f(w) = α(u−w) for all u,w ∈ v12 (even when [u,w] is not an edge of g2). since e1 ⊆ e12∪e, this implies that f|v1 is also a decomposing function, so by hypothesis must also be the restriction of a similarity. thus there are a scalar β and a vector y such that f(v) = βv + y for all v ∈ v1. now, fix distinct u,w ∈ v12. consistency requires αu + x = βu + y and αw+x = βw+y, which quickly forces x = y and α = β. thus f is a similarity. (ii) in the notation of part (i), we clearly have (e1 \e12) ∪e2 ⊆ e. note that we make no assumption about whether the edge [u,w] belongs to either g1 or g2. recall that a graph g is called a cycle if |v | = k ≥ 3 and v can be ordered as {v1, . . . ,vk}, so that e = {{v1,v2}, . . . ,{vk−1,vk},{vk,v1}}. the number k is said to be the length of the cycle. the next result is a rewording of [14, proposition 2]. once formulated it is easy to prove, yet surprisingly useful. the 3-dimensional case has already been used in [14, §4]. we state it explicitly here, since we will use both the 3-dimensional and higher dimensional versions in the next sections. proposition 3. any cycle, whose vertices are affinely independent, is an indecomposable geometric graph. in particular, a polytope will be indecomposable, if its skeleton contains a cycle, whose vertices are not contained in any affine hyperplane, and which touches every facet. the next result indicates further how indecomposability of a graph can be established by considering smaller subgraphs. theorem 4. (i) let h = (v,e) be an indecomposable geometric graph and for each e = [u,v] ∈ e, let ge = (ve,ee) be an indecomposable geometric graph containing both vertices u,v. then the union ⋃ e ge is an indecomposable geometric graph. (ii) let g1 = (v1,e1), g2 = (v2,e2), . . . , gn = (vn,en) be indecomposable geometric graphs and let v1,v2, . . . ,vn be a collection of affinely indemore indecomposable polyhedra 175 pendent vertices. set g0 = gn, v0 = vn and suppose vi ∈ vi ∩vi−1 for each i. then the union g1 ∪g2 ∪·· ·∪gn is indecomposable. (iii) let p be a polytope, and let g1 = (v1,e1), g2 = (v2,e2) be two indecomposable subgraphs of the skeleton of p , with v1 ∩ v2 6= ∅, and suppose that v1 ∪ v2 contains all but at most d − 2 vertices of p . then p is indecomposable. proof. (i) just apply theorem 2(ii) successively, replacing each edge e of h with the graph ge. (ii) the graph with vertices v1, . . . ,vn and edges [v1,v2], . . ., [vn−1,vn], [vn,v1] is indecomposable by proposition 3. now we just apply (i). (iii) if v1 ∩v2 contains two or more elements, the conclusion follows from theorem 2(i). so we assume that v1 ∩v2 contains a unique element, say v2. set v ′i = vi \ {v2}, and c = v \ (v ′ 1 ∪ v ′ 2). then c contains at most d − 1 vertices, so their removal from the graph of p will not disconnect it. since v ′1 and v ′2 are disjoint, there must then be an edge between them, say between v1 ∈ v ′1 and v3 ∈ v ′ 2. letting g3 be the graph with the single edge [v1,v3], we can apply (ii) with n = 3. (we cannot claim g1 ∪g2 is indecomposable.) it is easy to see that [9, theorem 9] is precisely the case n = 3 of part (ii), and that [9, theorem 10] is implied by the case n = 4. part (iii) is a strengthening of [16, corollary 8.6], where it is assumed that every vertex of p lies in v1 ∪ v2. we will indicate the strength of this with another two examples. in both polyhedra whose graphs are shown in figure 3, we can take g1 and g2 as chains of triangles. only part (iii) of the preceding theorem is capable of proving the indecomposability of these two polyhedra. sufficient conditions for decomposability are not so common. the following result was proved without statement by shephard [15, result (15)]. more precisely, he made the stronger assumption that every vertex in f had degree d; however, his proof also works in the formulation presented here. it may be interesting to present a proof using decomposing functions. proposition 5. a polytope p is decomposable whenever there is a facet f such that every vertex in f has a unique neighbor outside f , and p has at least two vertices outside f. proof. let y be a support functional for f . we may suppose that y(f) = {1} and that y(x) < 1 for all other x in the polytope. label the vertices of f as v1, . . . ,vn. for each vertex vi of f, denote by wi the unique vertex which 176 k. przes lawski, d. yost figure 3: another two examples is adjacent to vi but not in f. set α = max n i=1 y(wi); clearly α < 1. for each i, let xi be the unique point on the edge [vi,wi] satisfying y(xi) = α. now define a function f by f(vi) = xi and f(v) = v for all other vertices. clearly f(v) − f(w) = v − w whenever both vertices are outside f . since f(vi) − f(wi) = xi − wi and xi is a convex combination of vi and wi, the condition for a decomposing function is also satisfied when one vertex lies in f. what if both vertices lie in f? fix two adjacent vertices vi,vj in f, and consider a 2-face containing them but not contained in f. this face must contain xi and xj. since y(vi) = y(vj) 6= y(xi) = y(xj), the line segments [vi,vj] and [xi,xj] must be parallel; we do not claim that [xi,xj] is an edge of p . then f(vi) −f(vj) = xi −xj is a non-negative multiple of vi −vj. so f is a decomposing function, as anticipated. finally, f is not a similarity, because it coincides with the identity function at all the vertices outside f but is not equal to the identity function. decomposability follows. the next section requires the 3-dimensional case of the following result. it is not difficult, but appears to be new, so we state it in full generality. proposition 6. let f be a facet of a polytope p . suppose that f is indecomposable. let q be obtained from p by stacking a pyramid on f. then p is decomposable if and only q is decomposable. proof. let v be the vertex set of p , u the unique vertex of q not in p , s the pyramid being glued onto f and x the ambient vector space. it is easy more indecomposable polyhedra 177 to show that any decomposing function defined on f has a unique extension to s. if p is indecomposable, so is its graph, g(p). let f : v ∪{u} → x be a decomposing function for q. then f|v is a decomposing function for p , so we find α,x so that f(v) = αv + x for all v ∈ v . by the previous paragraph, αu + x is the only conceivable value for f(v). thus q is indecomposable. conversely, suppose q is indecomposable. let g : v → x be any decomposing function for p . again by the first paragraph, there a unique decomposing function f : v ∪{u} → x which extends g. since f must be the restriction of a homothety on x, so must g. 2. polyhedra with 15 edges if a given polyhedron has v vertices, e edges and f faces, then euler’s relation e = v +f−2 suggests that the number of edges is a reasonable measure of its complexity. accordingly, we gave in [19] the complete classification, in terms of decomposability, of the 58 combinatorial types of polyhedra with 14 edges. the classification of the 44 types of polyhedra with 13 or fewer edges was essentially known [17, §6]. indeed, there are only four types of polyhedra with 6, 8 or 10 edges, and they are easily seen to be indecomposable. no polyhedron can have 7 edges. besides the triangular prism, the only other polyhedron with 9 edges is the triangular bipyramid, which is obviously indecomposable. a thorough study of this topic had already been made by smilansky, who showed [17, theorem 6.7] that a polyhedron is decomposable if there are more vertices than faces; and that a polyhedron is indecomposable if f ≥ 2v − 6. as remarked in [19, p 719], simply knowing the values of f and v is then enough to decide decomposability in all cases when e ≤ 11 or e = 13. the examples with 12 edges were discussed in more detail in [19], but the results were obviously known to smilansky. (the only example whose indecomposability is not clear from classical triangle arguments is [19, figure 2], and its indecomposability is guaranteed by [16, corollary 8.6].) the classification of polyhedra with 14 or fewer edges incidentally completed the classification of all polyhedra with 8 or fewer faces. we should recall that two polytopes are said to be combinatorially equivalent if their face lattices are isomorphic. in three dimensions, steinitz’s theorem assures us that two polyhedra are combinatorially equivalent as soon as we know that 178 k. przes lawski, d. yost their graphs are isomorphic. in many cases, two polytopes with the same combinatorial type will either both be decomposable or both be indecomposable. smilansky [17, §6] first announced that this is not so for polyhedra with 14 edges, and some explicit examples were given in [19]. we push this project a bit further in this section by completing the classification of the 158 combinatorial types of polyhedra with 15 edges. this also completes the classification of the 301 combinatorial types of polyhedra with 8 or fewer vertices. we also note the indecomposability of all higher dimensional polytopes with 15 or fewer edges. the aforementioned results of smilansky imply that a polyhedron is decomposable if (v,f) is either (10, 7) or (9, 8) and that a polyhedron is indecomposable if (v,f) = (7, 10); these three cases account for 84 combinatorial types. our assumption that v + f = 17 then tells us that the only case remaining is v = 8,f = 9. there are 74 combinatorial types of polyhedra with 8 vertices and 9 faces, which were first described verbally, but not visually, by kirkman [10, pp 362– 364]. it is possible to use computers to generate diagrams of such polyhedra, but we are dealing with a relatively small number of polyhedra, so it is simpler to use a published catalogue. the only one for this class seems to be that of britton and dunitz [4]. they exhibited diagrams of all the 301 combinatorially distinct types of polyhedra with up to 8 vertices. on their list, those with 8 vertices and 9 faces are numbers 129 to 202 in [4, fig. 5]. of these, we will see that most are indecomposable because they have sufficiently many triangles, and 2 are obviously decomposable thanks to proposition 5 (in the simplest geometric realizations, because they have a segment as a summand). the remaining 6 are also indecomposable but arguments using triangular faces alone don’t work; we need to use the results from §1 to establish their indecomposability. theorem 7. of the 74 types of polyhedra with 9 faces and 8 vertices, only 2 types are decomposable, 66 types are indecomposable by classical arguments, and the remaining 6 require some results from §1 to establish their indecomposability. more precisely: (i) polyhedra numbers 182 and 198 (on the list of britton and dunitz) are decomposable. (ii) altogether, 66 are indecomposable by virtue of having a connected chain of triangular faces. specifically, we mean those numbered 129–172, 174–178, 180, 181, 183–186, 188, 189, 191, 193–197, 200–202. more indecomposable polyhedra 179 figure 4: the capped prism, bd182 and bd198 (iii) the other 6, namely numbers 173, 179, 187, 190, 192 and 199, are indecomposable thanks to either proposition 1 or 3 or theorem 2. proof. we begin with the decomposable examples. as remarked in the opening paragraph, a triangular prism is the minkowski sum of a triangle and a line segment. if we glue a tetrahedron onto one end of a prism, we obtain the capped prism, which is decomposable as the sum of a tetrahedron and a line segment. better still, proposition 5 guarantees that any polyhedron combinatorially equivalent to this will be decomposable. there are two ways to glue a second tetrahedron onto the capped prism. either we glue it onto one face of the first tetrahedron, or we glue it onto the remaining triangular face of the original prism. in both cases, we obtain a polyhedron with 8 vertices and 9 faces. all three are pictured here, the latter two being numbers 182 and 198 respectively from the list of [4]. in case the three edges which lie between pairs of quadrilateral faces are all parallel, each of the latter two polyhedra will be decomposable, being the sum of a triangular bipyramid and a line segment. since one of them has vertices of degree 5 and the other does not, they are not combinatorially equivalent. this exemplifies the fact that the combinatorial type of two polyhedra does not determine the type of their sum. our diagrams are not identical to those in [4]; we have drawn them slightly differently to emphasize their decomposability. it is true, but not totally obvious, that any polyhedron combinatorially equivalent to these two will be decomposable. let us prove it. proposition 5 clearly implies that [4, 182] is decomposable (but not necessarily that a line segment will be a summand). for [4, 198], recall that the capped prism is decomposable, and then apply proposition 6. now let us look at the indecomposable examples. numbers 129, 130 and 180 k. przes lawski, d. yost figure 5: bd173 and bd179 131 each have one hexagonal face and 8 triangular faces. each of examples 132–155 has one pentagonal face, one quadrilateral face and 7 triangular faces. indecomposability of all these examples is assured by our remarks in the previous section, because they have at most two non-triangular faces. examples 156–181, 183–197 and 199–202 all have three quadrilateral and six triangular faces. by inspection, all but six of them (namely 173, 179, 187, 190, 192 and 199) are indecomposable because (some of) their triangular faces can be ordered into a chain whose union touches every face. we note also that for some examples, the chain of indecomposable triangles does not contain every vertex. (in particular, 157 and several others each have a vertex which does not lie in any triangular face.) thus the weakness of the assumption, that the chain only touches every face, is significant. proposition 3 implies the indecomposability of examples 173, 179, 187, 190 and 192 from britton and dunitz. alternatively, their indecomposability can also be established by proposition 1. none of these examples contains a connected sequence of triangular faces touching every face, so some new technique was needed. we present here their diagrams, with an appropriate 4-cycle highlighted. in each case, three vertices of the 4-cycle lie in one face, while the fourth does not, so the 4-cycle cannot be coplanar. the diagrams make it clear that the 4-cycle touches every face. this time, we have used the same diagrams as in [4], except that for aesthetic reasons we have reversed the front and back faces of 190 and 192. finally, we recall from §1 that 199 is also indecomposable. we remark that all higher dimensional polytopes with 15 or fewer edges are indecomposable; this extends [19, proposition 2.11]. the “smallest” ddimensional polytope, the simplex, obviously has exactly 1 2 d(d + 1) edges. so more indecomposable polyhedra 181 figure 6: bd187, bd190 and bd192 in dimensions 6 and higher, there are in fact no polytopes with 15 or fewer edges. in dimension 5, the only polytope with 15 or fewer edges is the simplex. two more examples exist in dimension 4, but the next result shows they are both indecomposable. proposition 8. the assertion “every 4-dimensional polytope with n edges is indecomposable” is true if and only if n ≤ 15 or n = 17. proof. any polytope satisfying these restrictions on n will have at most 7 vertices [7, 10.4.2]. this condition forces indecomposability by [18, proposition 6]. for the converse, we need to consider various possible values for n. we will simply describe the examples, and not verify all the details. a particularly simple decomposable polytope with 18 edges is the sum of two triangles lying in orthogonal planes. the sum of a 4-dimensional simplex with a line segment, which is parallel to one 2-face but not parallel to any edge of the simplex, will have 19 edges. the sum of a 4-dimensional simplex with a line segment, which is not parallel to any proper face of the simplex, will have 20 edges. denoting by ei the usual basis vectors, let p be the convex hull of {0,e1,e2, e3,e4,e3 + e4}. then the sum of p with the segment [0,e1] has 22 edges. the sum of the cyclic polytope c(6, 4) with a line segment which is parallel to one of its edges, will have 25 edges. the sum of a 4-dimensional simplex with a triangle, which is parallel to one of its 2-faces but has the opposite orientation, will have 27 edges. if p is a polyhedron with e edges and v vertices, then the sum of a p with a line segment (not parallel to the affine hull of p) is easily seen to have 182 k. przes lawski, d. yost 2e +v edges. (this is equally true in higher dimensions.) the possible values of e and v for polyhedra are well known [7, §10.3], and the corresponding values of 2e + v account for all remaining values of n. in particular, the sum of a tetrahedron with a line segment has 16 edges. in the next section, we will see that this is (up to combinatorial equivalence) the only example with 16 edges. 3. polytopes with not too many edges a simplicial prism, i.e. the sum of a segment with a (d− 1)-dimensional simplex, has 2d vertices, d2 edges and d + 2 facets. these numbers turn out be the minimum possible, for a d-dimensional decomposable polytope. in the case of vertices or edges, the prism is (up to combinatorial equivalence) the unique minimiser. in d dimensions, any polytope has at least d+1 facets, and only the simplex has d + 1 facets. so no non-trivial bound on the number of facets will imply indecomposability. nor can uniqueness be expected; a (d − 2)-fold pyramid over a quadrilateral also has d + 2 facets. for further examples, see lemma 10 below. the conclusions regarding the numbers of vertices and edges are more interesting; for edges, this extends proposition 8 to higher dimensions. proposition 3 is an essential tool for these. so also is gale’s result [15, (14)] that any pyramid, i.e. the convex hull of a maximal face and a single point, is indecomposable. this is clear, because every 2-face outside the base must be triangular. as noted in [18, proposition 6], a d-dimensional polytope with strictly fewer than 2d vertices is automatically indecomposable, and this estimate is the best possible. we will prove now that the simplicial prism is the only decomposable d-polytope with 2d or fewer vertices, before the corresponding result about edges. we have learnt recently that this result was first proved by kallay [8, theorem 7.1, page 39] but never published; his argument is different, using balinski’s theorem. recall that a d-polytope p is simple if every vertex is simple, i.e. has degree d. clearly every simple d-polytope, other than a simplex, is decomposable. theorem 9. let p be a decomposable d-dimensional polytope with 2d or fewer vertices. then p is combinatorially equivalent to the sum of a line more indecomposable polyhedra 183 segment and a (d−1)-dimensional simplex (and hence has precisely d2 edges). proof. the 2-dimensional case is almost obvious and the 3-dimensional case is quite easy, from §2. we proceed by induction on d. so let p be a decomposable (d + 1)-dimensional polytope with 2(d + 1) or fewer vertices. then some d-dimensional facet, say f, must be decomposable. since p is not a pyramid, there must be (at least) two vertices of p outside f; this implies that f has at most 2d vertices. by the inductive hypothesis, f is combinatorially equivalent to the sum of a line segment and a (d−1)-simplex. this means that f has two faces which are simplices, whose vertex sets {v1,v2, . . . ,vd} and {w1,w2, . . . ,wd} can be labelled in such a way that vi is adjacent to wi for each i. in particular f has 2d vertices and d 2 edges. furthermore there must be precisely two vertices of p outside f , say x and y. suppose that one of them is adjacent to vertices in both simplices, say [x,vi] and [x,wj] are both edges of p for some i and j. a routine degree argument shows that x is adjacent to at least two vertices in one simplex, so without loss of generality i 6= j. we may renumber the vertices so that i = 1,j = d. but then {v1,v2, . . . ,vd,wd,x} will be an affinely independent (d+2)-cycle. it touches every facet, since p has only 2d+ 2 vertices. this contradicts our assumption that p is decomposable. thus each of x,y is adjacent to vertices in only one simplex, say x is not adjacent to any wj and y is not adjacent to any vi. since all vertices have degree at least d + 1, it follows that x is adjacent to each vi, y is adjacent to each wj, and x and y are adjacent to each other. this means that the skeleton of p is isomorphic to the skeleton of the sum of a line segment and a simplex. now observe that p is simple and so is in fact combinatorially equivalent to the sum of a line segment and a simplex, thanks to a result of blind and mani [2]. proposition 5 implies that if we cut any vertex from any polytope, the resulting polytope will be decomposable. this makes it easy to construct decomposable polytopes with any number of vertices greater than 2d. on the other hand, proposition 8 asserts that there are gaps in the possibe numbers of edges of decomposable polytopes, at least in dimension 4. we show now that this is also true in higher dimensions. in fact, a decomposable 184 k. przes lawski, d. yost d-dimensional polytope with strictly less than d2 + 1 2 d edges must be combinatorially equivalent to a prism; this is an easy consequence of theorem 9. with some additional material, we can prove a stronger result. we will first examine the existence of simple polytopes with less than 3d vertices. being decomposable, theorem 9 implies that no simple d-polytope has between d + 1 and 2d vertices. this also follows from barnette’s lower bound theorem. for results concerning higher numbers of vertices, see [13] and the references therein. we denote by ∆m,n the sum of an m-dimensional simplex and an ndimensional simplex lying in complementary subspaces. it is routine to check that ∆m,n is a simple (m + n)-dimensional polytope with (m + 1)(n + 1) vertices, 1 2 (m + n)(m + 1)(n + 1) edges and m + n + 2 facets. we denote by wd the result of cutting a vertex from a d-dimensional simplicial prism ∆1,d−1. this simple polytope has 3d− 1 vertices, 1 2 d(3d− 1) edges, and d + 3 facets, comprising 2 simplices, 2 prisms and d − 1 copies of wd−1. in dimension 3, w3 is simply the 5-wedge. lemma 10. (i) the (combinatorial types of) simple d-dimensional polytopes with d + 2 facets are precisely the polytopes ∆k,d−k for 1 ≤ k ≤ 12d. (ii) up to combinatorial equivalence, the only simple d-dimensional polytopes with fewer than 3d vertices are the simplex ∆0,d, the simplicial prism ∆1,d−1, the polytope ∆2,d−2, the 6-dimensional polytope ∆3,3, the polytope wd, the 3-dimensional cube ∆1,1,1 and the 7-dimensional polytope ∆3,4. (iii) for every d 6= 6, the smallest vertex counts of simple d-polytopes are d+1, 2d, 3d−3 and 3d−1. in dimension 6 only, there is also a simple polytope with 3d− 2 vertices. proof. (i) the simplicial polytopes with d + 2 vertices are described in detail by grünbaum [7, §6.1], and these are their duals. (ii) obviously the simplex is the only polytope with d+ 1 (or fewer) facets. barnette, [1] or [5, §19], showed that a polytope with d + 4 or more facets has at least 4d− 2 ≥ 3d vertices. he also showed that a polytope with d + 3 facets has at least 3d − 1 vertices, and that if d > 3 the only such example with precisely 3d−1 vertices arises from truncating a vertex from a simplicial prism, i.e. it is wd. if d = 3, the cube ∆1,1,1 is the unique other example. we are left with the case of d + 2 facets. clearly ∆1,d−1 and ∆2,d−2 have respectively 2d and 3d− 3 vertices. if 3 ≤ k ≤ 1 2 d, then d ≥ 6. if d ≥ 8, then ∆k,d−k has at least (3 + 1)(d−3 + 1) > 3d−1 vertices. if d = 7, we have the example ∆3,4, which has 20 = 3d−1 more indecomposable polyhedra 185 vertices. if d = 6, we must also consider ∆3,3, which has 16 = 3d−2 vertices. (iii) this follows immediately from (ii). theorem 11. let p be a decomposable d-dimensional polytope with no more than d2 + 1 2 d edges. then either p is combinatorially equivalent to a simplicial prism ∆1,d−1 (and hence has precisely d 2 edges), or d = 4 and p is combinatorially equivalent to ∆2,2. proof. a d-dimensional polytope with 2d + 1 or more vertices must have at least 1 2 (2d + 1)d edges. so if p has 2d + 1 vertices, it must be simple, and lemma 10 implies that 2d + 1 ≥ 3d− 3. thus d = 4 and p is ∆2,2. otherwise, p has at most 2d vertices and the conclusion follows from theorem 9. in particular, a polychoron with 17 edges is necessarily indecomposable. grünbaum [7, p 193] showed that there is no polychoron at all with 8 vertices and 17 edges. we finish by using the preceding results to show that this is not an isolated curiosity: in fact, there is no d-dimensional polytope with 2d vertices and d2 + 1 edges for any higher value of d. (there are two easy examples when d = 3; see [4, fig. 3].) lemma 12. the polytope ∆2,d−3 cannot be a facet of any decomposable d-dimensional polytope with 3d− 4 vertices. proof. we can realize ∆2,d−3 as the convex hull of three (d−3)-simplices, say s,t,u, all translates of one another, so that the convex hull of any two of them is a facet therein, combinatorially equivalent to ∆1,d−3. moreover in each such facet, e.g. co(s,t), each of the d − 2 edges joining s and t also belongs to a triangular face whose third vertex lies in u. suppose that this copy of ∆2,d−3 is a facet of a decomposable polytope p with 3d − 4 vertices. denote v,w the two vertices of p lying outside this facet. then co(s,t) is a ridge in p ; denote by f the other facet containing it. then f contains at least one of v,w. in particular, f omits at most d − 1 vertices of p . these d − 1 vertices cannot form a facet, so f touches every facet. decomposability of p then implies that f is also decomposable. since f has at most 2d − 2 vertices, it can only be a copy of the prism ∆1,d−2, with one of v,w adjacent to every vertex in s and no vertex in t, 186 k. przes lawski, d. yost and the other adjacent to every vertex in t and no vertex in s. the same argument applied to co(t,u) and co(s,u) quickly yields a contradiction. theorem 13. let p be a d-dimensional polytope with 2d vertices and d2 + 1 or fewer edges. then either p is combinatorially equivalent to the prism ∆1,d−1 (and hence has precisely d 2 edges), or d = 3. proof. if d is 1 or 2, the conclusion is obvious. in case we can establish decomposability, the conclusion will follow from theorem 11. if p has exactly d2 edges, then it is simple, hence decomposable by shephard’s result, proposition 5. since every vertex has degree at least d, p cannot have fewer than d2 edges. we are forced to contemplate the possibility that p has precisely d2 + 1 edges. then p is indecomposable by theorem 11. since 2e −dv = 2, there are at most two vertices which are not simple. now suppose that some vertex has degree d + 2, and choose a facet f not containing v. then p must be a pyramid over f, otherwise it would be decomposable by shephard’s result. then f has v = 2d − 1 = 2(d − 1) + 1 vertices, and hence at least 1 2 (d− 1)v = (d− 1)2 + 1 2 (d− 1) edges. hence p will have at least (d − 1)2 + 1 2 (d − 1) + (2d − 1) = d2 + 1 2 d − 1 2 edges. the hypothesis then implies that 1 2 d− 1 2 ≤ 1. we conclude that d = 3 and p is a pentagonal prism. next consider the case that one vertex v has degree d + 1 and that all its neighbors are simple vertices. if we cut this vertex from p , the resulting facet will be simple and contain d + 1 vertices. this facet cannot be a simplex, so lemma 10 implies that d + 1 ≥ 2(d− 1), i.e. d ≤ 3. finally consider the case that p has two adjacent vertices of degree d + 1. we can find a hyperplane which has this edge on one side, and all other vertices of p on the other side. this divides p into two polytopes, say q and r respectively, with a common facet f. all other vertices are simple, so f will be simple and contain 2d vertices. now f cannot be a simplex or a prism, because it has more than (d− 1) + 1 or 2(d− 1) vertices. lemma 10(iii) then forces 2d ≥ 3(d− 1) − 3, i.e. d ≤ 6. if d = 6, then f has 12 vertices, and can only be ∆2,3. but q has 14 vertices, which is impossible according to lemma 12. if d = 5, then f is simple and has 10 = 3(d − 1) − 2 vertices, which according to lemma 10 is impossible unless d − 1 = 6. grünbaum [7, p 193] showed that the case d = 4 is impossible. the only remaining possibility is that d = 3 and p is combinatorially equivalent to the second last example in [4, fig. 3]. more indecomposable polyhedra 187 acknowledgements we thank eran nevo for assistance in translating reference [8], and vladimir fonf for assistance in translating reference [17]. the second author records his thanks to the university of zielona góra, for hospitality during his visits in 2008 and 2011. references [1] d. w. barnette, the minimum number of vertices of a simple polytope, israel j. math. 10 (1971), 121 – 125. [2] r. blind, p. mani-levitska, puzzles and polytope isomorphisms, aequationes math. 34 (2-3) (1987), 287 – 297. [3] d. briggs, d. yost, polyhedra with 16 edges, in preparation. [4] d. britton, j. d. dunitz, a complete catalogue of polyhedra with eight or fewer vertices, acta cryst. ser. a 29 (4) (1973), 362 – 371. [5] a. brøndsted, “an introduction to convex polytopes”, graduate texts in mathematics, 90, springer-verlag, new york-berlin, 1983. [6] e. j. friedman, finding a simple polytope from its graph in polynomial time, discrete comput geom. 41 (2) (2009), 249 – 256. [7] b. grünbaum, “convex polytopes”, second edition, graduate texts in mathematics, 221, springer-verlag, new york, 2003. [8] m. kallay, “decomposability of convex polytopes”, ph.d. dissertation, the hebrew university of jerusalem, 1979. [9] m. kallay, indecomposable polytopes, israel j. math. 41 (3) (1982), 235 – 243. [10] t. p. kirkman, applications of the theory of the polyedra to the enumeration and registration of results, proc. roy. soc. london 12 (1863), 341 – 380. [11] p. mcmullen, indecomposable convex polytopes, israel j. math. 58 (3) (1987), 321 – 323. [12] w. j. meyer, indecomposable polytopes, trans. amer. math. soc. 190 (1974), 77 – 86. [13] n. prabhu, hamiltonian simple polytopes, discrete comput. geom. 14 (3) (1995), 301 – 304. [14] k. przes lawski, d. yost, decomposability of polytopes, discrete comput. geom. 39 (1-3) (2008), 460 – 468. [15] g. c. shephard, decomposable convex polyhedra, mathematika 10 (1963), 89 – 95. [16] z. smilansky, “decomposability of polytopes and polyhedra”, ph.d. dissertation, hebrew university of jerusalem, 1986. [17] z. smilansky, decomposability of polytopes and polyhedra, geom. dedicata, 24 (1) (1987), 29 – 49. 188 k. przes lawski, d. yost [18] d. yost, irreducible convex sets, mathematika 38 (1) (1991), 134 – 155. [19] d. yost, some indecomposable polyhedra, optimization 56 (5-6) (2007), 715 – 724. e extracta mathematicae vol. 31, núm. 2, 123 – 144 (2016) weighted spaces of holomorphic functions on banach spaces and the approximation property manjul gupta, deepika baweja department of mathematics and statistics, iit kanpur, india 208016 manjul@iitk.ac.in, dbaweja@iitk.ac.in presented by manuel maestre received april 4, 2016 abstract: in this paper, we study the linearization theorem for the weighted space hw(u; f) of holomorphic functions defined on an open subset u of a banach space e with values in a banach space f . after having introduced a locally convex topology τm on the space hw(u; f), we show that (hw(u; f), τm) is topologically isomorphic to (l(gw(u); f), τc) where gw(u) is the predual of hw(u) consisting of all linear functionals whose restrictions to the closed unit ball of hw(u) are continuous for the compact open topology τ0. finally, these results have been used in characterizing the approximation property for the space hw(u) and its predual for a suitably restricted weight w. key words: holomorphic mappings, weighted spaces of holomorphic functions, linearization, approximation property. ams subject class. (2010): 46g20, 46e50, 46b28. 1. introduction approximation properties for various classes of holomorphic functions have been studied earlier by using linearization techniques in [6], [7], [8], [18], etc. if e and f are banach spaces and u is an open subset of e, then the linearization results help in identifying a given class of holomorphic functions defined on u with values in f, with the space of continuous linear mappings from a certain banach space g to f; indeed, a holomorphic mapping is being identified with a linear operator through linearization results. this study for various classes of holomorphic mappings have been carried out by beltran [2], galindo, garcia and maestre [11], mazet [17], mujica [18, 19, 20] and several other mathematicians. on the other hand, whereas the weighted spaces of holomorphic functions defined on an open subset of the finite dimensional space cn, n ∈ n (set of natural numbers) have been investigated in [3], [4], [5], [24], etc., the infinite dimensional case was considered by garcia, maestre and rueda [12], jorda [15], rueda [25]. the present paper is an attempt to study approximation 123 124 m. gupta, d. baweja properties for weighted spaces of holomorphic mappings. indeed, after having given preliminaries in section 2, we prove in section 3 a linearization theorem for the weighted space hw(u; f) of holomorphic functions defined on u with values in f. as an application of this result, we show that e is topologically isomorphic to a complemented subspace of gw(u) for those weights w for which hw(u) contains all the polynomials. in case of a weight being given by an entire function with positive coefficients, we also obtain estimates for the norm of the topological isomorphism. in section 4 we define a locally convex topology τm on the space hw(u; f) and show the topological isomorphism between the spaces (hw(u; f), τm) and (l(gw(u); f), τc) for a weight w on an open set u. finally, in section 5 we consider the applications of results proved in sections 3 and 4 to obtain characterizations of the approximation property for the space hw(u) and its predual gw(u); for instance, we prove that hw(u) has the approximation property if and only if it satisfies the holomorphic analogue of theorem 2.4(iv), i.e, for any banach space f, each mapping in hw(u; f) with relatively compact range belongs to the ∥ · ∥w-closure of the subspace of hw(u; f) consisting of finite dimensional holomorphic mappings. besides, it is proved that for a suitably restricted w and u, gw(u) has the approximation property if and only if e has the approximation property. 2. preliminaries throughout this paper, the symbols n, n0 and c respectively denote the set of natural numbers, n∪{0} and the complex plane. the letters e and f are used for complex banach spaces. the symbols e′ and e∗ denote respectively the algebraic dual and topological dual of e. we denote by u a non-empty open subset of e; and by ue and be, the open and closed unit ball of e. for a locally convex space x, we denote by x∗β and x ∗ c , the topological dual x∗ of x equipped respectively with the strong topology, i.e., the topology of uniform convergence on all bounded subsets of x, and the compact open topology. for each m ∈ n, l(me; f) is the banach space of all continuous m-linear mappings from e to f endowed with its natural sup norm. for m=1, we write l(e, f) for l(me; f). a mapping p : e → f is said to be a continuous m-homogeneous polynomial if there exists a continuous m-linear map a ∈ l(me; f) such that p(x) = a(x, . . . , x), x ∈ e. weighted spaces of holomorphic functions 125 in this case, we also write p = â. the space of all continuous m-homogeneous polynomials from e to f is denoted by p(me; f) which is a banach space endowed with the sup norm. a continuous polynomial p is a mapping from e into f which can be represented as a sum p = p0 + p1 + · · · + pk with pm ∈ p(me; f) for m = 0, 1, . . . , k. the vector space of all continuous polynomials from e into f is denoted by p(e; f). a polynomial p ∈ p(me; f) is said to be of finite type if it is of the form p(x) = k∑ j=1 ϕmj (x)yj, x ∈ e, where ϕj ∈ e∗ and yj ∈ f, 1 ≤ j ≤ k. we denote by pf(me; f) the space of finite type polynomials from e into f . a continuous polynomial p from e into f is said to be of finite type if it has a representation as a sum p = p0 + p1 + · · · + pk with pm ∈ pf(me; f) for m = 0, 1, . . . , k. the vector space of continuous polynomials of finite type from e into f is denoted by pf(e; f). a mapping f : u → f is said to be holomorphic, if for each ξ ∈ u, there exists a ball b(ξ, r) with center at ξ and radius r > 0, contained in u and a sequence {pm}∞m=1 of polynomials with pm ∈ p( me; f), m ∈ n0 such that f(x) = ∞∑ m=0 pm(x − ξ), (2.1) where the series converges uniformly for x ∈ b(ξ, r). the series in (2.1) is called the taylor series of f at ξ and in analogy with complex variable case, it is written as f(x) = ∞∑ m=0 1 m! d̂mf(ξ)(x − ξ), (2.2) where pm = 1 m! d̂mf(ξ). the space of all holomorphic mappings from u to f is denoted by h(u; f). it is usually endowed with the topology τ0 of uniform convergence on compact subsets of u and (h(u; f), τ0) is a fréchet space when u is an open subset of a finite dimensional banach space. in case u = e, the class h(e; f) is the space of entire mappings from e into f. for f = c, we write h(u) for h(u; c). we refer to [1], [9], [19] and [22] for notations and various results on infinite dimensional holomorphy. 126 m. gupta, d. baweja if f ∈ h(u; f) and n ∈ n0, we write snf(x) = ∑n m=0 1 m! d̂mf(0)(x) and cnf(x) = 1 n+1 ∑n k=0 skf(x). it has been shown in [18] that sn(f)(x) = 1 π ∫ π −π f(eitx)dn(t)dt and cn(f)(x) = 1 π ∫ π −π f(eitx)kn(t)dt, where dn(t) and kn(t) are respectively the dirichlet and fejer kernels given as follows: dn(t) = 1 2 + n∑ k=1 cos kt and kn(t) = 1 n + 1 n∑ k=0 dk(t). a subset a of u is called u-bounded if a is bounded and dist(a, ∂u) > 0, where ∂u denotes the boundary of u. a mapping f in h(u; f) is of bounded type if it maps u-bounded sets to bounded sets. the space of holomorphic mappings of bounded type is denoted by hb(u; f). the space hb(u; f) endowed with the topology τb, the topology of uniform convergence on ubounded sets, is a fréchet space, cf. [1, p. 81]. for u = ue, the following result is quoted from [27]. theorem 2.1. if {xn} is a sequence of distinct points in ue such that lim n→∞ dist({xn}, ∂ue) = 0 and {un} is a sequence of vectors in f then there exists f ∈ hb(ue; f) such that f(xn) = un, n = 1, 2, . . . a weight w on u is a continuous and strictly positive function satisfying 0 < inf a w(x) ≤ sup a w(x) < ∞ (2.3) for each u-bounded set a. a weight w defined on an open balanced subset u of e is said to be radial if w(tx) = w(x) for all x ∈ u and t ∈ c, with |t| = 1; and on e it is said to be rapidly decreasing if supx∈e w(x)∥x∥m < ∞ for each m ∈ n0. corresponding to a weight function w, the weighted space of holomorphic functions is defined as hw(u; f) = { f ∈ h(u; f) : ∥f∥w = sup x∈u w(x)∥f(x)∥ < ∞ } . weighted spaces of holomorphic functions 127 the space (hw(u; f), ∥·∥w) is a banach space and bw denotes its closed unit ball. for f = c, we write hw(u) = hw(u; c). it can be easily seen that the norm topology τ∥·∥w on hw(u; f) is finer than the topology induced by τ0. in case, p(e) ⊂ hw(u), we have the following result from [12]. proposition 2.2. the topology τ∥·∥w restricted to p( me) coincides with the norm topology. since the closed unit ball bw of hw(u) is τ0-compact by the ascoli’s theorem, the predual of hw(u) is given by gw(u) = { ϕ ∈ hw(u)′ : ϕ|bw is τ0 − continuous } by the ng theorem; cf. [23]. further, we consider the locally convex topology τbc on hw(u) for which a set a ⊂ hw(u) is τbc open if and only if a ∩ b is open in (b, b|τ0) for each ∥ · ∥w-bounded subset b of hw(u). concerning this topology, we have the following result from [25]. proposition 2.3. let u be an open subset of a banach space e and w be a weight on u. then (i) (hw(u), ∥ · ∥w) and (hw(u), τbc) have the same bounded sets. (ii) gw(u) = (hw(u), τbc)∗β. (iii) (hw(u), τbc) = gw(u)∗c. an operator t in l(e; f) is said to have a finite rank if the range of t is finite dimensional and, an operator t in l(e; f) is called compact if t(be) is a relatively compact subset of f. we denote by f(e; f) and k(e; f), respectively, the space of all finite rank operators and compact operators from e into f. a banach space e is said to have the approximation property if for every compact set k of e and ϵ > 0, there exists an operator t ∈ f(e; e) such that sup x∈k ∥t(x) − x∥ < ϵ. the following characterization of the approximation property due to grothendieck, is given in [16]. 128 m. gupta, d. baweja theorem 2.4. for a banach space e, the following are equivalent: (i) e has the approximation property. (ii) for every banach space f, f(e; f) τc = l(e; f). (iii) for every banach space f, f(f; e) τc = l(f ; e). (iv) for every banach space f, f(f; e) ∥·∥ = k(f; e). proposition 2.5. let e be a banach space. then e∗ has the approximation property if and only if f(e; f) ∥·∥ = k(e; f), for every banach space f. proposition 2.6. let e be a banach space with the approximation property. then each complemented subspace of e also has the approximation property. 3. linearization theorem for hw(u; f) and its applications in this section, we consider the linearization theorem for hw(u; f) and some of its applications. let us begin with theorem 3.1. (linearization theorem) for an open subset u of a banach space e and a weight w on u, there exists a banach space gw(u) and a mapping ∆w ∈ hw(u; gw(u)) with the following property: for each banach space f and each mapping f ∈ hw(u; f), there is a unique operator tf ∈ l(gw(u); f) such that tf ◦ ∆w = f. the correspondence ψ between hw(u; f) and l(gw(u); f) given by ψ(f) = tf is an isometric isomorphism. the space gw(u) is uniquely determined up to an isometric isomorphism by these properties. proof. though the proof of this result is similar to the one given in [2], we sketch the same for the sake of completeness. let bw be the closed unit ball of hw(u). then it is τ0-compact by ascoli’s theorem. hence by the ng’s theorem, hw(u) is a dual banach space, its predual being given by gw(u) = {h ∈ hw(u)′ : h|bw is τ0-continuous}. weighted spaces of holomorphic functions 129 further the mapping jwu : hw(u) → gw(u) ∗, jwu (f) = f̂ with f̂(h) = h(f), f ∈ hw(u) and h ∈ gw(u), is an isometric isomorphism. now define ∆w : u → gw(u) as ∆w(x) = δx, where δx(f) = f(x), f ∈ hw(u). since for x ∈ u and f ∈ hw(u), jwu (f) ◦ ∆w(x) = j w u (f)(δx) = f(x) and jwu (hw(u)) = gw(u) ∗, ∆w is weakly holomorphic and hence holomorphic, cf. [1, p.66]. in order to show that ∆w ∈ hw(u; gw(u)), fix x0 ∈ u. then for f ∈ hw(u), |δx0(f)| = |f(x0)| ≤ 1 w(x0) ∥f∥w implies ∥δx0∥ ≤ 1 w(x0) . hence ∥∆w∥w = sup x∈u w(x)∥δx∥ ≤ 1. consequently, ∆w ∈ hw(u; gw(u)). corresponding to f in hw(u; f), we now define tf. for the case f = c, define tf = j w u (f). then tf ◦ ∆w(f) = f and ∥tf∥ = ∥f∥w. in case of an arbitrary banach space f, we first define tf : gw(u) → f ∗∗ as tf(h)(ϕ) = h(ϕ ◦ f), h ∈ gw(u), ϕ ∈ f ∗. note that tf is, indeed, f-valued; for tf(δx) = f(x) ∈ f and span{δx : x ∈ u} = gw(u). further, ∥f∥w = sup x∈u w(x)∥f(x)∥ = sup x∈u w(x)∥tf(δx)∥ ≤ ∥tf∥ and ∥tf(h)(ϕ)∥ ≤ ∥h∥∥ϕ∥∥f∥w, h ∈ gw(u), ϕ ∈ f ∗. thus ∥tf∥ = ∥f∥w and ψ is an isometric isomorphism. remark 3.2. if (w∆w)(x) = w(x)∆w(x), x ∈ u, then jwu (bw) = { (w∆w)(x) : x ∈ u }◦ . consequently, (jwu (bw)) ◦ = bgw(u) = γ{(w∆w)(x) : x ∈ u}, where γ(a) denotes the absolutely convex closed hull of a. in case the weight w is given by an entire function γ with positive coefficients, i.e., w(x) = 1 γ(∥x∥), x ∈ e, we write hγ for hw; and the above linearization theorem takes the following form: theorem 3.3. let γ be an entire function with positive coefficients. then for an open subset u of a banach space e and weight w, w(x) = 1 γ(∥x∥), x ∈ u, there exists a banach space gγ(u) and a mapping ∆γ ∈ hγ(u; gγ(u)), ∥∆γ∥ = 1 with the following property: for each banach space f and each 130 m. gupta, d. baweja mapping f ∈ hγ(u; f), there is a unique operator tf ∈ l(gγ(u); f) such that tf ◦ ∆γ = f. the correspondence ψ between hγ(u; f) and l(gγ(u); f) given by ψ(f) = tf is an isometric isomorphism. the space gγ(u) is uniquely determined up to an isometric isomorphism by these properties. proof. it suffices to prove here that ∥∆γ∥ = 1. let γ(z) = ∑∞ n=0 anz n with an > 0 for each n ∈ n0. fix x0 ∈ e. choose ϕ ∈ e∗ with ∥ϕ∥ = 1 and |ϕ(x0)| = ∥x0∥. define f : e → c as f(x) = ∞∑ n=1 anϕ n(x), x ∈ e. clearly, f ∈ hγ(e) and ∥f∥γ ≤ 1. since |f(x0)| = γ(∥x0∥), we have ∥δx0∥ = sup ∥h∥γ≤1 |h(x0)| = γ(∥x0∥). thus ∥∆γ∥ = 1. before we consider the applications of the above linearization theorem, let us prove results related to the inclusion of polynomials in the weighted space of holomorphic mappings. proposition 3.4. let w be a weight defined on an open subset u of a banach space e. then, for each m ∈ n, the following are equivalent: (a) p(me; f) ⊂ hw(u; f) for each banach space f. (b) p(me) ⊂ hw(u). proof. (a)⇒(b). immediate. (b)⇒(a). consider q ∈ p(me; f). for x ∈ u, choose ϕx ∈ f ∗ such that ∥ϕx∥ = 1 and ϕx(q(x)) = ∥q(x)∥. write a = {ϕx ◦ q : x ∈ u}. then a is a ∥·∥-bounded subset of p(me) since ∥ϕx◦q∥ ≤ ∥q∥. hence by proposition 2.2, a is ∥·∥w-bounded. consequently, ∥q∥w = sup x∈u w(x)|ϕx(q(x))| ≤ sup x∈u sup y∈u w(y)|ϕx(q(y))| < ∞. thus q ∈ hw(u; f) and (a) follows. weighted spaces of holomorphic functions 131 proposition 3.5. let w be a weight on an open subset u of a banach space e. then (a) if u is bounded, p(e) ⊂ hw(u) if and only if w is bounded. (b) for u = e, p(e) ⊂ hw(e) if and only if w is rapidly decreasing. proof. (a) since constant functions are in p(e), the proof follows. (b) this is a particular case of a result proved in [12, p. 6], by taking the family v consisting of a single weight. in the remaining part of this section, we consider weights w defined on an open subset u of e so that the space p(e, f) is contained in hw(u, f), for which it suffices to consider the scalar case in view of proposition 3.4. proposition 3.6. let w be a weight defined on an open subset u of a banach space e such that p(e) ⊂ hw(u). then e is topologically isomorphic to a complemented subspace of gw(u). proof. since the inclusion map i from u to e is a member of hw(u; e), by theorem 3.1, there exists t ∈ l(gw(u); e) and ∆w ∈ hw(u; gw(u)) such that t ◦ ∆w(x) = iw(x) = x, x ∈ u. fix a ∈ u and write s = d1∆w(a). note that s ∈ l(e; gw(u)). further, by cauchy’s integral formula, s(t) = 1 2πi ∫ |ζ|=r ∆w(a + ζt) ζ2 dζ, t ∈ e, where r > 0 is chosen so that {a + ζt : |ζ| ≤ r} ⊂ u. now t ◦ s(t) = 1 2πi ∫ |ζ|=r (a + ζt) ζ2 dζ = t, t ∈ e. this gives ∥s(t)∥ ≥ 1∥t∥∥t∥ and so, s is injective and s −1 is continuous. define p = s ◦t . then p is a projection map from gw(u) into itself. also s(e) = p(gw(u)). hence s is a topological isomorphism between e and a complemented subspace of gw(u). for the weight w as considered in theorem 3.3, we have 132 m. gupta, d. baweja proposition 3.7. let γ be an entire function with positive coefficients and t0 be a positive real satisfying the equation γ(t) = tγ ′ (t). assume that u is an open subset of a banach space e for which {x ∈ e : ∥x∥ ≤ t0} ⊂ u. then there exists a topological isomorphism s between e and a complemented subspace of gγ(u) with ∥s∥ = γ(t0) t0 . proof. since the weight given by γ is bounded, i ∈ hγ(u; e). by theorem 3.3, there exists t ∈ l(gγ(u); e) and ∆γ ∈ hγ(u; gγ(u)) such that t ◦ ∆γ = i and ∥t∥ = ∥i∥γ. but ∥t∥ = ∥i∥γ = sup x∈u ∥x∥ γ(∥x∥) = t0 γ(t0) . (3.1) writing s for d1∆γ(0), by cauchy’s inequality, we get ∥s∥ = ∥d1∆γ(0)∥ ≤ 1 t0 sup ∥x∥=t0 ∥∆γ(x)∥ = 1 t0 sup ∥x∥=t0 ∥δx∥ = γ(t0) t0 . (3.2) now proceeding as in the proof of proposition 3.4, we have t ◦ s(t) = t, ∀t ∈ e. consequently, by (3.1) and (3.2), we get ∥t∥ = ∥t ◦ s(t)∥ ≤ t0 γ(t0) ∥s(t)∥ ≤ ∥t∥, t ∈ e. hence, ∥s∥ = γ(t0) t0 . illustrating the above result, we have example 3.8. let γ(z) = eτz, τ > 0. one can easily find that t0 = 1 τ . in this case ∥i∥γ = 1τe and ∥s∥ = τe. if τ = 1 e , s becomes an isometric isomorphism. for our next result, we make use of the following linearization theorem quoted from [18] and proved by using tensor product techniques for locally convex spaces in [26]. weighted spaces of holomorphic functions 133 theorem 3.9. let e be a banach space and m ∈ n. then there exists a banach space q(me) and a polynomial qm ∈ p(me; q(me)) such that for any banach space f and each polynomial p ∈ p(me; f), there is a unique operator tp ∈ l(q(me); f) satisfying tp ◦ qm = p. the correspondence φ : p(me; f) → l(q(me); f), φ(p) = tp is an isometric isomorphism and the space q(me) is uniquely determined up to an isometric isomorphism. in the statement of the above result, the space q(me) is defined as the predual of p(me), i.e., {h ∈ p(me)′ : h|bm is τ0-continuous}, where bm is the closed unit ball of p(me). the map qm : e → q(me) is given by qm(x) = δx, where δx(p) = p(x), p ∈ p(me) or equivalently qm(x) = x ⊗ · · · ⊗ x, cf. [10, p. 29]. for w and u as in proposition 3.6, we prove proposition 3.10. the space q(me) is topologically isomorphic to a complemented subspace of gw(u). proof. consider qm ∈ p(me; q(me)). by theorem 3.1, there exist tm ∈ l(gw(u); q(me)) and ∆w ∈ hw(u; gw(u)) such that tm ◦∆w = qm. let sm be the m-th taylor series coefficient of ∆w around ’a’, .i.e., sm = 1 m! d̂m∆w(a). as sm ∈ p(me; gw(u)), by theorem 3.9 there exists rm ∈ l(q(me); gw(u)) such that rm ◦ qm = sm. now, tm ◦ rm ◦ qm = tm ◦ sm = 1 m! d̂m(tm ◦ ∆w)(a) = 1 m! d̂mqm(a). as span{qm(x) : x ∈ e} = q(me), it follows that tm ◦ rm(u) = u, u ∈ q(me). let pm = rm ◦ tm. then pm is a projection map from gw(u) into itself and rm is the topological isomorphism between q( me) and a complemented subspace of gw(u). proposition 3.11. for m ∈ n, there exists a topological isomorphism rm between the space q( me) and a complemented subspace of gγ(u), for any open subset u of e containing the set {x ∈ e : ∥x∥ ≤ r0}, r0 being a positive real number satisfying the equation rγ′(r) − mγ(r) = 0 and r0 > m. further ∥rm∥ = γ(r0) rm0 . proof. as qm ∈ hγ(u; q(me)), by theorem 3.3, there exist tm ∈ l(gγ(u); q(me)) and ∆γ ∈ hγ(u; gγ(u)) such that tm ◦ ∆γ = qm. since supx∈u ∥x∥m γ(∥x∥) = rm0 γ(r0) , we have ∥qm∥γ = ∥tm∥ = rm0 γ(r0) . (3.3) 134 m. gupta, d. baweja now by cauchy’s inequality, we get∥∥∥ 1 m! d̂m∆γ(0) ∥∥∥ ≤ 1 rm0 sup ∥x∥=r0 ∥∆γ(x)∥ = γ(r0) rm0 . continuing as in the proof of the above result, we have tm ◦ rm(u) = u, u ∈ q(me). (3.4) by using (3.3) and (3.4), we get ∥u∥ = ∥tm ◦ rm(u)∥ ≤ rm0 γ(r0) ∥rm(u)∥ ≤ ∥u∥ for every u ∈ q(me). thus ∥rm∥ = γ(r0) rm0 . considering the function given in example 3.8, we have the following, illustrating the above result example 3.12. if γ(z) = eτz, τ > 0, we find r0 = m τ and, so ∥rm∥ = τmem mm . also, by using the same argument as in proposition 3.11, one can easily check example 3.13. for n ∈ n, define w : ue → (0, ∞) by w(x) = (1 − ∥x∥)n, x ∈ ue. then ∥rm∥ = ( n m + n )n for any m ∈ n. 4. the topology τm in this section we introduce a locally convex topology τm on hw(u; f) of which the particular cases have been considered in [18] and [25]. for a finite set a and r > 0, let us define n(a, r) = {f ∈ hw(u; f) : inf x∈a w(x) sup y∈a ∥f(y)∥ ≤ r}. consider the class u = { ∞∩ j=1 n(aj, rj) : (aj) varies over all sequences of finite subsets of u and (rj) varies over all positive sequences diverging to infinity } weighted spaces of holomorphic functions 135 it can be easily checked that each member of u is balanced, convex and absorbing. thus it forms a fundamental neighborhood system at 0 for a locally convex topology, which we denote by τm. equivalently, this topology is generated by the family{ pα,a : α = (αj) ∈ c + 0 , a = (aj), aj being finite subset of u for each j } of seminorms given by pα,a(f) = sup j∈n ( αj inf x∈aj w(x) sup y∈aj ∥f(y)∥ ) . these are the minkowski functionals of members in u. for f = c, τm = τbc, cf. [25, p. 350]. for our results in the sequel, we make use of the following lemma 4.1. let m be a compact subset of gw(u). then there exist sequences α = (αj) ∈ c+0 and a = (aj) of finite subsets of u such that m ⊂ γ ( ∪ j≥1 { αj inf x∈aj w(x)∆w(y) : y ∈ aj }) . proof. since m◦ is a τc-neighborhood of 0 in gw(u)∗, it is τbc-neighborhood of 0 by proposition 2.3(iii). consequently, there exist sequences (αj) ∈ c+0 and a = (aj) of finite subsets of u such that {f ∈ hw(u) : pα,a(f) ≤ 1} ⊂ m◦, where m◦ = {f ∈ hw(u) : supu∈m | < f, u > | ≤ 1}. writing b = ∪ j≥1{αj infx∈aj w(x)∆w(y) : y ∈ aj}, we get b ◦ ⊂ m◦. therefore, by the bipolar theorem, we have m ⊂ γ ( ∪ j≥1 { αj inf x∈aj w(x)∆w(y) : y ∈ aj }) . relating τm with τ0 and τ∥.∥w, and bounded sets with respect to these topologies, we prove proposition 4.2. for a weight w on an open subset u of a banach space e, the following hold: (i) τ0 ≤ τm ≤ τ∥.∥w on hw(u; f). (ii) τm and ∥·∥w-bounded sets are the same. (iii) τm|b = τ0|b for any ∥·∥w-bounded set b. 136 m. gupta, d. baweja proof. (i) let k be a compact subset of u. then by lemma 4.1, there exist sequences (αj) ∈ c+0 and a = (aj) of finite subsets of u such that ∆w(k) ⊂ γ ( ∪ j≥1 { αj inf x∈aj w(x)∆w(y) : y ∈ aj }) . hence, for f ∈ hw(u; f), we have sup x∈k ∥f(x)∥ = sup x∈k ∥tf ◦ ∆w(x)∥ ≤ pα,a(f). thus τm ≥ τ0 on hw(u; f). the inequality τm ≤ τ∥·∥w clearly holds. (ii) as every ∥·∥w-bounded set is τm-bounded, it suffices to prove the other implication. assume that there exists a τm-bounded set a which is not ∥·∥wbounded. then for each k ∈ n, there exist fk ∈ a such that ∥fk∥w > k2. therefore, w(xk)∥fk(xk)∥ > k2 for some sequence {xk} ⊂ u. consider the τm-continuous semi-norm p on hw(u; f) defined by the sequences {1j } and {xj} obtained as above, namely p(f) = sup j∈n 1 j w(xj)∥f(xj)∥. then p( fk k ) > 1, for each k. this contradicts the τm-boundedness of a as 1 k → 0 and {fk} ⊂ a, cf. [14, p. 161] . (iii) let b be a bounded set in (hw(u; f), ∥·∥w). then there exists a constant m > 0 such that ∥f∥w ≤ m, for every f ∈ b. in order to show that τm|b ≤ τ0|b, consider a τm-continuous semi-norm p given by p(f) = sup j∈n ( αj inf x∈aj w(x) sup y∈aj ∥f(y)∥ ) , f ∈ hw(u; f), where (αj) ∈ c+0 and (aj) is a sequence of finite subsets of u. fix ϵ > 0 arbitrarily. then there exists k0 ∈ n such that αj < ϵ 2m , ∀j > k0. write k = ∪ j≤k0 aj. then k is a compact subset of u. for f, g ∈ b, p(f − g) < ϵ whenever pk(f − g) < δ, weighted spaces of holomorphic functions 137 where δ = ϵ ∥α∥∞ sup 1≤j≤k0 ( inf x∈aj w(x) ); indeed sup j≤k0 ( αj inf x∈aj w(x) sup y∈aj ∥(f − g)(y)∥ ) ≤ ∥α∥∞ sup 1≤j≤k0 ( inf x∈aj w(x) ) pk(f − g). this completes the proof as the other implication is obviously true. proceeding on the lines similar to [25, remark 3.32], it can be proved that the topology τm may be strictly finer than τ0 on hw(u; f). however, for the sake of convenience of the reader, we give example 4.3. let e be a banach space and w be a bounded weight on ue. assume that τm = τ0 on hw(ue; f). choose a sequence {xn} in ue such that ∥xn∥ → 1 and {un} in f with ∥un∥ = n, n ∈ n. then by theorem 2.1, there exists a function f ∈ hb(u; f) such that f(xn) = un w(xn) , n ∈ n. since ∥f∥w = supx∈u w(x)∥f(x)∥ > n for all n ∈ n, f /∈ hw(ue; f). consequently, the set a = { n∑ m=0 1 m! d̂mf(0) : n = 0, 1, 2, . . . } is not ∥·∥w bounded. but the convergence of the series ∑∞ m=0 1 m! d̂mf(0) to f in τ0 topology yields that the set a is τ0-bounded. as τm and ∥·∥w-bounded sets are the same by proposition 4.2(ii), it follows that τm ̸= τ0, i.e., τ0 < τm. one can easily establish the following observation which we write as proposition 4.4. let (aj) be a sequence of finite sets in e and a =∪ j∈n aj. then a is bounded if and only if the set k = ( ∪ j∈n αjaj) ∪ {0} is compact for each α = (αj) ∈ c0. proof. immediate. 138 m. gupta, d. baweja proposition 4.5. let e and f be banach spaces. for a weight w on an open subset u of e with p(e) ⊂ hw(u), τm coincides with τ0 on p(me; f) for each m ∈ n. proof. let p be a τm-continuous semi-norm on hw(u; f). then there exist sequences α = (αj) ∈ c+0 and a = (aj) of finite subsets of u such that p(f) = sup j∈n ( αj inf x∈aj w(x) sup y∈aj ∥f(y)∥ ) , f ∈ hw(u; f). define k = ∪ j∈n { (αj infx∈aj w(x)) 1 m y : y ∈ aj } ∪ {0}. for each y ∈ u, choose ϕy ∈ e∗ with ∥ϕy∥ = 1 and ϕy(y) = ∥y∥. then the set b = {ϕmy : y ∈ u} is a norm bounded subset of p(me) and hence ∥·∥w-bounded by proposition 2.2. therefore sup j∈n sup y∈aj w(y)∥y∥m ≤ sup y∈u sup x∈u w(x)∥ϕmy (x)∥ < ∞. then by proposition 4.4, k is a compact subset of e. since p(p) = sup j∈n sup y∈aj ∥∥∥p((αj inf x∈aj w(x) ) 1 m y )∥∥∥ = pk(p). for any p ∈ p(me; f), the proof follows. next, we prove proposition 4.6. let e and f be banach spaces. for a radial weight w on a balanced open subset u of e with p(e) ⊂ hw(u), the space p(e; f) is τm-dense in hw(u; f). proof. recalling the notations sn(f) and cn(f), and their integral representations for f ∈ hw(u; f) from section 2, we have ∥cn(f)(x)∥ = ∥∥∥ 1 π ∫ π −π f(eitx)kn(t)dt ∥∥∥ ≤ sup t∈[−π,π] ∥f(eitx)∥ since ∫ π −π kn(t)dt = 1, cf. [28, p. 45]. consequently, for each n ∈ n0, ∥cn(f)(x)∥w ≤ sup x∈u w(x) sup |t|=1 ∥f(tx)∥ = sup x∈u sup |t|=1 w(tx)∥f(tx)∥ ≤ ∥f∥w < ∞. thus, for given f ∈ hw(u; f), the set {cn(f) : n ∈ n0} is ∥·∥w-bounded in hw(u; f). as cnf → f in (h(u; f), τ0), the result follows by proposition 4.2(iii). weighted spaces of holomorphic functions 139 finally in this section, we consider an analogue of theorem 3.1 on hw(u;f) when it is equipped with the topology τm. this result will be useful for our study of approximation properties in the next section. indeed, we prove theorem 4.7. let e and f be banach spaces, and w be a weight on an open subset u of e. then the mapping ψ : ( hw(u; f), τm ) → ( l(gw(u); f), τc ) is a topological isomorphism. proof. let m be a compact subset of gw(u). then by lemma 4.1, there exist sequences (αj) ∈ c+0 and a = (aj) of finite subsets of u such that m ⊂ γ ( ∪ j≥1 { αj inf x∈aj w(x)∆w(y) : y ∈ aj }) . hence for f ∈ hw(u; f), pm(ψ(f)) = sup u∈m ∥tf(u)∥ ≤ sup j∈n ( αj inf x∈aj w(x) sup y∈aj ∥f(y)∥ ) = pα,a(f). thus ψ is τm − τc continuous. in order to show the continuity of the inverse map ψ−1, let us note that sup j∈n sup y∈aj ( inf x∈aj w(x)∥∆w(y)∥ ) ≤ 1. hence by proposition 4.4, the set k = γ ( ∪ j≥1 { αj inf x∈aj w(x)∆w(y) : y ∈ aj }) ∪ {0} is a compact subset of gw(u), which immediately yields the τc−τm continuity of the inverse mapping ψ−1. 5. the approximation properties this section is devoted to the study of the approximation property for the space e, the weighted space hw(u) of holomorphic mappings and its predual gw(u). we write hw(u) ⊗ f = {f ∈ hw(u; f) : f has finite dimensional range} 140 m. gupta, d. baweja and hcw(u; f) = {f ∈ hw(u; f) : wf has a relatively compact range}. in the next proposition we establish the interplay between the properties of a mapping f ∈ hw(u; f) and the corresponding operator tf ∈ l(gw(u); f). proposition 5.1. let u be an open subset of a banach space e and w be a weight on u. then for any banach space f, (a) f ∈ hw(u) ⊗ f if and only if tf ∈ f(gw(u); f), (b) f ∈ hcw(u; f) if and only if tf ∈ k(gw(u); f). proof. (a) note that for (gi) n i=1 ⊂ hw(u) and (yi) n i=1 ⊂ f, f(x) = n∑ i=1 gi(x)yi ⇔ tf(δx) = n∑ i=1 < δx, gi > yi for each x ∈ u. as gw(u)∗ = hw(u) and span{δx : x ∈ u} = gw(u), the result follows. (b) by remark 3.2, bgw(u) = γ(w∆w)(u), the result follows from (wf)(u) = tf ( (w∆w)(u) ) ⊂ tf ( γ(w∆w)(u) ) = γ ( (wf)(u) ) . proposition 5.2. let w be a weight on an open subset u of a banach space e. then f(gw(u); f) ∥·∥ = k(gw(u); f) if and only if hw(u) ⊗ f ∥·∥w = hcw(u; f) for each banach space f. proof. assume that f(gw(u); f) ∥·∥ = k(gw(u); f). consider f ∈ hcw(u; f). then tf ∈ k(gw(u); f) by proposition 5.1(b). hence there exists a net (tα) ⊂ f(gw(u); f) such that tα ∥·∥ −−→ tf. now, corresponding to each α, we have fα ∈ hw(u) ⊗ f such that tfα = tα by proposition 5.1(a). apply theorem 3.1 to get fα ∥·∥w−−−→ f, thereby proving hw(u) ⊗ f ∥·∥w = hw(u; f). conversely, for t ∈ k(gw(u); f), there exists f ∈ hcw(u; f) such that t = tf by proposition 5.1(b). then there exists a net {fα} ⊂ hw(u) ⊗ f such that fα ∥·∥w−−−→ f. thus (tfα) ⊂ f(gw(u); f) by proposition 5.1(a) and tα ∥·∥ −−→ tf = t by proposition 3.1. weighted spaces of holomorphic functions 141 proposition 5.3. let w be a weight on an open subset u of a banach space e. then f(gw(u); f) τc = l(gw(u); f) if and only if hw(u) ⊗ f τm = hw(u; f) for each banach space f. proof. the proof follows analogously by using theorem 4.7 and proposition 5.1(b). characterizing the approximation property for the space e, we have theorem 5.4. let e be a banach space. then for each banach space f, the following are equivalent: (i) e has the approximation property. (ii) hw(v ) ⊗ e τm = hw(v ; e), for each open subset v of f and weight w on v . (iii) hw(v ) ⊗ e ∥·∥w = hcw(v ; e), for each open subset v of f and weight w on v . proof. (i) ⇒ (ii): assume that e has the approximation property. then by theorem 2.4, f(gw(u); e) τc = l(gw(u); e). thus hw(v ) ⊗ e τm = hw(v ; e) by proposition 5.3. (ii) ⇒(i): we claim that f(f; e) τc = l(f; e) for each banach space f. let a ∈ l(f; e). applying proposition 3.4 , there exist operators s ∈ l(f; gw(uf )) and t ∈ l(gw(uf ); f) such that t ◦ s(y) = y, y ∈ f. since gw(uf )∗ ⊗ e τm = hw(uf ; e) by (ii), in view of proposition 5.3 there exists a net (aα) ⊂ f(gw(uf ); e) such that aα τc−→ a◦t . thus aα ◦s τc−→ a◦t ◦s = a. as aα ◦ s ⊂ f(f; e), our claim holds and (i) follows by theorem 2.4. (i) ⇒ (iii): again using theorem 2.4, f(gw(u); e) ∥·∥ = k(gw(u); e) by (i). therefore hw(u) ⊗ f ∥·∥w = hcw(u; f) by proposition 5.2. (iii) ⇒(i): let a ∈ k(f; e) and t , s be the operators as above. then a ◦ t ∈ k(gw(uf ); e). by hypothesis and proposition 5.2, there exists a sequence (an) ⊂ f(gw(uf ); e) such that an ∥·∥ −−→ a ◦ t . thus an ◦ s ∥·∥ −−→ a and we have, f(f; e) ∥·∥ = k(f; e). this proves (i). next, we characterize the approximation property for the weighted space hw(u). 142 m. gupta, d. baweja theorem 5.5. for an open subset u of a banach space e, hw(u) has the approximation property if and only if hw(u) ⊗ f is ∥·∥w-dense in hcw(u; f) for each banach space f . proof. by proposition 2.5, gw(u)∗ has the approximation property if and only if f(gw(u); f) is ∥·∥-dense in k(gw(u); f) for each banach space f. as hw(u) = gw(u)∗, the result follows by proposition 5.2. we now cite the following known result, cf. [18]; along with the proof for convenience. proposition 5.6. if a banach space e has the approximation property, then for every banach space f and m ∈ n, pf(me; f) τc = p(me; f). proof. let p ∈ p(me; f). then for a compact subset k of e and ϵ > 0, there exists a δ > 0 such that ∥p(x) − p(y)∥ < ϵ whenever x ∈ k and y ∈ e with ∥y − x∥ < δ. since e has the approximation property, there is a t ∈ f(e; e) such that supx∈k ∥t(x) − x∥ < δ. thus, supx∈k ∥p ◦ t(x) − p(x)∥ < ϵ. making use of the above proposition, we finally prove theorem 5.7. let e be a banach space and w be a radial weight on a balanced open subset u of e such that hw(u) contains all the polynomials. then the following assertions are equivalent: (i) e has the approximation property. (ii) pf(e; f) τm = hw(u; f) for each banach space f. (iii) hw(u) ⊗ f τm = hw(u; f) for each banach space f. (iv) gw(u) has the approximation property. proof. (i) ⇒ (ii): let p be a τm continuous semi-norm on hw(u; f). then for f ∈ hw(u; f), there exists p ∈ p(e; f) such that p(f − p) < ϵ2 by proposition 4.6. let p = p0+p1+· · ·+pk, pm ∈ p(me; f), 0 ≤ m ≤ k. then by using proposition 5.6 and proposition 4.5, there exist qm in pf(me; f), 0 ≤ m ≤ k such that p(pm − qm) < ϵ 2(k + 1) . write q = q0 + q1 + · · · + qk. clearly q ∈ pf(e; f) and p(f − q) < ϵ. weighted spaces of holomorphic functions 143 (ii) ⇒ (iii): it suffices to prove that pf(e; f) ⊂ hw(u) ⊗ f. consider p ∈ pf(e; f). then there exist ϕj ∈ e∗ and yj ∈ f, 1 ≤ j ≤ k such that p = k∑ j=1 ϕmj ⊗ yj . now, ϕmj ∈ hw(u) for each 1 ≤ j ≤ k as w is bounded. thus p ∈ hw(u)⊗f. (iii) ⇒ (iv): note that ∆w ∈ hw(u) ⊗ gw(u) τm by taking f = gw(u) in (iii). now hw(u) ⊗ gw(u) τm can be identified with f(gw(u); gw(u)) τc via the map ψ by proposition 5.1(a) and theorem 4.7 . since t∆w ◦ ∆w = ∆w, we get ψ(∆w) = i, the identity map on gw(u). thus i ∈ f(gw(u); gw(u)) τc . 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[28] a. zygmund, “trignometrical series”, dover, new york, 1955. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 35, num. 2 (2020), 137 – 184 doi:10.17398/2605-5686.35.2.137 available online june 20, 2020 unitary skew-dilations of hilbert space operators v. agniel∗ univ. lille, cnrs, umr 8524 laboratoire paul painlevé, france vidal.agniel@univ-lille.fr received february 29, 2020 presented by mostafa mbekhta accepted may 12, 2020 abstract: the aim of this paper is to study, for a given sequence (ρn)n≥1 of complex numbers, the class of hilbert space operators possessing (ρn)-unitary dilations. this is the class of bounded linear operators t acting on a hilbert space h, whose iterates tn can be represented as tn = ρnphu n|h, n ≥ 1, for some unitary operator u acting on a larger hilbert space, containing h as a closed subspace. here ph is the projection from this larger space onto h. the case when all ρn’s are equal to a positive real number ρ leads to the class cρ introduced in the 1960s by foias and sz.-nagy, while the case when all ρn’s are positive real numbers has been previously considered by several authors. some applications and examples of operators possessing (ρn)-unitary dilations, showing a behavior different from the classical case, are given in this paper. key words: hilbert space operators, dilations, compressions of linear operators, functional calculi, numerical radius, ρ-radii, ρ-classes, (ρn)-classes if there are too many of them, you can remove ρ-radii and (ρn)-classes. ams subject class. (2010): 47a12, 47a20, 47a30, 47a60. 1. introduction classes cρ have been introduced by b. sz-nagy and c. foias [22] in 1966. for a complex hilbert space h and a real number ρ > 0, a bounded linear operator t ∈l(h) is said to be in the class cρ(h) if all powers of t can be skew-dilated to powers of a unitary operator on a hilbert space k, containing h as a closed subspace. this means that tn = ρphu n|h, for all n ≥ 1, where u ∈ l(k) is a suitable unitary operator, and ph ∈ l(k) denotes the orthogonal projection onto h. such an operator t is called a ρ-contraction, while the unitary operator u is called a ρ-dilation, or a ρ-unitary dilation, of t. ∗ http://perso.eleves.ens-rennes.fr/~vaign357/index.html issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.35.2.137 mailto:vidal.agniel@univ-lille.fr http://perso.eleves.ens-rennes.fr/~vaign357/index.html https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 138 v. agniel the famous sz.-nagy dilation theorem (see [22]) shows that c1(h) is exactly the class of all hilbert space contractions i.e., operators of norm no greater than one. it is also known (see [6]) that the class c2(h) coincides with the class of all operators t with numerical range w(t) included in the closed unit disk; equivalently, those t satisfying w(t) ≤ 1. here the numerical range w(t) and the numerical radius w(t) of t are defined by w(t) = { 〈tx,x〉 : ‖x‖ = 1 } ; w(t) = sup { |λ| : λ ∈ w(t) } . let t be an operator in the class cρ. then (i) t is power-bounded. more precisely, we have ‖tn‖≤ max(1,ρ), for all n ≥ 0. in particular, the spectral radius r(t) of t satisfies r(t) ≤ 1; (ii) tk is in cρ(h) for every k ≥ 1; (iii) for a closed subspace, f , of h which is stable by t (i.e., t(f) ⊂ f), the restriction t |f is in cρ(f); (iv) the functional calculus map f 7→ f(t) that sends a polynomial f into f(t) can be extended in a well-defined manner to the disk algebra a(d) := c0(d) ∩ hol(d). it is a morphism of banach algebras, and satisfies ‖f(t)‖≤ max(1,ρ)‖f‖l∞(d); (v) t is similar to a contraction: there is an invertible operator l ∈ l(h) such that ‖ltl−1‖≤ 1. we refer the reader to [11, 12, 23, 17, 15] for proofs of these results, which mainly use several characterizations of classes cρ(h). we record the principal ones in the following theorem. theorem. let t be an operator in l(h) and let ρ > 0. the following are equivalent: (i) t ∈ cρ(h); (ii) r(t) ≤ 1 and, for all z ∈ d, we have ( 1 − 2 ρ ) i + 2 ρ re ( (i −zt)−1 ) ≥ 0; (iii) for all z ∈ d and all h ∈ h we have ( 2 ρ −1 ) ‖zth‖2 + ( 2− 2 ρ ) 〈zth,h〉≤ ‖h‖2. we remark that these characterization can be expressed in terms of classes of operator-valued holomorphic functions. for instance, (ii) says that the unitary skew-dilations of hilbert space operators 139 map z 7→ (1 − 2 ρ )i + 2 ρ (i − zt)−1 is in the caratheodory class of operatorvalued holomorphic functions on d, having all real parts positive-definite operators. item (iii) can be equivalently expressed by the membership of z 7→ zt((ρ− 1)zt −ρi)−1 to the schur class of holomorphic maps f : d → l(h) having all norms no greater than one (i.e., ‖f(z)‖≤ 1 for every z ∈ d). j.a.r. holbrook [11] and j.p. williams [24] introduced the notion of ρradius of an operator t ∈l(h) as follows: wρ(t) := inf { u > 0 : 1 u t ∈ cρ(h) } . this ρ-radius is a quasi-norm on the banach space l(h), equivalent to the operator norm, whose closed unit ball is exactly cρ(h). recall ([13]) that a quasi-norm satisfies all properties of a norm, except that the triangular inequality holds true up to a multiplicative constant. for ρ > 2, the quasinorm wρ satisfies ([23, 14]) wρ(t1 + t2) ≤ ρ ( wρ(t1) + wρ(t2) ) . therefore the ρ-contractions are exactly the contractions for the ρ-radius, and many relationships between classes cρ can be expressed more easily using the associated ρ-radii. the ρ-radius is a usual banach-space norm for 0 < ρ ≤ 2. some generalizations of classes cρ have been studied, like classes ca(h) introduced by h. langer (see [23, page 53] and its references, and [20]), or the classes c(ρn)(h) considered by several authors (see [17, 4, 18]). this latter generalization will be the main topic of study in this paper, with the novelty that we consider the general case when the ρn’s are non-zero complex scalars. this will lead to classes of operators with several new features and different behavior. 2. hilbert space operators with (ρn)-dilations definition and first properties. in light of the preceding discussion we introduce the following definition. definition 2.1. (classes c(ρn)) let (ρn)n≥1 be a sequence of complex numbers, with ρn 6= 0 for each n. we write (ρn)n≥1 ∈ (c∗)n ∗ . let h be a complex hilbert space. define now c(ρn)(h) := { t ∈l(h) : there exists a hilbert space k and a unitary operator u ∈l(k) with h ⊂ k and tn = ρnphun|h, ∀n ≥ 1 } . 140 v. agniel here ph ∈l(k) is the orthogonal projection from k onto its closed subspace h. we say in this case that t possesses (ρn)-dilations. in other words, an operator t is in the class c(ρn)(h) if and only if all its powers admit dilations of the form ρnu n for a certain unitary operator u acting on a larger hilbert space. for the rest of this paper, we will suppose that the hilbert space h on which t acts is fixed. if there is no ambiguity, c(ρn)(h) will be abbreviated as c(ρn). note also that the sequence (ρn) = (ρn)n≥1 starts at n = 1: for n = 0 we have of course t 0 = ih = phu 0|h. in the papers [17, 4, 18], the case when the ρn’s are non-negative real numbers is considered. we went for a broader choice of sequences as the main ideas do not rely heavily on the fact that ρn are in r∗+ and as this eventually allows for some interesting new phenomena for the classes c(ρn). one first difference is recorded in the following remark. remark 2.2. the definition of c(ρn) easily gives that t ∈ c(ρn) if and only if t∗ ∈ c(ρn). therefore, when the ρn are real scalars, the class c(ρn) is stable under the adjoint map t 7→ t∗. this is no longer true in the general case. remark 2.3. as another basic remark, we note that if t is in c(ρn), then we have ‖tn‖≤ |ρn|. thus, r(t) ≤ lim infn ( |ρn| 1 n ) . this relationship implies that two different cases appear in the study of the classes c(ρn): (i) 0 < lim infn ( |ρn| 1 n ) ≤ +∞; (ii) lim infn ( |ρn| 1 n ) = 0. although many of the proofs below work the same way in both cases, most of the results will be stated in the case (i). the study of the case (ii) is more problematic. indeed, in case (ii), the class c(ρn) will only contain quasinilpotent operators, that is operators whose spectra reduce to {0}. we also note that when lim infn ( |ρn| 1 n ) = +∞, we trivially have r(t) < lim infn ( |ρn| 1 n ) for every operator t. we also note that the condition lim infn ( |ρn| 1 n ) = +∞ leads to small changes in the proofs below: the main difference between this condition and lim infn ( |ρn| 1 n ) < +∞ in case (i) is the fact that the quantity 1 lim infn ( |ρn| 1 n ), which exists when lim infn (|ρn|1n) ∈ ]0, +∞[, has to be replaced by 0 when lim infn ( |ρn| 1 n ) = +∞. this motivates the following convention. unitary skew-dilations of hilbert space operators 141 convention. for the rest of this paper, we assume that 1 lim infn ( |ρn| 1 n ) = 0 whenever lim inf n ( |ρn| 1 n ) = +∞. (2.1) one of the main tools to characterize the classes c(ρn) is the following herglotz-type theorem. theorem 2.4. let h a hilbert space. let f : d → h be an analytic function such that: (i) f(0) = i (ii) re(f(z)) ≥ 0, ∀z ∈ d. then, there exists a hilbert space k containing h and u ∈l(k) an unitary operator such that f(z) = ph(i + zu)(i −zu)−1|h, ∀z ∈ d a proof of this theorem can be found in [8, pages 65 – 69]. definition 2.5. for (ρn)n ∈ (c∗)n ∗ and for w in a complex banach algebra, f(ρn) denotes the entire series given by f(ρn)(w) = ∑ n≥1 2wn ρn . for a ∈ r, we denote re≥a the half-plane {z ∈ c, re(z) ≥ a}, while re>a is the half-plane {z ∈ c, re(z) > a}. proposition 2.6. let (ρn)n ∈ (c∗)n ∗ and let t ∈ l(h). the following are equivalent: (i) t ∈ c(ρn); (ii) the series f(ρn)(zt) = ∞∑ n=1 2 ρn zntn is absolutely convergent in l(h) and i + re ( f(ρn)(zt) ) ≥ 0, ∀z ∈ d. proof. (i) ⇒ (ii) let u be an unitary operator on a hilbert space k, with k containing h as a closed subspace, such that tn = ρnphu n|h, ∀n ≥ 1. 142 v. agniel for every polynomial p(x) = a0 + · · · + anxn and every z ∈ d, we have a0i+ a1 ρ1 zt +· · ·+ an ρn (zt)n = ph ( a0i+a1zu+· · ·+an(zu)n ) |h = ph p(zu)|h. since the series 1 + ∑ n≥1 2w n converges absolutely to f(w) = 1+w 1−w for all w ∈ d, and since u is unitary, the series i + ∑ n≥1 2(zu) n converges in norm to f(zu) = (i + zu)(i −zu)−1, ∀z ∈ d. thus, as ∥∥∥∥tnρn ∥∥∥∥ = ∥∥phun|h∥∥ ≤ ∥∥un∥∥ ≤ 1, the series ih + ∑ n≥1 2 ρn (zt)n is absolutely convergent and converges to ph[(i + zu)(i − zu)−1]|h for all z ∈ d. as u is unitary, f(zu) is normal, so the closure of its numerical range w(f(zu)) is the convex hull of its spectrum. we have σ ( f(zu) ) = f ( σ(zu) ) ⊂ f(d) ⊂ re>0. thus, w ( (i + zu)(i −zu)−1 ) = w ( f(zu) ) ⊂ hull ( σ(f(zu)) ) ⊂ re≥0. furthermore, w(phf(zu)|h) ⊂ w(f(zu)), so the numerical range of ih + f(ρn)(zt) is included in re≥0. this is equivalent to re(ih + f(ρn)(zt)) ≥ 0, so (ii) is true. (ii) ⇒ (i) we define f(z) := ih +f(ρn)(zt). thus, f is analytic on d, f(0) = ih, and re(f(z)) ≥ 0 for all z ∈ d. by applying theorem 2.4, we obtain a hilbert space k and a unitary operator u ∈l(k), such that f(z) = ph(i + zu)(i − zu)−1|h, for all z ∈ d. by developing both analytic expressions in entire series, and identifying their coefficients, we obtain 2 ρn tn = 2phu n|h for all n ≥ 1. therefore t ∈ c(ρn). we will obtain most of the following results by applying proposition 2.6. we can directly see by applying this proposition that any class c(ρn) contains 0, the null operator, so none of these classes is empty. one remark is in order. we did not consider the case where ρn = 0 for some n in definition 2.1. indeed, this condition does not go well with computations similar to the ones in the proof of proposition 2.6. having ρn = 0 unitary skew-dilations of hilbert space operators 143 implies tn = 0, but it does not give any information on phu n|h. this prevents us from showing that certain sums of powers of t and t∗ are positive, which is a crucial tool when dealing with operators in the class c(ρn). if we were to denote m := inf{n : ρn = 0}, then any operator t in c(ρn) would need to be nilpotent of order at most m. the following corollary treats this nilpotent case and gives a characterization that was the one we expected in the case ρm = 0. see also [5, proposition 6.1] for another use of the positivity condition (ii) below. corollary 2.7. let (ρn)n ∈ (c∗)n ∗ and m ≥ 1. let t ∈ l(h) be such that tm = 0. then, the following are equivalent: (i) t ∈ c(ρn); (ii) i + re (∑m−1 n=1 z n 2 ρn tn ) ≥ 0 for all z ∈ d. thus, for any sequence (τn) such that ρk = τk, for all 1 ≤ k < m, we have t ∈ c(τn) if and only if t ∈ c(ρn). proof. a direct application of proposition 2.6 with the extra condition tm = 0 gives the equivalence. now we come back to proposition 2.6. when lim infn(|ρn| 1 n ) > 0, we can see that the series ∑∞ n=1 2 ρn zntn is absolutely convergent if and only if |z|r(t) < lim infn(|ρn| 1 n ). we can thus reformulate proposition 2.6 as follows. theorem 2.8. let (ρn)n∈(c∗)n ∗ with lim infn(|ρn| 1 n )>0. let t ∈l(h). then, the following assertions are equivalent: (i) t ∈ c(ρn); (ii) r(t) ≤ lim infn ( |ρn| 1 n ) and, for f(ρn)(zt) := ∑∞ n=1 2 ρn zntn, we have i + re ( f(ρn)(zt) ) ≥ 0, ∀z ∈ d. remark 2.9. replacing the condition of absolute convergence of a series by a condition concerning the spectral radius of t is useful in several instances. we can first notice that if we take v > 0 small enough, then vt will satisfy the spectral radius condition. however, if lim infn ( |ρn|1/n ) = 0, this condition must be replaced by lim supn (‖tn‖ |ρn| 1/n) ≤ 1, which can only be satisfied by 144 v. agniel certain quasinilpotent operators. hence, aside from nilpotent operators and corollary 2.7, knowing which operators can be “near” operators belonging to a class c(ρn) is a difficult problem. in this case, the map f(ρn) also has convergence radius 0, so we cannot use analytic or geometric properties related to the images of certain disks by f(ρn). many of the following results, related to specific operators or to f(ρn) will have no meaning in this case, but others will be true under the additional condition lim sup n ( ‖tn‖ |ρn| )1 n ≤ 1. we look now at the closure of the class c(ρn) for the operator norm. corollary 2.10. let (ρn)n ∈ (c∗)n ∗ with lim infn ( |ρn| 1 n ) > 0. then the class c(ρn) is closed for the operator norm: if (tm)m a sequence of operators converging in l(h) to t, such that tm ∈ c(ρn), then t ∈ c(ρn). proof. let (tm)m a sequence of operators converging to t such that tm ∈ c(ρn). we have r(t) = lim m ( r(tm) ) ≤ lim inf n ( |ρn| 1 n ) . thus, for any z ∈ d, the series f(ρn)(zt) converges absolutely and f(ρn)(zt) = limm f(ρn)(ztm). hence, for any h ∈ h, we have re (〈( i + f(ρn)(zt) ) h,h 〉) = re [ lim m 〈( i + f(ρn)(ztm) ) h,h 〉] ≥ 0. this implies that i + re ( f(ρn)(zt) ) ≥ 0, and the proof is complete by using theorem 2.8. operator radii. the condition in theorem 2.8 will be useful when studying the (ρn)-radius, which is introduced in the following definition. definition 2.11. let (ρn)n ∈ (c∗)n ∗ . let t ∈ l(h). we define the (ρn)-radius of t as: w(ρn)(t) := inf { u > 0 : t u ∈ c(ρn) } ∈ [0, +∞]. unitary skew-dilations of hilbert space operators 145 the definition of the (ρn)-radius is similar to the definition of the ρ-radius that can be found in [11, 1, 3, 2]. as the classes c(ρn) and cρ share the same type of definition, the (ρn)-radius and the ρ-radius will share the same role with some slight different variations. we will for now focus on properties of the (ρn)-radius. lemma 2.12. let (ρn)n ∈ (c∗)n ∗ with lim infn ( |ρn| 1 n ) > 0. then, the map t 7→ w(ρn)(t) takes values in [0, +∞[, is a quasi-norm, is equivalent as a quasi-norm to the operator norm ‖·‖, and its closed unit ball is the class c(ρn). we also have w(ρn)(t) ≥ ( ‖tm‖ |ρm| ) 1 m and w(ρn)(t) ≥ r(t) lim infn ( |ρn| 1 n ). proof. we start off by showing that the (ρn)-radius is finite while obtaining its equivalence with the operator norm ‖·‖. let t ∈l(h). let u > 0 be such that t u ∈ c(ρn). for any m ≥ 1, we have ‖tm‖ um ≤ |ρm|, that is u ≥ ( ‖tm‖ |ρm| ) 1 m . therefore, by taking the infimum over u such that t u ∈ c(ρn), we get w(ρn)(t) ≥ ( ‖tm‖ |ρm| ) 1 m . for m = 1 we obtain w(ρn)(t) ≥ (‖t‖ |ρ1| ) . if we also take the lim sup of the right-hand side quantity, we get w(ρn)(t) ≥ r(t) lim infn ( |ρn| 1 n ) . now, let r < lim infn ( |ρn| 1 n ) . therefore, the series f(|ρn|)(rz) := ∑∞ n=1 2 |ρn| rnzn is absolutely convergent for all z ∈ d, thus analytic on d. since f(|ρn|)(0) = 0, there is a radius r0, with 1 > r0 > 0, such that |f(|ρn|)(r0w)| ≤ 1 for all |w| ≤ r. let u > 0 be such that ‖t‖ u < r0r. thus, we have r ( t u ) < r0r < lim inf n ( |ρn| 1 n ) , 146 v. agniel and for all z ∈ d we have∥∥∥∥f(ρn) ( z t u )∥∥∥∥≤ ∞∑ n=1 2 |ρn| |z|n ( t u )n ≤ ∞∑ n=1 2 ρn |z|n(r0r)n = ∣∣f(|ρn|)(r0|z|r)∣∣ ≤ 1. we recall that for any b ∈l(h) we have re(b) ≥−‖re(b)‖i = − ∥∥∥∥b + b∗2 ∥∥∥∥i ≥−‖b‖i. thus, for any z ∈ d, f(ρn) ( zt u ) converges absolutely and we have i + re ( f(ρn) ( z t u )) ≥ i − ∥∥∥∥f(ρn) ( z t u )∥∥∥∥i ≥ 0. this means that t u ∈ c(ρn) according to proposition 2.6, so w(ρn)(t) ≤ u < +∞. furthermore, since t u ∈ c(ρn) for every u such that u > ‖t‖ r0r , we get w(ρn)(t) ≤ ‖t‖ r0r . hence, we have ‖t‖ |ρ1| ≤ w(ρn)(t) ≤ ‖t‖ r0r . with these inequalities we immediately get w(ρn)(t) = 0 ⇔ t = 0. these inequalities also imply that, for s,t ∈l(h), we have w(ρn)(s + t) ≤ ‖s + t‖ r0r ≤ ‖s‖ + ‖t‖ r0r ≤ |ρ1| r0r ( w(ρn)(s) + +w(ρn)(t) ) . in order to show that w(ρn)(·) is a quasi-norm, we still have to show that it is homogeneous, that is w(ρn)(zt) = |z|w(ρn)(t) for any z ∈ c. let z ∈ c. the cases z = 0 and t = 0 have been treated, so we now consider z = eit|z| 6= 0 and t 6= 0. let u ≥ w(ρn)(zt) be such that zt u ∈ c(ρn). denote u ′ = u|z|. we can see that r ( zt u ) = r ( t u′ ) and that f(ρn) ( wzt u ) = f(ρn) ( eitw t u′ ) for any w ∈ d. thus, the series f(ρn) ( eitw t u′ ) converges absolutely and i+re ( f(ρn)(e itw t u′ ) ) ≥ 0, for any w ∈ d. hence t u′ ∈ c(ρn), so u′ = u |z| ≥ w(ρn)(t). unitary skew-dilations of hilbert space operators 147 thus, by taking the infimum for u ≥ w(ρn)(zt), we get w(ρn)(zt) ≥ |z|w(ρn)(t). applying the same result to t ′ = zt and z′ = 1 z , we obtain w(ρn)(t) = w(ρn)(z ′t ′) ≥ |z′|w(ρn)(t ′) = 1 |z| w(ρn)(zt), which proves the desired equality. we will now prove that the closed unit ball for the (ρn)-radius is exactly c(ρn). notice again that w(ρn)(t) = 0 reduces to t = 0. if t ∈ c(ρn), then w(ρn)(t) ≤ 1 1 = 1. conversely, suppose that w(ρn)(t) ≤ 1 and let (um)m be a sequence, with um > 0, converging to w(ρn)(t) such that t um ∈ c(ρn). using the fact that the class c(ρn) is closed for the operator norm, as proved in corollary 2.10, we get t w(ρn)(t) ∈ c(ρn). therefore, we have r(t) ≤ r ( t w(ρn)(t) ) ≤ lim inf n ( |ρn| 1 n ) and i + re ( f(ρn)(zt) ) ≥ 0 for every z with |z| ≤ 1 w(ρn)(t) . since 1 w(ρn)(t) ≥ 1, we can conclude that t ∈ c(ρn). the proof is now complete. remark 2.13. in the case when lim infn ( |ρn| 1 n ) = 0, we have w(ρn)(t) = +∞ unless t is quasinilpotent and the sequence of ‖tn‖ 1 n decreases to 0 fast enough. remark 2.14. since the (ρn)-radius is homogeneous and w(ρn)(t) ≤ 1 ⇔ t ∈ c(ρn), whenever t 6= 0, we have{ u > 0: t u ∈ c(ρn) } = [ w(ρn)(t), +∞ [ . corollary 2.15. let (ρn)n ∈ (c∗)n ∗ with lim infn ( |ρn| 1 n ) > 0. let t ∈l(h). we have 148 v. agniel (i) for any z 6= 0, 1|z|w(ρn)(t) = w(ρn)( 1 z t) = w(znρn)(t); (ii) if t ∈ c(ρn)(h), then t k ∈ c(ρkn)(h), for all k ≥ 1; (iii) w(ρkn)n(t k) ≤ w(ρn)(t) k, for all k ≥ 1; (iv) w(ρn)(t) = w(ρn)(t ∗). proof. (i) the left-hand equality is given by the homogeneity of w(ρn)(·). for the right-hand one, we can see that( t z )n = ρnphu n|h if and only if tn = znρnphun|h. thus t z ∈ c(ρn) if and only if t ∈ c(znρn). lemma 2.12 implies that w(ρn) ( 1 z t ) = w(znρn)(t). (ii) by definition of the class c(ρn), if t ∈ c(ρn), then (tk)m = ρkmph(u k)m|h, so tk ∈ c(ρkn)(h). (iii) the result is true when t = 0. when t 6= 0, consider t ′ = t w(ρn)(t) . by homogeneity of w(ρn)(·), we have w(ρn)(t ′) = 1, so t ′ ∈ c(ρn) according to lemma 2.12. thus, for any k ≥ 1, (t ′)k ∈ c(ρkn)(h). using again the homogeneity of the (ρn)-radius, we obtain w(ρkn)n(t k) w(ρn)(t) k = w(ρkn) ( (t ′)k ) ≤ 1. this completes the proof. (iv) we use remark 2.2 and lemma 2.12 to obtain the equivalence w(ρn)(t) ≤ 1 ⇔ w(ρn)(t ∗) ≤ 1. since the (ρn)-radii are homogenous, these quantities must be equal. corollary 2.16. let (ρn)n ∈ (c∗)n ∗ with lim infn ( |ρn| 1 n ) > 0. let t ∈l(h). the following assertions are true: unitary skew-dilations of hilbert space operators 149 (i) let f be an invariant closed subspace of t. then w(ρn)(t |f ) ≤ w(ρn)(t); (ii) for any isometry v we have w(ρn)(v tv ∗) ≤ w(ρn)(t), with equality if v is unitary; (iii) for a hilbert space k we have w(ρn)(t ⊗ ik) = w(ρn)(t); (iv) for tm ∈l(hm), m ≥ 1 with supm(‖tm‖) < +∞, we have w(ρn)(⊕m≥1tm) = sup m≥1 ( w(ρn)(tm) ) ; (v) if t (∞) denotes the countable orthogonal sum t ⊕ t ⊕ ··· , then w(ρn) ( t (∞) ) = w(ρn)(t). proof. (i) we have r(t|f ) ≤ r(t). if i + re ( f(ρn)(zt) ) is positive, then i + re ( f(ρn)(zt |f ) ) is positive too. thus, by using lemma 2.12 we obtain w(ρn)(t) ≤ 1 ⇒ w(ρn)(t|f ) ≤ 1. the homogeneity of the (ρn)-radius gives the result. (ii) we have r(v tv ∗)≤r(t) and (v tv ∗)n =v tnv ∗. thus, f(ρn)(zv tv ∗) = v f(ρn)(zt)v ∗. hence, for any h ∈ h and any z ∈ d, we have re (〈( i + f(ρn)(zv tv ∗) ) h,h 〉) = re (〈( i + f(ρn)(zt) ) v ∗h,v ∗h 〉) . by applying theorem 2.8 and lemma 2.12, we get w(ρn)(t) ≤ 1 ⇒ w(ρn)(v tv ∗) ≤ 1. the homogeneity of the (ρn)-radii gives the desired inequality. when the isometry v is also invertible, the converse inequality is true, so both quantities are equal. (iii) since ‖tn‖ = ‖(t ⊗ ik)n‖, we have r(t) = r(t ⊗ ik). let u > 0 be such that u ≥ r(t) lim infn ( |ρn| 1 n ). thus the series f(ρn)(zt⊗iku ) is absolutely convergent for all z ∈ d, and f(ρn) ( zt⊗ik u ) = f(ρn) ( zt u ) ⊗ ik. since for any h1 ⊗k1,h2 ⊗k2 ∈ h ⊗k we have 〈h1 ⊗k1,h2 ⊗k2〉 = 〈h1,h2〉〈k1,k2〉, we can see that the condition〈( i + re ( f(ρn) ( z t ⊗ ik u ))) (h⊗k),h⊗k 〉 ≥ 0, ∀h⊗k ∈ h ⊗k, 150 v. agniel is equivalent to〈( i + re ( f(ρn) ( z t u ))) (h),h 〉 ≥ 0, ∀h ∈ h. hence, t⊗ik u ∈ c(ρn)(h ⊗ k) is equivalent to t u ∈ c(ρn)(h), which implies that w(ρn)(t) = w(ρn)(t ⊗ ik). (iv) since supm(‖tm‖) < +∞, the linear map t = ⊕m≥1tm is bounded on the hilbert space h = ⊕m≥1hm, and ‖t‖ = supm(‖tm‖). thus, r(t) = supm(r(tm)). let u > 0 be such that u ≥ r(t) lim infn ( |ρn| 1 n ). we have r ( tm u ) ≤ r ( t u ) ≤ lim inf n ( |ρn| 1 n ) . thus, the series f(ρn) ( zt u ) and f(ρn) ( ztm u ) are absolutely convergent for all z ∈ d, and f(ρn) ( z t u ) = ⊕m≥1f(ρn) ( z tm u ) . since for any h = (hm)m ∈ h, we have[ i + re ( f(ρn) ( z t u ))] (h) = (( i + re ( f(ρn) ( z tm u ))) (hm) ) m , this implies that〈( i + re ( f(ρn) ( z t u ))) (h),h 〉 ≥ 0, ∀h ∈ h, is equivalent to〈( i + re ( f(ρn) ( z tm u ))) (hm),hm 〉 ≥ 0, ∀hm ∈ hm, ∀m ≥ 1. hence, the assertion t u ∈ c(ρn)(h) is equivalent to tm u ∈ c(ρn)(hm), ∀n ≥ 1, which implies that w(ρn)(t) = supm ( w(ρn)(tm) ) . (v) the proof is a consequence of item (iii) and [5, remark 1.1]. the items (i) and (ii) of this corollary show that the classes c(ρn) are unitarily invariant, and stable under the restriction to an invariant closed subspace. the item (iv) is a generalization of a known property of direct unitary skew-dilations of hilbert space operators 151 sums of operators in the class c(ρ). items (i), (ii) and (v) show that, under the condition of corollary 2.16, the radius w(ρn) is an admissible radius in the terminology of [5, definition 1.1]. thus, all the results proved in [5] for admissible radii are valid for w(ρn) when lim infn ( |ρn| 1 n ) > 0. in particular, the following result is true. corollary 2.17. let t ∈ l(h), with ‖t‖ ≤ 1 and tn = 0 for some n ≥ 2. then, for each polynomial p with complex coefficients, we have w(ρn) ( p(t) ) ≤ w(ρn) ( p(s∗n) ) . here s∗n is the nilpotent jordan cell s∗n =   0 1 0 · · · 0 0 0 0 1 · · · 0 0 ... ... ... . . . ... ... 0 0 0 · · · 0 1 0 0 0 · · · 0 0   on the standard euclidean space cn. some other consequences of the condition lim infn ( |ρn| 1 n ) > 0 are proved in the next proposition. proposition 2.18. let (ρn)n ∈ (c∗)n ∗ with lim infn ( |ρn| 1 n ) > 0. the following assertions are true: (i) we have w(ρn)(i) = min ({ r ≥ lim inf n ( |ρn| 1 n )−1 : f(ρn) ( d ( 0, 1 r )) ⊂ re≥−1 }) ; (ii) for any t ∈l(h), we have w(ρn)(t) ≥ r(t)w(ρn)(i); (iii) if t is normal, then w(ρn)(t) = ‖t‖w(ρn)(i). proof. (i) take u = w(ρn)(i) such that i u ∈ c(ρn). we have r( i u ) ≤ lim infn ( |ρn| 1 n ) , so 1 u is no greater than the convergence radius of f(ρn). for any z ∈ d, we have f(ρn) ( z i u ) = f(ρn)( z u )i. thus, i + re ( f(ρn) ( z i u )) ≥ 0 for any z ∈ d if and only if f(ρn) ( d ( 0, 1 u )) ∈ re≥−1. (ii) let t ∈l(h). there is nothing to prove if t = 0 or r(t) = 0. otherwise, let u = w(ρn)(t) be such that t u ∈ c(ρn) (cf. lemma 2.12). since i + 152 v. agniel re ( f(ρn) ( zt u )) ≥ 0, the spectrum of i +f(ρn) ( zt u ) lies in re≥0. this spectrum is the set { 1 + f(ρn)(zw), w ∈ σ ( t u )} . the union of these spectra, when z describes d, is { 1 + f(ρn)(w), |w| < r(t) u } . since r(t) u > 0, we obtain from item (i) that u r(t) ≥ w(ρn)(i). hence w(ρn)(t) ≥ r(t)w(ρn)(i). (iii) let t be a normal operator with t 6= 0. for u = ‖t‖.w(ρn)(i), we have r ( t u ) = ‖t‖ u = 1 w(ρn)(i) ≤ lim inf n ( |ρn| 1 n ) . thus, we obtain that ⋃ z∈d σ ( i + f(ρn) ( z t u )) = { 1 + f(ρn)(w), |w| < 1 w(ρn)(i) } . item (i) of this proposition tells us that this set is included in re≥0. as t is normal, i + f(ρn) ( zt u ) is also normal, so w ( i + f(ρn) ( z t u )) ⊂ hull ( σ ( i + f(ρn) ( z t u ))) ⊂ re≥0, ∀z ∈ d. hence, i + re ( f(ρn) ( zt u )) ≥ 0, and t u ∈ c(ρn). by lemma 2.12, we then have w(ρn)(t) ≤ u = ‖t‖w(ρn)(i). the inequality of item (ii) provides the desired equality. remark 2.19. since we also have w(ρn)(i) ≥ 1 lim infn ( |ρn| 1 n ), the inequality in proposition 2.18 is better than the last one of lemma 2.12. thus, if there is t such that w(ρn)(t) = r(t) lim infn ( |ρn| 1 n ), the same must be true for the identity operator i. in the case when ρn = ρ, ρ > 0, this can only happen when ρ ≥ 1. unitary skew-dilations of hilbert space operators 153 3. classes c(ρ) for ρ 6= 0 in this section, we will focus on the case where ρn = ρ, for some ρ ∈ c∗. this is an intermediate class between the classical case considered by sz.-nagy and foias (classes cτ for τ > 0) and the the general c(ρn)-classes. thus the obtained results are already known when ρ > 0, but the generalization to the case ρ ∈ c∗ seems to be new. nevertheless, we acknowledge the influence of [23, 1, 2, 14] for the results of this section. the results obtained here will turn out to be useful when we will look again at c(ρn)-classes in the next section. some characterizations. lemma 3.1. let ρ 6= 0 and ρn = ρ, ∀n ≥ 1. let t ∈l(h). the following are equivalent: (i) t ∈ c(ρ)(h); (ii) r(t) ≤ 1 and re (( 1 − 2 ρ ) i + 2 ρ (i −zt)−1 ) ≥ 0, ∀z ∈ d; (iii) r(t) ≤ 1 and re ( 2 ρ (i−zt) ) + re ( 1− 2 ρ ) (i−zt)∗(i−zt) ≥ 0, ∀z ∈ d; (iv) re ( 2 ρ (i −zt) ) + re ( 1 − 2 ρ ) (i −zt)∗(i −zt) ≥ 0, ∀z ∈ d. proof. (i) ⇔ (ii) we have lim infn ( |ρn| 1 n ) = 1. when r(t) ≤ 1, for z ∈ d, we have i + ∑ n≥1 2 ρ (zt)n = ( 1 − 2 ρ ) i + 2 ρ (i −zt)−1. apply now proposition 2.6. (ii) ⇔ (iii) we will use several times the known fact that for a,b ∈ l(h), with a invertible, re(b) ≥ 0 ⇔ re(a∗ba) ≥ 0. we obtain the equivalence (ii) ⇔ (iii) by choosing a = (i −zt), b = ( 1 − 2 ρ ) i + 2 ρ (i −zt)−1 and by rearranging the expression, using that (i −zt)∗(i −zt) is a positive self-adjoint operator and re(a∗) = re(a). (iii) ⇒ (iv) it is immediate. 154 v. agniel (iv) ⇒ (iii) suppose that r(t) > 1. thus, there exists γ ∈ c such that |γ| = r(t) > 1, and there is a sequence (hn) of vectors hn ∈ h such that ‖hn‖ = 1 and ‖(t − γi)hn‖ → 0 as n → ∞. let 0 < � < |γ| − 1 and set gn := (t −γi)hn. let also η = �eit, for some t that will be chosen later on. let z := 1+η γ . then, |z| < 1+(|γ|−1)|γ| = 1. furthermore, we have (i −zt)hn = ( i − 1 γ t ) hn − η γ thn + ηhn −ηhn = −zgn −ηhn. thus, we obtain re (〈[ 2 ρ (i −zt) + ( 1 − 2 ρ ) (i −zt)∗(i −zt) ] hn, hn 〉) ≥ 0 ⇒ re ( 2 ρ [ −η‖hn‖2 −〈zgn,hn〉 ] + ( 1 − 2 ρ ) ‖(i −zt)hn‖2 ) ≥ 0 ⇒ re ( 2 ρ [−η −〈zgn,hn〉] + ( 1 − 2 ρ )[ |η|2 + 2re(〈zgn,hn〉) + |z|2‖gn‖2 ]) ≥ 0. hence, by taking the limit as n → +∞, we obtain re ( 2 ρ (−η) + ( 1 − 2 ρ ) |η|2 ) = re ( −2 ρ eit ) � + re ( 1 − 2 ρ ) �2 ≥ 0. we can then choose t ∈ r depending on arg(ρ) and sgn ( re ( 1− 2 ρ )) to obtain either −2 |ρ| � + ∣∣∣∣re ( 1 − 2 ρ )∣∣∣∣�2 ≥ 0 or 2|ρ|�− ∣∣∣∣re ( 1 − 2 ρ )∣∣∣∣�2 ≤ 0. but since −2|ρ| < 0, there is some � > 0 such that −2 |ρ| + ∣∣re(1 − 2 ρ )∣∣� is strictly negative, which is impossible. this contradiction shows that r(t) ≤ 1, which concludes the proof. lemma 3.2. let ρ 6= 0 and α > 0 be two scalars. let t ∈ l(h). the following assertions are equivalent: (i) w(ρ)(t) ≤ α; (ii) r(t) ≤ α, ( (ρ − 1)zt − ραi ) is invertible and ∥∥(zt)((ρ − 1)zt − ραi )−1∥∥ ≤ 1, ∀z ∈ d; (iii) r(t) ≤ α, ( (ρ−1)t−ρwi ) is invertible and ∥∥t((ρ−1)t−ρwi)−1∥∥ ≤ 1, ∀|w| > α. unitary skew-dilations of hilbert space operators 155 proof. (i) ⇒ (ii) when replacing t with t α , all expressions in (i) and (ii) are reduced to the case α = 1. now, as w(ρn)(t) ≤ α = 1, we use lemma 3.1 to have r(t) ≤ 1 and re (( 1 − 2 ρ ) i + 2 ρ (i −zt)−1 ) ≥ 0, ∀z ∈ d. we denote cz := ( 1 − 2 ρ ) i + 2 ρ (i − zt)−1, for z ∈ d. we recall that since re(cz) ≥ 0, we have (cz + i) invertible and∥∥(cz − i)(cz + i)−1∥∥ ≤ 1. a computation gives cz − i = 2 ρ zt(i −zt)−1 and cz + i = [ 2i + ( 2 ρ − 2 ) zt ] (i −zt)−1. thus, (cz − i)(cz + i)−1 = 1 ρ zt [ i + ( 1 ρ − 1 ) zt ]−1 = zt [ ρi + (1 −ρ)zt ]−1 = −zt [ −ρi + (ρ− 1)zt ]−1 . this means that all the conditions of (ii) are fulfilled. (ii) ⇒ (i) we again reduce to the case α = 1. we denote dz = zt [ ρi − (ρ− 1)zt ]−1 , for z ∈ d. since ‖dz‖≤ 1, we have dz ∈ c(1), so r(dz) ≤ 1 and re ( (i + wdz)(i −wdz)−1 ) ≥ 0, for all w ∈ d. we obtain: i + wdz = [ ρi + (w + 1 −ρ)zt ][ ρi − (ρ− 1)zt ]−1 and i −wdz = [ ρi + (−w + 1 −ρ)zt ][ ρi − (ρ− 1)zt ]−1 . thus, (i + wdz)(i −wdz)−1 = [ ρi + (w + 1 −ρ)zt ][ ρi + (−w + 1 −ρ)zt ]−1 . 156 v. agniel since r(t) ≤ 1, (i − zt) is invertible so [ρi + (−w + 1 −ρ)zt]−1 converges to 1 ρ (i −zt)−1 when w tends to 1, by continuity of the inverse map. thus, lim w→1, w∈d (i + wdz)(i −wdz)−1 = 1 ρ ( ρi + (2 −ρ)zt ) (i −zt)−1 = cz. hence, re(cz) ≥ 0 for all z ∈ d and r(t) ≤ 1, so t ∈ c(ρ). (ii) ⇔ (iii) for z 6= 0, we take w = α z to obtain the result. the converse gives the result for all z ∈ d, z 6= 0, which extends to d by continuity. reducing to the case ρ > 0. with this characterization of c(ρ) classes, we are now able to obtain the main relationship between (ρ)-radii and (τ)-radii, ρ ∈ c∗, τ > 0. this relationship extends the “symmetric” relationship τw(τ)(t) = (2 − τ)w(τ)(t), 0 < τ < 2, that was already known (see [2, theorem 3]). proposition 3.3. let ρ 6= 0 and α > 0 be two scalars. let t ∈ l(h). the following assertions are equivalent: (i) w(ρ)(t) ≤ α; (ii) ( (ρ − 1)zt − ραi ) is invertible and ∥∥(zt)((ρ − 1)zt − ραi)−1∥∥ ≤ 1, ∀z ∈ d; (iii) ( (ρ−1)t−ρwi ) is invertible and ∥∥t((ρ−1)t−ρwi)−1∥∥ ≤ 1, ∀|w| > α. furthermore, we have: |ρ|w(ρ)(t) = ( 1 + |ρ− 1| ) w1+|ρ−1|(t). (3.1) hence, the map ρ ∈ c∗ 7→ |ρ|w(ρ)(t) is constant on circles of center 1, is continuous on c∗ and can be extended continuously to 2w(2)(t) at 0. proof. using the results of lemma 3.2, we can see that items (ii) and (iii) are equivalent and that item (i) implies item (ii). we only need to show that item (ii) implies r(t) ≤ α. we can reduce the proof to the case α = 1 by considering t α instead of t. we also recall that if ρ > 0, the result is valid (see [19, thm. 1] or [7] for a proof). let ρ 6= 0. we denote s = 1+|ρ−1||ρ| t. suppose unitary skew-dilations of hilbert space operators 157 that [(ρ − 1)zt − ρi]−1 exists and that ‖zt[(ρ − 1)zt − ρi]−1‖ ≤ 1, for all z ∈ d. with ρ− 1 = |ρ− 1|eit, ρ = |ρ|eis and w = z.e−is+it we then have∥∥∥zt[(ρ− 1)zt −ρi]−1∥∥∥ ≤ 1 ⇔ ∥∥∥zt[|ρ− 1|eitzt −|ρ|eisi]−1∥∥∥ ≤ 1 ⇔ ∥∥∥ze−iseite−itt[|ρ− 1|ze−iseitt −|ρ|i]−1∥∥∥ ≤ 1. ⇔ ∣∣e−it∣∣∥∥∥wt[|ρ− 1|wt −|ρ|i]−1∥∥∥ ≤ 1 ⇔ ∥∥∥wt[(1 + |ρ− 1|− 1)wt −|ρ|i]−1∥∥∥ ≤ 1 ⇔ ∥∥∥∥∥w1 + |ρ− 1||ρ| t [( 1 + |ρ− 1|− 1 ) w 1 + |ρ− 1| |ρ| t − ( 1 + |ρ− 1| ) i ]−1∥∥∥∥∥ ≤ 1 ⇔ ∥∥∥ws[(1 + |ρ− 1|− 1)ws −(1 + |ρ− 1|)i]−1∥∥∥ ≤ 1. since w describes d when z does, this is true for all w ∈ d. therefore w(1+|ρ−1|)(s) ≤ 1 as 1 + |ρ − 1| > 0 (see the beginning of the proof and lemma 3.2). thus, r(s) ≤ 1, which implies r(t) ≤ |ρ| 1+|ρ−1| ≤ 1. now that we have showed that the condition about the spectral radius of t is not necessary, we can see that the equivalences in the previous computations give w(ρ)(t) ≤ 1 ⇔ w(1+|ρ−1|) ( 1 + |ρ− 1| |ρ| t ) ≤ 1. by homogeneity of the (ρn)-radii, this is equivalent to |ρ|w(ρ)(t) = ( 1 + |ρ− 1| ) w(1+|ρ−1|)(t). the properties of the map ρ ∈ c∗ 7→ |ρ|w(ρ)(t) can now be obtained from its restriction to [1, +∞[, which is known to be continuous (see [2, corollary 2] for example). equation (3.1) gives a simple geometric understanding of a formula that was previously known only for real numbers ρ between 0 and 2. it also implies the following relationship between cρ classes. corollary 3.4. we have c(ρ) = 1 + |ρ− 1| |ρ| c(1+|ρ−1|). 158 v. agniel we conclude that complex (ρ)-radii of an operator t can be expressed in terms of the real positive ones. corollary 3.5. let ρ 6= 0 and let t ∈l(h). we have: (i) w(ρ)(i) = 1+|ρ−1| |ρ| , ∀ρ 6= 0; (ii) if t is normal, then w(ρ)(t) = ‖t‖ 1+|ρ−1| |ρ| ; (iii) if t 2 = 0, then w(ρn)(t) = w(ρ1)(t) = 2w(t) |ρ1| = ‖t‖ |ρ1| ; (iv) it t 2 = bi, b ∈ c, then |ρ|w(ρ)(t) = w(t)+ √ w2(t)2 + |b|(|ρ− 1|2 − 1); (v) it t 2 = at, a ∈ c, then |ρ|wρ(t) = 2w(t) + |a||ρ− 1|. proof. (i) it is known that w(ρ)(i) = 1 when 1 ≤ ρ. the relationship of proposition 3.3 gives the result. (ii) when t is normal, we have w(ρn)(t) = ‖t‖w(ρn)(i). (iii) if t 2 = 0, then t ∈ c(ρn) if and only if i + re ( 2 ρ1 zt ) ≥ 0 for all z ∈ d. by corollary 2.7, this is equivalent to t|ρ1| ∈ c(1), to 2t |ρ1| ∈ c(2) and to t ∈ c(ρ1). thus, lemma 2.12 and the following facts w(2)(t) = w(t) and w(1)(t) = ‖t‖ imply that w(ρn)(t) = w(ρ1)(t) = 2w(t) |ρ1| = ‖t‖ |ρ1| . (iv), (v) we can reduce these cases to t 2 = i (respectively t 2 = t) by taking δ to be a square root of b (respectively a) and looking at t δ (respectively t δ2 ). then, [2, theorem 6] gives the result when ρ > 0, and we extend it to ρ ∈ c∗ by using proposition 3.3. computations and some applications. for the next auxiliary result we need some notation. for an operator t acting on h and for h ∈ h, define vh := span ( tn(h), n ≥ 0 ) and th := t|vh ∈l(vh). lemma 3.6. let t ∈l(h). let ρ 6= 0. then, with the previous notation, we have w(ρ)(t) = sup h∈h ( w(ρ)(th) ) . unitary skew-dilations of hilbert space operators 159 if we also have p(t) = 0 for some p ∈ c[x] with deg(p) = n, then th can be identified as some matrix s ∈ mn(c) such that p(s) = 0, and the computation of w(ρ)(th) can be obtained from the computation of w(ρ)(s). proof. let h ∈ h. we already proved in corollary 2.16 that w(ρ)(th) ≤ w(ρ)(t). conversely, for 1 u = suph∈h ( w(ρ)(th) ) , ( i − zt u ) is invertible as( i−zth u ) is invertible for all h ∈ h and we have re (〈( i +f(ρ) ( zt u )) g,g 〉) ≥ 0 for all g ∈ h. thus t u ∈ c(ρ), which implies suph∈h ( w(ρ)(th) ) ≥ w(ρ)(t) and concludes the proof. remark 3.7. here is an attempt to compute w(ρ)(t), ρ > 1, when t satisfies the quadratic equation t 2 + at + bi = 0. we use some ideas from [2], which allows one to obtain an expression of w(ρ)(t) depending on w(2)(t) when a = 0 or b = 0. up to considering reitt, we can assume that |b| = 1 and re(āb) = 0. with α,β the roots of x2 + ax + b and η ∈ c, we want to compute w(ρ)(m), for m =  α η 0 β  . using lemma 3.2 (ii) and proposition 3.3, we obtain that w(ρ)(m) is the largest (in modulus) z that is solution of 2(ρ− 1)2 + ρ2|z|2 ( |a|2 + |a2 − 4b| 2 ) + |η|2ρ2|z|2 = ∣∣∣∣(ρ− 1)2 ia|a| + a(ρ− 1)ρz + ρ2z2 ∣∣∣∣2 + 1. in the case ρ = 2, the equation simplifies into 2 + 2|z|2 ( |a|2 + |a2 − 4b| ) + 4|η|2|z|2 = ∣∣∣∣4z2 + 2az + ia|a| ∣∣∣∣2 + 1. however, unlike the case where a = 0 or b = 0 in [2], we couldn’t find a way to have an algebraic expression of w(ρ)(t) in terms of w(2)(t). using proposition 3.3, we can also generalize some results of [1] about characterizing unitary operators through their ρ-radii. proposition 3.8. let t ∈l(h) be invertible. then (i) t is unitary if and only if σ(t) ⊂ ∂d and there exists ρ ∈ c∗ such that w(ρ)(t) ≤ w(ρ)(i). 160 v. agniel (ii) t = ‖t‖u for u unitary if and only if there exists ρ ∈ c∗ and m > 0 such that w(ρ)(t −m) w(ρ)(i) = ( w(ρ)(t) w(ρ)(i) )−m . proof. (i) the formula of proposition 3.3 can be rewritten as w(ρ)(s) = w(ρ)(i)w(1+|ρ−1|)(s). it allows us to obtain the same relationship between t and i for w(1+|ρ−1|), and we can then apply [1, theorem 2.1] to get the result. (ii) the formula of proposition 3.3 allows us to obtain the same relationship for w(1+|ρ−1|), which simplifies into: w1+|ρ−1|(t −m) = w1+|ρ−1|(t) −m. we can now apply [1, theorem 1.1], and the proof is complete. proposition 3.9. let ρ 6= 0 be a complex number. then (i) the ρ-radius w(ρ)(·) is a norm on l(h) if and only if |ρ− 1| ≤ 1; (ii) if |ρ− 1| > 1, then, for all operators t1 and t2 in l(h), we have wρ(t1 + t2) ≤ ( 1 + |ρ− 1| )( wρ(t1) + wρ(t2) ) . proof. for two operators t1,t2, we have w(ρ)(t1 + t2) ≤ c ( w(ρ)(t1) + w(ρ)(t2) ) if and only if the same is true for w(1+|ρ−1|). it is known [23, 14] that for τ > 0, w(τ) is a norm if and only if 0 < τ ≤ 2. we conclude that w(ρ)(·) is a norm if and only if ρ lies in the closed circle of center 1 and radius 1. moreover, when τ > 2, w(τ) is a quasi-norm with multiplicative constant (also called the modulus of concavity of the quasi-norm [13]) lower or equal to τ. we thus obtain (ii). for the next proposition we recall that for r > 0 the disc algebra over the disc d(0,r), a(d(0,r)), is the set of holomorphic functions on d(0,r) that are continuous on d(0,r). proposition 3.10. let ρ 6= 0 be a complex number. let t ∈ c(ρ). then the functional calculus map f 7→ f(t) that sends a polynomial f into f(t) unitary skew-dilations of hilbert space operators 161 can be extended continuously to the disk algebra a ( d ( 0, 1 w(ρ)(i) )) . it is a morphism of banach algebras, and satisfies ‖f(t)‖≤ ( 1 + |ρ− 1| ) ‖f‖ l∞ ( d ( 0, 1 w(ρ)(i) )). furthermore, for f ∈ a ( d ( 0, 1 w(ρ)(i) )) such that f(0) = 0, we have w(ρ) ( f(t) ) ≤ w(ρ)(i)‖f‖l∞ ( d ( 0, 1 w(ρ)(i) )). if f ∈ a(d) with f(0) = 0, we also have w(ρ) ( f(t) ) ≤‖f‖l∞(d). the constants in these three inequalities are optimal. proof. we notice first that t ∈ c(ρ) is equivalent to w(ρ)(t) ≤ 1, which is equivalent to w(1+|ρ−1|)(t) ≤ |ρ| 1 + |ρ− 1| = 1 w(ρ)(i) ≤ 1. hence, w(ρ)(i)t lies in c(1+|ρ−1|), so there exists a hilbert space k and an unitary operator u over k such that( w(ρ)(i)t )n = ( 1 + |ρ− 1| ) phu n|h, ∀n ≥ 1. therefore, if we denote v := u w(ρ)(i) , for any polynomial p we get p(t) = ph [( 1 + |ρ− 1| ) p(v ) −|ρ− 1|p(0)i ] |h. since v is a normal operator with spectral radius 1 w(ρ)(i) , we then have ‖p(t)‖≤ ∥∥(1 + |ρ− 1|)p(v ) −|ρ− 1|p(0)i∥∥ ≤ ∥∥(1 + |ρ− 1|)p −|ρ− 1|p(0)∥∥ l∞ ( d ( 0, 1 w(ρ)(i) )). as the polynomials are dense in the algebra a ( d ( 0, 1 w(ρ)(i) )) , the morphism of algebras p 7→ p(t) extends continuously on a ( d ( 0, 1 w(ρ)(i) )) . 162 v. agniel let us estimate the norm of this map. for f in the algebra we denote g(z) := f ( z w(ρ)(i) ) . hence, g ∈ a(d), and we have f(t) = g ( w(ρ)(i)t ) . applying a reformulation of theorem 2 in [15] by ando and okubo, we obtain ‖f(t)‖ = ∥∥g(w(ρ)(i)t)∥∥ ≤ max (1, 1 + |ρ− 1|)‖g‖l∞(d) = ( 1 + |ρ− 1| ) ‖f‖ l∞ ( d ( 0, 1 w(ρ)(i) )). we will now prove the two remaining inequalities. the fact that v is normal implies that f 7→ f(v ) is well defined and bounded on a ( d ( 0, 1 w(ρ)(i) )) . therefore f(t) = ph [( 1 + |ρ− 1| ) f(v ) −|ρ− 1|f(0)i ] |h, ∀f ∈ a ( d ( 0, 1 w(ρ)(i) )) . we now suppose that f satisfies f(0) = 0. if f ≡ 0, then f(t) = 0 and the statements are true. otherwise, up to dividing f by its norm, we may assume that ‖f‖ l∞ ( d ( 0, 1 w(ρ)(i) )) = 1. for a fixed n ≥ 1, we get f(t)n = fn(t) = ( 1 + |ρ− 1| ) phf n(v )|h = ( 1 + |ρ− 1| ) phf(v ) n|h. as we have ‖f(v )‖≤ ‖f‖ l∞ ( d ( 0, 1 w(ρ)(i) )) = 1, the operator f(v ) lies in c(1) which in turns implies that f(v ) can be dilated on a larger hilbert space as follows f(v )m = pkw m|k, ∀m ≥ 1, with w a suitable unitary operator. combining the two dilations, we obtain f(t)n = ( 1 + |ρ− 1| ) phw n|h, ∀n ≥ 1. therefore f(t) lies in c(1+|ρ−1|), which is equivalent to w(1+|ρ−1|)(f(t)) ≤ 1. this inequality is in turn equivalent to w(ρ)(f(t)) ≤ w(ρ)(i), which proves the second inequality of this proposition. lastly, if f ∈ a(d) with f(0) = 0, we can use schwarz’s lemma to obtain ‖f‖ l∞ ( d ( 0, 1 w(ρ)(i) )) ≤ 1 w(ρ)(i) ‖f‖l∞(d), which in turn gives w(ρ)(f(t)) ≤‖f‖l∞(d). unitary skew-dilations of hilbert space operators 163 for the optimality of these inequalities, let us take t such that t 2 = 0 and ‖t‖ = |ρ|, and f(z) = z. we then have w(ρ)(t) = ‖t‖ |ρ| = 1 = w(ρ)(i)‖f‖l∞ ( d ( 0, 1 w(ρ)(i) )) = ‖f‖l∞(d) and ‖f(t)‖ = |ρ| = ( 1 + |ρ− 1| ) ‖f‖ l∞ ( d ( 0, 1 w(ρ)(i) )). the proof is complete. when ρ does not lie in [1, +∞[, the algebra where the functional calculus is defined strictly contains the disc algebra a(d). for 0 < ρ < 1, the norm of this map is then 2 − ρ. this result differs from [15, theorem 2] as ando and okubo looked in [15] at the map f 7→ f(t) on a(d) and not on a larger algebra. 4. inequalities and parametrizations for (ρn)-radii operator radii of products and tensor products. a useful tool, used to study the behavior of a product or sum of double-commuting operators, is the following result, proved in [11, theorem.4.2]. proposition 4.1. let tn,sn ∈ l(h), n ∈ z, be such that for all 0 ≤ r < 1, t ∈ r, the series ∑∞ n=−∞r |n|einttn and ∑∞ n=−∞r |n|eintsn converge absolutely and have self-adjoint non-negative sums. if, moreover, we have tn ·sm = sm ·tn, ∀m,n ∈ z, then the series ∑∞ n=−∞r |n|einttn ·sn converges absolutely and has a self-adjoint non-negative sum, for all 0 ≤ r < 1, t ∈ r. using proposition 4.1 we can easily obtain the following auxiliary result. lemma 4.2. let t,s ∈l(h) be two operators that are double-commuting (i.e., ts =st, ts∗=s∗t). let (ρn)n, (τn)n∈ (c∗)n ∗ with lim infn ( |ρn| 1 n ) > 0 and lim infn ( |τn| 1 n ) > 0. then, we have w(ρnτn)(st) ≤ w(ρn)(s)w(τn)(t). proof. if s = 0 or t = 0, then st = 0 and both sides of the inequality are equal to zero. if s 6= 0 and t 6= 0, then, up to dividing s and t by their 164 v. agniel respective radius, we can consider that w(ρn)(s) = w(τn)(t) = 1. thus, we need to prove that w(ρnτn)(s.t) ≤ 1. we define tm :=   1 ρm tm if m ≥ 1, i if m = 0, 1 ρ|m| (t∗)|m| if m ≤−1, sm :=   1 τm sm if m ≥ 1, i if m = 0, 1 τ|m| (s∗)|m| if m ≤−1. the condition w(ρn)(s) = w(τn)(t) = 1, together with lemma 2.12 and proposition 2.6, ensure us that the conditions of proposition 4.1 are fulfilled, since i + re ( f(ρn)(re its) ) = ∑ m∈z r |m|eimtsm, for all 0 ≤ r < 1, t ∈ r. thus, ∑ m∈z r |m|eimtsmtm converges absolutely, is self-adjoint, and has a positive sum, for all 0 ≤ r < 1, t ∈ r. this implies that the series ∑ n≥1 2 ρnτn ( reitst )n = f(ρnτn) ( reitst ) is absolutely convergent and that i + re ( f(ρnτn)(re itst) ) ≥ 0 for all 0 ≤ r < 1, t ∈ r. thus st ∈ c(ρnτn) and w(ρnτn)(st) ≤ 1, which concludes the proof. corollary 4.3. let t,s ∈ l(h) and let (ρn)n, (τn)n ∈ (c∗)n ∗ with lim infn ( |ρn| 1 n ) > 0 and lim infn ( |τn| 1 n ) > 0. (i) if t and s double-commute, then w(ρn)(st) ≤ w(1)(s)w(ρn)(t) ≤ |τ1|w(τn)(s)w(ρn)(t). this inequality is optimal when dim(h) ≥ 4. (ii) we have w(1)(st) ≤ w(1)(s)w(1)(t) ≤ |τ1||ρ1|w(τn)(s)w(ρn)(t). this inequality is optimal when dim(h) ≥ 2. (iii) for r ∈l(h′), we have w(ρnτn)(t ⊗r) ≤ w(ρn)(t)w(τn)(r). unitary skew-dilations of hilbert space operators 165 proof. (i) we use lemma 4.2 for s,t and (1)n, (ρn)n to get the lefthand side inequality. the right-hand side inequality comes from the fact w(τn)(s) ≥ ‖s‖ |τ1| (cf. lemma 2.12). by taking s =   0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0   , t =   0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0   such that st =   0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0   , some computation show that s and t double-commute, and that ‖s‖ = ‖t‖ = ‖st‖, s2 = t 2 = (st)2 = 0. corollary 3.5(iii) shows that all three quantities are equal to ‖st‖ |ρ1| . (ii) the inequality on the right-hand side follows again from lemma 2.12. by taking s = ( 0 1 0 0 ) , t = ( 0 0 1 0 ) such that st = ( 1 0 0 0 ) , we have ‖s‖ = ‖t‖ = ‖st‖ = 1, s2 = t 2 = 0, and st is self-adjoint. thus, w(τn)(s) = w(ρn)(t) = 1 ρ and w(1)(st) = 1, so all quantities are equal to 1. (iii) as ih,t double-commute and ih′,r double-commute too, we can apply lemma 4.2 to (t ⊗ ih′)(ih ⊗ r) = t ⊗ r. we then apply corollary 2.16(iii). although these inequalities are optimal for some operators, they tend to lose a good part of the information in the general case. for example, we have w(3)(i) = 1 ≤ w(−1)(i)w(−3)(i) = 5. such a loss of information on the radius of the identity operator i also impacts almost every estimate of radii for other operators in l(h). corollary 4.4. let (ρn)n ∈ (c∗)n ∗ with lim infn ( |ρn| 1 n ) > 0. then, ‖t‖ |ρ1| ≤ w(ρn)(t) ≤‖t‖w(ρn)(i). furthermore, the coefficients in this equivalence of quasi-norms are optimal. 166 v. agniel proof. the left-hand side inequality ‖t‖ |ρ1| ≤ w(ρn)(t) has been obtained in lemma 2.12. the equality case is obtained for t such that t 2 = 0, as seen in corollary 3.5. the right-hand side inequality comes from lemma 4.2, with s = i and τn = 1. it is an improvement of the one that was obtained in lemma 2.12. the equality case is obtained for any t normal of norm 1. operator radii as 1-parameter families. to better understand the behavior of the associated radii associated with classes of operators, it is useful to look at (ρn)-radii as 1-parameter families. this is obtained by studying the map z 7→ w(zρn). we will present results for the real parameter case (r ∈]0, +∞[) and for the complex one (z ∈ c∗). the two main ingredients we are using are the double-commuting inequality of lemma 4.2 for t,i and (ρn)n,(1)1, and the fact that f(zρn) = 1 z f(ρn). proposition 4.5. let t ∈ l(h) and consider (ρn)n ∈ (c∗)n ∗ with lim infn ( |ρn| 1 n ) > 0. (i) for all z 6= 0, we have: |z| 1 + |z − 1| w(zρn)(t) ≤ w(ρn)(t) ≤ w(zρn)(t) ( |z| + |z − 1| ) . (ii) the map z 7→ w(zρn)(t) is continuous on c ∗, and r 7→ w(reitρn)(t) is continuous and decreasing on ]0, +∞[, for all t ∈] −π,π]. (iii) we have 1 3 w(zρn)(t) ≤ w(|z|ρn)(t) ≤ 3w(zρn)(t), and these inequalities are optimal. proof. (i) we use lemma 4.2 to obtain w(zρn)(t) ≤ w(z)(i)w(ρn)(t) and w(ρn)(t) ≤ w(z−1)(i)w(zρn)(t). as w(z)(i) = 1+|z−1| |z| and w(z−1)(i) = |z| + |z − 1|, we obtain the desired inequalities. (ii) up to changing (ρn)n by (wρn)n, the continuity must only be shown at the point w = 1, that is when z → 1. as we have w(ρn)(t) ≤ w(zρn)(t) ( |z| + |z − 1| ) ≤ w(ρn)(t) ( |z| + |z − 1| )1 + |z − 1| |z| unitary skew-dilations of hilbert space operators 167 and as (|z| + |z − 1|), 1+|z−1||z| both tend to 1 from above as z → 1, we obtain lim z→1 w(zρn)(t) = w(ρn)(t). for any t ∈ r and 0 < r < r, we have w(reitρn)(t) ≤ w(rr−1)(i)w(reitρn)(t) = w(reitρn)(t). thus, r 7→ w(reitρn)(t) is decreasing on ]0, +∞[. (iii) we use the fact that w(eit)(i) = 1 + |eit − 1| has a maximum of 3 when eit = −1. the equality case for the inequality on the left-hand side is attained at t = i, ρn = 1 and z = −1, whereas the equality case for the one on the right-hand side is attained at t = i, ρn = −1, z = −1. since r 7→ w(rρn)(t) is decreasing, the classes c(rρn) are increasing (for the usual order of inclusion of sets), for r ∈]0, +∞[. by using nilpotent operators of order 2, and item (iii) of corollary 3.5, we can also immediately show that these inclusions are always strict. for the following propositions, we recall that 1 lim infn ( |ρn| 1 n ) = 0 if lim infn ( |ρn| 1 n ) = +∞. proposition 4.6. let (ρn)n ∈ (c∗)n ∗ and t ∈ l(h) be such that lim infn ( |ρn| 1 n ) > r(t) ≥ 0. then, there is r > 0 such that for all z with |z| = r, r(t) lim infn ( |ρn| 1 n ) ≤ w(zρn)(t) ≤ 1. proof. let s > 1 be such that r(st) < lim infn ( |ρn| 1 n ) . as lim sup n→∞ ( 2sn‖tn‖ |ρn| )1 n = r(st) lim infn ( |ρn| 1 n ) < 1, there is b > 0 such that 2sn‖tn‖ |ρn| ≤ b. thus, for all w ∈ d, we have ∥∥f(zρn)(wt)∥∥ ≤ ∑ n≥1 2‖tn‖ |z||ρn| ≤ ∑ n≥1 b |z|sn = 1 |z| sb 1 −s < +∞. by taking |z| large enough, we have ‖f(zρn)(wt)‖ < 1, which implies that i + re ( f(zρn)(wt) ) ≥ 0, ∀w ∈ d. 168 v. agniel thus w(zρn)(t) ≤ 1. the left-hand side inequality comes from items (i) and (ii) of proposition 2.18: we have w(zρn)(t) ≥ r(t)w(zρn)(i) and w(zρn)(i) ≥ 1 lim infn ( |zρn| 1 n ). proposition 4.7. let t ∈ l(h) and let (ρn)n ∈ (c∗)n ∗ be such that lim infn ( |ρn| 1 n ) > 0. then lim |z|→+∞ ( w(zρn)(t) ) = r(t) lim infn ( |ρn| 1 n ). proof. according to proposition 4.5 and proposition 2.18, the map r 7→ w(reitρn)(t) is decreasing on ]0, +∞[ and w(ρn)(t) ≥ r(t)w(ρn)(i) ≥ r(t) lim infn ( |ρn| 1 n ). we will show that w(zρn)(t) is as close to this lower bound as we want when z is large enough. let � > 0. if r(t) = 0, then r ( 1 � t ) = 0, so proposition 4.6 implies the existence of r > 0 such that w(zρn) ( t � ) ≤ 1 for all z with |z| = r. thus, w(zρn)(t) ≤ �. if r(t) 6= 0, for 0 < r < lim infn ( |ρn| 1 n ) we have r ( rt (1 + �)r(t) ) ≤ lim inf n ( |ρn| 1 n ) . thus, by proposition 4.6, there exists r > 0 such that w(zρn) ( rt (1+�)r(t) ) ≤ 1 for all z with |z| = r. hence, r(t) lim infn ( |ρn| 1 n ) ≤ w(zρn)(t) ≤ (1 + �)r(t)r . we then obtain the result by taking r = lim infn ( |ρn| 1 n ) (1−�) if lim infn ( |ρn| 1 n ) is finite, or r = 1 � if lim infn ( |ρn| 1 n ) = +∞. proposition 4.8. let t ∈ l(h). let (ρn)n ∈ (c∗)n ∗ be such that lim infn ( |ρn| 1 n ) > 0. we have: unitary skew-dilations of hilbert space operators 169 (i) z 7→ w(zρn)(t) is uniformly continuous on c\d(0,�), for all � > 0. this maps tends to +∞ as |z|→ 0, and to r(t) lim infn ( |ρn| 1 n ) as |z|→ +∞; (ii) for any t ∈ r, the map r 7→ w(reitρn)(t) is log-convex on ]0, +∞[. proof. (i) on the closed set c\d(0,�), the function z 7→ w(zρn)(t) is continuous, decreasing on every half-line of the form eit[�, +∞[, and converges to r(t) lim infn ( |ρn| 1 n ) as |z|→ +∞. thus, a standard argument (considering two cases, � ≤ |z| ≤ r and |z| ≥ r) shows that this map is uniformly continuous. one can also use the double-commuting inequality of lemma 4.2 for t and ih, as well as the uniform continuity of the map z 7→ w(z)(i) on c \ d(0,η), in order to prove the uniform continuity of z 7→ w(zρn)(t). the limit as |z| → +∞ has been obtained in proposition 4.7, while the limit as |z| → 0 comes from the fact that w(zρn)(t) ≥ ‖t‖ |z||ρ1| , as remarked in lemma 2.12. (ii) let t ∈ r. denote g′(z) := −e−itf(ρn)(zt). for any α > 0, we have w(reitρn)(t) ≤ α if and only if f(eitρn) ( zt α ) is analytic on d and i + re ( 1 r f(eitρn) ( zt α )) ≥ 0, for all z ∈ d. by taking w = z α , this is equivalent to g′(w) being analytic on d ( 0, 1 α ) and re(g′(w)) ≤ ri, for all w ∈ d ( 0, 1 α ) . the result is then obtained by mimicking the proof of [2, theorem 1] by ando and nishio and replacing g with g′. even though the expression of f(zρn) is more complex than f(z)(w) = 2 z w 1−w , the main regularity properties remain valid due to its analyticity. proposition 4.9. let (ρn)n ∈ (c∗)n ∗ be such that lim infn ( |ρn| 1 n ) > 0. if one of the following assertions is true (i) lim infn ( |ρn| 1 n ) < 1; (ii) |ρn| < 1 for some n ≥ 1; (iii) w(ρn)(i) > 1; (iv) ρn = m + xn, (xn)n ∈ `2(c), 170 v. agniel then all operators in c(ρn)(h) are similar to contractions. if, on the contrary, we have: (i’) w(ρn)(i) < 1, then c(ρn)(h) contains operators that are not similar to contractions. both statements remain true if the conditions are only fulfilled for the subsequence (ρkn)n, for some fixed k ≥ 1. proof. (i), (ii), (iii) we can see that (i) ⇒ (ii) ⇒ (iii). if (iii) is true, then for t ∈ c(ρn), we have r(t) ≤ w(ρn)(t) w(ρn)(i) < 1, so t is similar to a contraction. (iv) it has been shown in [17, chapter 2] (see also [4, corollary 5.2.1]) that when ρn = m + xn, (xn)n ∈ `2(c), all operators in c(ρn) are similar to contractions. (i’) on the contrary, when w(ρn)(i) < 1, 1 w(ρn)(i) i ∈ c(ρn) and this operator is not similar to a contraction. the last assertion of the theorem follows from two facts. the first one is that t ∈ c(ρn) implies t k ∈ c(ρkn). the second one is that t k is similar to a contraction if and only if t is similar to a contraction: see [10, problem 6 (ii)] for a proof when k = 2 that extends to any k by taking ((f,g)) := ∑k−1 j=1〈a jf,ajg〉. proposition 4.10. let (ρn)n ∈ (c∗)n ∗ be such that lim infn ( |ρn| 1 n ) > 0. (i) if lim infn ( |ρn| 1 n ) = +∞, then ⋃ r>0 c(rρn)(h) = l(h). (ii) if lim infn ( |ρn| 1 n ) < +∞, then we have{ t : r(t) < lim inf n ( |ρn| 1 n )} ⊂ ⋃ r>0 c(rρn)(h) ⊂ { t : r(t) ≤ lim inf n ( |ρn| 1 n )} . (iii) moreover, we have{ t : r(t) < lim inf n ( |ρn| 1 n )} = ⋃ r>0 c(rρn)(h) unitary skew-dilations of hilbert space operators 171 if and only if w(rρn) ( lim inf n ( |ρn| 1 n ) i ) > 1, ∀r > 0. proof. (i) by using proposition 4.6, for any t there exists r > 0 such that w(rρn)(t) ≤ 1. (ii) we use again proposition 4.6 in order to obtain the left-hand side inclusion. the other inclusion follows from proposition 2.6. (iii) suppose that there is a number r > 0 and an operator t with r(t) = lim infn ( |ρn| 1 n ) such that w(rρn)(t) ≤ 1. then 1 ≥ w(rρn)(t) ≥ r(t)w(rρn)(i) ≥ r(t) lim infn ( |ρn| 1 n ) = 1. thus all inequalities are equalities, and w(rρn) ( lim infn(|ρn| 1 n ).i ) = 1. hence, the union of all c(rρn) strictly contains { t : r(t) < lim infn ( |ρn| 1 n )} if and only if it contains lim infn ( |ρn| 1 n ) i. the proof is complete. remark 4.11. replacing (ρn)n by (e itρn)n leaves unchanged the quantity lim infn ( |ρn| 1 n ) . however, the union of all classes c(rρn) can become a different set. with ρn = ρ, we have lim infn ( |ρn| 1 n ) = 1 and w(ρ)(i) = 1 if and only if ρ ∈ [1, +∞[. thus,⋃ r>0 c(reit)n(h) = { t : r(t) < 1 } if t 6= 0 [2π]. this is not an equality if t = 0 (look at the identity operator i). however, the set ⋃ r>0 c(r)(h) does not contain all operators with spectral radius one. indeed, it has been proven in [17, chapter 2] (see also [4, corollary 5.2.1]) that all operators contained in this union are all similar to contractions. furthermore, all operators similar to a contraction are not in this union. for a counterexample, any non-orthogonal projection t (that is t 2 = t and ‖t‖ > 1) is not in this union since corollary 4.3,(v), says that w(ρ)(t) = ‖t‖+|ρ−1| |ρ| > 1. for a sequence (ρn)n that satisfies α = lim infn ( |ρn| 1 n ) ∈]0, +∞[, we can go back to the case lim infn ( |ρn| 1 n ) = 1 by considering the sequence ( ρn αn ) n . as this normalization is equivalent to a dilation by a factor 1 α on the class 172 v. agniel c(ρn), we can then try to see if in this case the class c(ρn) is always included in the set of operators that are similar to contractions. this question is motivated by properties 4.10 and 4.9. the answer is true when w(ρn)(i) > 1, but corollary 4.18 will give a negative answer in many remaining cases, even if we consider the inclusion in the set of power-bounded operators. at this point we would like to mention that, for every k ≥ 2, there is ([9]) a hilbert space operator t /∈∪ρ>0cρ but with tk belonging to c(τ) for every τ ≥ 1. related results are given in the next proposition. proposition 4.12. let (ρn)n ∈ (c∗)n ∗ be such that lim infn ( |ρn| 1 n ) > 0. let h a hilbert space of dimension at least 2. (i) for t ∈ l(h) with t 2 = 0 and ‖t‖ > |ρ1|, tk is in the class c(ρn) for every k ≥ 2, but t is not. (ii) if lim infn ( |ρn| 1 n ) > 1, then tk ∈ ⋃ r>0 c(rρn) for some k ≥ 2 implies that t ∈ ⋃ r>0 c(rρn). (iii) if lim infn ( |ρn| 1 n ) < 1, then there exists t ∈l(h) such that tk lies in⋃ r>0 c(rρn) for every k ≥ 2 whereas t does not. (iv) if lim infn ( |ρn| 1 n ) = 1 and i /∈ ⋃ r>0 c(rρn), then t k ∈ ⋃ r>0 c(rρn) for some k ≥ 2 implies that t ∈ ⋃ r>0 c(rρn). proof. (i) as we have ‖t‖ > |ρ1|, t cannot lie in c(ρn), whereas t k = 0 does. (ii) let t be such that tk ∈ ⋃ r>0 c(rρn) for some k ≥ 2. then, r(t k) ≤ lim infn ( |ρn| 1 n ) . hence, r(t) ≤ lim inf n ( |ρn| 1 n )1 k < lim inf n ( |ρn| 1 n ) , that is t ∈ ⋃ r>0 c(rρn) according to proposition 4.10, (i) and (ii). (iii) take r > 0 such that lim inf n ( |ρn| 1 n ) < r < lim inf n ( |ρn| 1 n )1 2 , and denote t = ri. thus, using item (ii) of proposition 4.10, we can see that since for every k ≥ 2 we have r(tk) ≤ r(t 2) < lim inf n ( |ρn| 1 n ) < r(t), unitary skew-dilations of hilbert space operators 173 t doesn’t lie in ⋃ r>0 c(rρn) whereas t k does. (iv) if tk ∈ ⋃ r>0 c(rρn), then r(t k) < 1 according to item (iii) of proposition 4.10. this implies that r(t) < 1, which implies in turn that t ∈ ⋃ r>0 c(rρn). remark 4.13. as the classes c(rρn) are increasing for the inclusion of sets, the assertion t ∈ ⋃ r>0 c(rρn) is equivalent to the existence of r > 0 such that t ∈ c(rρn) for every r ≥ r. when lim infn ( |ρn| 1 n ) = 1 and i ∈ ⋃ r>0 c(rρn), which is the case when ρn = ρ > 0, we do not know if the result of găvruţa [9] stays true, as the type of operators he used in his proof is not suited in this setting: since there are sequences (ρn) such that ⋃ r>0 c(rρn) contains all power-bounded operators (see corollary 4.18), taking a t such that tk = i will not work. example 4.14. for ρn = 2(n!), we have i + f(ρn)(zt) = exp(zt), and a quick computation gives w(2(n!))(i) = 2 π < 1 (see item (iv) of example 4.20 for another proof). therefore π 2 i ∈ c(2(n!))(h) and this class contains an operator not similar to a contraction. we can also try to obtain some relationships between the (γnρn)-radii of an operator, for sequences (γn)n ∈ ∂dn ∗ , in order to see for which sequences (γn)n the maximal or minimal radii are attained. the following lemma answers the question for the maximal radii when t = i. lemma 4.15. let (ρn)n ∈ (c∗)n ∗ be such that α = lim infn ( |ρn| 1 n ) > 0. if limx→α− f(|ρn|)(x) > 1, then f(|ρn|)(x) = 1 has a unique solution, r1, on ]0,α[. otherwise, denote r1 = α. we then have: (i) w(−|ρn|)(i) = 1 r1 ; (ii) w(−|ρn|)(i) ≥ w(γnρn)(i) ≥ 1 α , for any (γn)n ∈ ∂dn ∗ ; (iii) the condition w(rγnρn)(i) = 1 α , ∀r ≥ 1, ∀γn ∈ ∂d is equivalent to lim x→α− f(|ρn|)(x) ≤ 1 and to w(−|ρn|)(i) = 1 α . 174 v. agniel proof. (i), (ii) the right-hand side inequality of (ii) is the last inequality of lemma 2.12. for any z ∈ d(0,α) and γn ∈ ∂d, we have ∣∣f(γnρn)(z)∣∣ ≤ ∑ n≥1 2|z|n |ρn| = f(|ρn|)(|z|). also, the map x 7→ f(|ρn|)(x) is strictly increasing on ]0,α[, as f(|ρn|) is nonconstant with positive taylor coefficients, so if limx→α− f(|ρn|)(x) > 1 the real number r1 is indeed unique. let u > 0 be such that u ≥ 1r1 ≥ 1 α . then 1 u ≤ r1 and lim x→1 u − f|ρn|(x) ≤ 1. since we have f(γnρn) ( d ( 0, 1 u )) ⊂ d ( 0, lim x→1 u − f(|ρn|)(x) ) ⊂ d, proposition 2.18 implies that w(γnρn)(i) ≤ 1 r1 . when γn = − ρn|ρn|, we have f(γnρn)(x) = f(−|ρn|)(x) = −f(|ρn|)(x). thus, the negative number lim x→1 u − ( − f(|ρn|)(x) ) lies in the adherence of f(−|ρn|) ( d ( 0, 1 u )) , and the smallest u ≥ 1 α such that f(−|ρn|) ( d ( 0, 1 u )) ⊂ re≥−1 is 1 r1 . hence, w(−|ρn|)(i) = 1 r1 ≥ w(γnρn)(i). (iii) by (ii) and using that r 7→ w(rγnρn)(i) is decreasing, we have w(rγnρn)(i) = 1 α , ∀r ≥ 1, ∀(γn)n ∈ ∂dn ∗ if and only if w(−|ρn|)(i) = 1 α . this equation is equivalent to r1 = α, that is limx→α− ( f(|ρn|)(x) ) ≤ 1. unitary skew-dilations of hilbert space operators 175 we do not know if the (|ρn|)-radius of i is always the minimal one. the idea of the proof of lemma 4.15 can be transported to any operator t if we add a summability condition to the sequence (ρn)n. proposition 4.16. let a = (an)n ∈ (c∗)n ∗ be such that ∑ n≥1 1 |an| ≤ 1. let t ∈l(h) and define ρn := { 2an‖tn‖ if tn 6= 0, 1 otherwise, (i) if r(t) > 0 or if t is nilpotent, then t ∈ c(ρn). (ii) if r(t) > 0 and lim infn ( |an| 1 n ) = 1, then w(znρn)(t) = 1, for all zn such that |zn| ≥ 1 and limn ( |zn| 1 n ) = 1. proof. (i) suppose first that r(t) > 0. since ∑ n 1 |an| < +∞, we have lim infn(|an| 1 n ) ≥ 1, thus lim infn ( |ρn| 1 n ) ≥ r(t) > 0. we also have: ∥∥f(ρn)(zt)∥∥ ≤ ∑ n≥1 2|z|n‖tn‖ 2|an|‖tn‖ ≤ ∑ n≥1 1 |an| ≤ 1. thus, i + re ( f(ρn)(zt) ) ≥ ( 1 − ∥∥f(ρn)(zt)∥∥)i ≥ 0 for all z ∈ d, so t ∈ c(ρn). if t is nilpotent then f(ρn)(zt) becomes a finite sum and the same computation gives the result, as lim infn ( |ρn| 1 n ) > 0. (ii) when r(t) > 0 and lim infn ( |an| 1 n ) = 1, we have r(t) = lim infn ( |ρn| 1 n ) , so 1 ≥ w(ρn)(t) ≥ r(t) lim infn ( |ρn| 1 n ) = 1. thus w(ρn)(t) = 1. if we multiply each an by a complex number zn with |zn| ≥ 1 and limn ( |zn| 1 n ) = 1, the sum ∑ n≥1 1 |znan| decreases, while lim infn ( |znan| 1 n ) = 1. thus, we can apply the previous result to (znρn)n and obtain w(znρn)(t) = 1. remark 4.17. for any t with r(t) > 0, if we take a sequence (ρn)n as in item (ii) of the previous proposition, then the result says that z 7→ w(zρn)(t) is constant and equal to 1 on c\d. the choice of (ρn)n only depends on ‖tn‖. for example, with any t normal 176 v. agniel with ‖t‖ = 1, by taking an = π 2 6 n2, we have w(2anzn)(t) = 1 for any sequence (zn)n such that 1 ≤ |zn| and sup |zn| < +∞. if t is quasinilpotent but not nilpotent, we have lim infn ( |ρn| 1 n ) = 0. however, the statement of item (i) holds true for such a t, with a very similar proof. using the ideas in the proof of proposition 4.16, we can show that some sets⋃ r>0 c(rρn) largely differ from ⋃ ρ>0 c(ρ) or {t : r(t) < 1} even if lim infn ( |ρn| 1 n ) = 1. corollary 4.18. let (ρn)n be such that lim infn ( |ρn| 1 n ) = 1. the following assertions are true: (i) if ( 1 ρn ) ∈ `1, then ⋃ r>0 c(rρn) contains all power-bounded operators. (ii) if f(ρn) ∈ h ∞(d) and f ′ (ρn) ∈ h∞(d), then ⋃ r>0 c(rρn) contains an operator that is not power-bounded. (iii) if nk+1+� = o(|ρn|) for k ∈ n∗ and some � > 0, then ⋃ r>0 c(rρn) contains all operators t such that ‖tn‖ = o(nk). proof. (i) let t be a power-bounded operator with ‖tn‖≤ c. let r > 0 and z ∈ d. we have ∥∥f(rρn)(zt)∥∥ ≤ ∑ n≥1 2 r|ρn| |z|n‖tn‖≤ 2c r ∑ n≥1 1 |ρn| < +∞. hence, for r large enough, we have ∥∥f(rρn)(zt)∥∥ ≤ 1 for every z ∈ d. this implies that i + re ( f(rρn)(zt) ) ≥ 0, ∀z ∈ d. this in turn implies that t ∈ c(rρn) since we also know that r(t) ≤ 1 = lim infn ( |rρn| 1 n ) . (ii) we first note that both the entire series f( ρn n )(z) = ∑ n≥1 2n ρn zn and f(ρn) have radii of convergence 1, so their sum is analytic on d. we also have f( ρn n )(z) = zf ′ (ρn) (z). let n be a nilpotent operator of order 2 and set t = i +n. since tn = i +nn, we have ‖tn‖' n‖n‖ so t is not power-bounded. we will show that t belongs to a class c(rρn) for large enough r > 0. let unitary skew-dilations of hilbert space operators 177 r > 0 and z ∈ d. we have: ∥∥f(rρn)(zt)∥∥ = ∥∥∥∥∥∑ n≥1 2 rρn zn(i + nn) ∥∥∥∥∥ = ∥∥∥∥1rf(ρn)(z)i + 1rzf ′(ρn)(z)n ∥∥∥∥ ≤ 1 r (∥∥f(ρn)∥∥h∞ + ∥∥f ′(ρn)∥∥h∞‖n‖) < +∞. hence, for r large enough, we have ∥∥f(rρn)(zt)∥∥ ≤ 1 for every z ∈ d, which implies that i + re ( f(rρn)(zt) ) ≥ 0, ∀z ∈ d. this in turn implies that t ∈ c(rρn) since we also know that r(t) = 1 = lim infn ( |rρn| 1 n ) . (iii) let t be such that ‖tn‖ = o(nk) and let z ∈ d. we have ‖t n‖ |ρn| = o ( 1 n1+� ) , so this sequence is in `1. if t is nilpotent, then t is power-bounded and we can apply (i) to get a positive r > 0 such that t ∈ c(rρn). otherwise, we can consider the complex numbers an := ρn ‖tn‖ ∥∥∥∥ ( ‖tn‖ |ρn| ) n ∥∥∥∥ `1 . we have ∑ n≥1 1 |an| = ∥∥∥∥ ( ‖tn‖ |ρn| ) n ∥∥∥∥−1 `1 ∑ n≥1 ‖tn‖ |ρn| = 1. thus, for τn := 2an‖tn‖, we can use proposition 4.16 to obtain t ∈ c(τn). since τn = 2ρn ∥∥∥(‖tn‖|ρn| )n ∥∥∥ `1 , we have τn = rρn for some r > 0, which concludes the proof. the condition f ′ (ρn) ∈ h∞(d) implies that the sequence ( n ρn ) n is bounded, but it does not imply the condition ( 1 ρn ) ∈ `1 from (i). thus, for a sequence (ρn) satisfying the conditions of item (ii), the set ⋃ r>0 c(rρn) may not contain every power-bounded operator. some examples. we conclude this paper by providing a computation of w(zρn)(i) in two examples, where sequences (ρn)n were chosen to match some common analytic maps. the difficulty lies in the computation of the boundary of f(zρn) ( d ( 1, 1 u )) , as some specific points on the boundary do not always have an explicit expression. 178 v. agniel example 4.19. let r > 0 and −π < t ≤ π. we have: (i) i + f(reitn)(zi) = i − 2 reit log(1 −zi); (ii) w(reitn)(i) = 1 if t = 0 and r ≥ 2 log(2); (iii) w(reitn)(i) = 1 exp( r 2 )−1 > 1 if t = 0 and 0 < r < 2 log(2); (iv) w(reitn)(i) = 1 1−exp(−r 2 ) > 1 if t = π; (v) w(reitn)(i) = 1 if t = ± π 2 and r ≥ π; (vi) w(reitn)(i) = 1 sin( r 2 ) if t = ±π 2 and 0 < r < π; (vii) w(reitn)(i) = 1 if 0 < |t| < π 2 and r ≥ 2 cos(t) log(2 cos(t)) + 2 sin(t)t; (viii) if we have 0 < |t| < π 2 and 0 < r < 2 cos(t) log(2 cos(t)) + 2 sin(t)t, then w(reitn)(i) = inf { u > 1 : 1 − 2 r gt(u) ≥ 0 } > 1 with gt(u) := cos(t) log (√ u2 − sin(t)2 + cos(t) u ) + arcsin ( sin(t) u ) sin(t). the same holds if π 2 < |t| < π. proof. let r > 0, t∈ ]−π,π]. as n ∈ r, we have w(re−itn)(i) = w(reitn)(i), so we can restrict the study to t ∈ [0,π]. a direct computation gives: f(reitn)(zt) = − 2 reit log(1 −zt). as lim infn ( |n| 1 n ) = 1, we have w(reitn)(i) ≥ 1. thus, we consider those u > 1 such that i + re ( f(reitn) ( zi u )) is positive for every z ∈ d. it is equivalent to look at the positivity of 1 + re ( f(reitn) (z u )) = 1 − 2 r re ( e−it log ( 1 − z u )) . we start off by studying the boundary of log ( d ( 1, 1 u )) . by analyticity, we have ∂ log ( d ( 1, 1 u )) ⊂ log ( ∂d ( 1, 1 u )) . as log ( eisr ∩ d ( 1, 1 u )) is a horizontal interval that is non-empty if and only if |s| ≤ arcsin ( 1 u ) , the previous sets unitary skew-dilations of hilbert space operators 179 are equal and log ( d ( 1, 1 u )) is convex. thus, the set log ( ∂d ( 1, 1 u )) can be parameterized by two arcs depending on the imaginary part of its elements: s 7→ log ( cos(s) ± 1 u √ 1 − sin(s)2u2 ) + is, s ∈ [ − arcsin (1 u ) ; arcsin (1 u )] . we want to compute the minimum of 1 − 2 r re ( e−it log ( 1 − e is u )) in order to find for which u > 1 this minimum is non-negative. for the cases t = 0, t = π, and t = π 2 , computing this minimum amounts to finding the extrema of the real or imaginary part of the elements in log ( ∂d ( 1, 1 u )) . as these extrema are log ( 1 ± 1 u ) for the real part and ±arcsin ( 1 u ) for the imaginary part, an easy computation gives all the u > 1 such that inf w∈r ( 1 − 2 r re ( e−it log ( 1 − eiw u ))) ≥ 0 in all three cases, which proves the items (ii), (iii), (iv), (v), (vi). for 0 < t < π 2 , computing this minimum leads to searching the lower bound of f1(s) := cos(π − t) log ( cos(s) − 1 u √ 1 − sin(s)2u2 ) −s sin(π − t). for π 2 < t < π, computing this minimum leads to searching the lower bound of f2(s) := cos(π − t) log ( cos(s) + 1 u √ 1 − sin(s)2u2 ) −s sin(π − t). the derivatives of these maps are: f ′1(s) = sin(s)u cos(π − t)√ 1 − sin(s)2u2 −sin(π−t), f ′2(s) = − sin(s)u cos(π − t)√ 1 − sin(s)2u2 −sin(π−t). both of them only have one zero, at s = −arcsin (sin(t) u ) . and in both cases the searched minimum for 1 − 2 r re ( e−it log ( 1 − e is u )) is: 1− 2 r [ cos(t) log (√ u2 − sin(t)2 + cos(t) u ) +arcsin (sin(t) u ) sin(t) ] = 1− 2 r gt(u). if 0 < t < π 2 , this minimum decreases towards 1 − 2 r gt(1) := 1 − 2 r [ cos(t) log ( 2 cos(t) ) + t sin(t) ] 180 v. agniel when u → 1+. so i u ∈ c(reitn) for every u > 1 if and only if 1 − 2 r gt(1) ≥ 0, that is r ≥ 2gt(1). this proves item (vii) and half of item (viii). if π 2 < t < π, this minimum decreases towards −∞ as u → 1+, so the smallest u for which this minimum is non-negative verifies u > 1 and w(reitn)(i) = u. this gives the other half of item (viii) and concludes the proof. example 4.20. let r > 0 and −π < t ≤ π. we have: (i) i + f(reitn!)(zi) = i + 2 r.eit ( exp(zi) − i ) ; (ii) w(reitn!)(i) = 1 log( r 2 +1) if t = π; (iii) w(reitn!)(i) = 1 log( 2 2−r ) if t = 0 and 0 < r ≤ 2 − 2 e ; (iv) w(reitn!)(i) = 1 π 2 −t if 0 ≤ |t| < π 2 and r = 2 cos(t); (v) w(reitn!)(i) ≤ 1 log( r 2 −cos(t)) for r > 2 + 2 cos(t); (vi) w(reitn!)(i) ≥ 1√ π2+log ( r 2 cos(t) −1 )2 if 0 ≤ |t| < π2 and r > 4 cos(t); (vii) in general, we have w(reit.n!)n(i) = inf { u > 0 : ∀θ ∈ [−π,π] with θ + sin(θ) u = t + kπ, k ∈ z, we have (−1)ke cos(θ) u cos(θ) ≥ cos(t) − r 2 } . for r ≥ 2eπ/2 − 2, we can restrict the infimum after u in ]0, 2 π ] and to the smallest θ ∈ ]−π 2 , 0] such that θ + sin(θ) u = t + kπ. proof. let r > 0, t ∈ [−π,π] and u > 0. as n ∈ r, we have w(re−itn!)(i) = w(reitn!)(i), so we restrict the study to t ∈ [0,π]. a computation gives i + f(reitn!)(zi) = i + 2 r.eit ( exp(zi) − i ) . we will first use lemma 4.15 to compute w(−rn!)(i) and rule out the case t = π. as f(rn!)(x) = 2 r ( exp(x) − 1 ) , we get f(rn!)(x) = 1 ⇔ x = log ( r 2 + 1 ) . unitary skew-dilations of hilbert space operators 181 hence, w(−rn!)(i) = 1 log( r 2 +1) and item (ii) is proved. as lim infn ( |n!| 1 n ) = +∞, we have i u ∈ c(reitn!) if and only if u ≥ w(reitn!)(i), if and only if i + re ( f(reitn!) ( z i u )) for every z ∈ d. thus, we need to study the positivity of 1 + re ( f(reitn!) (z u )) = 1 + 2 r re ( e−it ( exp (z u ) − 1 )) , for every z ∈ d and for any u > 0. by analyticity, we only need to make the computations for z ∈ ∂d. we have 1 + 2 r re ( exp (z u − it ) −e−it ) ≥ 0 ⇔ exp ( re (z u )) cos ( im(z) u − t ) ≥− r 2 + cos(t). denote, for s ∈ [−π,π], gu(s) := e cos(s) u cos ( t− sin(s) u ) . thus, i u ∈ c(reitn!) is equivalent to min s∈[−π,π] (gu(s)) ≥− r 2 + cos(t). therefore, this inequality will be verified if and only if u ≥ w(reitn!)(i). also, since min s ( gu(s) ) = min |w|= 1 u ( re ( exp(w − it) )) = min |w|< 1 u ( re ( exp(w − it) )) , we can see that mins ( gu(s) ) is the minimum of a harmonic non-constant map over the disc d ( 0, 1 u ) . the maximum principle implies that the map u 7→ mins ( gu(s) ) is strictly increasing. hence, w(reitn!)(i) is the only number u > 0 such that mins∈[−π,π] ( gu(s) ) = −r 2 + cos(t). let us focus now on the minimum of gu. the derivative of gu is g′u(s) = 1 u e cos(s) u sin ( t− sin(s) u −s ) . 182 v. agniel hence, the minimum of gu will be reached for a s0 such that hu(s0) := t − s0 − sin(s0) u = kπ, for some k ∈ z. for such a s0, we will also have gu(s0) = (−1)ke cos(s0) u cos(s0). if u ≥ 1, the map hu is strictly decreasing, with range [t−π,t + π]. hence, there will only be 2 (resp. 3) values of s such that hu(s) = kπ if t ∈]0,π[ (resp. t = 0). if t = 0 and u ≥ 1, these values of s will be −π, 0, π, and the minimum of gu will be gu(π) = exp (−1 u ) . thus, if t = 0 and w(rn!)(i) ≥ 1, we will have exp ( −1 w(rn!)(i) ) = − r 2 + 1, which is equivalent to 0 < r ≤ 2 − 2 e . thus w(rn!)(i) = 1 log( 2 2−r ) , proving item (iii). when t ∈ ]0,π[ and u ≥ 1, we have however no explicit formula for the two values of s mentioned above. for t ∈ [0, π 2 [ and r = 2 cos(t), we will have mins ( gw (reitn!) (i)(s) ) = 0. as e cos(s) u cos(s) = 0 if and only if s = ±π 2 , this minimum will be attained at π 2 or −π 2 , and w(reitn!)(i) will be the largest u > 0 such that gu( π 2 ) = 0 or gu(−π2 ) = 0. the latter condition is equivalent to 1 u ± t = π 2 + kπ, that is 1 u = π 2 ± t + kπ. since we have 0 ≤ t < π 2 , the integer k needs to be positive. by looking at the smallest possible value for 1 u we get w(reitn!)(i) = 1 π 2 −t, proving item (iv). in general, we can see that mins ( gu(s) ) ≥ −e 1 u . when r > 2 + 2 cos(t), the inequality −e 1 u ≥ cos(t)−r 2 is equivalent to u ≥ 1 log( r 2 −cos(t)) , which proves item (v). if uπ < 1, we have uπ = sin(α) for some α > 0, and gu(α) = −cos(t)e √ 1−u2π2 u . when r > 4 cos(t), the inequality gu(α) ≤ cos(t) − r2 is equivalent to u ≤ 1√ π2 + log ( r 2 cos(t) − 1 )2 . item (vi) is now proved. unitary skew-dilations of hilbert space operators 183 taking r ≥ 2eπ/2 − 2 = r0, we get −r2 + 1 ≤ 2 −e π/2 < −1 and w(reitn!)(i) ≤ w(−rn!)(i) ≤ w(−r0n!)(i) = 2 π , for every 0 ≤ t ≤ π, according to lemma 4.15. we can then see that for u = w(reitn!)(i) and for a number s0 such that gu(s0) = mins ( gu(s) ) and hu(s0) = kπ, the relationship (−1)ke cos(s0) u cos(s0) = gu(s0) = − r 2 + cos(t) < −1 implies that cos(s0) > 0 and that k is odd. in this case, |s0| will be the smallest real s in [0, π 2 [ such that hu(s) or hu(−s) is equal to kπ with k odd. as we also have hu( −π 2 ) = t + π 2 + 1 u ≥ π ≥ t = hu(0) ≥ 0, we can see that s0 lies in ]−π 2 , 0]. this gives all the announced results. acknowledgements this work was supported in part by the project front of the french national research agency (grant anr-17-ce40-0021) and by the labex cempi (anr-11-labx-0007-01). the author would like to thank catalin badea for his help and encouragement while writing this article. references [1] t. ando, c.k. li, operator radii and unitary operators, oper. matrices 4 (2) (2010), 273 – 281. 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[24] j.p. williams, schwarz norms for operators, pacific j. math. 24 (1968), 181 – 188. introduction hilbert space operators with (n)-dilations classes c() for =0 inequalities and parametrizations for (n)-radii e extracta mathematicae vol. 32, núm. 1, 25 – 54 (2017) trace inequalities of lipschitz type for power series of operators on hilbert spaces s.s. dragomir mathematics, school of engineering & science victoria university, po box 14428, melbourne city, mc 8001, australia sever.dragomir@vu.edu.au , http://rgmia.org/dragomir dst-nrf centre of excellence in mathematical and statistical sciences school of computer science & applied mathematics, university of the witwatersrand private bag 3, johannesburg 2050, south africa presented by alfonso montes received may 23, 2016 abstract: let f(z) = ∑∞ n=0 αnz n be a function defined by power series with complex coefficients and convergent on the open disk d(0, r) ⊂ c, r > 0. we show, amongst other that, if t, v ∈ b1(h), the banach space of all trace operators on h, are such that ∥t∥1 , ∥v ∥1 < r, then f(v ), f(t), f′((1 − t)t + tv ) ∈ b1(h) for any t ∈ [0, 1] and tr [f(v )] − tr [f(t)] = ∫ 1 0 tr [ (v − t)f′ ( (1 − t)t + tv )] dt. several trace inequalities are established. applications for some elementary functions of interest are also given. key words: banach algebras of operators on hilbert spaces, power series, lipschitz type inequalities, jensen’s type inequalities, trace of operators, hilbert-schmidt norm. ams subject class. (2010): 47a63, 47a99. 1. introduction let b(h) be the banach algebra of bounded linear operators on a separable complex hilbert space h. the absolute value of an operator a is the positive operator |a| defined as |a| := (a∗a)1/2. it is known that [4] in the infinite-dimensional case the map f(a) := |a| is not lipschitz continuous on b(h) with the usual operator norm, i.e., there is no constant l > 0 such that∥∥ |a| − |b|∥∥ ≤ l∥a − b∥ for any a,b ∈ b(h). 25 26 s.s. dragomir however, as shown by farforovskaya in [17], [18] and kato in [22], the following inequality holds ∥∥ |a| − |b|∥∥ ≤ 2 π ∥a − b∥ ( 2 + log ( ∥a∥ + ∥b∥ ∥a − b∥ )) (1.1) for any a,b ∈ b(h) with a ̸= b. if the operator norm is replaced with hilbert-schmidt norm ∥c∥hs := (trc∗c)1/2 of an operator c, then the following inequality is true [2]∥∥ |a| − |b|∥∥ hs ≤ √ 2 ∥a − b∥hs (1.2) for any a,b ∈ b(h). the coefficient √ 2 is best possible for a general a and b. if a and b are restricted to be selfadjoint, then the best coefficient is 1. it has been shown in [4] that, if a is an invertible operator, then for all operators b in a neighborhood of a we have∥∥ |a| − |b|∥∥ ≤ a1 ∥a − b∥ + a2 ∥a − b∥2 + o(∥a − b∥3 ) (1.3) where a1 = ∥∥a−1∥∥∥a∥ and a2 = ∥∥a−1∥∥ + ∥∥a−1∥∥3 ∥a∥2 . in [3] the author also obtained the following lipschitz type inequality ∥f(a) − f(b)∥ ≤ f ′(a) ∥a − b∥ (1.4) where f is an operator monotone function on (0,∞) and a,b ≥ aih > 0. one of the central problems in perturbation theory is to find bounds for ∥f(a) − f(b)∥ in terms of ∥a − b∥ for different classes of measurable functions f for which the function of operator can be defined. for some results on this topic, see [5], [19] and the references therein. by the help of power series f(z) = ∑∞ n=0 anz n we can naturally construct another power series which will have as coefficients the absolute values of the coefficient of the original series, namely, fa(z) := ∑∞ n=0 |an|z n. it is obvious that this new power series will have the same radius of convergence as the original series. we also notice that if all coefficients an ≥ 0, then fa = f. trace inequalities of lipschitz type 27 we notice that if f(z) = ∞∑ n=1 (−1)n n zn = ln 1 1 + z , z ∈ d(0,1) ; (1.5) g(z) = ∞∑ n=0 (−1)n (2n)! z2n = cosz , z ∈ c ; h(z) = ∞∑ n=0 (−1)n (2n + 1)! z2n+1 = sinz , z ∈ c ; l(z) = ∞∑ n=0 (−1)nzn = 1 1 + z , z ∈ d(0,1) ; where d(0,1) is the open disk centered in 0 and of radius 1, then the corresponding functions constructed by the use of the absolute values of the coefficients are fa(z) = ∞∑ n=1 1 n! zn = ln 1 1 − z , z ∈ d(0,1) ; (1.6) ga(z) = ∞∑ n=0 1 (2n)! z2n = coshz , z ∈ c ; ha(z) = ∞∑ n=0 1 (2n + 1)! z2n+1 = sinhz , z ∈ c ; la(z) = ∞∑ n=0 zn = 1 1 − z , z ∈ d(0,1) . other important examples of functions as power series representations with nonnegative coefficients are: 28 s.s. dragomir exp(z) = ∞∑ n=0 1 n! zn , z ∈ c ; (1.7) 1 2 ln ( 1 + z 1 − z ) = ∞∑ n=1 1 2n − 1 z2n−1 , z ∈ d(0,1) ; sin−1(z) = ∞∑ n=0 γ(n + 1 2 ) √ π(2n + 1)n! z2n+1 , z ∈ d(0,1) ; tanh−1(z) = ∞∑ n=1 1 2n − 1 z2n−1 , z ∈ d(0,1) ; 2f1(α,β,γ,z) = ∞∑ n=0 γ(n + α)γ(n + β)γγ) n!γ(α)γ(β)γ(n + γ) zn , α,β,γ > 0 , z ∈ d(0,1) ; where γ is gamma function. we recall the following result that provides a quasi-lipschitzian condition for functions defined by power series and operator norm ∥·∥ [14]: theorem 1. let f(z) := ∑∞ n=0 anz n be a power series with complex coefficients and convergent on the open disk d(0,r), r > 0. if t,v ∈ b(h) are such that ∥t∥ ,∥v ∥ < r, then ∥f(t) − f(v )∥ ≤ f ′a ( max {∥t∥ ,∥v ∥} ) ∥t − v ∥ . (1.8) if ∥t∥ ,∥v ∥ ≤ m < r, then from (1.8) we have the simpler inequality ∥f(t) − f(v )∥ ≤ f ′a(m) ∥t − v ∥ (1.9) in the recent paper [13] we improved the inequality (1.8) as follows: theorem 2. let f(z) := ∑∞ n=0 anz n be a power series with complex coefficients and convergent on the open disk d(0,r), r > 0. if t,v ∈ b(h) are such that ∥t∥ ,∥v ∥ < r, then ∥f(t) − f(v )∥ ≤ ∥t − v ∥ ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥ ) dt . (1.10) in order to obtain similar results for the trace of bounded linear operators on complex infinite dimensional hilbert spaces we need some preparations as follows. trace inequalities of lipschitz type 29 2. some preliminary facts on trace for operators let ( h,⟨·, ·⟩ ) be a complex hilbert space and {ei}i∈i an orthonormal basis of h. we say that a ∈ b(h) is a hilbert-schmidt operator if∑ i∈i ∥aei∥2 < ∞ . (2.1) it is well know that, if {ei}i∈i and {fj}j∈j are orthonormal bases for h and a ∈ b(h) then ∑ i∈i ∥aei∥2 = ∑ j∈i ∥afj∥2 = ∑ j∈i ∥a∗fj∥2 (2.2) showing that the definition (2.1) is independent of the orthonormal basis and a is a hilbert-schmidt operator iff a∗ is a hilbert-schmidt operator. let b2(h) the set of hilbert-schmidt operators in b(h). for a ∈ b2(h) we define ∥a∥2 := (∑ i∈i ∥aei∥2 )1/2 (2.3) for {ei}i∈i an orthonormal basis of h. this definition does not depend on the choice of the orthonormal basis. using the triangle inequality in l2(i), one checks that b2(h) is a vector space and that ∥·∥2 is a norm on b2(h), which is usually called in the literature as the hilbert-schmidt norm. denote the modulus of an operator a ∈ b(h) by |a| := (a∗a)1/2. because ∥|a|x∥ = ∥ax∥ for all x ∈ h, a is hilbert-schmidt iff |a| is hilbert-schmidt and ∥a∥2 = ∥|a|∥2. from (2.2) we have that if a ∈ b2(h), then a∗ ∈ b2(h) and ∥a∥2 = ∥a ∗∥2. the following theorem collects some of the most important properties of hilbert-schmidt operators: theorem 3. we have: (i) (b2(h),∥·∥2) is a hilbert space with inner product ⟨a,b⟩2 := ∑ i∈i ⟨aei,bei⟩ = ∑ i∈i ⟨b∗aei,ei⟩ (2.4) and the definition does not depend on the choice of the orthonormal basis {ei}i∈i; 30 s.s. dragomir (ii) we have the inequalities ∥a∥ ≤ ∥a∥2 (2.5) for any a ∈ b2(h) and ∥at∥2 , ∥ta∥2 ≤ ∥t∥ ∥a∥2 (2.6) for any a ∈ b2(h) and t ∈ b(h); (iii) b2(h) is an operator ideal in b(h), i.e., b(h)b2(h)b(h) ⊆ b2(h) ; (iv) bfin(h), the space of operators of finite rank, is a dense subspace of b2(h); (v) b2(h) ⊆ k(h), where k(h) denotes the algebra of compact operators on h. if {ei}i∈i an orthonormal basis of h, we say that a ∈ b(h) is trace class if ∥a∥1 := ∑ i∈i ⟨|a|ei,ei⟩ < ∞ . (2.7) the definition of ∥a∥1 does not depend on the choice of the orthonormal basis {ei}i∈i. we denote by b1(h) the set of trace class operators in b(h). the following proposition holds: proposition 1. if a ∈ b(h), then the following are equivalent: (i) a ∈ b1(h); (ii) |a|1/2 ∈ b2(h); (iii) a (or |a|) is the product of two elements of b2(h). the following properties are also well known: theorem 4. with the above notations: (i) we have ∥a∥1 = ∥a ∗∥1 and ∥a∥2 ≤ ∥a∥1 (2.8) for any a ∈ b1(h). trace inequalities of lipschitz type 31 (ii) b1(h) is an operator ideal in b(h), i.e., b(h)b1(h)b(h) ⊆ b1(h) . (iii) we have b2(h)b2(h) = b1(h). (iv) we have ∥a∥1 = sup { |⟨a,b⟩2| : b ∈ b2(h), ∥b∥ ≤ 1 } . (v) (b1(h),∥·∥1) is a banach space. (vi) we have the following isometric isomorphisms b1(h) ∼= k(h)∗ and b1(h)∗ ∼= b(h) , where k(h)∗ is the dual space of k(h) and b1(h)∗ is the dual space of b1(h). we define the trace of a trace class operator a ∈ b1(h) to be tr(a) := ∑ i∈i ⟨aei,ei⟩ , (2.9) where {ei}i∈i an orthonormal basis of h. note that this coincides with the usual definition of the trace if h is finite-dimensional. we observe that the series (2.9) converges absolutely and it is independent from the choice of basis. the following result collects some properties of the trace: theorem 5. we have: (i) if a ∈ b1(h) then a∗ ∈ b1(h) and tr(a∗) = tr(a) ; (2.10) (ii) if a ∈ b1(h) and t ∈ b(h), then at , ta ∈ b1(h) and tr(at) = tr(ta) and |tr(at)| ≤ ∥a∥1 ∥t∥ ; (2.11) (iii) tr(·) is a bounded linear functional on b1(h) with ∥tr∥ = 1 ; (iv) if a, b ∈ b2(h) then ab, ba ∈ b1(h) and tr(ab) = tr(ba) ; (v) bfin(h) is a dense subspace of b1(h). 32 s.s. dragomir utilising the trace notation we obviously have that ⟨a,b⟩2 = tr(b ∗a) = tr(ab∗) , ∥a∥22 = tr(a ∗a) = tr ( |a|2 ) for any a, b ∈ b2(h). the following hölder’s type inequality has been obtained by ruskai in [28] |tr(ab)| ≤ tr(|ab|) ≤ [ tr ( |a|1/α )]α [ tr ( |b|1/(1−α) )]1−α (2.12) where α ∈ (0,1) and a,b ∈ b(h) with |a|1/α , |b|1/(1−α) ∈ b1(h). in particular, for α = 1 2 we get the schwarz inequality |tr(ab)| ≤ tr(|ab|) ≤ [ tr ( |a|2 )]1/2 [ tr ( |b|2 )]1/2 (2.13) with a,b ∈ b2(h). if a ≥ 0 and p ∈ b1(h) with p ≥ 0, then 0 ≤ tr(pa) ≤ ∥a∥ tr(p) . (2.14) indeed, since a ≥ 0, then ⟨ax,x⟩ ≥ 0 for any x ∈ h. if {ei}i∈i is an orthonormal basis of h, then 0 ≤ ⟨ ap1/2ei,p 1/2ei ⟩ ≤ ∥a∥ ∥∥∥p1/2ei∥∥∥2 = ∥a∥ ⟨pei,ei⟩ for any i ∈ i. summing over i ∈ i we get 0 ≤ ∑ i∈i ⟨ ap1/2ei,p 1/2ei ⟩ ≤ ∥a∥ ∑ i∈i ⟨pei,ei⟩ = ∥a∥ tr(p) , and since ∑ i∈i ⟨ ap1/2ei,p 1/2ei ⟩ = ∑ i∈i ⟨ p1/2ap1/2ei,ei ⟩ = tr ( p1/2ap1/2 ) = tr(pa) , we obtain the desired result (2.14). this obviously imply the fact that, if a and b are selfadjoint operators with a ≤ b and p ∈ b1(h) with p ≥ 0, then tr(pa) ≤ tr(pb) . (2.15) trace inequalities of lipschitz type 33 now, if a is a selfadjoint operator, then we know that |⟨ax,x⟩| ≤ ⟨|a|x,x⟩ for any x ∈ h. this inequality follows from jensen’s inequality for the convex function f(t) = |t| defined on a closed interval containing the spectrum of a. if {ei}i∈i is an orthonormal basis of h, then |tr(pa)| = ∣∣∣∣∣∑ i∈i ⟨ ap1/2ei,p 1/2ei ⟩∣∣∣∣∣ ≤ ∑ i∈i ∣∣∣⟨ap1/2ei,p1/2ei⟩∣∣∣ ≤ ∑ i∈i ⟨ |a|p1/2ei,p1/2ei ⟩ = tr(p |a|) , (2.16) for any a a selfadjoint operator and p ∈ b1(h) with p ≥ 0. for the theory of trace functionals and their applications the reader is referred to [31]. for some classical trace inequalities see [9], [11], [26] and [35], which are continuations of the work of bellman [7]. for related works the reader can refer to [1], [8], [9], [20], [23], [24], [25], [29] and [32]. 3. trace inequalities we have the following representation result: theorem 6. let f(z) := ∑∞ n=0 anz n be a power series with complex coefficients and convergent on the open disk d(0,r), r > 0. if t,v ∈ b1(h) are such that tr(|t |),tr(|v |) < r, then f(v ),f(t),f ′ ( (1−t)t +tv ) ∈ b1(h) for any t ∈ [0,1] and tr [f(v )] − tr [f(t)] = ∫ 1 0 tr [ (v − t)f ′ ( (1 − t)t + tv )] dt . (3.1) proof. we use the identity (see for instance [6, p. 254]) an − bn = n−1∑ j=0 an−1−j(a − b)bj (3.2) that holds for any a,b ∈ b(h) and n ≥ 1. 34 s.s. dragomir for t,v ∈ b(h) we consider the function φ : [0,1] → b(h) defined by φ(t) = [(1 − t)t + tv ]n. for t ∈ (0,1) and ε ̸= 0 with t + ε ∈ (0,1) we have from (3.2) that φ(t + ε) −φ(t) = [(1 − t − ε)t + (t + ε)v ]n − [(1 − t)t + tv ]n = ε n−1∑ j=0 [(1 − t − ε)t + (t + ε)v ]n−1−j (v − t) [(1 − t)t + tv ]j . dividing with ε ̸= 0 and taking the limit over ε → 0 we have in the norm topology of b that φ′(t) = lim ε→0 1 ε [φ(t + ε) − φ(t)] (3.3) = n−1∑ j=0 [(1 − t)t + tv ]n−1−j (v − t) [(1 − t)t + tv ]j . integrating on [0,1] we get from (3.3) that ∫ 1 0 φ′(t) dt = n−1∑ j=0 ∫ 1 0 [(1 − t)t + tv ]n−1−j (v − t) [(1 − t)t + tv ]j dt , and since ∫ 1 0 φ′(t) dt = φ(1) − φ(0) = v n − tn, then we get the following equality of interest in itself v n − tn = n−1∑ j=0 ∫ 1 0 [(1 − t)t + tv ]n−1−j (v − t) [(1 − t)t + tv ]j dt (3.4) for any t,v ∈ b(h) and n ≥ 1. trace inequalities of lipschitz type 35 if t,v ∈ b1(h) and we take the trace in (3.4) we get tr(v n) − tr(tn) = n−1∑ j=0 ∫ 1 0 tr ( [(1 − t)t + tv ]n−1−j (v − t) [(1 − t)t + tv ]j ) dt = n−1∑ j=0 ∫ 1 0 tr ( [(1 − t)t + tv ]n−1 (v − t) ) dt (3.5) = n ∫ 1 0 tr ( [(1 − t)t + tv ]n−1 (v − t) ) dt = n ∫ 1 0 tr ( (v − t) [(1 − t)t + tv ]n−1 ) dt for any n ≥ 1. let m ≥ 1. then by (3.5) we have have tr ( m∑ n=0 anv n ) − tr ( m∑ n=0 ant n ) = m∑ n=0 an [ tr ( v n ) − tr ( tn )] = m∑ n=1 an [ tr ( v n ) − tr ( tn )] (3.6) = m∑ n=1 nan ∫ 1 0 tr ( (v − t) [(1 − t)t + tv ]n−1 ) dt = ∫ 1 0 tr ( (v − t) m∑ n=1 nan [(1 − t)t + tv ]n−1 ) dt for any t,v ∈ b1(h). since tr(|t |),tr(|v |) < r with t,v ∈ b1(h) then the series ∑∞ n=0 anv n,∑∞ n=0 ant n and ∑∞ n=1 nan [(1 − t)t + tv ] n−1 are convergent in b1(h) and ∞∑ n=0 anv n = f(v ) , ∞∑ n=0 ant n = f(t) and ∞∑ n=1 nan (1 − t)t + tv ]n−1 = f ′ ( (1 − t)t + tv ) 36 s.s. dragomir where t ∈ [0,1]. moreover, we have f(v ) , f(t) , f ′ ( (1 − t)t + tv ) ∈ b1(h) for any t ∈ [0,1]. by taking the limit over m → ∞ in (3.6) we get the desired result (3.1). in addition to the power identity (3.5), for any t,v ∈ b1(h) we have other equalities as follows tr [exp(v )] − tr [exp(t)] = ∫ 1 0 tr ( (v − t) exp((1 − t)t + tv ) ) dt , (3.7) tr [sin(v )] − tr [sin(t)] = ∫ 1 0 tr ( (v − t) cos((1 − t)t + tv ) ) dt , (3.8) tr [sinh(v )] − tr [sinh(t)] = ∫ 1 0 tr ( (v − t) cosh((1 − t)t + tv ) ) dt . (3.9) if t,v ∈ b1(h) with tr(|t |),tr(|v |) < 1 then tr [ (1h − v )−1 ] − tr [ (1h − t)−1 ] (3.10) = ∫ 1 0 tr ( (v − t)(1h − (1 − t)t − tv )−2 ) dt , and tr [ ln(1h − v )−1 ] s − tr [ ln(1h − t)−1 ] (3.11) = ∫ 1 0 tr ( (v − t)(1h − (1 − t)t − tv )−1 ) dt . we have the following result: corollary 1. with the assumptions in theorem 6 we have the inequalities trace inequalities of lipschitz type 37 ∣∣tr [f(v )] − tr [f(t)]∣∣ ≤ min{∥v − t∥∫ 1 0 ∥∥f ′((1 − t)t + tv )∥∥ 1 dt , (3.12) ∥v − t∥1 ∫ 1 0 ∥∥f ′((1 − t)t + tv )∥∥dt} ≤ min { ∥v − t∥ ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥1 ) dt, ∥v − t∥1 ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥ ) dt } , where ∥·∥ is the operator norm and ∥·∥1 is the 1-norm introduced for trace class operators. proof. from (3.1), we have by taking the modulus ∣∣tr [f(v )] − tr [f(t)] ∣∣ ≤ ∫ 1 0 ∣∣tr((v − t)f ′((1 − t)t + tv ))∣∣dt . (3.13) utilising the inequality (2.11) twice, for any t ∈ [0,1] we get∣∣tr((v − t)f ′((1 − t)t + tv ))∣∣ ≤ ∥v − t∥∥∥f ′((1 − t)t + tv )∥∥ 1 ,∣∣tr((v − t)f ′((1 − t)t + tv ))∣∣ ≤ ∥v − t∥1 ∥∥f ′((1 − t)t + tv )∥∥ . by integrating these inequalities, we get the first part of (3.12). we have, by the use of ∥·∥1 properties that ∥∥f ′((1 − t)t + tv )∥∥ 1 = ∥∥∥∥∥ ∞∑ n=1 nan [(1 − t)t + tv ]n−1 ∥∥∥∥∥ 1 ≤ ∞∑ n=1 n |an| ∥∥∥[(1 − t)t + tv ]n−1∥∥∥ 1 ≤ ∞∑ n=1 n |an| ∥∥(1 − t)t + tv ∥∥n−1 1 = f ′a ( ∥(1 − t)t + tv ∥1 ) for any t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r. 38 s.s. dragomir this proves the first part of the second inequality. since ∥x∥ ≤ ∥x∥1 for any x ∈ b1(h) then ∥(1 − t)t + tv ∥ < r for any t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r which shows that f ′ a ( ∥(1 − t)t + tv ∥ ) is well defined. the second part of the second inequality follows in a similar way and the details are omitted. remark 1. we observe that f ′a is monotonic nondecreasing and convex on the interval [0,r) and since the function ψ(t) := ∥(1 − t)t + tv ∥ is convex on [0,1] we have that f ′a ◦ ψ is also convex on [0,1]. utilising the hermitehadamard inequality for convex functions (see for instance [16, p. 2]) we have the sequence of inequalities∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥ ) dt ≤ 1 2 [ f ′a (∥∥∥∥t + v2 ∥∥∥∥ ) + f ′a(∥t∥) + f ′a(∥v ∥) 2 ] ≤ 1 2 [ f ′a(∥t∥) + f ′ a(∥v ∥) ] ≤ max { f ′a(∥t∥),f ′ a(∥v ∥) } . (3.14) we also have∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥ ) dt ≤ ∫ 1 0 f ′a ( (1 − t) ∥t∥ + t∥v ∥ ) dt ≤ 1 2 [ f ′a ( ∥t∥ + ∥v ∥ 2 ) + f ′a(∥t∥) + f ′a(∥v ∥) 2 ] ≤ 1 2 [ f ′a(∥t∥) + f ′ a(∥v ∥) ] ≤ max { f ′a(∥t∥),f ′ a(∥v ∥) } . (3.15) we observe that if ∥v ∥ ̸= ∥t∥, then by the change of variable s = (1−t) ∥t∥+ t∥v ∥ we have ∫ 1 0 f ′a ( (1 − t) ∥t∥ + t∥v ∥ ) dt = 1 ∥v ∥ − ∥t∥ ∫ ∥v ∥ ∥t∥ f ′a(s) ds = fa(∥v ∥) − fa(∥t∥) ∥v ∥ − ∥t∥ . trace inequalities of lipschitz type 39 if ∥v ∥ = ∥t∥, then ∫ 1 0 f ′a ( (1 − t) ∥t∥ + t∥v ∥ ) dt = f ′a(∥t∥) . utilising these observations we then get the following divided difference inequality for t ̸= v ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥ ) dt ≤   fa(∥v ∥)−fa(∥t∥) ∥v ∥−∥t∥ if ∥v ∥ ̸= ∥t∥ , f ′a(∥t∥) if ∥v ∥ = ∥t∥ . (3.16) similar comments apply for the 1-norm ∥·∥1 when t,v ∈ b1(h). if we use the first part in the inequalities (3.12) and the above remarks, then we get the following string of inequalities ∣∣tr [f(v )] − tr [f(t)]∣∣ ≤ ∥v − t∥∫ 1 0 ∥∥f ′((1 − t)t + tv )∥∥ 1 dt ≤ ∥v − t∥ ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥1 ) dt (3.17) ≤ ∥v − t∥ ×   1 2 [ f ′a (∥∥t+v 2 ∥∥ 1 ) + f′a(∥t∥1)+f ′ a(∥v ∥1) 2 ] ,  fa(∥v ∥1)−fa(∥t∥1) ∥v ∥1−∥t∥1 if ∥v ∥1 ̸= ∥t∥1 , f ′a(∥t∥1) if ∥v ∥1 = ∥t∥1 , ≤ 1 2 ∥v − t∥ [ f ′a(∥t∥1) + f ′ a(∥v ∥1) ] ≤ ∥v − t∥ max { f ′a(∥t∥1),f ′ a(∥v ∥1) } provided t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r. if ∥t∥1 ,∥v ∥1 ≤ m < r, then we have from (3.17) the simple inequality∣∣tr [f(v )] − tr [f(t)]∣∣ ≤ ∥v − t∥f ′a(m) . a similar sequence of inequalities can also be stated by swapping the norm ∥·∥ with ∥·∥1 in (3.17). we omit the details. 40 s.s. dragomir if we use the inequality (3.17) for the exponential function, then for any t,v ∈ b1(h) we have the inequalities ∣∣tr [exp(v )] − tr [exp(t)]∣∣ ≤ ∥v − t∥∫ 1 0 ∥∥exp((1 − t)t + tv )∥∥ 1 dt ≤ ∥v − t∥ ∫ 1 0 exp ( ∥(1 − t)t + tv ∥1 ) dt (3.18) ≤ ∥v − t∥ ×   1 2 [ exp (∥∥t+v 2 ∥∥ 1 ) + exp(∥t∥1)+exp(∥v ∥1) 2 ] ,  exp(∥v ∥1)−exp(∥t∥1) ∥v ∥1−∥t∥1 if ∥v ∥1 ̸= ∥t∥1 , exp(∥t∥1) if ∥v ∥1 = ∥t∥1 , ≤ 1 2 ∥v − t∥ [ exp(∥t∥1) + exp(∥v ∥1) ] ≤ ∥v − t∥ max { exp(∥t∥1),exp(∥v ∥1) } . if ∥t∥1 ,∥v ∥1 < 1, then we have the inequalities∣∣tr[ ln(1h − v )−1] − tr[ln(1h − t)−1]∣∣ (3.19) ≤ ∥v − t∥ ∫ 1 0 ∥∥(1h − (1 − t)t − tv )−1∥∥1 dt ≤ ∥v − t∥ ∫ 1 0 (1 − ∥(1 − t)t + tv ∥1) −1dt ≤ ∥v − t∥ ×   1 2 [( 1 − ∥∥t+v 2 ∥∥ 1 )−1 + (1−∥t∥1) −1+(1−∥v ∥1) −1 2 ] ,  ln(1−∥v ∥1) −1−ln(1−∥t∥1) −1 ∥v ∥1−∥t∥1 if ∥v ∥1 ̸= ∥t∥1 , (1 − ∥t∥1) −1 if ∥v ∥1 = ∥t∥1 , ≤ 1 2 ∥v − t∥ [ (1 − ∥t∥1) −1 + (1 − ∥v ∥1) −1] ≤ ∥v − t∥ max { (1 − ∥t∥1) −1,(1 − ∥v ∥1) −1} . the following result for the hilbert-schmidt norm ∥·∥2 also holds: theorem 7. let f(z) := ∑∞ n=0 anz n be a power series with complex coefficients and convergent on the open disk d(0,r), r > 0. if t,v ∈ b2(h) are trace inequalities of lipschitz type 41 such that tr(|t |2),tr(|v |2) < r2, then f(v ),f(t),f ′((1 − t)t + tv ) ∈ b2(h) for any t ∈ [0,1] and tr [f(v )] − tr [f(t)] = ∫ 1 0 tr((v − t)f ′ ( (1 − t)t + tv ) ) dt . (3.20) moreover, we have the inequalities∣∣tr [f(v )] − tr [f(t)]∣∣ ≤ ∥v − t∥2 ∫ 1 0 ∥∥f ′((1 − t)t + tv )∥∥ 2 dt , ≤ ∥v − t∥2 ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥2 ) dt (3.21) ≤ ∥v − t∥2 ×   1 2 [ f ′a (∥∥t+v 2 ∥∥ 2 ) + f′a(∥t∥2)+f ′ a(∥v ∥2) 2 ] ,  fa(∥v ∥2)−fa(∥t∥2) ∥v ∥2−∥t∥2 if ∥v ∥2 ̸= ∥t∥2 , f ′a(∥t∥2) if ∥v ∥2 = ∥t∥2 , ≤ 1 2 ∥v − t∥2 [ f ′a(∥t∥2) + f ′ a(∥v ∥2) ] ≤ ∥v − t∥2 max { f ′a(∥t∥2),f ′ a(∥v ∥2) } . proof. the proof of the first part of the theorem follows in a similar manner to the one from theorem 6. taking the modulus in (3.20) and using the schwarz inequality for trace (2.13) we have |tr [f(v )] − tr [f(t)]| ≤ ∫ 1 0 ∣∣tr((v − t)f ′((1 − t)t + tv ))∣∣dt (3.22) ≤ ∫ 1 0 ∥v − t∥2 ∥∥f ′((1 − t)t + tv )∥∥ 2 dt . the rest follows in a similar manner as in the case of 1-norm and the details are omitted. we notice that similar examples to (3.18) and (3.19) may be stated where both norms ∥·∥ and ∥·∥1 are replaced by ∥·∥2. we also observe that, if t,v ∈ b2(h) with ∥t∥2 ,∥v ∥2 ≤ k < r, then we have from (3.17) the simple inequality∣∣tr [f(v )] − tr [f(t)]∣∣ ≤ ∥v − t∥2 f ′a(k) . 42 s.s. dragomir 4. norm inequalities we have the following norm inequalities: theorem 8. let f(z) := ∑∞ n=0 anz n be a power series with complex coefficients and convergent on the open disk d(0,r), r > 0. (i) if t,v ∈ b1(h) are such that tr(|t |),tr(|v |) < r, then we have the norm inequalities ∥f(v ) − f(t)∥1 ≤   ∥v − t∥1 ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥ ) dt , ∥v − t∥ ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥1 ) dt . (4.1) (ii) if t,v ∈ b2(h) are such that tr(|t |2),tr(|v |2) < r2, then we also have the norm inequalities ∥f(v ) − f(t)∥2 ≤   ∥v − t∥2 ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥ ) dt , ∥v − t∥ ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥2 ) dt . (4.2) proof. we use the equality v n − tn = n−1∑ j=0 ∫ 1 0 [(1 − t)t + tv ]n−1−j (v − t) [(1 − t)t + tv ]j dt , (4.3) for any t,v ∈ b(h) and n ≥ 1. (i) if t,v ∈ b1(h) are such that tr(|t |),tr(|v |) < r, then by taking the ∥·∥1 norm and using its properties, for any n ≥ 1 we have successively ∥v n − tn∥1 ≤ n−1∑ j=0 ∫ 1 0 ∥∥∥[(1 − t)t + tv ]n−1−j (v − t) [(1 − t)t + tv ]j∥∥∥ 1 dt ≤ n−1∑ j=0 ∫ 1 0 ∥∥∥[(1 − t)t + tv ]n−1−j (v − t)∥∥∥ 1 ∥∥∥[(1 − t)t + tv ]j∥∥∥dt ≤ n−1∑ j=0 ∫ 1 0 ∥v − t∥1 ∥∥∥[(1 − t)t + tv ]n−1−j∥∥∥∥∥∥[(1 − t)t + tv ]j∥∥∥dt trace inequalities of lipschitz type 43 ≤ ∥v − t∥1 n−1∑ j=0 ∥(1 − t)t + tv ∥n−1−j ∥(1 − t)t + tv ∥j dt = n∥v − t∥1 ∫ 1 0 ∥(1 − t)t + tv ∥n−1 dt . (4.4) let m ≥ 1. by (4.4) we have ∥∥∥∥∥ m∑ n=0 anv n − m∑ n=0 ant n ∥∥∥∥∥ 1 = ∥∥∥∥∥ m∑ n=1 an(v n − tn) ∥∥∥∥∥ 1 (4.5) ≤ m∑ n=1 |an| ∥v n − tn∥1 ≤ ∥v − t∥1 m∑ n=1 |an|n ∫ 1 0 ∥(1 − t)t + tv ∥n−1 dt = ∥v − t∥1 ∫ 1 0 ( m∑ n=1 n |an| ∥(1 − t)t + tv ∥n−1 ) dt . also, we observe that ∥(1 − t)t + tv ∥ ≤ ∥(1 − t)t + tv ∥1 ≤ (1 − t) ∥t∥1 + t∥v ∥1 < r for any t ∈ [0,1], which implies that the series ∑∞ n=1 n |an| ∥(1 − t)t + tv ∥ n−1 is convergent and ∞∑ n=1 n |an| ∥(1 − t)t + tv ∥n−1 = f ′a ( ∥(1 − t)t + tv ∥ ) for any t ∈ [0,1]. since the series ∑∞ n=0 anv n and ∑∞ n=0 ant n are convergent in( b1(h),∥·∥1 ) , then by letting m → ∞ in the inequality (4.5) we get the first inequality in (4.1). 44 s.s. dragomir for any n ≥ 1 we also have ∥v n − tn∥1 ≤ n−1∑ j=0 ∫ 1 0 ∥∥∥[(1 − t)t + tv ]n−1−j (v − t) [(1 − t)t + tv ]j∥∥∥ 1 dt ≤ n−1∑ j=0 ∫ 1 0 ∥∥∥[(1 − t)t + tv ]n−1−j (v − t)∥∥∥∥∥∥[(1 − t)t + tv ]j∥∥∥ 1 dt ≤ n−1∑ j=0 ∫ 1 0 ∥v − t∥ ∥∥∥[(1 − t)t + tv ]n−1−j∥∥∥∥∥∥[(1 − t)t + tv ]j∥∥∥ 1 dt ≤ ∥v − t∥ n−1∑ j=0 ∥(1 − t)t + tv ∥n−1−j ∥(1 − t)t + tv ∥j1 dt ≤ ∥v − t∥ n−1∑ j=0 ∥(1 − t)t + tv ∥n−1−j1 ∥(1 − t)t + tv ∥ j 1 dt = n∥v − t∥ ∫ 1 0 ∥(1 − t)t + tv ∥n−11 dt , which by a similar argument produces the second inequality in (4.1). (ii) follows in a similar way by utilizing the inequality ∥ta∥2 ≤ ∥t∥ ∥a∥2 that holds for t ∈ b(h) and a ∈ b2(h). the details are omitted. remark 2. from the first inequality in (4.1) we have the sequence of inequalities ∥f(v ) − f(t)∥1 ≤ ∥v − t∥1 ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥ ) dt (4.6) ≤ ∥v − t∥1 ×   1 2 [ f ′a (∥∥t+v 2 ∥∥) + f′a(∥t∥)+f′a(∥v ∥) 2 ] ,  fa(∥v ∥)−fa(∥t∥) ∥v ∥−∥t∥ if ∥v ∥ ̸= ∥t∥ , f ′a(∥t∥) if ∥v ∥ = ∥t∥ , ≤ 1 2 ∥v − t∥1 [ f ′a(∥t∥) + f ′ a(∥v ∥) ] ≤ ∥v − t∥1 max { f ′a(∥t∥),f ′ a(∥v ∥) } trace inequalities of lipschitz type 45 for t,v ∈ b1(h) such that tr(|t |),tr(|v |) < r and a similar result by swapping in the right hand side of (4.6) the norm ∥·∥ with ∥·∥1. in particular, if tr(|t |),tr(|v |) ≤ m < r, then we have the simpler inequality ∥f(v ) − f(t)∥1 ≤ f ′ a(m) ∥v − t∥1 . (4.7) if t,v ∈ b2(h) are such that tr(|t |2),tr(|v |2) < r2, then we have the norm inequalities ∥f(v ) − f(t)∥2 ≤ ∥v − t∥2 ∫ 1 0 f ′a ( ∥(1 − t)t + tv ∥ ) dt (4.8) ≤ ∥v − t∥2 ×   1 2 [ f ′a (∥∥t+v 2 ∥∥) + f′a(∥t∥)+f′a(∥v ∥) 2 ] ,  fa(∥v ∥)−fa(∥t∥) ∥v ∥−∥t∥ if ∥v ∥ ̸= ∥t∥ , f ′a(∥t∥) if ∥v ∥ = ∥t∥ , ≤ 1 2 ∥v − t∥2 [ f ′a(∥t∥) + f ′ a(∥v ∥) ] ≤ ∥v − t∥2 max { f ′a(∥t∥),f ′ a(∥v ∥) } and a similar result by swapping in the right hand side of (4.6) the norm ∥·∥ with ∥·∥2. in particular, if tr(|t | 2 ),tr(|v |2) ≤ k2 < r2, then we have the simpler inequality ∥f(v ) − f(t)∥2 ≤ f ′ a(k) ∥v − t∥2 . (4.9) 5. applications for jensen’s difference we have the following representation: lemma 1. let f(z) := ∑∞ n=0 anz n be a power series with complex coefficients and convergent on the open disk d(0,r), r > 0. if either t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r, or t,v ∈ b2(h) with ∥t∥2 ,∥v ∥2 < r then f(v ), f(t), f ( v +t 2 ) ∈ b1(h) or f(v ), f(t), f ( v +t 2 ) ∈ b2(h), respectively and tr [f(v )] + tr [f(t)] 2 − tr [ f ( v + t 2 )] (5.1) = 1 4 ∫ 1 0 tr ( (v − t) [ f′ ( (1 − t) v + t 2 + tv ) − f′ ( (1 − t) v + t 2 + tt )]) dt . 46 s.s. dragomir proof. the first part of the theorem follows from theorem 6. from the identity (3.1), for t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r we have tr [f(v )] − tr [ f ( v + t 2 )] (5.2) = ∫ 1 0 tr [( v − v + t 2 ) f ′ ( (1 − t) v + t 2 + tv )] dt = 1 2 ∫ 1 0 tr [ (v − t)f ′ ( (1 − t) v + t 2 + tv )] dt and tr [f(t)] − tr [ f ( v + t 2 )] (5.3) = ∫ 1 0 tr [( t − v + t 2 ) f ′ ( (1 − t) v + t 2 + tt )] dt = 1 2 ∫ 1 0 tr [ (t − v )f ′ ( (1 − t) v + t 2 + tt )] dt = − 1 2 ∫ 1 0 tr [ (v − t)f ′ ( (1 − t) v + t 2 + tt )] dt . if we add the above inequalities (5.2) and (5.3) and divide by 2 we get the desired result (5.1). theorem 9. let f(z) := ∑∞ n=0 anz n be a power series with complex coefficients and convergent on the open disk d(0,r), r > 0. if t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r, then∣∣∣∣tr [f(v )] + tr [f(t)]2 − tr [ f ( v + t 2 )]∣∣∣∣ (5.4) ≤ 1 2 ∥v − t∥2 ∫ 1 0 ∣∣∣∣t − 12 ∣∣∣∣f ′′a(∥(1 − t)t + tv ∥1 )dt ≤ 1 24 ∥v − t∥2 [ f ′′a (∥∥∥∥v + t2 ∥∥∥∥ 1 ) + f ′′a (∥v ∥1) + f ′′ a (∥t∥1) 2 ] ≤ 1 12 ∥v − t∥2 [ f ′′a (∥v ∥1) + f ′′ a (∥t∥1) ] ≤ 1 6 ∥v − t∥2 max { f ′′a (∥v ∥1),f ′′ a (∥t∥1) } . trace inequalities of lipschitz type 47 proof. taking the modulus in (5.1), for t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r we have∣∣∣∣tr [f(v )] + tr [f(t)]2 − tr [ f ( v + t 2 )]∣∣∣∣ (5.5) ≤ 1 4 ∫ 1 0 ∣∣∣∣tr ( (v − t) [ f′ ( (1 − t) v + t 2 + tv ) − f′ ( (1 − t) v + t 2 + tt )])∣∣∣∣dt . using the properties of trace, for t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r and t ∈ [0,1] we have∣∣∣∣tr ( (v − t) [ f ′ ( (1 − t) v + t 2 + tv ) − f ′ ( (1 − t) v + t 2 + tt )])∣∣∣∣ (5.6) ≤ ∥v − t∥ ∥∥∥∥ [ f ′ ( (1 − t) v + t 2 + tv ) − f ′ ( (1 − t) v + t 2 + tt )]∥∥∥∥ 1 . from (4.1), for a,b ∈ b1(h) with ∥a∥1 ,∥b∥1 < r we have ∥f(a) − f(b)∥1 ≤ ∥a − b∥ ∫ 1 0 f ′a ( ∥(1 − t)b + ta∥1 ) dt (5.7) ≤ 1 2 ∥a − b∥ [ f ′a (∥∥∥∥a + b2 ∥∥∥∥ 1 ) + f ′a(∥a∥1) + f ′ a(∥b∥1) 2 ] ≤ 1 2 ∥a − b∥ [ f ′a(∥a∥1) + f ′ a(∥b∥1) ] ≤ ∥a − b∥ max { f ′a(∥a∥1),f ′ a(∥b∥1) } . applying the second and third inequalities in (5.7) for f ′, a = (1−t)v +t 2 +tv and b = (1 − t)v +t 2 + tt we get∥∥∥∥ [ f ′ ( (1 − t) v + t 2 + tv ) − f ′ ( (1 − t) v + t 2 + tt )]∥∥∥∥ 1 (5.8) ≤ 1 2 t∥v − t∥ [ f ′′a (∥∥∥∥v + t2 ∥∥∥∥ 1 ) + f ′′a (∥∥(1 − t)v +t 2 + tv ∥∥ 1 ) + f ′′a (∥∥(1 − t)v +t 2 + tt ∥∥ 1 ) 2 ] ≤ 1 2 t∥v − t∥ × [ f ′′a (∥∥∥∥(1 − t)v + t2 + tv ∥∥∥∥ 1 ) + f ′′a (∥∥∥∥(1 − t)v + t2 + tt ∥∥∥∥ 1 )] 48 s.s. dragomir for t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r and t ∈ [0,1]. since f ′′a is convex and monotonic nondecreasing and ∥·∥1 is convex, then f ′′a (∥∥(1 − t)v +t 2 + tv ∥∥ 1 ) + f ′′a (∥∥(1 − t)v +t 2 + tt ∥∥ 1 ) 2 (5.9) ≤ (1 − t)f ′′a (∥∥∥∥v + t2 ∥∥∥∥ 1 ) + t f ′′a (∥v ∥1) + f ′′ a (∥t∥1) 2 for t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r and t ∈ [0,1]. from (5.8) and (5.9) we get∥∥∥∥ [ f ′ ( (1 − t) v + t 2 + tv ) − f ′ ( (1 − t) v + t 2 + tt )]∥∥∥∥ 1 (5.10) ≤ 1 2 t∥v − t∥ × [ f ′′a (∥∥∥∥(1 − t)v + t2 + tv ∥∥∥∥ 1 ) + f ′′a (∥∥∥∥(1 − t)v + t2 + tt ∥∥∥∥ 1 )] ≤ t∥v − t∥ [ (1 − t)f ′′a (∥∥∥∥v + t2 ∥∥∥∥ 1 ) + t f ′′a (∥v ∥1) + f ′′ a (∥t∥1) 2 ] for t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r and t ∈ [0,1]. integrating (5.10) over t on [0,1] we get∫ 1 0 ∥∥∥∥ [ f ′ ( (1 − t) v + t 2 + tv ) − f ′ ( (1 − t) v + t 2 + tt )]∥∥∥∥ 1 dt ≤ 1 2 ∥v − t∥ × [∫ 1 0 tf ′′a (∥∥∥∥(1 − t)v + t2 + tv ∥∥∥∥ 1 ) dt + ∫ 1 0 tf ′′a (∥∥∥∥(1 − t)v + t2 + tt ∥∥∥∥ 1 dt )] ≤ ∥v − t∥ [ f ′′a (∥∥∥∥v + t2 ∥∥∥∥ 1 )∫ 1 0 t(1 − t) dt + f ′′a (∥v ∥1) + f ′′ a (∥t∥1) 2 ∫ 1 0 t2 dt ] = 1 6 ∥v − t∥ [ f ′′a (∥∥∥∥v + t2 ∥∥∥∥ 1 ) + f ′′a (∥v ∥1) + f ′′ a (∥t∥1) 2 ] , trace inequalities of lipschitz type 49 which together with (5.5) and (5.6) produce the inequality∣∣∣∣tr [f(v )] + tr [f(t)]2 − tr [ f ( v + t 2 )]∣∣∣∣ (5.11) ≤ 1 8 ∥v − t∥2 × [∫ 1 0 tf ′′a (∥∥∥∥(1 − t)v + t2 + tv ∥∥∥∥ 1 ) dt + ∫ 1 0 tf ′′a (∥∥∥∥(1 − t)v + t2 + tt ∥∥∥∥ 1 dt )] ≤ 1 24 ∥v − t∥2 [ f ′′a (∥∥∥∥v + t2 ∥∥∥∥ 1 ) + f ′′a (∥v ∥1) + f ′′ a (∥t∥1) 2 ] . now, observe that∫ 1 0 tf ′′a (∥∥∥∥(1 − t)v + t2 + tv ∥∥∥∥ 1 ) dt (5.12) = ∫ 1 0 tf ′′a (∥∥∥∥1 − t2 t + 1 + t2 v ∥∥∥∥ 1 ) dt , ∫ 1 0 tf ′′a (∥∥∥∥(1 − t)v + t2 + tt ∥∥∥∥ 1 ) dt (5.13) = ∫ 1 0 tf ′′a (∥∥∥∥1 − t2 v + 1 + t2 t ∥∥∥∥ 1 ) dt . using the change of variable u = 1+t 2 , then we get∫ 1 0 tf ′′a (∥∥∥∥1 − t2 t + 1 + t2 v ∥∥∥∥ 1 ) dt (5.14) = 2 ∫ 1 1 2 (2u − 1)f ′′a ( ∥(1 − u)t + uv ∥ ) du. also, by changing the variable v = 1−t 2 , we get∫ 1 0 tf ′′a (∥∥∥∥1 − t2 v + 1 + t2 t ∥∥∥∥ 1 ) dt (5.15) = 2 ∫ 1 2 0 (1 − 2v)f ′′a ( ∥(1 − v)t + vv ∥ ) dv . 50 s.s. dragomir utilising the equalities (5.12)-(5.15) we obtain∫ 1 0 tf ′′a (∥∥∥∥(1 − t)v + t2 + tv ∥∥∥∥ 1 ) dt + ∫ 1 0 tf ′′a (∥∥∥∥(1 − t)v + t2 + tt ∥∥∥∥ 1 ) dt = 2 ∫ 1 1 2 (2t − 1)f ′′a ( ∥(1 − t)t + tv ∥ ) dt + 2 ∫ 1 2 0 (1 − 2t)f ′′a ( ∥(1 − t)t + tv ∥ ) dt = 2 ∫ 1 0 |2t − 1|f ′′a ( ∥(1 − t)t + tv ∥ ) dt = 4 ∫ 1 0 ∣∣∣∣t − 12 ∣∣∣∣f ′′a(∥(1 − t)t + tv ∥)dt for t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 < r. making use of (5.11) we deduce the first two inequalities in (5.4). the rest is obvious. corollary 2. let f(z) := ∑∞ n=0 anz n be a power series with complex coefficients and convergent on the open disk d(0,r), r > 0. if t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 ≤ m < r, then∣∣∣∣tr [f(v )] + tr [f(t)]2 − tr [ f ( v + t 2 )]∣∣∣∣ ≤ 18 ∥v − t∥2 f ′′a (m) . (5.16) the constant 1 8 is best possible in (5.16). proof. from the first inequality in (5.4) we have∣∣∣∣tr [f(v )] + tr [f(t)]2 − tr [ f ( v + t 2 )]∣∣∣∣ ≤ 1 2 ∥v − t∥2 ∫ 1 0 ∣∣∣∣t − 12 ∣∣∣∣f ′′a(∥(1 − t)t + tv ∥1 )dt ≤ 1 2 ∥v − t∥2 f ′′a (m) ∫ 1 0 ∣∣∣∣t − 12 ∣∣∣∣dt = 18 ∥v − t∥2 f ′′a (m) for t,v ∈ b1(h) with ∥t∥1 ,∥v ∥1 ≤ m < r, and the inequality is proved. if we consider the scalar case and take f(z) = z2, v = a, t = b with a,b ∈ r then we get in both sides of (5.16) the same quantity 1 4 (b − a)2. trace inequalities of lipschitz type 51 remark 3. a similar result holds by swapping the norm ∥·∥ with ∥·∥1 in the right hand side of (5.4). the case of hilbert-schmidt norm may also be stated, however the details are not presented here. if we write the inequality (5.4) for the exponential function, then we get∣∣∣∣tr [exp(v )] + tr [exp(t)]2 − tr [ exp ( v + t 2 )]∣∣∣∣ (5.17) ≤ 1 2 ∥v − t∥2 ∫ 1 0 ∣∣∣∣t − 12 ∣∣∣∣exp(∥(1 − t)t + tv ∥1 )dt ≤ 1 24 ∥v − t∥2 [ exp (∥∥∥∥v + t2 ∥∥∥∥ 1 ) + exp(∥v ∥1) + exp(∥t∥1) 2 ] ≤ 1 12 ∥v − t∥2 [ exp(∥v ∥1) + exp(∥t∥1) ] ≤ 1 6 ∥v − t∥2 max { exp(∥v ∥1),exp(∥t∥1) } for any for t,v ∈ b1(h). if t,v ∈ b1(h) with ∥v ∥1 ,∥t∥1 ≤ m, then∣∣∣∣tr [exp(v )] + tr [exp(t)]2 − tr [ exp ( v + t 2 )]∣∣∣∣ (5.18) ≤ 1 8 ∥v − t∥2 exp(m) . if we write the inequality (5.4) for the function f(z) = (1 − z)−1, then we get∣∣∣∣∣tr [ (1h − v )−1 ] + tr [ (1h − t)−1 ] 2 − tr [( 1h − v + t 2 )−1]∣∣∣∣∣ (5.19) ≤ ∥v − t∥2 ∫ 1 0 ∣∣∣∣t − 12 ∣∣∣∣(1 − ∥(1 − t)t + tv ∥1 )−3 dt ≤ 1 12 ∥v − t∥2 × [( 1 − ∥∥∥∥v + t2 ∥∥∥∥ 1 )−3 + (1 − ∥v ∥1) −3 + (1 − ∥t∥1) −3 2 ] ≤ 1 6 ∥v − t∥2 [ (1 − ∥v ∥1) −3 + (1 − ∥t∥1) −3] ≤ 1 3 ∥v − t∥2 max { (1 − ∥v ∥1) −3,(1 − ∥t∥1) −3} , 52 s.s. dragomir for any for t,v ∈ b1(h) with ∥v ∥1 ,∥t∥1 < 1. moreover, if ∥v ∥1 ,∥t∥1 ≤ m < 1, then∣∣∣∣∣tr [ (1h − v )−1 ] + tr [ (1h − t)−1 ] 2 − tr [( 1h − v + t 2 )−1]∣∣∣∣∣ (5.20) ≤ 1 4 ∥v − t∥2 (1 − m)−3. the interested reader may choose other examples of power series to get similar results. however, the details are not presented here. references [1] t. ando, matrix young inequalities, in “ operator theory in function spaces and banach lattices ”, oper. theory adv. appl., 75, birkhäuser, basel, 1995, 33 – 38. 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[36] c.-j. zhao, w.-s. cheung, on multivariate grüss inequalities, j. inequal. appl. 2008, art. id 249438, 8 pp. e extracta mathematicae vol. 32, núm. 2, 213 – 238 (2017) invariant control systems on lie groups: a short survey rory biggs, claudiu c. remsing department of mathematics and applied mathematics, university of pretoria, 0002 pretoria, south africa rory.biggs@up.ac.za department of mathematics, rhodes university, 6140 grahamstown, south africa c.c.remsing@ru.ac.za presented by dikran dikranjan received march 6, 2016 abstract: this is a short survey of our recent research on invariant control systems (and their associated optimal control problems). we are primarily concerned with equivalence and classification, especially in three dimensions. key words: invariant control affine systems, detached feedback equivalence, optimal control. ams subject class. (2010): 93b27, 22e60, 49j15, 53c17. 1. introduction geometric control theory began in the late 1960s with the study of (nonlinear) control systems by using concepts and methods from differential geometry (cf. [9,50,73]). a smooth control system may be viewed as a family of vector fields (or dynamical systems) on a manifold, smoothly parametrized by a set of controls. an integral curve corresponding to some admissible control function (from some time interval to the set of controls) is called a trajectory of the system. the first basic question one asks of a control system is whether or not any two points can be connected by a trajectory: this is known as the controllability problem. once one has established that two points can be connected by a trajectory, one may wish to find a trajectory that minimizes some (practical) cost function: this is known as the optimality problem. the research leading to these results has received funding from the european union’s seventh framework programme (fp7/2007-2013) under grant agreement no. 317721. also, the first author would like to acknowledge the financial support of the claude leon foundation towards this research. 213 214 r. biggs, c. c. remsing a significant subclass of control systems rich in symmetry are those evolving on lie groups and invariant under left translations; for such a system the left translation of any trajectory is a trajectory. this class of systems was first considered in 1972 by brockett [35] and by jurdjevic and sussmann [53]; it forms a natural geometric framework for various (variational) problems in mathematical physics, mechanics, elasticity, and dynamical systems (cf. [9,33,50,51]). in the last few decades substantial work on applied nonlinear control has drawn attention to invariant control affine systems evolving on matrix lie groups of low dimension (see, e.g., [52,67,69,70] and the references therein). this paper serves as a short survey of our recent research on (the equivalence of) left-invariant control systems and the associated optimal control problems. ideas and key results from several papers published over the last couple of years are reexamined and restructured; some elements are also reinterpreted. the first aspect we address is the equivalence of control systems. both state space and (detached) feedback equivalence are characterized in simple algebraic terms. the classification problem in three dimensions is revisited. the second aspect we address is the equivalence of invariant optimal control problems, or rather, their associated cost-extended systems. one associates to each cost-extended system, via the pontryagin maximum principle, a quadratic hamilton–poisson system on the associated lie–poisson space. equivalence of cost-extended systems implies equivalence of the associated hamilton–poisson systems. additionally, the subclass of drift-free systems with homogeneous cost are reinterpreted as invariant sub-riemannian structures. an extended version of this survey will appear in [32]. throughout, we make use of the classification of three-dimensional lie groups and lie algebras; relevant details are given in the appendix. 2. invariant control systems and their equivalence 2.1. invariant control affine systems. a ℓ-input left-invariant control affine system σ on a (real, finite-dimensional, connected) lie group g consists of a family of left-invariant vector fields ξu on g, affinely parametrized by controls u ∈ rℓ. such a system is written as ġ = ξu(g) = ξ(g,u) = g(a + u1b1 + · · · + uℓbℓ), g ∈ g, u ∈ rℓ. here a,b1, . . . ,bℓ are elements of the lie algebra g with b1, . . . ,bℓ linearly independent. the “product” ga denotes the left translation t1lg · a of invariant control systems on lie groups 215 a ∈ g by the tangent map of lg : g → g, h 7→ gh. (when g is a matrix lie group, this product is simply a matrix multiplication.) note that the dynamics ξ : g×rℓ → tg are invariant under left translations, i.e., ξ (g,u) = g ξ (1,u). σ is completely determined by the specification of its state space g and its parametrization map ξ (1, ·). when g is fixed, we specify σ by simply writing σ : a + u1b1 + · · · + uℓbℓ. the trace γ of a system σ is the affine subspace γ = a + γ0 = a + ⟨b1, . . . ,bℓ⟩ of g. (here γ0 = ⟨b1, . . . ,bℓ⟩ is the subspace of g spanned by b1, . . . ,bℓ.) a system σ is called homogeneous if a ∈ γ0, and inhomogeneous otherwise; σ is said to be drift free if a = 0. also, σ is said to have full rank if its trace generates the whole lie algebra, i.e., lie(γ) = g. the admissible controls are piecewise continuous maps u(·) : [0,t ] → rℓ. a trajectory for an admissible control u(·) is an absolutely continuous curve g(·) : [0,t ] → g such that ġ(t) = g(t) ξ (1,u(t)) for almost every t ∈ [0,t]. we say that a system σ is controllable if for any g0,g1 ∈ g, there exists a trajectory g(·) : [0,t ] → g such that g(0) = g0 and g(t) = g1. if σ is controllable, then it has full rank. for more details about invariant control systems see, e.g., [9,50,53,71]. 2.2. equivalence of systems. the most natural equivalence relation for control systems is equivalence up to coordinate changes in the state space. this is called state space equivalence (see [47]). state space equivalence is well understood. it establishes a one-to-one correspondence between the trajectories of the equivalent systems. however, this equivalence relation is very strong. in the (general) analytic case, krener characterized local state space equivalence in terms of the existence of a linear isomorphism preserving iterated lie brackets of the system’s vector fields ([58], see also [9,72,73]). another fundamental equivalence relation for control systems is that of feedback equivalence. two feedback equivalent control systems have the same set of trajectories (up to a diffeomorphism in the state space) which are parametrized differently by admissible controls. feedback equivalence has been extensively studied in the last few decades (see [68] and the references therein). there are a few basic methods used in the study of feedback equivalence. these methods are based either on (studying invariant properties of) associated distributions or on cartan’s method of equivalence ([41]) or inspired by the hamiltonian formalism ([47]); also, another fruitful approach is closely related to poincaré’s technique for linearization of dynamical systems. 216 r. biggs, c. c. remsing feedback transformations play a crucial role in control theory, particularly in the important problem of feedback linearization ([48]). the study of feedback equivalence of general control systems can be reduced, by a simple trick, to the case of control affine systems ([47]). for a thorough study of the equivalence and classification of (general) control affine systems, see [39]. we consider state space equivalence and feedback equivalence in the context of left-invariant control affine systems ([29], see also [18]). characterizations of state space equivalence and (detached) feedback equivalence are obtained in terms of lie group isomorphisms. furthermore, the classification of systems on the three-dimensional lie groups is treated. 2.2.1. state space equivalence. two systems σ and σ′ are called state space equivalent if there exists a diffeomorphism ϕ : g → g′ such that, for each control value u ∈ rℓ, the vector fields ξu and ξ′u are ϕ-related, i.e., tgϕ · ξ (g,u) = ξ′ (ϕ(g),u) for g ∈ g and u ∈ rℓ. we have the following simple algebraic characterization of this equivalence. theorem 2.1. ([29], see also [58]) two full-rank systems σ and σ′ are state space equivalent if and only if there exists a lie group isomorphism ϕ : g → g′ such that t1ϕ · ξ (1,u) = ξ′ (1,u) for all u ∈ rℓ. proof sketch. suppose that σ and σ′ are state space equivalent. by composition with a left translation, we may assume ϕ(1) = 1. as the elements ξu(1), u ∈ rℓ generate g and the push-forward ϕ∗ξu of the left-invariant vector fields ξu are left invariant, it follows that ϕ is a lie group isomorphism satisfying the requisite property (cf. [18]). conversely, suppose that ϕ : g → g′ is a lie group isomorphism as prescribed. then tgϕ · ξ(g,u) = t1(ϕ ◦ lg) · ξ(1,u) = t1(lϕ(g) ◦ ϕ) · ξ(1,u) = ξ′(ϕ(g),u). remark. if ϕ is defined only between some neighbourhoods of identity of g and g′, then σ and σ′ are said to be locally state space equivalence. a characterization similar to that given in theorem 2.1, in terms of lie algebra automorphisms, holds ([29]). in the case of simply connected lie groups, local and global equivalence are the same (as daut(g) = aut(g)). state space equivalence is quite a strong equivalence relation. hence, there are so many equivalence classes that any general classification appears to be very difficult if not impossible. however, there is a chance for some reasonable classification in low dimensions. we give an example to illustrate this point. invariant control systems on lie groups 217 example 2.1. ([1]) any two-input inhomogeneous full-rank control affine system on the euclidean group se (2) is state space equivalent to exactly one of the following systems σ1,αβγ : αe3 + u1(e1 + γ1e2) + u2(βe2) σ2,αβγ : βe1 + γ1e2 + γ2e3 + u1(αe3) + u2e2 σ3,αβγ : βe1 + γ1e2 + γ2e3 + u1(e2 + γ3e3) + u2(αe3). here α > 0,β ̸= 0 and γ1,γ2,γ3 ∈ r, with different values of these parameters yielding distinct (non-equivalent) class representatives. note. a full classification (under state space equivalence) of systems on se (2) appears in [1], whereas a classification of systems on se (1,1) appears in [11]. for a classification of systems on so (2,1)0, see [30]. 2.2.2. detached feedback equivalence. we specialize feedback equivalence in the context of invariant systems by requiring that the feedback transformations are compatible with the lie group structure (cf. [18]). two systems σ and σ′ are called detached feedback equivalent if there exist diffeomorphisms ϕ : g → g′ and φ : rℓ → rℓ such that, for each control value u ∈ rℓ, the vector fields ξu and ξ′φ(u) are ϕ-related, i.e., tgϕ · ξ (g,u) = ξ′ (ϕ(g),φ(u)) for g ∈ g and u ∈ rℓ. we have the following simple algebraic characterization of this equivalence in terms of the traces γ = im ξ(1, ·) and γ′ = im ξ′(1, ·) of σ and σ′. theorem 2.2. ([29]) two full-rank systems σ and σ′ are detached feedback equivalent if and only if there exists a lie group isomorphism ϕ : g → g′ such that t1ϕ · γ = γ′. proof sketch. suppose σ and σ′ are detached feedback equivalent. by composing ϕ with an appropriate left translation, we may assume ϕ(1) = 1′. hence t1ϕ · ξ(1,u) = ξ′(1′,φ(u)) and so t1ϕ · γ = γ′. moreover, as the elements ξu(1), u ∈ rℓ generate g and the push-forward of the left-invariant vector fields ξu are left invariant, it follows that ϕ is a group isomorphism (cf. [18]). on the other hand, suppose there exists a group isomorphism ϕ : g → g′ such that t1ϕ · γ = γ′. then there exists a unique affine isomorphism φ : rℓ → rℓ ′ such that t1ϕ · ξ(1,u) = ξ′(1′,φ(u)). as with state space equivalence, by left-invariance and the fact that ϕ is a lie group isomorphism, it then follows that tgϕ · ξ(g,u) = ξ′(ϕ(g),φ(u)). 218 r. biggs, c. c. remsing remark. if ϕ is defined only between some neighbourhoods of identity of g and g′, then σ and σ′ are said to be locally detached feedback equivalent. a characterization similar to that given in theorem 2.2, in terms of lie algebra automorphisms, holds. as for state space equivalence, in the case of simply connected lie groups local and global equivalence are the same (as daut(g) = aut(g)). detached feedback equivalence is notably weaker than state space equivalence. to illustrate this point, we give a classification, under detached feedback equivalence, of the same class of systems considered in example 2.1. example 2.2. ([23]) any two-input inhomogeneous full-rank control affine system on se (2) is detached feedback equivalent to exactly one of the following systems σ1 : e1 + u1e2 + u2e3 σ2,α : αe3 + u1e1 + u2e2. here α > 0 parametrizes a family of class representatives, each different value corresponding to a distinct non-equivalent representative. 2.2.3. classification in three dimensions. we exhibit a classification, under detached feedback equivalence, of the full-rank systems evolving on unimodular three-dimensional lie groups (i.e., the classical abelian, heisenberg, euclidean, semi-euclidean, pseudo-orthogonal and orthogonal groups). we shall restrict our discussion to the simply connected groups. a representative is identified for each equivalence class. systems on the euclidean group and the orthogonal group are discussed as typical examples. details on the classification of three-dimensional lie groups and their lie algebras (along with standard ordered bases), as well as the corresponding automorphisms groups, can be found in appendix a. note. a classification, under detached feedback equivalence, of all (fullrank) control systems on three-dimensional lie groups appears in [27] (see also [19,21–23] and [24,26]). on higher dimensional lie groups, a classification of control systems on the orthogonal group so (4) was obtained in [4] (see also [2]). controllability of the respective systems is also addressed in these papers. invariant control systems on lie groups 219 we start with the solvable groups; the classification procedure is as follows. firstly, the group of automorphisms is determined (see appendix a). equivalence class representatives are then constructed by considering the action of an automorphism on the trace of a typical system. lastly, one verifies that none of the representatives are equivalent. theorem 2.3. ([22,23]) suppose σ is a full-rank system evolving on a simply connected unimodular solvable lie group g. then g is isomorphic to one of the groups listed below and σ is detached feedback equivalent to exactly one of accompanying (full-rank) systems on that group. 1. on r3, we have the systems σ(2,1) : e1 + u1e2 + u2e3 σ (3,0) : u1e1 + u2e2 + u3e3. 2. on h3, we have the systems σ(1,1) : e2 + ue3 σ (2,0) : u1e2 + u2e3 σ (2,1) 1 : e1 + u1e2 + u2e3 σ (2,1) 2 : e3 + u1e1 + u2e2 σ(3,0) : u1e1 + u2e2 + u3e3. 3. on se (1,1), we have the systems σ (1,1) 1 : e2 + ue3 σ (1,1) 2,α : αe3 + ue2 σ(2,0) : u1e2 + u2e3 σ (2,1) 1 : e1 + u1e2 + u2e3 σ (2,1) 2 : e1 + u1(e1 + e2) + u2e3 σ (2,1) 3,α : αe3 + u1e1 + u2e2 σ(3,0) : u1e1 + u2e2 + u3e3. 4. on s̃e (2), we have the systems σ (1,1) 1 : e2 + ue3 σ (1,1) 2,α : αe3 + ue2 σ(2,0) : u1e2 + u2e3 σ (2,1) 1 : e1 + u1e2 + u2e3 σ (2,1) 2,α : αe3 + u1e1 + u2e2 σ (3,0) : u1e1 + u2e2 + u3e3. here α > 0 parametrizes families of distinct (non-equivalent) class representatives. 220 r. biggs, c. c. remsing proof. we treat, as typical case, only item (4). the group of linearized automorphisms of s̃e (2) is given by daut(s̃e (2)) =     x y u−σy σx v 0 0 σ   : x,y,u,v ∈ r, x2 + y2 ̸= 0, σ = ±1   . let σ be a single-input inhomogeneous system with trace γ = a + γ0 ⊂ s̃e (2). suppose e∗3(γ 0) ̸= {0}. (here e∗3 is the corresponding element of the dual basis.) then γ = a1e1 + a2e2 + ⟨b1e1 + b2e2 + e3⟩. thus ψ =   a1 a1 b1−a1 a2 b2 0 0 1   is an automorphism such that ψ · γ(1,1)1 = γ. so σ is equivalent to σ (1,1) 1 . on the other hand, suppose e∗3(γ 0) = {0}. then γ = a1e1 + a2e2 + a3e3 + ⟨b1e1 + b2e2⟩ with a3 ̸= 0 (as lie(γ) = s̃e(2)). hence ψ =   b2 sgn(a3) b1 a1 a3 sgn(a3) −b1 sgn(a3) b2 a2a3 sgn(a3) 0 0 sgn(a3)   is an automorphism such that ψ · γ(1,1)2,α = γ, where α = a3 sgn(a3). let σ be a two-input homogeneous system with trace γ = ⟨b1,b2⟩. then σ̂ : b1 +⟨b2⟩ is a (full-rank) single-input inhomogeneous system. therefore, there exists an automorphism ψ such that ψ · (b1 + ⟨b2⟩) equals either e2+⟨e3⟩ or αe3+⟨e2⟩. hence, in either case, we get ψ·⟨b1,b2⟩ = ⟨e2,e3⟩. thus σ is equivalent to σ(2,0). the classification for the two-input inhomogeneous systems follows similarly. if σ is a three-input system, then clearly it is equivalent to σ(3,0). most pairs of systems cannot be equivalent due to different homogeneities or different number of inputs. as the subspace ⟨e1,e2⟩ is invariant (under the action of automorphisms), σ (1,1) 1 is not equivalent to any system σ (1,1) 2,α . for a ∈ s̃e (2) and ψ ∈ daut(se) (2), we have that e∗3(ψ · αe3) = ±α. thus σ (1,1) 2,α and σ (1,1) 2,α′ are equivalent only if α = α ′. for the two-input inhomogeneous systems, similar arguments hold. we now proceed to the semisimple lie groups; the procedure for classification is similar to that of the solvable groups. however, here we employ an invariant control systems on lie groups 221 invariant bilinear product ω (the lorentzian product and the dot product, respectively); the inhomogeneous systems are (partially) characterized by the level set {a ∈ g : ω(a,a) = α} that their trace is tangent to. theorem 2.4. ([21]) suppose σ is a full-rank system evolving on a simply connected semisimple lie group g. then g is isomorphic to one of the groups listed below and σ is detached feedback equivalent to exactly one of accompanying (full-rank) systems on that group. 1. on ã = s̃l (2,r), we have the systems σ (1,1) 1 : e3 + u(e2 + e3) σ (1,1) 2,α : αe2 + ue3 σ (1,1) 3,α : αe1 + ue2 σ (1,1) 4,α : αe3 + ue2 σ (2,0) 1 : u1e1 + u2e2 σ (2,0) 2 : u1e2 + u2e3 σ (2,1) 1 : e3 + u1e1 + u2(e2 + e3) σ (2,1) 2,α : αe1 + u1e2 + u2e3 σ (2,1) 3,α : αe3 + u1e1 + u2e2 σ (3,0) : u1e1 + u2e2 + u3e3. 2. on su (2), we have the systems σ(1,1)α : αe2 + ue3 σ (2,0) : u1e2 + u2e3 σ(2,1)α : αe1 + u1e2 + u2e3 σ (3,0) : u1e1 + u2e2 + u3e3. here α > 0 parametrizes families of distinct (non-equivalent) class representatives. proof. we consider only item (2), i.e., systems on the unitary group su (2). (the proof for item (1), although more involved, is similar.) the group of linearized automorphisms of su (2) is daut(su (2)) = so (3) = {g ∈ r3×3 : gg⊤ = 1, detg = 1}. the dot product • on su (2) is given by a • b = a1b1 + a2b2 + a3b3. (here a = ∑3 i=1 aiei and b = ∑3 i=1 biei.) the level sets sα = {a ∈ su (2) : a • a = α} are spheres of radius √ α (and are preserved by automorphisms). the group of automorphisms acts transitively on each sphere sα. the critical point c•(γ) (at which an inhomogeneous affine subspace is tangent to a sphere sα) is given by c•(γ) = a − a • b b • b b c•(γ) = a − [ b1 b2 ] [b1 • b1 b1 • b2 b1 • b2 b2 • b2 ]−1 [ a • b1 a • b2 ] . 222 r. biggs, c. c. remsing critical points behave well under the action of automorphisms, i.e., ψ·c•(γ) = c•(ψ · γ) for any automorphism ψ. (the critical point of γ is well defined as it is independent of parametrization.) let σ be a single-input inhomogeneous system with trace γ. there exists an automorphism ψ such that ψ · γ = αsinθe1 + αcosθe2 + ⟨e3⟩, where α = √ c•(γ) • c•(γ). hence ψ′ =  cosθ − sinθ 0sinθ cosθ 0 0 0 1   is an automorphism such that ψ′ · ψ · γ = γ(1,1)α . let σ be a two-input homogeneous system with trace γ = ⟨b1,b2⟩. then σ̂ : b1 +⟨b2⟩ is a (full-rank) single-input inhomogeneous system. therefore, there exists an automorphism ψ such that ψ · (b1 + ⟨b2⟩) = αe2 + ⟨e3⟩. hence ψ · ⟨b1,b2⟩ = ⟨e2,e3⟩. thus σ is equivalent to σ(2,0). let σ be a two-input inhomogeneous system with trace γ. we have c•(γ) • c•(γ) = α2 for some α > 0. as c•(γ1,α) • c•(γ1,α) = α2, there exists an automorphism ψ such that ψ · c•(γ) = c•(γ1,α). hence ψ · γ and γ1,α are both equal to the tangent plane of sα2 at ψ · c•(γ), and are therefore identical. if σ is a three-input system, then it is equivalent to σ(3,0). lastly, we note that none of the representatives obtained are equivalent. (again, we first distinguish representatives in terms of homogeneity and number of inputs.) as α2 = c•(γ (1,1) α ) • c•(γ (1,1) α ) (resp. α 2 = c•(γ (2,1) α ) • c•(γ (2,1) α )) is an invariant quantity, the systems σ (1,1) α and σ (1,1) α′ (resp. σ (2,1) α and σ (2,1) α′ ) are equivalent only if α = α ′. 3. invariant optimal control we consider the class of left-invariant optimal control problems on lie groups with fixed terminal time, affine dynamics, and affine quadratic cost. formally, such problems are given by ġ = g (a + u1b1 + · · · + uℓbℓ) , g ∈ g, u ∈ rℓ (1) g(0) = g0, g(t) = g1 (2) j = ∫ t 0 (u(t) − µ)⊤ q(u(t) − µ) dt −→ min. (3) invariant control systems on lie groups 223 here g is a (real, finite-dimensional) connected lie group with lie algebra g, a, b1, . . . , bℓ ∈ g (with b1, . . . , bℓ linearly independent), u = (u1, . . . ,uℓ) ∈ rℓ, µ ∈ rℓ, and q is a positive definite ℓ × ℓ matrix. to each such problem, we associate a cost-extended system (σ,χ). here σ is the control system (1) and the cost function χ : rℓ → r has the form χ(u) = (u − µ)⊤ q(u − µ). each cost-extended system corresponds to a family of invariant optimal control problems; by specification of the boundary data (g0,g1,t), the associated problem is uniquely determined. optimal control problems of this kind have received considerable attention in the last few decades. various physical problems have been modelled in this manner, such as optimal path planning for airplanes, motion planning for wheeled mobile robots, spacecraft attitude control, and the control of underactuated underwater vehicles ([61,67,75]); also, the control of quantum systems and the dynamic formation of dna ([37,43]). many problems (as well as sub-riemannian structures) on various low-dimensional matrix lie groups have been considered by a number of authors (see, e.g., [15, 16, 34, 49, 52, 54, 63,65,66,69,70]). we introduce a form of equivalence for problems of the form (1)–(2)–(3), or rather, the associated cost-extended systems (cf. [20, 28]). cost equivalence establishes a one-to-one correspondence between the associated optimal trajectories, as well as the associated extremal curves. via the pontryagin maximum principle, we associate to each cost-extended systems a quadratic hamilton–poisson systems on the associated lie–poisson space. we show that cost equivalence of cost-extended systems implies equivalence of the associated hamiltonian systems. in addition, we reinterpret drift-free cost-extended systems (with homogeneous cost) as invariant sub-riemannian structures. 3.1. pontryagin maximum principle. the pontryagin maximum principle provides necessary conditions for optimality which are naturally expressed in the language of the geometry of the cotangent bundle t∗g of g (see [9,40,50]). the cotangent bundle t∗g can be trivialized (from the left) such that t∗g = g × g∗; here g∗ is the dual of the lie algebra g. to an optimal control problem (1)–(2)–(3) we associate, for each real number λ and each control parameter u ∈ rℓ a hamiltonian function on t∗g = g × g∗: hλu (ξ) = λχ(u) + ξ (ξu(g)) = λχ(u) + p(ξu(1)), ξ = (g,p) ∈ t∗g. (4) 224 r. biggs, c. c. remsing we denote by h⃗λu the corresponding hamiltonian vector field (with respect to the symplectic structure on t∗g). in terms of the above hamiltonians, the maximum principle can be stated as follows. maximum principle. suppose the controlled trajectory (ḡ(·), ū(·)) defined over the interval [0,t] is a solution for the optimal control problem (1)–(2)–(3). then, there exists a curve ξ(·) : [0,t] → t∗g with ξ(t) ∈ t∗ ḡ(t) g, t ∈ [0,t ], and a real number λ ≤ 0, such that the following conditions hold for almost every t ∈ [0,t ] : (λ,ξ(t)) ̸≡ (0,0) (5) ξ̇(t) = h⃗λū(t)(ξ(t)) (6) hλū(t) (ξ(t)) = maxu hλu (ξ(t)) = constant. (7) an optimal trajectory, g(·) : [0,t] → g is the projection of an integral curve ξ(·) of the (time-varying) hamiltonian vector field h⃗λ ū(t) . a trajectory-control pair (ξ(·),u(·)) is said to be an extremal pair if ξ(·) satisfies the conditions (5), (6), and (7). the projection ξ(·) of an extremal pair is called an extremal. an extremal curve is called normal if λ < 0 and abnormal if λ = 0. for the class of optimal control problems under consideration, the maximum condition (7) eliminates the parameter u from the family of hamiltonians (hu); as a result, we obtain a smooth g-invariant function h on t∗g = g × g∗. this hamilton–poisson system on t∗g can be reduced to a hamilton–poisson system on the (minus) lie–poisson space g∗−, with poisson bracket given by {f,g} = −p([df(p),dg(p)]). here f,g ∈ c∞(g∗) and df(p),dg(p) are elements of the double dual g∗∗ which is canonically identified with the lie algebra g. 3.2. equivalence of cost-extended systems. let (σ,χ) and (σ′,χ′) be two cost-extended systems. (σ,χ) and (σ′,χ′) are said to be cost equivalent if there exist a lie group isomorphism ϕ : g → g′ and an affine isomorphism φ : rℓ → rℓ such that tgϕ · ξ(g,u) = ξ′(ϕ(g),φ(u)) and χ′ ◦ φ = rχ for g ∈ g, u ∈ rℓ and some r > 0. equivalently, (σ,χ) and (σ′,χ′) are cost equivalent if and only if there exist a lie group isomorphism ϕ : g → g′ and an affine isomorphism φ : rℓ → rℓ such that t1ϕ·ξ(1,u) = ξ′(1′,φ(u)) and χ′ ◦ φ = rχ for some r > 0. accordingly: invariant control systems on lie groups 225 • if (σ,χ) and (σ′,χ′) are cost equivalent, then σ and σ′ are detached feedback equivalent. • if two full-rank systems σ and σ′ are state space equivalent, then (σ,χ) and (σ′,χ) are cost equivalent for any cost χ. • if two full-rank systems σ and σ′ are detached feedback equivalent with respect to a feedback transformation φ, then (σ,χ◦φ) and (σ′,χ) are cost equivalent for any cost χ. remark. the cost-preserving condition χ′ ◦φ = rχ is partially motivated by the following considerations. each cost χ on rℓ induces a strict partial ordering u < v ⇐⇒ χ(u) < χ(v). it turns out that χ and χ′ induce the same strict partial ordering on rℓ if and only if χ = rχ′ for some r > 0. the dynamics-preserving condition tgϕ · ξ(g,u) = ξ′(ϕ(g),φ(u)) is just that of detached feedback equivalence (on full-rank systems). let (g(·),u(·)) be a controlled trajectory, defined over an interval [0,t ], of a cost-extended system (σ,χ). we say that (g(·),u(·)) is a virtually optimal controlled trajectory (shortly voct) if it is a solution for the associated optimal control problem with boundary data (g(0),g(t),t). similarly, we say that (g(·),u(·)) is an extremal controlled trajectory (shortly ect) if it satisfies the necessary conditions of the pontryagin maximum principle (with λ ≤ 0). clearly, any voct is an ect. a map ϕ × φ defining a cost equivalence between two cost-extended systems establishes a one-to-one correspondence between their respective vocts (and ects). proposition 3.1. ([20,28]) suppose ϕ×φ defines a cost equivalence between (σ,χ) and (σ′,χ′). then 1. (g(·),u(·)) is a voct if and only if (ϕ ◦ g(·),φ ◦ u(·)) is a voct; 2. (g(·),u(·)) is an ect if and only if (ϕ ◦ g(·),φ ◦ u(·)) is an ect. one can classify the cost-extended systems corresponding to a given invariant control system by use of the following result. (we denote by aff (rℓ) the group of affine isomorphisms of rℓ.) proposition 3.2. ([20,28]) let (σ,χ) and (σ,χ′) are two cost-extended systems (with identical underlying control system σ) and let tς = { φ ∈ aff (rℓ) : ∃ψ ∈ daut(g), ψ · γ = γ, ψ · ξ(1,u) = ξ(1,φ(u)) } 226 r. biggs, c. c. remsing be the group of feedback transformations leaving σ invariant. (σ,χ) and (σ,χ′) are cost equivalent if and only if there exists an element φ ∈ tς such that χ′ = rχ ◦ φ for some r > 0. example 3.1. ([20], cf. theorem 2.3, item 4) on se (2) , any full-rank two-input drift-free cost-extended system (σ,χ) with homogeneous cost (i.e., ξ(1,0) = 0 and χ(0) = 0) is cost equivalent to (σ(2,0),χ(2,0)) : { σ : u1e2 + u2e3 χ(u) = u21 + u 2 2. example 3.2. (cf. theorem 2.3, item 2) any controllable cost-extended system on h3 is c-equivalent to exactly one of the cost-extended systems ( σ(2,0),χ (2,0) 1 ) : { σ(2,0) : u1e2 + u2e3 χ (2,0) 1 (u) = u 2 1 + u 2 2( σ(2,0),χ (2,0) 2 ) : { σ(2,0) : u1e2 + u2e3 χ (2,0) 2 (u) = (u1 − 1) 2 + u22 ( σ(2,1),χ(2,1)α ) : { σ(2,1) : e1 + u1e2 + u2e3 χ(2,1)α (u) = (u1 − α) 2 + u22( σ(3,0),χ (3,0) α ) : { σ(3,0) : u1e1 + u2e2 + u3e3, χ (3,0) α (u) = (u1 − α1)2 + (u2 − α2)2 + u23. here α,α1,α2 ≥ 0 parametrize families of (non-equivalent) class representatives. note. several examples of classification under cost-equivalence can be found in [12,14,17,28] 3.3. pontryagin lift. to any cost-extended system (σ,χ) on a lie group g we associate, via the pontryagin maximum principle, a hamilton– poisson system on the associated lie–poisson space g∗− (cf. [9, 50, 71]). we show that equivalence of cost-extended systems implies equivalence of the associated hamilton–poisson systems. invariant control systems on lie groups 227 note. the pontryagin lift may be realized as a contravariant functor between the category of cost-extended control systems and the category of hamilton–poisson systems ([28], see also [40]). a quadratic hamilton–poisson system (g∗−,ha,q) is specified by ha,q : g ∗ → r, p 7→ p(a) + q(p). here a ∈ g and q is a quadratic form on g∗. if a = 0, then the system is called homogeneous; otherwise, it is called inhomogeneous. (when g∗− is fixed, a system (g∗−,ha,q) is identified with its hamiltonian ha,q.) to each function h ∈ c∞(g∗), we associate a hamiltonian vector field h⃗ on g∗ specified by h⃗[f] = {f,h}. a function c ∈ c∞(g∗) is a casimir function if {c,f} = 0 for all f ∈ c∞(g∗), or equivalently c⃗ = 0. a linear map ψ : g∗ → h∗ is a linear poisson morphism if {f,g} ◦ ψ = {f ◦ ψ,g ◦ ψ} for all f,g ∈ c∞(h∗). linear poisson morphisms are exactly the dual maps of lie algebra homomorphisms. let (e1, . . . ,en) be an ordered basis for the lie algebra g and let (e∗1, . . . ,e ∗ n) denote the corresponding dual basis for g ∗. we write elements b ∈ g as column vectors and elements p ∈ g∗ as row vectors. whenever convenient, linear maps will be identified with their matrices. if we write elements u ∈ rℓ as column vectors as well, then we can express ξu(1) = a+u1b1+· · ·+uℓbℓ as ξu(1) = a+bu, where b = [ b1 · · · bℓ ] is a n × ℓ matrix. the equations of motion for the integral curve p(·) of the hamiltonian vector field h⃗ corresponding to h ∈ c∞(g∗) then take the form ṗi = −p([ei,dh(p)]). let (σ,χ) be a cost-extended system with ξu(1) = a + bu, χ(u) = (u − µ)⊤q(u − µ). by the pontryagin maximum principle we have the following result. proposition 3.3. (cf. [20,50,59]) any normal ect (g(·),u(·)) of (σ,χ) is given by ġ(t) = ξ(g(t),u(t)), u(t) = q−1 b⊤ p(t) ⊤ + µ where p(·) : [0,t] → g∗ is an integral curve for the hamilton–poisson system on g∗− specified by h(p) = p (a + bµ) + 1 2 p b q−1 b⊤ p⊤. (8) 228 r. biggs, c. c. remsing we say that two quadratic hamilton–poisson systems (g∗−,g) and (h ∗ −,h) are linearly equivalent if there exists a linear isomorphism ψ : g∗ → h∗ such that the hamiltonian vector fields g⃗ and h⃗ are ψ-related, i.e., tpψ ·g⃗(p) = h⃗(ψ(p)) for p ∈ g∗. proposition 3.4. the following pairs of hamilton–poisson systems (on g∗−, specified by their hamiltonians) are linearly equivalent: 1. ha,q ◦ ψ and ha,q, where ψ : g∗− → g∗− is a linear lie–poisson automorphism; 2. ha,q and ha,rq, where r > 0; 3. ha,q and ha,q + c, where c is a casimir function. theorem 3.1. ([28]) if two cost-extended systems are cost equivalent, then their associated hamilton–poisson systems, given by (8), are linearly equivalent. proof. let (σ,χ) and (σ′,χ′) be cost-extended systems with ξu(1) = a + bu and ξ′u(1) = a ′ + b′ u′, respectively. the associated hamilton– poisson systems (on g∗− and (g ′)∗−, respectively) are given by h(σ,χ)(p) = p(a + bµ) + 1 2 pbq−1 b⊤ p⊤ h(σ′,χ′)(p) = p(a ′ + b′ µ′) + 1 2 pb′ q′−1 b′⊤ p⊤. suppose ϕ × φ defines a cost equivalence between (σ,χ) and (σ′,χ′), where φ(u) = ru + φ0 and r ∈ rℓ×ℓ. we have χ′ ◦ φ = rχ for some r > 0. a simple calculation yields t1ϕ·a = a′+b′ φ0, rµ+φ0 = µ′, t1ϕ·b = b′r, rq−1 r⊤ = r (q′)−1. thus (h(σ,χ)◦(t1ϕ)∗)(p) = p(a′+b′ µ′)+ r2 pb ′ (q′)−1 b′⊤ p⊤. here (t1ϕ) ∗ : (g′)∗ → g∗ is the dual of the linear map t1ϕ. hence, the vector fields associated with h(σ′,χ′) and h(σ,χ)◦(t1ϕ)∗, respectively, are related by the dilation δ1/r : (g ′)∗ → (g′)∗, p 7→ 1 r p (proposition 3.4). moreover, the vector fields associated with h(σ,χ) ◦ (t1ϕ)∗ and h(σ,χ), respectively, are related by the linear poisson isomorphism (t1ϕ) ∗ (proposition 3.4). consequently 1 r (t1ϕ) ∗ defines a linear equivalence between ((g′)∗−, h(σ′,χ′)) and (g ∗ −, h(σ,χ)). invariant control systems on lie groups 229 remark. the converse of theorem 3.1 is not true in general. in fact, one can construct cost-extended systems with different number of inputs but equivalent hamiltonians (see, e.g., [28]). in example 3.2 we gave a classification of the cost extended systems on h3. each hamiltonian system ((h3) ∗ −,h), where h is a positive definite quadratic form, can be realized as the hamiltonian system (8) associated to some cost-extended system. hence, by theorem 3.1, we get the following result. example 3.3. any quadratic hamilton–poisson systems ((h3) ∗ −,h), where h is a positive definite quadratic form, is linearly equivalent to the system on (h3) ∗ − with hamiltonian h ′(p) = 1 2 (p21 + p 2 2 + p 2 3). 3.4. sub-riemannian structures. left-invariant sub-riemannian (and, in particular, riemannian) structures on lie groups can naturally be associated to drift-free cost-extended systems with homogeneous cost. we show that if two cost-extended systems are cost equivalent, then the associated sub-riemannian structures are isometric up to rescaling. a left-invariant sub-riemannian manifold is a triplet (g,d,g), where g is a (real, finite-dimensional) connected lie group, d is a nonintegrable leftinvariant distribution on g, and g is a left-invariant riemannian metric on d. more precisely, d(1) is a linear subspace of the lie algebra g of g and d(g) = gd(1); the metric g1 is a positive definite symmetric bilinear from on g and gg(ga,gb) = g1(a,b) for a,b ∈ g, g ∈ g. when d = tg (i.e., d(1) = g) then one has a left-invariant riemannian structure. an absolutely continuous curve g(·) : [0,t ] → g is called a horizontal curve if ġ(t) ∈ d(g(t)) for almost all t ∈ [0,t ]. we shall assume that d satisfies the bracket generating condition, i.e., d(1) has full rank; this condition is necessary and sufficient for any two points in g to be connected by a horizontal curve. a standard argument shows that the length minimization problem ġ(t) ∈ d(g(t)), g(0) = g0, g(t) = g1,∫ t 0 √ g(ġ(t), ġ(t)) −→ min is equivalent to the energy minimization problem, or invariant optimal control 230 r. biggs, c. c. remsing problem: ġ = ξu(g), u ∈ rℓ g(0) = g0, g(t) = g1∫ t 0 χ(u(t))dt −→ min. (9) here ξu(1) = u1b1 +· · ·+uℓbℓ where b1, . . . ,bℓ are some linearly independent elements of g such that ⟨b1, . . . ,bℓ⟩ = d(1); χ(u(t)) = u(t)⊤qu(t) = g1(ξu(t)(1),ξu(t)(1)) for some ℓ × ℓ positive definite (symmetric) matrix q. more precisely, energy minimizers are exactly those length minimizers which have constant speed. in other words, the vocts of the cost-extended system (σ,χ) associated with (9) are exactly the (constant speed) minimizing geodesics of the sub-riemannian structure (g,d,g); the normal (resp. abnormal) ects of (σ,χ) are the normal (resp. abnormal) geodesics of (g,d,g). accordingly, to a (full-rank) cost-extended system (σ,χ) on g of the form σ : u1b1 + · · · + uℓbℓ, χ(u) = u⊤qu we associate a sub-riemannian structure (g,d,g) specified by d(1) = γ = ⟨b1, . . . ,bℓ⟩ , g1(u1b1+· · ·+uℓbℓ,u1b1+· · ·+uℓbℓ) = χ(u). let (g,d,g) and (g′,d′,g′) be two sub-riemannian structures associated to (σ,χ) and (σ′,χ′), respectively. theorem 3.2. (σ,χ) and (σ′,χ′) are cost equivalent if and only if there exists a lie group isomorphism ϕ : g → g′ such that ϕ∗d = d′ and g = rϕ∗g′ for some r > 0. proof. suppose ϕ × φ defines a cost equivalence between (σ,χ) and (σ′,χ′), i.e., ϕ∗ξu = ξ ′ φ(u) and χ′ ◦φ = rχ for some r > 0. as t1ϕ·ξu(1) = ξφ(u)(1), it follows that t1ϕ · d(1) = d′(1). hence, as ϕ is a lie group isomorphism, by left invariance we have ϕ∗d = d′. furthermore rχ(u) = χ′(φ(u)) ⇐⇒ r g1(ξu(1),ξu(1)) = g′1(ξ ′ φ(u)(1),ξ ′ φ(u)(1)) (10) ⇐⇒ r g1(ξu(1),ξu(1)) = g′1(t1ϕ · ξu(1),t1ϕ · ξu(1)). hence, as ϕ is a lie group isomorphism, by left invariance we have r g = ϕ∗g′. conversely, suppose ϕ∗d = d′ and g = rϕ∗g′. we have t1ϕ · d(1) = d′(1) and so t1ϕ·γ = γ′. hence there exists a unique linear map φ : rℓ → rℓ invariant control systems on lie groups 231 such that t1ϕ · ξu(1) = ξ′φ(u)(1). thus ϕ × φ defines a detached feedback equivalence between σ and σ′. by (10), it follows that χ′ ◦ φ = rχ. thus (σ,χ) and (σ′,χ′) are cost equivalent. remark. for (sub-riemannian) carnot groups and invariant riemannian structures on nilpotent lie groups, any isometry is the composition of a left translation and a lie group isomorphism (see [36,45,55] and [60,76], respectively). recently, this has been shown to generalize to any nilpotent metric lie group ([56]). hence, at least for these classes, if (g,d,g) and (g′,d′,g′) are isometric, then (σ,χ) and (σ′,χ′) are cost equivalent. analogous to example 3.1, we have the following classification of subriemannian structures on the euclidean group se (2). example 3.4. on se (2), any left-invariant sub-riemannian structure (d,g) isometric up to rescaling to the structure (d̄, ḡ) with orthonormal frame (e2,e3); here e2 and e3 are viewed as left-invariant vector fields. 4. final remarks as already mentioned, a complete classification of the invariant control affine systems in three dimensions was obtained in [27] (see also [21–23]). there is no complete classification of the cost-extended systems in three dimensions. however, there are classifications of the invariant sub-riemannian structures ([8]) and invariant riemannian structures ([44]). classifications in four dimensions (and beyond) are also topics for future research. in order to find the extremal trajectories for a cost-extended system, one needs to integrate the associated hamilton–poisson system (see proposition 3.3). in the last decade or so several authors have considered quadratic hamilton–poisson systems on low-dimensional lie–poisson spaces (see, e.g., [3,6,10,16,74]). to our knowledge there is currently no general classification of the quadratic hamilton–poisson systems in three dimensions. a first attempt towards such a classification appears in [31] (see also [5,7,13,25,38]). a. three-dimensional lie algebras and groups there are eleven types of three-dimensional real lie algebras; in fact, nine algebras and two parametrized infinite families of algebras (see, e.g., [57, 62, 64]). in terms of an (appropriate) ordered basis (e1,e2,e3), the commutation 232 r. biggs, c. c. remsing operation is given by [e2,e3] = n1e1 − ae2 [e3,e1] = ae1 + n2e2 [e1,e2] = n3e3. the structure parameters a,n1,n2,n3 for each type are given in table 1. a n1 n2 n3 u n im o d u la r n il p o te n t c o m p l. s o lv . e x p o n en ti a l s o lv a b le s im p le connected groups 3g1 0 0 0 0 • • • • • r3, r2 × t, r × t2, t3 g2.1 ⊕ g1 1 1 −1 0 • • • aff (r)0 × r, aff (r)0 × t g3.1 0 1 0 0 • • • • • h3, h∗3 g3.2 1 1 0 0 • • • g3.2 g3.3 1 0 0 0 • • • g3.3 g03.4 0 1 −1 0 • • • • se (1,1) ga3.4 a>0 a ̸=1 1 −1 0 • • • g a 3.4 g03.5 0 1 1 0 • • s̃e (2), sen(2), se (2) ga3.5 a>0 1 1 0 • • g a 3.5 g3.6 0 1 1 −1 • • ã, an, sl (2,r), so (2,1)0 g3.7 0 1 1 1 • • su (2), so (3) table 1: three-dimensional lie algebras a classification of the three-dimensional (real, connected) lie groups can be found in [42]. let g be a three-dimensional (real, connected) lie group with lie algebra g. 1. if g is abelian, i.e., g ∼= 3g1, then g is isomorphic to r3, r2 × t, r × t, or t3. 2. if g ∼= g2.1 ⊕ g1, then g is isomorphic to aff (r)0 × r or aff (r)0 × t. 3. if g ∼= g3.1, then g is isomorphic to the heisenberg group h3 or the lie group h∗3 = h3/z(h3(z)), where z(h3(z)) is the group of integer points in the centre z(h3) ∼= r of h3. invariant control systems on lie groups 233 4. if g ∼= g3.2, g3.3 , g03.4, g a 3.4, or g a 3.5, then g is isomorphic to the simply connected lie group g3.2, g3.3, g 0 3.4 = se (1,1), g a 3.4, or g a 3.5, respectively. 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[76] e.n. wilson, isometry groups on homogeneous nilmanifolds, geom. dedicata 12 (83) (1982), 337 – 346. e extracta mathematicae vol. 31, núm. 1, 47 – 67 (2016) quasi grüss type inequalities for complex functions defined on unit circle with applications for unitary operators in hilbert spaces s. s. dragomir mathematics, college of engineering & science, victoria university, po box 14428, melbourne city, mc 8001, australia and school of computational & applied mathematics, university of the witwatersrand, private bag 3, johannesburg 2050, south africa sever.dragomir@vu.edu.au http://rgmia.org/dragomir presented by alfonso montes received april 14, 2015 abstract: some quasi grüss type inequalities for the riemann-stieltjes integral of continuous complex valued integrands defined on the complex unit circle c(0, 1) and various subclasses of integrators are given. natural applications for functions of unitary operators in hilbert spaces are provided. key words: grüss type inequalities, riemann-stieltjes integral inequalities, unitary operators in hilbert spaces, spectral theory, quadrature rules. ams subject class. (2010): 26d15, 47a63. 1. introduction the concept of riemann-stieltjes integral ∫ b a f(t) du(t), where f is called the integrand and u is called the integrator, plays an important role in mathematics, for instance in the definition of complex integral, the representation of bounded linear functionals on the banach space of all continuous functions on an interval [a, b], in the spectral representation of selfadjoint operators on complex hilbert spaces and other classes of operators such as the unitary operators, etc. one can approximate the riemann-stieltjes integral ∫ b a f(t) du(t) with the following simpler quantity: 1 b − a [u(b) − u(a)] · ∫ b a f(t) dt ([11], [12]). (1.1) in order to provide a priory sharp bounds for the approximation error, 47 48 s. s. dragomir consider the functionals: d(f, u; a, b) := ∫ b a f(t) du(t) − 1 b − a [u(b) − u(a)] · ∫ b a f(t) dt. if the integrand f is riemann integrable on [a, b] and the integrator u : [a, b] → r is l−lipschitzian, i.e., |u(t) − u(s)| ≤ l|t − s| for each t, s ∈ [a, b], (1.2) then the riemann-stieltjes integral ∫ b a f(t) du(t) exists and, as pointed out in [11], the following quasi grüss type inequality holds |d(f, u; a, b)| ≤ l ∫ b a ∣∣∣∣f(t) − ∫ b a 1 b − a f(s) ds ∣∣∣∣dt. (1.3) the inequality (1.3) is sharp in the sense that the multiplicative constant c = 1 in front of l cannot be replaced by a smaller quantity. moreover, if there exists the constants m, m ∈ r such that m ≤ f(t) ≤ m for a.e. t ∈ [a, b], then [11] |d(f, u; a, b)| ≤ 1 2 l(m − m)(b − a). (1.4) the constant 1 2 is best possible in (1.4). we call this type of inequalities of quasi grüss type since for integrators of integral form u(t) := 1 b−a ∫ t a g(s)ds the left hand side becomes ∣∣∣∣ 1b − a ∫ b a f(t) du(t) − 1 b − a ∫ b a f(t) dt · 1 b − a ∫ b a g(s) ds ∣∣∣∣ that is related with the well known grüss inequality. a different approach in the case of integrands of bounded variation were considered by the same authors in 2001, [12], where they showed that |d(f, u; a, b)| ≤ max t∈[a,b] ∣∣∣∣f(t) − 1b − a ∫ b a f(s)ds ∣∣∣∣ b∨ a (u), (1.5) provided that f is continuous and u is of bounded variation. here ∨b a(u) denotes the total variation of u on [a, b]. the inequality (1.5) is sharp. quasi grüss type inequalities 49 if we assume that f is k−lipschitzian, then [12] |d(f, u; a, b)| ≤ 1 2 k(b − a) b∨ a (u), (1.6) with 1 2 the best possible constant in (1.6). for various bounds on the error functional d(f, u; a, b) where f and u belong to different classes of function for which the stieltjes integral exists, see [9], [8], [7], and [6] and the references therein. for other inequalities for the riemann-stieltjes integral see [1] [4], [5] [10], [14] and the references therein. for continuous functions f : c(0, 1) → c, where c(0, 1) is the unit circle from c centered in 0 and u : [a, b] ⊆ [0, 2π] → c is a function of bounded variation on [a, b], we can define the following functional of quasi grüss type as well: dc(f; u, a, b) := ∫ b a f(eit)du(t) − 1 b − a [u(b) − u(a)] · ∫ b a f(eit)dt. (1.7) in this paper we establish some bounds for the magnitude of sc(f; u, a, b) when the integrand f : c(0, 1) → c satisfies some hölder’s type conditions on the circle c(0, 1) while the integrator u is of bonded variation. it is shown that this functional can be naturally connected with continuous functions of unitary operators on hilbert spaces. we recall here some basic facts on unitary operators and spectral families that will be used in the sequel. we say that the bounded linear operator u : h → h on the hilbert space h is unitary iff u∗ = u−1. it is well known that (see for instance [13, p. 275-p. 276]), if u is a unitary operator, then there exists a family of projections {eλ}λ∈[0,2π], called the spectral family of u with the following properties: a) eλ ≤ eµ for 0 ≤ λ ≤ µ ≤ 2π; b) e0 = 0 and e2π = 1h (the identity operator on h); c) eλ+0 = eλ for 0 ≤ λ < 2π; d) u = ∫ 2π 0 eiλdeλ, where the integral is of riemann-stieltjes type. moreover, if {fλ}λ∈[0,2π] is a family of projections satisfying the requirements a) d) above for the operator u, then fλ = eλ for all λ ∈ [0, 2π]. 50 s. s. dragomir also, for every continuous complex valued function f : c(0, 1) → c on the complex unit circle c(0, 1), we have f(u) = ∫ 2π 0 f ( eiλ ) deλ (1.8) where the integral is taken in the riemann-stieltjes sense. in particular, we have the equalities ⟨f(u)x, y⟩ = ∫ 2π 0 f ( eiλ ) d⟨eλx, y⟩ (1.9) and ∥f(u)x∥2 = ∫ 2π 0 ∣∣f(eiλ)∣∣2d∥eλx∥2 = ∫ 2π 0 ∣∣f(eiλ)∣∣2d⟨eλx, x⟩, (1.10) for any x, y ∈ h. from the above properties it follows that the function gx(λ) := ⟨eλx, x⟩ is monotonic nondecreasing and right continuous on [0, 2π] for any x ∈ h. such functions of unitary operators are exp(u) = ∫ 2π 0 exp ( eiλ ) deλ and un = ∫ 2π 0 einλdeλ for n an integer. we can also define the trigonometric functions for a unitary operator u by sin(u) = ∫ 2π 0 sin ( eiλ ) deλ and cos(u) = ∫ 2π 0 cos ( eiλ ) deλ and the hyperbolic functions by sinh(u) = ∫ 2π 0 sinh ( eiλ ) deλ and cosh(u) = ∫ 2π 0 cosh ( eiλ ) deλ where sinh(z) := 1 2 [exp z − exp(−z)] and cosh(z) := 1 2 [exp z + exp(−z)], z ∈ c. quasi grüss type inequalities 51 2. inequalities for riemann-stieltjes integral we say that the complex function f : c(0, 1) → c satisfies an h-r-hölder’s type condition on the circle c(0, 1), where h > 0 and r ∈ (0, 1] are given, if |f(z) − f(w)| ≤ h|z − w|r (2.1) for any w, z ∈ c(0, 1). if r = 1 and l = h then we call it of l-lipschitz type. consider the power function f : c\{0} → c, f(z) = zm where m is a nonzero integer. then, obviously, for any z, w belonging to the unit circle c(0, 1) we have the inequality |f(z) − f(w)| ≤ |m||z − w| which shows that f is lipschitzian with the constant l = |m| on the circle c(0, 1). for a ̸= ±1, 0 real numbers, consider the function f : c(0, 1) → c, fa(z) = 1 1−az . observe that |fa(z) − fa(w)| = |a||z − w| |1 − az||1 − aw| (2.2) for any z, w ∈ c(0, 1). if z = eit with t ∈ [0, 2π], then we have |1 − az|2 = 1 − 2a re(z̄) + a2|z|2 = 1 − 2a cos t + a2 ≥ 1 − 2|a| + a2 = (1 − |a|)2 therefore 1 |1 − az| ≤ 1 |1 − |a|| and 1 |1 − aw| ≤ 1 |1 − |a|| (2.3) for any z, w ∈ c(0, 1). utilising (2.2) and (2.3) we deduce |fa(z) − fa(w)| ≤ |a| (1 − |a|)2 |z − w| (2.4) for any z, w ∈ c(0, 1), showing that the function fa is lipschitzian with the constant la = |a| (1−|a|)2 on the circle c(0, 1). 52 s. s. dragomir theorem 1. let f : c(0, 1) → c satisfies an h-r-hölder’s type condition on the circle c(0, 1), where h > 0 and r ∈ (0, 1] are given. if u : [a, b] ⊆ [0, 2π] → c is a function of bounded variation on [a, b], then |dc(f; u, a, b)| ≤ 2rh b − a max t∈[a,b] br(a, b; t) b∨ a (u) ≤ h r + 1 (b − a)r b∨ a (u) (2.5) where br(a, b; t) := ∫ t a sinr ( t − s 2 ) ds + ∫ b t sinr ( s − t 2 ) ds ≤ 1 2r (t − a)r+1 + (b − t)r+1 r + 1 (2.6) for any t ∈ [a, b]. in particular, if f is lipschitzian with the constant l > 0, and [a, b] ⊂ [0, 2π] with b − a ̸= 2π, then we have the simpler inequality |dc(f; u, a, b)| ≤ 8l b − a sin2 ( b − a 4 ) b∨ a (u) ≤ 1 2 l(b − a) b∨ a (u). (2.7) if a = 0 and b = 2π and f is lipschitzian with the constant l > 0, then |dc(f; u, 0, 2π)| ≤ 4l π 2π∨ 0 (u). (2.8) proof. we have dc(f; u, a, b) = ∫ b a ( f ( eit ) − 1 b − a ∫ b a f ( eis ) ds ) du(t) = 1 b − a ∫ b a (∫ b a [ f ( eit ) − f ( eis )] ds ) du(t). (2.9) it is known that if p : [c, d] → c is a continuous function and v : [c, d] → c is of bounded variation, then the riemann-stieltjes integral ∫ d c p(t)dv(t) exists and the following inequality holds∣∣∣∣ ∫ d c p(t)dv(t) ∣∣∣∣ ≤ max t∈[c,d] |p(t)| d∨ c (v). (2.10) quasi grüss type inequalities 53 utilising this property and (2.9) we have |dc(f; u, a, b)| = 1 b − a ∣∣∣∣ ∫ b a (∫ b a [ f ( eit ) − f ( eis )] ds ) du(t) ∣∣∣∣ ≤ 1 b − a max x∈[a,b] ∣∣∣∣ ∫ b a [ f ( eit ) − f ( eis )] ds ∣∣∣∣ b∨ a (u). (2.11) utilising the properties of the riemann integral and the fact that f is of h-r-hölder’s type on the circle c(0, 1) we have∣∣∣∣ ∫ b a [ f ( eit ) − f ( eis )] ds ∣∣∣∣ ≤ ∫ b a ∣∣f(eit) − f(eis)∣∣ds ≤ h ∫ b a ∣∣eis − eit∣∣r ds (2.12) since ∣∣eis − eit∣∣2 = ∣∣eis∣∣2 − 2 re(ei(s−t)) + ∣∣eit∣∣2 = 2 − 2 cos(s − t) = 4 sin2 ( s − t 2 ) for any t, s ∈ r, then ∣∣eis − eit∣∣r = 2r ∣∣∣∣sin ( s − t 2 )∣∣∣∣r (2.13) for any t, s ∈ r. therefore∫ b a ∣∣eis − eit∣∣r ds = 2r ∫ b a ∣∣∣∣sin ( s − t 2 )∣∣∣∣r ds = 2r [∫ t a sinr ( t − s 2 ) ds + ∫ b t sinr ( s − t 2 ) ds ] (2.14) for any t ∈ [a, b]. on making use of (2.12) and (2.14) we have max x∈[a,b] ∣∣∣∣ ∫ b a [ f ( eit ) − f ( eis )] ds ∣∣∣∣ ≤ 2rh max t∈[a,b] br(a, b; t) and the first inequality in (2.5) is proved. 54 s. s. dragomir utilising the elementary inequality | sin(x)| ≤ |x|, x ∈ r we have br(a, b; t) ≤ ∫ t a ( t − s 2 )r ds + ∫ b t ( s − t 2 )r ds = 1 2r (t − a)r+1 + (b − t)r+1 r + 1 (2.15) for any t ∈ [a, b], and the inequality (2.6) is proved. if we consider the auxiliary function φ : [a, b] → r, φ(t) = (t − a)r+1 + (b − t)r+1, r ∈ (0, 1] then φ′(t) = (r + 1) [(t − a)r − (b − t)r] and φ′′(t) = (r + 1)r[(t − a)r−1 + (b − t)r−1]. we have φ′(t) = 0 iff t = a+b 2 , φ′(t) < 0 for t ∈ ( a, a+b 2 ) and φ′(t) > 0 for t ∈ (a+b 2 , b). we also have that φ′′(t) > 0 for any t ∈ (a, b) showing that φ is strictly decreasing on ( a, a+b 2 ) and strictly increasing on (a+b 2 , b). we also have that min t∈[a,b] φ(t) = φ ( a + b 2 ) = (b − a)r+1 2r and max t∈[a,b] φ(t) = φ(a) = φ(b) = (b − a)r+1. taking the maximum over t ∈ [a, b] in (2.15) we deduce the second inequality in (2.5). for r = 1 we have b(a, b; t) := ∫ t a sin ( t − s 2 ) ds + ∫ b t sin ( s − t 2 ) ds = 2 − 2 cos ( t − a 2 ) − 2 cos ( b − t 2 ) + 2 = 2 [ 1 − cos ( t − a 2 ) + 1 − cos ( b − t 2 )] = 2 [ 2 sin2 ( t − a 4 ) + 2 sin2 ( b − t 4 )] = 4 [ sin2 ( t − a 4 ) + sin2 ( b − t 4 )] quasi grüss type inequalities 55 for any t ∈ [a, b]. now, if we take the derivative in the first equality, we have b′(a, b; t) = sin ( t − a 2 ) − sin ( b − t 2 ) = 2 sin ( t − a+b 2 2 ) cos ( b − a 4 ) , for [a, b] ⊂ [0, 2π] and b − a ̸= 2π. we observe that b′(a, b; t) = 0 iff t = a+b 2 , b′(a, b; t) < 0 for t ∈ ( a, a+b 2 ) and b′(a, b; t) > 0 for t ∈ (a+b 2 , b). the second derivative is given by b′′(a, b; t) = cos ( t − a+b 2 2 ) cos ( b − a 4 ) and we observe that b′′(a, b; t) > 0 for t ∈ (a, b). therefore the function b(a, b; ·) is strictly decreasing on ( a, a+b 2 ) and strictly increasing on ( a+b 2 , b ) . it is also a strictly convex function on (a, b). we have min t∈[a,b] b(a, b; t) = b ( a, b; a + b 2 ) = 8 sin2 ( b − a 8 ) and max t∈[a,b] b(a, b; t) = b(a, b; a) = b(a, b; b) = 4 sin2 ( b − a 4 ) . this proves the bound (2.7). if a = 0 and b = 2π, then b(0, 2π; t) = 4 [ sin2 ( t 4 ) + sin2 ( 2π − t 4 )] = 4 and by (2.5) we get (2.8). the proof is complete. the following result also holds: theorem 2. let f : c(0, 1) → c satisfies an h-r-hölder’s type condition on the circle c(0, 1), where h > 0 and r ∈ (0, 1] are given. if u : [a, b] ⊆ [0, 2π] → c is a function of lipschitz type with the constant k > 0 on [a, b], then |dc(f; u, a, b)| ≤ 2rhk b − a cr(a, b) ≤ 2hk(b − a)r+1 (r + 1)(r + 2) (2.16) 56 s. s. dragomir where cr(a, b) := ∫ b a ∫ t a sinr ( t − s 2 ) ds dt + ∫ b a ∫ b t sinr ( s − t 2 ) ds dt ≤ (b − a)r+2 2r−1(r + 1)(r + 2) . (2.17) in particular, if f is lipschitzian with the constant l > 0, then we have the simpler inequality |dc(f; u, a, b)| ≤ 16lk b − a [ b − a 2 − sin ( b − a 2 )] ≤ lk(b − a)2 3 . (2.18) proof. it is well known that if p : [c, d] → c is a riemann integrable function and v : [c, d] → c is lipschitzian with the constant m > 0, then the riemann-stieltjes integral ∫ d c p(t)dv(t) exists and the following inequality holds ∣∣∣∣ ∫ d c p(t)dv(t) ∣∣∣∣ ≤ m ∫ d c |p(t)|dt. (2.19) utilising the equality (2.9) and this property we have |dc(f; u, a, b)| = 1 b − a ∣∣∣∣ ∫ b a (∫ b a [ f ( eit ) − f ( eis )] ds ) du(t) ∣∣∣∣ ≤ k b − a ∫ b a ∣∣∣∣ (∫ b a [ f ( eit ) − f ( eis )] ds )∣∣∣∣dt. (2.20) from (2.12) and (2.14) we have∣∣∣∣ ∫ b a [ f ( eit ) − f ( eis )] ds ∣∣∣∣ ≤ ∫ b a ∣∣f (eit) − f (eis)∣∣ds ≤ h ∫ b a ∣∣eis − eit∣∣r ds = 2rh [∫ t a sinr ( t − s 2 ) ds + ∫ b t sinr ( s − t 2 ) ds ] (2.21) quasi grüss type inequalities 57 and by (2.20) we deduce the first part of (2.16). since, by (2.15), we have ∫ t a ( t − s 2 )r ds + ∫ b t ( s − t 2 )r ds = 1 2r (t − a)r+1 + (b − t)r+1 r + 1 , then cr(a, b) ≤ ∫ b a [∫ t a ( t − s 2 )r ds + ∫ b t ( s − t 2 )r ds ] dt ≤ 1 2r ∫ b a (t − a)r+1 + (b − t)r+1 r + 1 dt = (b − a)r+2 2r−1(r + 1)(r + 2) , which proves the inequality (2.17). for r = 1, we have c1(a, b) := ∫ b a [∫ t a sin ( t − s 2 ) ds + ∫ b t sin ( s − t 2 ) ds ] dt = ∫ b a [ 2 − 2 cos ( t − a 2 ) − 2 cos ( b − t 2 ) + 2 ] dt = 4(b − a) − 4 sin ( b − a 2 ) − 4 sin ( b − a 2 ) = 8 [ b − a 2 − sin ( b − a 2 )] , which, by (2.16), produces the desired inequality (2.18). remark 1. in the case b = 2π and a = 0 the inequality (2.18) produces the simple inequality |dc(f; u, 0, 2π)| ≤ 8lk. (2.22) 58 s. s. dragomir the following result for monotonic integrators also holds. theorem 3. let f : c(0, 1) → c satisfies an h-r-hölder’s type condition on the circle c(0, 1), where h > 0 and r ∈ (0, 1] are given. if u : [a, b] ⊆ [0, 2π] → r is a monotonic nondecreasing function on [a, b], then |dc(f; u, a, b)| ≤ 2rh b − a dr(a, b) ≤ h (r + 1)(b − a) ∫ b a [ (t − a)r+1 + (b − t)r+1 ] du(t) ≤ h (r + 1) (b − a)r[u(b) − u(a)] (2.23) where dr(a, b) := ∫ b a br(a, b; t)du(t) (2.24) and br(a, b; t) is given by (2.6). in particular, if f is lipschitzian with the constant l > 0, then we have the simpler inequality |dc(f; u, a, b)| ≤ 8l b − a ∫ b a [ sin2 ( t − a 4 ) + sin2 ( b − t 4 )] du(t) ≤ l 2 (b − a)[u(b) − u(a)]. (2.25) proof. it is well known that if p : [c, d] → c is a continuous function and v : [c, d] → r is monotonic nondecreasing on [c, d], then the riemann-stieltjes integral ∫ d c p(t)dv(t) exists and the following inequality holds ∣∣∣∣ ∫ d c p(t)dv(t) ∣∣∣∣ ≤ ∫ d c |p(t)|dv(t). (2.26) utilising this property and the identity (2.9) we have quasi grüss type inequalities 59 |dc(f; u, a, b)| = 1 b − a ∣∣∣∣ ∫ b a (∫ b a [ f ( eit ) − f ( eis )] ds ) du(t) ∣∣∣∣ ≤ 1 b − a ∫ b a ∣∣∣∣ (∫ b a [ f ( eit ) − f ( eis )] ds )∣∣∣∣du(t) ≤ 1 b − a ∫ b a (∫ b a ∣∣(f (eit) − f (eis))∣∣ds)du(t) ≤ h b − a ∫ b a (∫ b a ∣∣eis − eit∣∣r ds)du(t) = 2rh b − a ∫ b a [∫ t a sinr ( t − s 2 ) ds + ∫ b t sinr ( s − t 2 ) ds ] du(t). (2.27) we also have that∫ b a [∫ t a sinr ( t − s 2 ) ds + ∫ b t sinr ( s − t 2 ) ds ] du(t) ≤ ∫ b a [∫ t a ( t − s 2 )r ds + ∫ b t ( s − t 2 )r ds ] du(t) = 1 2r ∫ b a (t − a)r+1 + (b − t)r+1 r + 1 du(t) = 1 2r(r + 1) ∫ b a [(t − a)r+1 + (b − t)r+1]du(t) and the first part of the inequality (2.23) is proved. since max t∈[a,b] [ (t − a)r+1 + (b − t)r+1 ] = (b − a)r+1 then the last part of (2.23) is also proved for r = 1 we have d1(a, b) := ∫ b a b1(a, b; t)du(t) = 4 ∫ b a [ sin2 ( t − a 4 ) + sin2 ( b − t 4 )] du(t) and the inequality (2.25) is obtained. 60 s. s. dragomir remark 2. the case a = 0, b = 2π can be stated as |dc(f; u, 0, 2π)| ≤ 4l π [u(2π) − u(0)]. (2.28) indeed, by (2.25) we have |dc(f; u, 0, 2π)| ≤ 8l 2π ∫ 2π 0 [ sin2 ( t 4 ) + sin2 ( 2π − t 4 )] du(t) = 4l π ∫ 2π 0 [ sin2 ( t 4 ) + sin2 ( π 2 − t 4 )] du(t) = 4l π ∫ 2π 0 [ sin2 ( t 4 ) + cos2 ( t 4 )] du(t) = 4l π [u(2π) − u(0)]. 3. applications for functions of unitary operators we have the following vector inequality for functions of unitary operators. theorem 4. assume that f : c(0, 1) → c satisfies an l-lipschitz type condition on the circle c(0, 1), where l > 0 is given. if the operator u : h → h on the hilbert space h is unitary and {eλ}λ∈[0,2π] is its spectral family, then∣∣∣∣⟨f(u)x, y⟩ − 12π ∫ 2π 0 f(eit)dt · ⟨x, y⟩ ∣∣∣∣ ≤ 4l π 2π∨ 0 (⟨e(·)x, y⟩) ≤ 4l π ∥x∥∥y∥ (3.1) for any x, y ∈ h. proof. for given x, y ∈ h, define the function u(λ) := ⟨eλx, y⟩, λ ∈ [0, 2π]. we will show that u is of bounded variation and 2π∨ 0 (u) =: 2π∨ 0 ( ⟨e(·)x, y⟩ ) ≤ ∥x∥∥y∥. (3.2) it is well known that, if p is a nonnegative selfadjoint operator on h, i.e., ⟨px, x⟩ ≥ 0 for any x ∈ h, then the following inequality is a generalization of quasi grüss type inequalities 61 the schwarz inequality in h |⟨px, y⟩|2 ≤ ⟨px, x⟩⟨py, y⟩, (3.3) for any x, y ∈ h. now, if d : 0 = t0 < t1 < · · · < tn−1 < tn = 2π is an arbitrary partition of the interval [0, 2π], then we have by schwarz’s inequality for nonnegative operators (3.3) that 2π∨ 0 (⟨ e(·)x, y ⟩) = sup d { n−1∑ i=0 ∣∣⟨(eti+1 − eti)x, y⟩∣∣ } ≤ sup d { n−1∑ i=0 [⟨( eti+1 − eti ) x, x ⟩1/2 ⟨( eti+1 − eti ) y, y ⟩1/2]} := i. (3.4) by the cauchy-buniakovski-schwarz inequality for sequences of real numbers we also have that i ≤ sup d   [ n−1∑ i=0 ⟨( eti+1 − eti ) x, x ⟩]1/2 [n−1∑ i=0 ⟨( eti+1 − eti ) y, y ⟩]1/2 ≤ sup d [ n−1∑ i=0 ⟨( eti+1 − eti ) x, x ⟩]1/2 sup d [ n−1∑ i=0 ⟨( eti+1 − eti ) y, y ⟩]1/2 = [ 2π∨ 0 (⟨ e(·)x, x ⟩)]1/2 [2π∨ 0 (⟨ e(·)y, y ⟩)]1/2 = ∥x∥∥y∥ (3.5) for any x, y ∈ h. utilising the inequality (2.8) we can write that∣∣∣∣ ∫ 2π 0 f(eit) d⟨etx, y⟩ − 1 2π [⟨e2πx, y⟩ − ⟨e0x, y⟩] · ∫ 2π 0 f(eit)dt ∣∣∣∣ ≤ 4l π 2π∨ 0 (⟨ e(·)x, y ⟩) , (3.6) for any x, y ∈ h. on making use of the representation theorem (1.9) and the inequality (3.2) we deduce the desired result (3.1). 62 s. s. dragomir remark 3. consider the function f : c(0, 1) → c, fa(z) = 11−az with a real and 0 < |a| < 1. we know that this function is lipschitzian with the constant l = |a| (1−|a|)2 . since |ae it| = |a| < 1, then∫ 2π 0 f ( eit ) dt = ∫ 2π 0 1 1 − aeit dt = ∫ 2π 0 ∞∑ n=0 ( aeit )n dt = ∞∑ n=0 an ∫ 2π 0 ( eit )n dt = ∫ 2π 0 dt = 2π, since for any natural number n ≥ 1 we have ∫ 2π 0 (eit)ndt = 0. applying the inequality (3.1) we have∣∣∣⟨(1h − au)−1 x, y⟩ − ⟨x, y⟩∣∣∣ ≤ 4|a| π(1 − |a|)2 2π∨ 0 (⟨ e(·)x, y ⟩) ≤ 4|a| π(1 − |a|)2 ∥x∥∥y∥ (3.7) for any x, y ∈ h. 4. a quadrature rule we consider the following partition of the interval [a, b] ∆n : a = x0 < x1 < · · · < xn−1 < xn = b. define hk := xk+1 − xk, 0 ≤ k ≤ n − 1 and ν(∆n) = max{hk : 0 ≤ k ≤ n − 1} the norm of the partition ∆n. for the continuous function f : c(0, 1) → c and the function u : [a, b] ⊆ [0, 2π] → c of bounded variation on [a, b], define the quadrature rule dn(f, u, ∆n) := n−1∑ k=0 u(xk+1) − u(xk) xk+1 − xk ∫ xk+1 xk f ( eit ) dt (4.1) and the remainder rn(f, u, ∆n) in approximating the riemann-stieltjes integral ∫ b a f(eit)du(t) by dn(f, u, ∆n). then we have∫ b a f ( eit ) du(t) = dn(f, u, ∆n) + rn(f, u, ∆n). (4.2) the following result provides a priory bounds for rn(f, u, ∆n) in several instances of f and u as above. quasi grüss type inequalities 63 proposition 1. assume that f : c(0, 1) → c satisfies the following lipschitz type condition |f(z) − f(w)| ≤ l|z − w| for any w, z ∈ c(0, 1), where l > 0 is given given. if [a, b] ⊆ [0, 2π] and the function u : [a, b] → c is of bounded variation on [a, b], then for any partition ∆n : a = x0 < x1 < · · · < xn−1 < xn = b with the norm ν(∆n) < 2π we have the error bound |rn(f, u, ∆n)| ≤ 8l n−1∑ k=0 1 xk+1 − xk sin2 ( xk+1 − xk 4 )xk+1∨ xk (u) ≤ 1 2 lν(∆n) b∨ a (u). (4.3) proof. since ν(∆n) < 2π, then on writing inequality (2.7) on each interval [xk, xk+1], where 0 ≤ k ≤ n − 1, we have∣∣∣∣ ∫ xk+1 xk f ( eit ) du(t) − u(xk+1) − u(xk) xk+1 − xk ∫ xk+1 xk f ( eit ) dt ∣∣∣∣ ≤ 8l xk+1 − xk sin2 ( xk+1 − xk 4 )xk+1∨ xk (u). utilising the generalized triangle inequality we then have |rn(f, u, ∆n)| = ∣∣∣∣∣ n−1∑ k=0 [∫ xk+1 xk f ( eit ) du(t) − u(xk+1) − u(xk) xk+1 − xk ∫ xk+1 xk f ( eit ) dt ]∣∣∣∣∣ ≤ n−1∑ k=0 ∣∣∣∣ [∫ xk+1 xk f ( eit ) du(t) − u(xk+1) − u(xk) xk+1 − xk ∫ xk+1 xk f ( eit ) dt ]∣∣∣∣ ≤ n−1∑ k=0 8l xk+1 − xk sin2 ( xk+1 − xk 4 )xk+1∨ xk (u) ≤ 8l max 0≤k≤n−1 { 1 xk+1 − xk sin2 ( xk+1 − xk 4 )}n−1∑ k=0 xk+1∨ xk (u) = 8l max 0≤k≤n−1 { 1 xk+1 − xk sin2 ( xk+1 − xk 4 )} b∨ a (u). 64 s. s. dragomir since 1 xk+1 − xk sin2 ( xk+1 − xk 4 ) ≤ 1 16 (xk+1 − xk) then max 0≤k≤n−1 { 1 xk+1 − xk sin2 ( xk+1 − xk 4 )} ≤ 1 16 ν(∆n) and the last part of (4.3) also holds. remark 4. the above proposition has some particular cases of interest. if we take for instance a = 0, x1 = π and b = 2π, then we have from (4.3) that∣∣∣∣ ∫ 2π 0 f ( eit ) du(t) − u(π) − u(0) π ∫ π 0 f ( eit ) dt − u(2π) − u(π) π ∫ 2π π f(eit)dt ∣∣∣∣ ≤ 8l π 2π∨ 0 (u). remark 5. we observe that the last bound in (4.3) provides a simple way to choose a division such that the accuracy in approximation is better that a given small ε > 0. indeed, if we want 1 2 lν(∆n) b∨ a (u) ≤ ε then we need to take ∆n such that ν(∆n) ≤ 2ε∨b a(u)l . the above proposition can be also utilized to approximate functions of unitary operators as follows. we consider the following partition of the interval [0, 2π] γn : 0 = λ0 < λ1 < · · · < λn−1 < λn = 2π where 0 ≤ k ≤ n − 1. if u is a unitary operator on the hilbert space h and {eλ}λ∈[0,2π], the spectral family of u, then we can introduce the following sums: dn(f, u,γn; x, y) := n−1∑ k=0 1 λk+1 − λk ∫ λk+1 λk f ( eit ) dt · ⟨( eλk+1 − eλk ) x, y ⟩ . (4.4) quasi grüss type inequalities 65 corollary 1. assume that f : c(0, 1) → c satisfies the following lipschitz type condition |f(z) − f(w)| ≤ l|z − w| for any w, z ∈ c(0, 1), where l > 0 is given. assume also that u is a unitary operator on the hilbert space h and {eλ}λ∈[0,2π] is the spectral family of u. if γn is a partition of the interval [0, 2π] with ν(γn) < 2π then we have the representation ⟨f(u)x, y⟩ = dn(f, u, γn; x, y) + rn(f, u, γn; x, y) (4.5) with the error rn(f, u, ∆n; x, y) satisfying the bounds |rn(f, u, γn; x, y)| ≤ 8l n−1∑ k=0 1 λk+1 − λk sin2 ( λk+1 − λk 4 )λk+1∨ λk (⟨ e(·)x, y ⟩) ≤ 1 2 lν(γn) 2π∨ 0 (⟨ e(·)x, y ⟩) ≤ 1 2 lν(γn)∥x∥∥y∥ (4.6) for any x, y ∈ h. remark 6. consider the exponential mean ez(p, q) := exp(pz) − exp(qz) p − q defined for complex numbers z and the real numbers p, q with p ̸= q. for the function f(z) = zm with m an integer we have∫ p q f ( eit ) dt = ∫ p q eimtdt = 1 im ( eimp − eimq ) = 1 im (p − q)eeim(p, q). for a partition γn as above, define the sum pn(u, γn; x, y) := 1 im n−1∑ k=0 eeim(λk+1, λk) ⟨( eλk+1 − eλk ) x, y ⟩ . (4.7) we can approximate the power m of an unitary operator as follows: ⟨umx, y⟩ = pn(u, γn; x, y) + tn(u, γn; x, y) (4.8) 66 s. s. dragomir where the error tn(u, γn; x, y) satisfies the bounds |tn(u, γn; x, y)| ≤ 8|m| n−1∑ k=0 1 λk+1 − λk sin2 ( λk+1 − λk 4 )λk+1∨ λk (⟨ e(·)x, y ⟩) ≤ 1 2 |m|ν(γn) 2π∨ 0 (⟨ e(·)x, y ⟩) ≤ 1 2 |m|ν(γn)∥x∥∥y∥ (4.9) for any vectors x, y ∈ h. references [1] m.w. alomari, a companion of ostrowski’s inequality for the riemann– stieltjes integral ∫ b a f(t)du(t), where f is of bounded variation and u is of rh-hölder type and applications, appl. math. comput. 219 (9) (2013), 4792 – 4799. [2] m.w. alomari, some grüss type inequalities for riemann-stieltjes integral and applications, acta math. univ. comenian. (n.s.) 81 (2) (2012), 211 – 220. [3] g.a. anastassiou, grüss type inequalities for the stieltjes integral, nonlinear funct. anal. appl. 12 (4) (2007), 583 – 593. [4] g.a. anastassiou, a new expansion formula, cubo mat. educ. 5 (1) (2003), 25 – 31. 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[11] s.s. dragomir, i.a. fedotov, an inequality of grüss type for the riemann-stieltjes integral and applications for special means, tamkang j. math., 29 (4) (1998), 287 – 292. quasi grüss type inequalities 67 [12] s.s. dragomir, i. fedotov, a grüss type inequality for mappings of bounded variation and applications to numerical analysis, nonlinear funct. anal. appl., 6 (3) (2001), 425 – 433. [13] g. helmberg, “introduction to spectral theory in hilbert space”, john wiley, new york, 1969. [14] z. liu, refinement of an inequality of grüss type for riemann-stieltjes integral, soochow j. math., 30 (4) (2004), 483 – 489. e extracta mathematicae vol. 32, núm. 1, 1 – 24 (2017) a note on some isomorphic properties in projective tensor products ioana ghenciu mathematics department, university of wisconsin-river falls, wisconsin, 54022, usa ioana.ghenciu@uwrf.edu presented by jesús m. f. castillo received may 23, 2016 abstract: a banach space x is sequentially right (resp. weak sequentially right) if every right subset of x∗ is relatively weakly compact (resp. weakly precompact). a banach space x has the l-limited (resp. the wl-limited) property if every l-limited subset of x∗ is relatively weakly compact (resp. weakly precompact). we study banach spaces with the weak sequentially right and the wl-limited properties. we investigate whether the projective tensor product of two banach spaces x and y has the sequentially right property when x and y have the respective property. key words: r-sets, l-limited sets, sequentially right spaces, l-limited property. ams subject class. (2010): 46b20, 46b25, 46b28. 1. introduction a bounded subset a of a banach space x is called a dunford-pettis (dp) (resp. limited) subset of x if every weakly null (resp. w∗-null) sequence (x∗n) in x∗ tends to 0 uniformly on a; i.e., lim n ( sup{|x∗n(x)| : x ∈ a} ) = 0. a sequence (xn) is dp (resp. limited) if the set {xn : n ∈ n} is dp (resp. limited). a subset s of x is said to be weakly precompact provided that every sequence from s has a weakly cauchy subsequence. every dp (resp. limited) set is weakly precompact [37, p. 377], [1] (resp. [4, proposition]). an operator t : x → y is called weakly precompact (or almost weakly compact) if t(bx) is weakly precompact and completely continuous (or dunford-pettis) if t maps weakly convergent sequences to norm convergent sequences. in [35] the authors introduced the right topology on a banach space x. it is the restriction of the mackey topology τ(x∗∗, x) to x and it is also the 1 2 i. ghenciu topology of uniform convergence on absolutely convex σ(x∗, x∗∗) compact subsets of x∗. further, τ(x∗∗, x) can also be viewed as the topology of uniform convergence on relatively σ(x∗, x∗∗) compact subsets of x∗ [26]. a sequence (xn) in a banach space x is right null if and only if it is weakly null and dp (see proposition 1). an operator t : x → y is called pseudo weakly compact (pwc) (or dunford-pettis completely continuous (dpcc)) if it takes right null sequences in x into norm null sequences in y ([35], [25]). every completely continuous operator t : x → y is pseudo weakly compact. if t : x → y is an operator with weakly precompact adjoint, then t is a pseudo weakly compact operator ([18, corollary 5]). a subset k of x∗ is called a right set (r-set) if each right null sequence (xn) in x tends to 0 uniformly on k [26]; i.e., lim n ( sup{|x∗(xn)| : x∗ ∈ k} ) = 0. a banach space x is said to be sequentially right (sr) (has property (sr)) if every pseudo weakly compact operator t : x → y is weakly compact, for any banach space y [35]. banach spaces with property (v ) are sequentially right ([35, corollary 15]). a subset a of a dual space x∗ is called an l-limited set if every weakly null limited sequence (xn) in x converges uniformly on a [39]; i.e., lim n ( sup{|x∗(xn)| : x∗ ∈ a} ) = 0. a banach space x has the l-limited property if every l-limited subset of x∗ is relatively weakly compact [39]. an operator t : x → y is called limited completely continuous (lcc) if t maps weakly null limited sequences to norm null sequences [40]. in this paper we introduce the weak sequentially right (wsr) and wllimited properties. a banach space x is said to have the weak sequentially right (wsr) (resp. the wl-limited) property if every right (resp. l-limited) subset of x∗ is weakly precompact. we obtain some characterizations of these properties with respect to some geometric properties of banach spaces, such as the gelfand-phillips property, the grothendieck property, and properties (wv ) and (wl). we generalize some results from [39]. we also show that property (sr) can be lifted from a certain subspace of x to x. we study whether the projective tensor product x ⊗ πy has the (sr) (resp. the l-limited) property if l(x, y ∗) = k(x, y ∗), and x and y have the respective property. we prove that in some cases, if x⊗π y has the (wsr) property, then l(x, y ∗) = k(x, y ∗). isomorphic properties in projective tensor products 3 2. definitions and notation throughout this paper, x, y , e, and f will denote banach spaces. the unit ball of x will be denoted by bx and x ∗ will denote the continuous linear dual of x. an operator t : x → y will be a continuous and linear function. we will denote the canonical unit vector basis of c0 by (en) and the canonical unit vector basis of ℓ1 by (e ∗ n). the set of all operators, weakly compact operators, and compact operators from x to y will be denoted by l(x, y ), w(x, y ), and k(x, y ). the projective tensor product of x and y will be denoted by x ⊗π y . a bounded subset a of x∗ is called an l-set if each weakly null sequence (xn) in x tends to 0 uniformly on a; i.e., lim n ( sup{|x∗(xn)| : x∗ ∈ a} ) = 0. a banach space x has the dunford-pettis property (dpp) if every weakly compact operator t : x → y is completely continuous, for any banach space y . schur spaces, c(k) spaces, and l1(µ) spaces have the dpp . the reader can check [8], [9], and [10] for a guide to the extensive classical literature dealing with the dpp . a banach space x has the dunford-pettis relatively compact property (dprcp) if every dunford-pettis subset of x is relatively compact [14]. schur spaces have the dprcp . the space x does not contain a copy of ℓ1 if and only if x∗ has the dprcp if and only if every l-set in x∗ is relatively compact ([14, theorem 1], [13, theorem 2]). the space x has the gelfand-phillips (gp) property if every limited subset of x is relatively compact. the following spaces have the gelfand-phillips property: schur spaces; spaces with w∗-sequential compact dual unit balls (for example subspaces of weakly compactly generated spaces, separable spaces, spaces whose duals have the radon-nikodým property, reflexive spaces, and spaces whose duals do not contain ℓ1); dual spaces x ∗ whith x not containining ℓ1; banach spaces with the separable complementation property, i.e., every separable subspace is contained in a complemented separable subspace (for example l1(µ) spaces, where µ is a positive measure) [42, p. 31], [4, proposition], [12, theorem 3.1 and p. 384], [11, proposition 5.2], [13, corollary 5]. a series ∑ xn in x is said to be weakly unconditionally convergent (wuc) if for every x∗ ∈ x∗, the series ∑ |x∗(xn)| is convergent. an operator t : x → y is called unconditionally converging if it maps weakly unconditionally convergent series to unconditionally convergent ones. 4 i. ghenciu a bounded subset a of x∗ is called a v -subset of x∗ provided that lim n ( sup{ |x∗(xn)| : x∗ ∈ a } ) = 0) for each wuc series ∑ xn in x. a banach space x has property (v ) if every v -subset of x∗ is relatively weakly compact [33]. a banach space x has property (v ) if every unconditionally converging operator t from x to any banach space y is weakly compact [33, proposition 1]. c(k) spaces and reflexive spaces have property (v ) ([33, theorem 1, proposition 7]). a banach space x has property (wv ) if every v -subset of x∗ is weakly precompact [41]. a banach space x has the reciprocal dunford-pettis property (rdpp) if every completely continuous operator t from x to any banach space y is weakly compact. the space x has the rdpp if and only if every l-set in x∗ is relatively weakly compact [28]. banach spaces with property (v ) have the rdpp [33]. a banach space x has property (wl) if every l-set in x∗ is weakly precompact [19]. a topological space s is called dispersed (or scattered) if every nonempty closed subset of s has an isolated point. a compact hausdorff space k is dispersed if and only if ℓ1 ̸↪→ c(k) [34, main theorem]. the banach-mazur distance d(x, y ) between two isomorphic banach spaces x and y is defined by inf(∥t∥∥t −1∥), where the infinum is taken over all isomorphisms t from x onto y . a banach space x is called an l∞-space (resp. l1-space) [5, p. 7] if there is a λ ≥ 1 so that every finite dimensional subspace of x is contained in another subspace n with d(n, ℓn∞) ≤ λ (resp. d(n, ℓn1) ≤ λ) for some integer n. complemented subspaces of c(k) spaces (resp. l1(µ)) spaces) are l∞-spaces (resp. l1-spaces) ([5, proposition 1.26]). the dual of an l1-space (resp. l∞-space) is an l∞-space (resp. l1space) ([5, proposition 1.27]). the l∞-spaces, l1-spaces, and their duals have the dpp ([5, corollary 1.30]). 3. the weak sequentially right and wl-limited properties the following result gives a characterization of right null sequences. proposition 1. a sequence (xn) in a banach space x is right null if and only if it is weakly null and dp. proof. suppose that (xn) is a right null sequence in x. then (xn) is weakly null, since the right topology is stronger than the weak topology. isomorphic properties in projective tensor products 5 let (x∗n) be a weakly null sequence in x ∗. since {x∗n : n ∈ n} is relatively weakly compact in x∗ and (xn) is right null, (xn) converges uniformly on {x∗n : n ∈ n}. therefore limn supi |x∗i (xn)| = 0, and thus limn |x ∗ n(xn)| = 0. hence {xn : n ∈ n} is a dp set. suppose that (xn) is a weakly null dp sequence. let k be a relatively weakly compact subset of x∗. suppose that (xn) does not converge uniformly on k. let ϵ > 0 and let (x∗n) be a sequence in k so that |x∗n(xn)| > ϵ for all n. without loss of generality suppose that (x∗n) converges weakly to x ∗, x∗ ∈ x∗. since (x∗n − x∗) is weakly null in x∗ and (xn) is dp, limn(x∗n − x∗)(xn) = 0. thus limn x ∗ n(xn) = 0, a contradiction. hence (xn) converges uniformly to zero on k, and thus (xn) is right null. a banach space x is sequentially right if and only if every right subset of x∗ is relatively weakly compact [26, theorem 3.25]. a banach space x has the l-limited property if and only if every limited completely continuous operator t : x → y is weakly compact, for every banach space y [39, theorem 2.8]. in the next theorem we give elementary operator theoretic characterizations of weak precompactness, relative weak compactness, and relative norm compactness for right sets and l-limited sets. the argument contains the theorems in [26] and [39] just cited. we say that a banach space x is weak sequentially right (wsr) or has the (wsr) property (resp. has the wl-limited property) if every right (resp. l-limited) subset of x∗ is weakly precompact. if ℓ1 ̸↪→ x∗, then x is weak sequentially right and has the wl-limited property, by rosenthal’s theorem ([8, ch. xi]). theorem 2. let x be a banach space. the following assertions are equivalent: 1. (i) for every banach space y , every pseudo weakly compact operator t : x → y has a weakly precompact (weakly compact, resp. compact) adjoint. (ii) every pseudo weakly compact operator t : x → ℓ∞ has a weakly precompact (weakly compact, resp. compact) adjoint. (iii) every right subset of x∗ is weakly precompact (relatively weakly compact, resp. relatively compact). 6 i. ghenciu 2. (i) for every banach space y , every limited completely continuous operator t : x → y has a weakly precompact (weakly compact, resp. compact) adjoint. (ii) every limited completely continuous operator t : x → ℓ∞ has a weakly precompact (weakly compact, resp. compact) adjoint. (iii) every l-limited subset of x∗ is weakly precompact (relatively weakly compact, resp. relatively compact). proof. we will show that 1.(i)⇒1.(ii)⇒1.(iii)⇒1.(i) in the weakly precompact case as well as 2.(i)⇒2.(ii)⇒2.(iii)⇒2.(i) in the compact case. these two arguments are similar, and the arguments for the remaining implications of the theorem follow the same pattern. 1. (weakly precompact) (i)⇒(ii) is clear. (ii)⇒(iii) let k be a right subset of x∗ and let (x∗n) be a sequence in k. define t : x → ℓ∞ by t(x) = (x∗i (x)). let (xn) be a right null sequence in x. since k is a right set, lim n ∥t(xn)∥ = lim n sup i |x∗i (xn)| = 0. therefore t is pseudo weakly compact, and thus t ∗ : ℓ∗∞ → x∗ is weakly precompact. hence (t ∗(e∗n)) = (x ∗ n) has a weakly cauchy subsequence. (iii)⇒(i) let t : x → y be a pseudo weakly compact operator. let (xn) be a right null sequence in x. if y∗ ∈ by ∗, ⟨t ∗(y∗), xn⟩ ≤ ∥t(xn)∥ → 0. then t ∗(by ∗) is a right subset of x ∗. therefore t ∗(by ∗) is weakly precompact, and thus t ∗ is weakly precompact. 2. (compact) (i)⇒(ii) is clear. (ii)⇒(iii) let k be an l-limited subset of x∗ and let (x∗n) be a sequence in k. define t : x → ℓ∞ as above and note that t is limited completely continuous. thus t ∗ : ℓ∗∞ → x∗ is compact, and (t ∗(e∗n)) = (x∗n) has a norm convergent subsequence. (iii)⇒(i) let t : x → y be a limited completely continuous operator. let (xn) be a weakly null limited sequence in x. if y ∗ ∈ by ∗, ⟨t ∗(y∗), xn⟩ ≤ ∥t(xn)∥ → 0. then t ∗(by ∗) is an l-limited subset of x∗. therefore t ∗(by ∗) is relatively compact, and thus t ∗ is compact. corollary 3. if x is weak sequentially right (has the wl-limited, resp. the l-limited property), then every quotient space of x has the same property. isomorphic properties in projective tensor products 7 proof. we only prove the result for the weak sequentially right property. the proofs for the other properties are similar. suppose that x is weak sequentially right. let z be a quotient space of x and q : x → z be a quotient map. let t : z → e be a pseudo weakly compact operator. then tq : x → e is pseudo weakly compact, and thus (tq)∗ is weakly precompact by theorem 2. since q∗t ∗(b∗e) is weakly precompact and q∗ is an isomorphism, t ∗(b∗e) is weakly precompact. apply theorem 2. corollary 4. suppose x is weak sequentially right and y is a banach space. then an operator t : x → y is pseudo weakly compact if and only if t ∗ : y ∗ → x∗ is weakly precompact. proof. if t : x → y is pseudo weakly compact, then t ∗ : y ∗ → x∗ is weakly precompact by theorem 2, since x is weak sequentially right. the converse follows from [18, corollary 5]. corollary 5. (i) if x is weak sequentially right (resp. has the wllimited property), then every pseudo weakly compact (resp. limited completely continuous) operator t : x → y is weakly precompact. (ii) if x is an infinite dimensional space with the schur property, then x is not weak sequentially right (resp. does not have the wl-limited property). (iii) if x is weak sequentially right (resp. has the wl-limited property), then ℓ1 ̸ c ↪→ x. proof. (i) suppose x is weak sequentially right (resp. has the wl-limited property). let t : x → y be pseudo weakly compact (resp. limited completely continuous). then t ∗ is weakly precompact by theorem 2. hence t is weakly precompact, by [2, corollary 2]. (ii) since x has the schur property, the identity operator i : x → x is pseudo weakly compact (resp. limited completely continuous). since x is an infinite dimensional space with the schur property, i is not weakly precompact. apply (i). (iii) apply corollary 3 and (ii). corollary 6. a banach space x has the l-limited property if every separable subspace of x has the same property. proof. let t : x → y be a limited completely continuous operator. then for every closed subspace z of x, t |z is limited completely continuous. let 8 i. ghenciu (xn) be a sequence in bx and let z = [xn : n ∈ n] be the closed linear span of (xn). since z is a separable subspace of x, z has the l-limited property. since t |z is limited completely continuous, it is weakly compact by theorem 2. then there is a subsequence (xnk) of (xn) so that (t(xnk)) is weakly convergent. thus t is weakly compact. apply theorem 2. example. corollary 6 cannot be reversed. indeed, consider ℓ1 as a subspace of ℓ∞. by [39, theorem 2.11], ℓ∞ has the l-limited property. however, ℓ1 does not have the l-limited property, by [39, corollary 2.9] (or corollary 5 (ii)). theorem 7. the banach space x has the dpp if and only if every right subset of x∗ is an l-set. proof. suppose x has the dpp . then every weakly null sequence (xn) is dp ([9, theorem 1]). therefore every right subset of x∗ is an l-set. conversely, let t : x → y be a pseudo weakly compact operator. then t ∗(by ∗) is a right subset of x ∗, hence an l-set. therefore t is completely continuous, and thus x has the dpp by [26, proposition 3.17], [25, theorem 1.5], [18, theorem 10]. corollary 8. suppose that x has the dpp. then the following are equivalent: (i) x does not contain a copy of ℓ1. (ii) every l-set in x∗ is relatively compact. (iii) every right subset of x∗ is relatively compact. (iv) x∗ has the schur property. proof. (i)⇔(ii) by [13, theorem 2]. (ii)⇔(iii) by theorem 7. (i)⇔(iv) by [9, p. 23]. corollary 9. x∗ has the schur property if and only if every right subset of x∗ is relatively compact. proof. if x∗ has the schur property, then x has the dpp and x does not contain a copy of ℓ1 ([9, p. 23]). hence every right subset of x ∗ is relatively compact by corollary 8. isomorphic properties in projective tensor products 9 conversely, let (x∗n) be a weakly cauchy sequence in x ∗. then (x∗n) is a right set, by the proof of [26, corollary 3.26]. thus (x∗n) is relatively compact, and x∗ has the schur property. corollary 10. (i) suppose x has the dpp and y has the dprcp. then any operator t : x → y is completely continuous. (ii) the space x has the dpp and the dprcp if and only if x has the schur property. proof. (i) let t : x → y be an operator. since y has the dprcp , t is pseudo weakly compact. then t ∗(by ∗) is a right set, thus an l-set in x ∗ (by theorem 7). hence t is completely continuous. (ii) suppose x has the dpp and the dprcp . then the identity operator i : x → x is completely continuous by (i). hence x has the schur property. if x has the schur property, then x has the dpp and the dprcp . corollary 10 (i) generalizes [13, corollary 6] when y is a dual space e∗ with e not containing ℓ1 (since e ∗ has the dprcp [14, theorem 1]). a bounded subset a of x∗ is called w∗sequentially compact if every sequence from a has a subsequence which converges to a point in the w∗topology of x∗. the following theorem generalizes [39, theorem 2.2 (b), (c)]. theorem 11. if (x∗n) is a w ∗-cauchy sequence in x∗, then {x∗n : n ∈ n} is an l-limited set. proof. supppse that (x∗n) is a w ∗-cauchy sequence in x∗ and {x∗n : n ∈ n} is not an l-limited set. by passing to a subsequence if necessary, there is an ϵ > 0 and a weakly null limited sequence (xn) in x such that |x∗n(xn)| > ϵ for all n. let k1 = 1 and choose k2 > k1 so that |x∗k1(xk2)| < ϵ/2. we can do this since (xn) is weakly null. continue inductively. choose kn > kn−1 so that |x∗kn−1(xkn)| < ϵ/2 for all n. then |(x∗kn − x ∗ kn−1 )(xkn)| = |x ∗ kn (xkn) − x ∗ kn−1 (xkn)| > ϵ/2. this is a contradiction, since (x∗kn − x ∗ kn−1 ) is w∗-null in x∗ and (xkn) is limited in x. a banach space x has the grothendieck property if every w∗convergent sequence in x∗ is weakly convergent [10, p. 179]. a space x is weakly sequentially complete if every weakly cauchy sequence in x is weakly convergent. 10 i. ghenciu corollary 12. (i) if x has the l-limited property, then x∗ is weakly sequentially complete. (ii) ([39, theorem 2.10]) if x has the l-limited property, then x is a grothendieck space. proof. (i) suppose that x has the l-limited property. let (x∗n) be a weakly cauchy sequence in x∗. by theorem 11, {x∗n : n ∈ n} is an l-limited set, and thus relatively weakly compact. hence (x∗n) is weakly convergent. (ii) let (x∗n) be a w ∗convergent sequence in x∗. by theorem 11, (x∗n) is an l-limited set, thus relatively weakly compact. hence (x∗n) is weakly convergent. corollary 13. (i) a banach space x with the gelfand-phillips property has the wl-limited property if and only if x∗ contains no copy of ℓ1. (ii) a banach space x with the dprcp has the (wsr) property if and only if x∗ contains no copy of ℓ1. (iii) if x has the wl-limited property, then c0 is not complemented in x. (iv) ([39, corollary 2.9]) a banach space x is reflexive if and only if it has the gelfand-phillips property and the l-limited property. (v) ([7, corollary 17]) a banach space x is reflexive if and only if it has the dprcp and the (sr) property. proof. (i) suppose that x has the gelfand-phillips property and the wllimited property. then the identity operator i : x → x is limited completely continuous (since x has the gelfand-phillips property) and i∗ : x∗ → x∗ is weakly precompact by theorem 2. hence x∗ contains no copy of ℓ1, by rosenthal’s ℓ1 theorem. the converse follows by rosenthal’s ℓ1 theorem. (ii) the proof is similar to that of (i). (iii) suppose that x has the wl-limited property. since c0 is separable, it has the gelfand-phillips property [4, proposition]. by (i), c0 does not have the wl-limited property. hence c0 is not complemented in x by corollary 3. (iv) if x is reflexive, then it has the gelfand-phillips property [4, proposition] and the l-limited property. conversely, x∗ contains no copy of ℓ1 by (i) and x∗ is weakly sequentially complete by corollary 12. then x∗, thus x, is reflexive. (v) suppose x is reflexive. then x has the (sr) property and x∗ does not contain a copy of ℓ1. hence x ∗∗, thus x, has the dprcp ([13, theorem 2]). conversely, x∗ contains no copy of ℓ1 by (i) and x ∗ is weakly sequentially complete by [26, corollary 3.26]. then x is reflexive. isomorphic properties in projective tensor products 11 example. the converse of corollary 12 (i) does not hold. let x be the first bourgain-delbaen space [5, p. 25]. then x has the schur property and x∗ is weakly sequentially complete. since x has the schur property, x does not have the l-limited property (by corollary 13 (iv)). corollary 14. (i) if x has property (wv ), then x is weak sequentially right. (ii) if x has the l-limited (resp. the wl-limited) property, then x is sequentially right (resp. weak sequentially right). (iii) if x is sequentially right (resp. weak sequentially right), then it has the rdpp (resp. property (wl)). (iv) if x is an infinite dimensional space with the l-limited property, then x∗ does not have the schur property. proof. (i) suppose x has property (wv ). let t : x → y be pseudo weakly compact. then t is unconditionally converging [35, proposition 14]. hence t ∗ is weakly precompact [19, theorem 1]. apply theorem 2. (ii) suppose x has the the l-limited (resp. the wl-limited) property. let (xn) be a weakly null limited sequence in x. then (xn) is a weakly null dp sequence. hence every right subset of x∗ is l-limited, thus relatively weakly compact (resp. weakly precompact). (iii) suppose x is sequentially right (resp. weak sequentially right). every l-set in x∗ is a right set, thus relatively weakly compact (resp. weakly precompact). hence x has the rdpp [28] (resp. property (wl)). (iv) suppose that x has the l-limited property. then x has the grothendieck property, by corollary 12 (ii). by the jossefson-nissezweig theorem, there is a w∗-null sequence (x∗n) in x ∗ of norm one. then (x∗n) is weakly null and not norm null, and x∗ does not have the schur property. the fact that a space with property (sr) has the rdpp was obtained in [26, corollary 3.3]. example. the converse of corollary 14 (i) is not true. let y be the second bourgain-delbaen space [5, p. 25]. the space y is a non-reflexive l∞space with the dpp that does not contain c0 or ℓ1 and such that y ∗ ≃ ℓ1. the space y is sequentially right by corollary 8. since y does not contain c0, the identity operator i : y → y is unconditionally converging ([8, p. 54]) and i∗ : y ∗ → y ∗ is not weakly precompact (since y ∗ ≃ ℓ1). thus y does not have property (wv ) by [19, theorem 1]. 12 i. ghenciu the converse of corollary 14 (ii) (strong properties) is not true. the second bourgain-delbaen space y is sequentially right and does not have the llimited property (by corollary 14 (iv)). the converse of corollary 14 (iii) (strong properties) is not true. let j be the original james space [24]. since j is separable and 1-codimensional in j∗∗, all duals of j are separable and ℓ1 fails to embed in any of them. moreover, none of these spaces can be weakly sequentially complete. thus j and its duals are weak sequentially right, but none of these spaces are sequentially right by [26, corollary 3.26], since their duals are not weakly sequentially complete. since j does not contain ℓ1, every completely continuous operator on j is compact (by a result of odell [37, p. 377]), and thus weakly compact. hence j has the rdpp . the following theorem shows that the space e has property (sr) if some subspace of it has property (sr). lemma 15. ([23, theorem 2.7]) let e be a banach space, f a reflexive subspace of e (resp. a subspace not containing copies of ℓ1), and q : e → e/f the quotient map. let (xn) be a bounded sequence in e such that (q(xn)) is weakly convergent (resp. weakly cauchy). then (xn) has a weakly convergent (resp. weakly cauchy) subsequence. let e be a banach space and f be a subspace of e∗. let ⊥f = { x ∈ e : y∗(x) = 0 for all y∗ ∈ f } . theorem 16. (i) let e be a banach space and f be a reflexive subspace of e∗. if ⊥f has property (sr) (resp. the l-limited property), then e has the same property. (ii) let e be a banach space and f be a subspace of e∗ not containing copies of ℓ1. if ⊥f has property (wsr) (resp. the wl-limited property), then e has the same property. proof. we only prove (i) for the (sr) property. the other proofs are similar. suppose that ⊥f has property (sr). let q : e∗ → e∗/f be the quotient map and i : e∗/f → (⊥f)∗ be the natural surjective isomorphism ([31, theorem 1.10.16]). it is known that iq : e∗ → (⊥f)∗ is w∗ − w∗ continuous, since iq(x∗) is the restriction of x∗ to ⊥f ([31, theorem 1.10.16]). then there is an operator s :⊥ f → e such that iq = s∗. isomorphic properties in projective tensor products 13 let t : e → g be a pseudo weakly compact operator. then ts :⊥ f → g is pseudo weakly compact. since ⊥f has property (sr), ts has a weakly compact adjoint, by theorem 2. since s∗t ∗ = iqt ∗ is weakly compact and i is a surjective isomorphism, qt ∗ is weakly compact. let (x∗n) be a sequence in bg∗. by passing to a subsequence, we can assume that (qt ∗(x∗n)) is weakly convergent. hence (t ∗(x∗n)) has a weakly convergent subsequence by lemma 15. thus e has property (sr). the w∗ − w continuous operators from x∗ to y will be denoted by lw∗(x ∗, y ). theorem 17. let x be a banach space and a be a bounded subset of x∗. the following are equivalent: (i) a is an l-limited set. (ii) every operator t ∈ lw∗(x∗, c0) that is w∗-norm sequentially continuous maps a into a relatively compact set. proof. (i)⇒(ii) let t ∈ lw∗(x∗, c0) be an operator so that t is w∗-norm sequentially continuous. note that t ∗ ∈ lw∗(ℓ1, x), (xn) = (t ∗(e∗n)) is a weakly null sequence in x, and t(x∗) = (x∗(xi))i. if (x ∗ n) is a w ∗-null sequence in x∗ and y ∈ bℓ1, then |⟨x∗n, t ∗(y)⟩| ≤ ∥t(x∗n)∥ → 0. hence t ∗(bℓ1), thus (xn), is limited. since a is an l-limited set, supx∗∈a |x∗(xn)| → 0. therefore t(a) is relatively compact in c0, by the characterization of relatively compact subsets of c0. (ii)⇒(i) let (xn) be a weakly null limited sequence in x. define t : x∗ → c0 by t(x∗) = (x∗(xn))n. note that t ∗(b) = ∑ bnxn, b = (bn) ∈ ℓ1, t ∗(ℓ1) ⊆ x, and t ∈ lw∗(x∗, c0). if (x∗n) is a w∗-null sequence in x∗, then ∥t(x∗n)∥ = sup i |x∗n(xi)| → 0, since (xi) is limited. hence t is w ∗-norm sequentially continuous operator, and t(a) is relatively compact in c0. by the characterization of relatively compact subsets of c0, supx∗∈a |x∗(xn)| → 0, and thus a is an l-limited subset of x∗. an operator t : x → y is called limited if t(bx) is a limited subset of y ([4]). the operator t is limited if and only if t ∗ : y ∗ → x∗ is w∗-norm sequentially continuous. 14 i. ghenciu corollary 18. let x be a banach space and a be a bounded subset of x∗. the following are equivalent: (i) a is an l-limited set. (ii) for every limited operator s ∈ lw∗(ℓ1, x), s∗(a) is relatively compact. proof. (i)⇒(ii) let s ∈ lw∗(ℓ1, x) be a limited operator. then s∗ ∈ lw∗(x ∗, c0) and s ∗ is w∗-norm sequentially continuous. by theorem 17, s∗(a) is relatively compact. (ii)⇒(i) let t ∈ lw∗(x∗, c0) be a w∗-norm sequentially continuous operator and let s = t ∗. then s ∈ lw∗(ℓ1, x), s is limited, and s∗(a) is relatively compact. by theorem 17, a is an l-limited set. corollary 19. suppose that a is a bounded subset of x∗ such that for every ϵ > 0, there is an l-limited subset aϵ of x ∗ such that a ⊆ aϵ + ϵbx∗. then a is an l-limited set. proof. suppose that a satisfies the hypothesis. let ϵ > 0 and aϵ as in the hypothesis. let t ∈ lw∗(x∗, c0) be an operator such that t is w∗-norm sequentially continuous and ∥t∥ ≤ 1. then t(a) ⊆ t(aϵ) + ϵbc0, and t(aϵ) is relatively compact by theorem 17. then t(a) is relatively compact [8, p. 5], and thus a is an l-limited set by theorem 17. 4. the (wsr) and wl-limited properties in projective tensor products in this section we consider the (sr) and l-limited properties in the projective tensor product x ⊗π y . we begin by noting that there are examples of banach spaces x and y such that x ⊗π y has the (sr) and l-limited properties. if 1 < q′ < p < ∞, then l(ℓp, ℓq′) = k(ℓp, ℓq′) ([36], [10, p. 247]). if q is the conjugate of q′, then ℓp ⊗π ℓq is reflexive (by [38, theorem 4.19], [10, p. 248]), and thus has the (sr) and l-limited properties. then the spaces x = ℓp and y = ℓq are as desired. if h ⊆ l(x, y ), x ∈ x and y∗ ∈ y ∗, let h(x) = {t(x) : t ∈ h} and h∗(y∗) = {t ∗(y∗) : t ∈ h}. in the proofs of theorems 23 and 25 we will need the following results. isomorphic properties in projective tensor products 15 theorem 20. ([20, theorem 1]) let h be a subset of k(x, y ) such that (i) h(x) is weakly precompact compact for all x ∈ x. (ii) h∗(y∗) is relatively weakly compact for all y∗ ∈ y ∗. then h is weakly precompact. theorem 21. ([20, theorem 3]) suppose that l(x, y ) = k(x, y ) and h is a subset of k(x, y ) such that: (i) h(x) is relatively weakly compact for all x ∈ x. (ii) h∗(y∗) is relatively weakly compact for all y∗ ∈ y ∗. then h is relatively weakly compact. lemma 22. suppose l(x, y ∗) = k(x, y ∗). if (xn) is a weakly null dp sequence in x and (yn) is a dp sequence in y , then (xn ⊗yn) is a weakly null dp sequence in x ⊗π y . proof. suppose that (xn) is weakly null dp in x and ∥yn∥ ≤ m for all n ∈ n. let t ∈ l(x, y ∗) ≃ (x ⊗π y )∗ ([10, p. 230]). since t is completely continuous, ⟨t, xn ⊗ yn⟩ ≤ m∥t(xn)∥ → 0. thus (xn ⊗ yn) is weakly null in x ⊗π y . let (an) be a weakly null sequence in (x ⊗π y )∗ ≃ l(x, y ∗) and let x∗∗ ∈ x∗∗. since the map γx∗∗ : l(x, y ∗) = k(x, y ∗) → y ∗, γx∗∗(t) = t ∗∗(x∗∗) is linear and bounded, (a∗∗n (x ∗∗)) is weakly null in y ∗. therefore ⟨x∗∗, a∗n(yn)⟩ = ⟨a ∗∗ n (x ∗∗), yn⟩ → 0, since (yn) is dp in y . hence (a ∗ n(yn)) is weakly null in x ∗. then ⟨an, xn ⊗ yn⟩ = ⟨a∗n(yn), xn⟩ → 0, since (xn) is dp in x. thus (xn ⊗ yn) is dp in x ⊗π y . theorem 23. ([7, theorem 18]) suppose that l(x, y ∗) = k(x, y ∗). if x and y are sequentially right, then x ⊗π y is sequentially right. 16 i. ghenciu proof. let h be a right subset of (x ⊗π y )∗ ≃ l(x, y ∗) = k(x, y ∗). we will use theorem 20. we will verify the conditions (i) and (ii) of this theorem. let (tn) be a sequence in h and let x ∈ x. we prove that {tn(x) : n ∈ n} is a right subset of y ∗. let (yn) be a right null sequence in y . thus (yn) is weakly null and dp. for each n, ⟨tn(x), yn⟩ = ⟨tn, x ⊗ yn⟩. we show that (x⊗yn) is right null in x⊗πy . if t ∈ (x⊗πy )∗ ≃ l(x, y ∗) ([10, p. 230]), then |⟨t, x ⊗ yn⟩| = |⟨t(x), yn⟩| → 0, since (yn) is weakly null. thus (x ⊗ yn) is weakly null. let (an) be a weakly null sequence in (x ⊗π y )∗ ≃ l(x, y ∗). since the map ϕx : l(x, y ∗) → y ∗, ϕx(t) = t(x) is linear and bounded, (an(x)) is weakly null in y ∗. therefore |⟨an, x ⊗ yn⟩| = |⟨an(x), yn⟩| → 0, since (yn) is dp in y . thus (x ⊗ yn) is dp and (x ⊗ yn) is right null. since (tn) is a right set, |⟨tn, x ⊗ yn⟩| = |⟨tn(x), yn⟩| → 0. thus {tn(x) : n ∈ n} is a right subset of y ∗, hence relatively weakly compact (by theorem 2). we thus verified (i) of theorem 20. let y∗∗ ∈ y ∗∗. we show that {t ∗n(y∗∗) : n ∈ n} is a right subset of x∗. let (xn) be a right null sequence in x. thus (xn) is weakly null and dp. for each n, ⟨t ∗n(y ∗∗), xn⟩ = ⟨y∗∗, tn(xn)⟩. it is enough to show that (tn(xn)) is weakly null in y ∗. let (yn) be a right null sequence in y . by lemma 22 and proposition 1, (xn ⊗yn) is right null in x ⊗π y . since (tn) is a right set, |⟨tn, xn ⊗ yn⟩| = |⟨tn(xn), yn⟩| → 0. therefore (tn(xn)) is a right subset of y ∗, thus relatively weakly compact (by theorem 2). by passing to a subsequence, we can assume that (tn(xn)) is weakly convergent. let y ∈ y . an argument similar to the one above shows that (xn ⊗ y) is right null in x ⊗π y . then |⟨tn, xn ⊗ y⟩| = |⟨tn(xn), y⟩| → 0, isomorphic properties in projective tensor products 17 since (tn) is a right set. hence (tn(xn)) is w ∗-null. since (tn(xn)) is also weakly convergent, (tn(xn)) is weakly null. then {t ∗n(y∗∗) : n ∈ n} is a right subset of x∗. hence {t ∗n(y∗∗) : n ∈ n} is relatively weakly compact (by theorem 2). by theorem 20, h is weakly precompact. we can assume without loss of generality that (tn) is weakly cauchy. since x and y are sequentially right, x∗ and y ∗ are both weakly sequentially complete [26, corollary 3.26], and thus l(x, y ∗) = k(x, y ∗) is weakly sequentially complete, by [22, theorem 3.10]. then (tn) is weakly convergent. remark. theorem 23 can also be proved as follows. let h be a right subset of (x ⊗π y )∗ ≃ l(x, y ∗) = k(x, y ∗) and let (tn) be a sequence in h. by the proof of theorem 23, {tn(x) : n ∈ n} and {t ∗n(y∗∗) : n ∈ n} are relatively weakly compact for all x ∈ x and y∗∗ ∈ y ∗∗. by theorem 21, h is relatively weakly compact. lemma 24. suppose l(x, y ∗) = k(x, y ∗). if (xn) is a weakly null limited sequence in x and (yn) is a limited sequence in y , then (xn ⊗ yn) is a weakly null limited sequence in x ⊗π y . proof. by lemma 22, (xn ⊗ yn) is a weakly null. let (an) be a w∗-null sequence in (x ⊗π y )∗ ≃ l(x, y ∗). then (a∗n(x)) is a w∗-null sequence in y ∗. if x ∈ x, then ⟨an(x), yn⟩ = ⟨a∗n(yn), x⟩ → 0, since (yn) is limited in y . hence (a∗n(yn)) is w ∗-null in x∗. since (xn) is limited, ⟨an, xn ⊗ yn⟩ = ⟨a∗n(yn), xn⟩ → 0. thus (xn ⊗ yn) is limited in x ⊗π y . theorem 25. ([7, theorem 25]) suppose that l(x, y ∗) = k(x, y ∗). if x and y have the l-limited property, then x⊗π y has the l-limited property. proof. the proof is similar to the proof of theorem 23 and uses lemma 24. remark. theorem 25 can also be proved with a method similar to the one in the previous remark. the fact that the (sr) and l-limited properties are inherited by quotients, immediately implies the following result. 18 i. ghenciu corollary 26. (i) suppose that l(x∗, y ∗) = k(x∗, y ∗), and x∗ and y are sequentially right. then the space n1(x, y ) of all nuclear operators from x to y is sequentially right. (ii) suppose that l(x∗, y ∗) = k(x∗, y ∗), and x∗ and y have the llimited property. then the space n1(x, y ) of all nuclear operators from x to y has the l-limited property. proof. it is known that n1(x, y ) is a quotient of x ∗ ⊗π y ([38, p. 41]). (i) apply theorem 23. (ii) apply theorem 25. observation 1. if t : y → x∗ be an operator such that t ∗|x is (weakly) compact, then t is (weakly) compact. to see this, let t : y → x∗ be an operator such that t ∗|x is (weakly) compact. let s = t ∗|x. suppose x∗∗ ∈ bx∗∗ and choose a net (xα) in bx which is w∗convergent to x∗∗. then (t ∗(xα)) w∗→ t ∗(x∗∗). now, (t ∗(xα)) ⊆ s(bx), which is a relatively (weakly) compact set. then (t ∗(xα)) → t ∗(x∗∗) (resp. (t ∗(xα)) w→ t ∗(x∗∗)). hence t ∗(bx∗∗) ⊆ s(bx), which is relatively (weakly) compact. therefore t ∗(bx∗∗) is relatively (weakly) compact, and thus t is (weakly) compact. it follows that if l(x, y ∗) = k(x, y ∗), then l(y, x∗) = k(y, x∗). the following result improves corollaries 19 and 21 of [7]. corollary 27. if x is sequentially right and y ∗ has the schur property (or y is sequentially right and x∗ has the schur property), then x ⊗π y is sequentially right. proof. since y ∗ has the schur property, every right set in y ∗ is relatively compact (by corollary 9). let t : x → y ∗ be an operator. then t is pseudo weakly compact (since y ∗ has the schur property), hence compact (by theorem 2). apply theorem 23. theorem 28. suppose that l(x, y ∗) = k(x, y ∗). the following statements are equivalent: 1. (i) x and y are sequentially right and at least one of them does not contain ℓ1. (ii) x ⊗π y is sequentially right. 2. (i) x and y have the l-limited property and at least one of them does not contain ℓ1. (ii) x ⊗π y has the l-limited property. isomorphic properties in projective tensor products 19 proof. we only prove 1. the other proof is similar. (i)⇒(ii) by theorem 23. (ii)⇒(i) suppose that x ⊗π y is sequentially right. then x and y are sequentially right, since the sequentialy right property is inherited by quotients [26, proposition 3.8]. we will show that ℓ1 ̸↪→ x or ℓ1 ̸↪→ y . suppose that ℓ1 ↪→ x and ℓ1 ↪→ y . hence l1 ↪→ x∗ ([32, theorem 3.4], [8, p. 212]). also, the rademacher functions span ℓ2 inside of l1, and thus ℓ2 ↪→ x∗. similarly ℓ2 ↪→ y ∗. then c0 ↪→ k(x, y ∗) ([15, theorem 3], [21, corollary 21]). thus ℓ1 c ↪→ x ⊗π y ([3, theorem 4], [8, theorem 10, p. 48]), a contradiction with corollary 5 (iii). observation 2. if ℓ1 ↪→ x and ℓ1 ↪→ y , then ℓ2 ↪→ x∗ and ℓ2 ↪→ y ∗, and c0 ↪→ k(x, y ∗) ([15, theorem 3], [21, corollary 21]). more generally, if ℓ1 ↪→ x and ℓp ↪→ y ∗, p ≥ 2, then c0 ↪→ k(x, y ∗) ([15], [21]). thus ℓ1 c ↪→ x ⊗π y ([3, theorem 4], [8, theorem 10, p. 48]). hence x ⊗π y is not weak sequentially right (and does not have the wl-limited property), by corollary 5 (iii). corollary 29. suppose that l(x, y ∗) = k(x, y ∗). 1. if x⊗πy is weak sequentially right, then x and y are weak sequentially right and at least one of them does not contain ℓ1. 2. if x ⊗π y has the wl-limited property, then x and y have the wllimited property and at least one of them does not contain ℓ1. proof. we only prove 1. the other proof is similar. if x ⊗π y is weak sequentially right, then x and y are weak sequentially right, since the weak sequentially right property is inherited by quotients (by corollary 3). apply observation 2. corollary 30. ([7, theorem 22]) suppose that x and y have the dpp . the following statements are equivalent: (i) x and y are sequentially right and at least one of them does not contain ℓ1. (ii) x ⊗π y is sequentially right. 20 i. ghenciu proof. (i)⇒(ii) suppose that x and y have the dpp . without loss of generality suppose that ℓ1 ̸↪→ x. then x∗ has the schur property [9]. apply corollary 27. (ii)⇒(i) by observation 2. by corollary 30, the space c(k1) ⊗π c(k2) is sequentially right if and only if either k1 or k2 is dispersed. next we present some results about the necessity of the condition l(x, y ∗) = k(x, y ∗). it is implicit in [6] that a banach space x has all bilinear forms weakly sequentially continuous if and only if every operator s : x → x∗ transforms weakly null sequences into l-sets. emmanuelle shows in [13] that a banach space x does not contain ℓ1 if and only if every l-set in x∗ is relatively compact. then, it is easy to see that if x and y are not containing ℓ1, then l(x, y ∗) = k(x, y ∗) if and only if every operator t : x → y ∗ transforms weakly null sequences into l-sets (for more details see [6]). a banach space x has the approximation property if for each norm compact subset m of x and ϵ > 0, there is a finite rank operator t : x → x such that ∥tx − x∥ < ϵ for all x ∈ m. if in addition t can be found with ∥t∥ ≤ 1, then x is said to have the metric approximation property. c(k) spaces, c0, ℓp, 1 ≤ p < ∞, lp(µ) (µ any measure), 1 ≤ p < ∞, and their duals have the metric approximation property [10, p. 238]. a separable banach space x has an unconditional compact expansion of the identity (u.c.e.i) if there is a sequence (an) of compact operators from x to x such that ∑ an(x) converges unconditionally to x for all x ∈ x [17]. in this case, (an) is called an (u.c.e.i.) of x. a sequence (xn) of closed subspaces of a banach space x is called an unconditional schauder decomposition of x if every x ∈ x has a unique representation of the form x = ∑ xn, with xn ∈ xn, for every n, and the series converges unconditionally [30, p. 48]. the space x has (rademacher) cotype q for some 2 ≤ q ≤ ∞ if there is a constant c such that for for every n and every x1, x2, . . . , xn in x, ( n∑ i=1 ∥xi∥q )1/q ≤ c (∫ 1 0 ∥ri(t)xi∥qdt )1/q , where (rn) are the radamacher functions. a hilbert space has cotype 2 [8, p. 118]. lp-spaces have cotype 2, if 1 ≤ p ≤ 2 [8, p. 118]. isomorphic properties in projective tensor products 21 theorem 31. assume one of the following holds: (i) if t : x → y ∗ is an operator which is not compact, then there is a sequence (tn) in k(x, y ∗) such that for each x ∈ x, the series ∑ tn(x) converges unconditionally to t(x). (ii) x is an l∞-space and y ∗ is a subspace of an l1-space. (iii) x = c(k), k a compact hausdorff space, and y ∗ is a space with cotype 2. (iv) either x or y ∗ has an (u.c.e.i.). (v) x has the dpp and ℓ1 ↪→ y . (vi) x and y have the dpp . if x ⊗π y is weak sequentially right, then l(x, y ∗) = k(x, y ∗). proof. suppose that x ⊗π y is weak sequentially right. then x and y are weak sequentially right. (i) let t : x → y ∗ be a noncompact operator. let (tn) be a sequence as in the hypothesis. by the uniform boundedness principle, { ∑ n∈a tn : a ⊆ n, a finite} is bounded in k(x, y ∗). then ∑ tn is wuc and not unconditionally convergent (since t is noncompact). hence c0 ↪→ k(x, y ∗) ([3, theorem 5]), ℓ1 c ↪→ x ⊗π y ([3, theorem 4]), and we have a contradiction with corollary 5 (iii). suppose (ii) or (iii) holds. it is known that any operator t : x → y ∗ is 2-absolutely summing ([8, p. 189]), hence it factorizes through a hilbert space. if l(x, y ∗) ̸= k(x, y ∗), then c0 ↪→ k(x, y ∗) (by [16, remark 3]), a contradiction. (iv) if l(x, y ∗) ̸= k(x, y ∗), then c0 ↪→ k(x, y ∗) (by [27, theorem 6]), a contradiction. (v) suppose that x has the dpp and ℓ1 ↪→ y . by observation 1, ℓ1 ̸↪→ x. then x∗ has the schur property ([9, theorem 3]). let t : y → x∗ be an operator. then t is pseudo weakly compact (since x∗ has the schur property), and thus weakly precompact (by corollary 5 (i)). then l(y, x∗) = k(y, x∗). hence l(x, y ∗) = k(x, y ∗), by observation 1. (vi) suppose that x and y have the dpp . then l(x, y ∗) = k(x, y ∗), either by (v) if ℓ1 ↪→ y , or since y ∗ has the schur property ([9, theorem 3]) if ℓ1 ̸↪→ y (by an argument similar to the one in (v)). assumption (i) of the previous theorem is satisfied, for instance, if x∗ (or y ∗) has an (u.c.e.i.). 22 i. ghenciu examples. by theorem 31, the space ℓp ⊗ ℓq, where 1 < p ≤ q′ < ∞ and q and q′ are conjugate, is not weak sequentially right, since the natural inclusion map i : ℓp → ℓq′ is not compact. the space c(k) ⊗π ℓp, with k not dispersed and 1 < p ≤ 2, is not weak sequentially right (by observation 2, since ℓ1 ↪→ c(k) and ℓ2 ↪→ ℓ∗p). for 1 < p1, p2 < ∞, lp1[0, 1] ⊗π lp2[0, 1] is not weak sequentially right by corollary 5 (iii), since ℓ1 c ↪→ lp1[0, 1] ⊗π lp2[0, 1] ([38, corollary 2.26]). theorem 32. (i) suppose y ∗ is complemented in a banach space z which has an unconditional schauder decomposition (zn), and l(x, zn) = k(x, zn) for all n. if x ⊗π y is weak sequentially right, then l(x, y ∗) = k(x, y ∗). (ii) suppose either x∗ or y ∗ has the metric approximation property. if x ⊗π y is sequentially right, then w(x, y ∗) = k(x, y ∗). proof. (i) let t : x → y ∗ be a noncompact operator, pn : z → zn, pn( ∑ zi) = zn, and let p be the projection of z onto y ∗. define tn : x → y ∗ by tn(x) = ppnt(x), x ∈ x, n ∈ n. note that pnt is compact since l(x, zn) = k(x, zn). then tn is compact for each n. for each z ∈ z,∑ pn(z) converges unconditionally to z; thus ∑ tn(x) converges unconditionally to t(x) for each x ∈ x. then ∑ tn is wuc and not unconditionally converging. hence c0 ↪→ k(x, y ∗) ([3, theorem 5]), and we obtain a contradiction. 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[42] t. schlumprecht, “limited sets in banach spaces”, ph.d. dissertation, ludwig-maximilians-universität, munich, 1987. e extracta mathematicae vol. 32, núm. 1, 55 – 81 (2017) characterizations of complete linear weingarten spacelike submanifolds in a locally symmetric semi-riemannian manifold jogli g. araújo, henrique f. de lima, fábio r. dos santos, marco antonio l. velásquez departamento de matemática, universidade federal de campina grande, 58.429 − 970 campina grande, paráıba, brazil jogli@mat.ufcg.edu.br henrique@mat.ufcg.edu.br fabio@mat.ufcg.edu.br marco.velasquez@mat.ufcg.edu.br presented by manuel de león received november 23, 2016 abstract: in this paper, we deal with n-dimensional complete spacelike submanifolds mn with flat normal bundle and parallel normalized mean curvature vector immersed in an (n + p)-dimensional locally symmetric semi-riemannian manifold ln+pp of index p obeying some standard curvature conditions which are naturally satisfied when the ambient space is a semi-riemannian space form. in this setting, we establish sufficient conditions to guarantee that, in fact, p = 1 and mn is isometric to an isoparametric hypersurface of ln+11 having two distinct principal curvatures, one of which is simple. key words: locally symmetric semi-riemannian manifold, complete linear weingarten spacelike submanifolds, isoparametric submanifolds. ams subject class. (2010): 53c42, 53a10, 53c20, 53c50. 1. introduction let l n+p p be an (n + p)-dimensional semi-riemannian space, that is, a semi-riemannian manifold of index p. an n-dimensional submanifold mn immersed in l n+p p is said to be spacelike if the metric on m n induced from that of l n+p p is positive definite. spacelike submanifolds with parallel normalized mean curvature vector field (that is, the mean curvature function is positive and that the corresponding normalized mean curvature vector field is parallel as a section of the normal bundle) immersed in semi-riemannian manifolds have been deeply studied for several authors (see, for example, [2, 3, 15, 19]). more recently, in [12] the second, third and fourth authors showed that complete linear weingarten spacelike submanifolds must be isometric to certain hyperbolic cylinders of a semi-riemannian space form qn+pp (c) of constant sectional curvature c, under suitable constraints on the values of the mean 55 56 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez curvature and of the norm of the traceless part of the second fundamental form. we recall that a spacelike submanifold is said to be linear weingarten when its mean and normalized scalar curvature functions are linearly related. now, let l n+p p be a locally symmetric semi-riemannian space, that is, the curvature tensor r̄ of l n+p p is parallel in the sense that ∇r̄ = 0, where ∇ denotes the levi-civita connection of l n+p p . in 1984, nishikawa [16] introduced an important class of locally symmetric lorentz spaces satisfying certain curvature constraints. in this setting, he extended the classical results of calabi [4] and cheng-yau [6] showing that the only complete maximal spacelike hypersurface immersed in such a locally symmetric space having nonnegative sectional curvature are the totally geodesic ones. this seminal nishikawa’s paper induced the appearing of several works approaching the problem of characterizing complete spacelike hypersurfaces immersed in such a locally symmetric space (see, for instance, [1, 10, 11, 13, 14]). our purpose in this paper is establish characterization results concerning complete linear weingarten submanifolds immersed in a locally symmetric manifold obeying certain curvature conditions which extend those ones due to nishikawa [16]. for this, we need to work with a cheng-yau modified operator l and we establish a generalized maximum principle. afterwards, under suitable constrains, we apply our omori-yau maximum principle to prove that such a submanifold must be isometric to an isoparametric hypersurface with two distinct principal curvatures, one of them being simple. our purpose in this work is to extend the results of [10] for the case that the ambient space is a locally symmetric semi-riemannian manifold l n+p p obeying certain geometric constraints. for this, in section 3 we develop a suitable simons type formula concerning spacelike submanifolds immersed in l n+p p and having certain positive curvature function. afterwards, in section 4 we prove an extension of the generalized maximum principle of omori [17] to a cheng yau modified operator l (see lemma 3). moreover, we use our simons type formula to obtain an appropriated lower estimate to the operator l acting on the mean curvature function of a linear weingarten spacelike submanifold (cf. proposition 1) and, next, we establish our characterization theorems (see theorems 1 and 2). linear weingarten spacelike submanifolds 57 2. preliminaries let mn be a spacelike submanifold immersed in a locally symmetric semiriemannian space l n+p p . in this context, we choose a local field of semiriemannian orthonormal frames e1, . . . , en+p in l n+p p , with dual coframes ω1, . . . , ωn+p, such that, at each point of m n, e1, . . . , en are tangent to m n. we will use the following convention of indices 1 ≤ a, b, c, . . . ≤ n+p, 1 ≤ i, j, k, . . . ≤ n and n+1 ≤ α, β, γ, . . . ≤ n+p. in this setting, the semi-riemannian metric of l n+p p is given by ds2 = ∑ a ϵa ω 2 a, where ϵi = 1 and ϵα = −1, 1 ≤ i ≤ n, n + 1 ≤ α ≤ n + p. denoting by {ωab} the connection forms of l n+p p , we have that the structure equations of l n+p p are given by: dωa = − ∑ b ϵb ωab ∧ ωb, ωab + ωba = 0, (2.1) dωab = − ∑ c ϵc ωac ∧ ωcb − 1 2 ∑ c,d ϵcϵdrabcd ωc ∧ ωd, (2.2) where, rabcd, rcd and r denote respectively the riemannian curvature tensor, the ricci tensor and the scalar curvature of the lorentz space l n+p p . in this setting, we have rcd = ∑ b εbrcbdb, r = ∑ a εaraa. (2.3) moreover, the components rabcd;e of the covariant derivative of the riemannian curvature tensor l n+p p are defined by∑ e εerabcd;eωe = drabcd − ∑ e εe ( rebcdωea + raecdωeb +rabedωec + rabceωed ) . next, we restrict all the tensors to mn. first of all, ωα = 0, n + 1 ≤ α ≤ n + p. 58 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez consequently, the riemannian metric of mn is written as ds2 = ∑ i ω 2 i . since − ∑ i ωαi ∧ ωi = dωα = 0, from cartan’s lemma we can write ωαi = ∑ j hαijωj, h α ij = h α ji. (2.4) this gives the second fundamental form of mn, b = ∑ α,i,j h α ijωi ⊗ ωjeα, and its square length from second fundamental form is s = |b|2 = ∑ α,i,j(h α ij) 2. furthermore, we define the mean curvature vector field h and the mean curvature function h of mn respectively by h = 1 n ∑ α (∑ i hαii ) eα and h = |h| = 1 n √√√√∑ α (∑ i hαii )2 . the structure equations of mn are given by dωi = − ∑ j ωij ∧ ωj, ωij + ωji = 0, dωij = − ∑ k ωik ∧ ωkj − 1 2 ∑ k,l rijklωk ∧ ωl, where rijkl are the components of the curvature tensor of m n. using the previous structure equations, we obtain gauss equation rijkl = rijkl − ∑ β ( h β ikh β jl − h β ilh β jk ) . (2.5) and n(n − 1)r = ∑ i,j rijij − n2h2 + s. (2.6) we also state the structure equations of the normal bundle of mn dωα = − ∑ β ωαβ ∧ ωβ, ωαβ + ωβα = 0, linear weingarten spacelike submanifolds 59 dωαβ = − ∑ γ ωαγ ∧ ωγβ − 1 2 ∑ k,l rαβklωk ∧ ωl. we suppose that mn has flat normal bundle, that is, r⊥ = 0 (equivalently rαβjk = 0), then rαβjk satisfy ricci equation rαβij = ∑ k ( hαikh β kj − h α kjh β ik ) . (2.7) the components hαijk of the covariant derivative ∇b satisfy∑ k hαijkωk = dh α ij − ∑ k hαikωkj − ∑ k hαjkωki − ∑ β h β ijωβα. (2.8) in this setting, from (2.4) and (2.8) we get codazzi equation rαijk = h α ijk − h α ikj. (2.9) the first and the second covariant derivatives of hαij are denoted by h α ijk and hαijkl, respectively, which satisfy∑ l hαijklωl = dh α ijk − ∑ l hαljkωli − ∑ l hαilkωlj − ∑ l hαijlωlk − ∑ β h β ijkωβα. thus, taking the exterior derivative in (2.8), we obtain the following ricci identity hαijkl − h α ijlk = − ∑ m hαimrmjkl − ∑ m hαmjrmikl. (2.10) restricting the covariant derivative rabcd;e of rabcd on m n, then rαijk;l is given by rαijkl = rαijk;l + ∑ β rαβjkh β il + ∑ β rαiβkh β jl + ∑ β rαijβh β kl + ∑ m,k rmijkh α lm. (2.11) where rαijkl denotes the covariant derivative of rαijk as a tensor on m n. 60 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez 3. locally symmetric spaces and some auxiliary results proceeding with the context of the previous section, along this work we will assume that there exist constants c1, c2 and c3 such that the sectional curvature k and the curvature tensor r of the ambient space l n+p p satisfies the following constraints: k(u, η) = c1 n , (3.1) for any u ∈ tm and η ∈ tm⊥; when p > 1, suppose that⟨ r(ξ, u)η, u ⟩ = 0, (3.2) for u ∈ tm and ξ, η ∈ tm⊥, with ⟨ξ, η⟩ = 0. suppose also k(u, v) ≥ c2, (3.3) for any u, v ∈ tm; and k(η, ξ) = c3 p , (3.4) for any η, ξ ∈ tm⊥. the curvature conditions (3.1) and (3.3), are natural extensions for higher codimension of conditions assumed by nishikawa [16] in context of hypersurfaces. obviously, when the ambient manifold l n+p p has constant sectional curvature c, then it satisfies conditions (3.1), (3.2), (3.3) and (3.4). on the other hand, the next example gives us a situation where the curvature conditions (3.1), (3.2), (3.3) and (3.4) are satisfied but the ambient space is not a space form. example 1. let l n+p p = r n1+p p × nn2κ be a semi-riemannian manifold, where rn1+pp stands for the (n1+p)-dimensional semi-euclidean space of index p and nn2κ is a n2-dimensional riemannian manifold of constant sectional curvature κ. we consider the spacelike submanifold mn = γn1 ×nn2κ of l n+p p , where γn1 is a spacelike submanifold of rn1+pp . taking into account that the normal bundle of γn1 ↪→ rn1+pp is equipped with p linearly independent timelike vector fields ξ1, ξ2, . . . , ξp, it is not difficult to verify that the sectional curvature k of l n+p p satisfies k (ξi, x) = ⟨ rrn1+pp (ξi, x1)ξ i, x1 ⟩ rn1+pp + ⟨ rnn2κ (0, x2)0, x2 ⟩ n n2 κ = 0, (3.5) linear weingarten spacelike submanifolds 61 for each i ∈ {1, . . . , p}, where rrn1+pp and rn n2 κ denote the curvature tensors of rn1+pp and nn2κ , respectively, ξi = (ξ i, 0) ∈ t ⊥m and x = (x1, x2) ∈ tm with ⟨ξi, ξi⟩ = ⟨x, x⟩ = 1. on the other hand, by a direct computation we obtain k(x, y ) = ⟨ rrn1+pp (x1, y1)x1, y1 ⟩ rn1+pp + ⟨ rnn2κ (x2, y2)x2, y2 ⟩ n n2 κ (3.6) for every x = (x1, x2), y = (y1, y2) ∈ tm such that ⟨x, y ⟩ = 0, ⟨x, x⟩ = ⟨y, y ⟩ = 1. consequently, from (3.6) we get k(x, y ) = κ ( |x2|2|y2|2 − ⟨x2, y2⟩2 ) ≥ min{κ, 0}. (3.7) moreover, we have that k(ξi, ξj) = 0, for all i, j ∈ {1, . . . , p} (3.8) and ⟨ r(ξi, x)ξj, x ⟩ = 0, for all i, j ∈ {1, . . . , p}. (3.9) we observe from (3.5), (3.7), (3.8) and (3.9) that the curvature constraints (3.1), (3.2), (3.3) and (3.4) are satisfied with c1 = c3 = 0 and c2 ≤ min{κ, 0}. denote by rcd the components of the ricci tensor of l n+p p , then the scalar curvature r of l n+p p is given by r = ∑ a εaraa = ∑ i,j rijij − 2 ∑ i,α riαiα + ∑ α,β rαβαβ. if l n+p p satisfies conditions (3.1) and (3.4), then r = ∑ i,j rijij − 2pc1 + (p − 1)c3. (3.10) but, it is well known that the scalar curvature of a locally symmetric lorentz space is constant. consequently, ∑ i,j rijij is a constant naturally attached to a locally symmetric lorentz space satisfying conditions (3.1) and (3.4). for sake of simplicity, in the course of this work we will denote the constant 1 n(n−1) ∑ i,j rijij by r. in order to establish our main results, we devote this section to present some auxiliary lemmas. using the ideas of the proposition 2.2 of [19] we have 62 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez lemma 1. let mn be a linear weingarten spacelike submanifold immersed in locally symmetric space l n+p p satisfying conditions (3.1) and (3.4), such that r = ah + b for some a, b ∈ r. suppose that (n − 1)a2 + 4n ( r − b ) ≥ 0. (3.11) then, |∇b|2 ≥ n2|∇h|2. (3.12) moreover, if the equality holds in (3.12) on mn, then h is constant on mn. proof. since we are supposing that r = ah + b and l n+p p satisfies the conditions (3.1) and (3.4) then from equation (2.6) we get 2 ∑ i,j,α hαijh α ijk = ( 2n2h + n(n − 1)a ) hk, (3.13) where hk stands for the k-th component of ∇h. thus, 4 ∑ k (∑ i,j,α hαijh α ijk )2 = ( 2n2h + n(n − 1)a )2|∇h|2. consequently, using cauchy-schwarz inequality, we obtain that 4s|∇b|2 = 4 ∑ i,j,α ( hαij )2 ∑ i,j,k,α ( hαijk )2 ≥ 4∑ k (∑ i,j,α hαijh α ijk )2 = ( 2n2h + n(n − 1)a )2|∇h|2. (3.14) on the other hand, since r = ah + b, from equation (2.6) we easily see that( 2n2h + n(n − 1)a )2 = n2(n − 1) [ (n − 1)a2 + 4n ( r − b )] + 4n2s. (3.15) thus, from (3.14) and (3.15) we have 4s|∇b|2 ≥ n2(n − 1) [ (n − 1)a2 + 4n ( r − b )] + 4n2s|∇h|2, (3.16) and taking account that since (n − 1)a2 + 4n ( r − b ) ≥ 0, from (3.16) we obtain s|∇b|2 ≥ sn2|∇h|2. linear weingarten spacelike submanifolds 63 therefore, either s = 0 and |∇b|2 = n2|∇h|2 = 0 or |∇b|2 ≥ n2|∇h|2. now suppose that |∇b|2 = n2|∇h|2. if (n − 1)a2 + 4n ( r − b ) > 0 then from (3.16) we have that h is constant. if (n − 1)a2 + 4n ( r − b ) = 0, then from (3.15) ( 2n2h + n(n − 1)a )2 − 4n2s = 0. (3.17) this together with (3.13) forces that s2k = 4n 2sh2k, k = 1, . . . , n, (3.18) where sk stands for the k-th component of ∇s. since the equality in (3.14) holds, there exists a real function ck on m n such that hn+1ijk = ckh n+1 ij ; h α ijk = ckh α ij, α > n + 1; i, j, k = 1, . . . , n. (3.19) taking the sum on both sides of equation (3.19) with respect to i = j, we get hk = ckh; h α k = 0, α > n + 1; k = 1, . . . , n. (3.20) from second equation in (3.20) we can see that en+1 is parallel. it follows from (3.19) that sk = 2 ∑ i,j,k,α hαijh α ijk = 2cks, k = 1, . . . , n. (3.21) multiplying both sides of equation (3.21) by h and using (3.20) we have hsk = 2hks, k = 1, . . . , n. (3.22) it follows from (3.18) and (3.22) that h2ks = h 2 kn 2h2, k = 1, . . . , n. (3.23) hence we have |∇h|2 ( s − n2h2 ) = 0. (3.24) we suppose that h is not constant on mn. in this case, |∇h| is not vanishing identically on mn. denote m0 = {x ∈ m; |∇h| > 0} and t = s − n2h2. it follows form (3.24) that m0 is open in m and t = 0 over m0. from the continuity of t , we have that t = 0 on the closure cl(m0) of m0. if m − cl(m0) ̸= ∅, then h is constant in m − cl(m0). it follows that s is constant and hence t is constant in m − cl(m0). from the continuity of t , we have that t = 0 and hence s = n2h2 on mn. it follows that h is constant on mn, which contradicts the assumption. hence we complete the proof. 64 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez in our next result, we will deal with submanifolds mn of l n+p p having parallel normalized mean curvature vector field, which means that the mean curvature function h is positive and that the corresponding normalized mean curvature vector field h h is parallel as a section of the normal bundle. extending the ideas of [9] we obtain the following simons type formula for locally symmetric spaces. lemma 2. let mn be an n-dimensional (n ≥ 2) submanifold with flat normal bundle and parallel normalized mean curvature vector field in a locally symmetric semi-riemannian space l n+p p . then, we have 1 2 ∆s = |∇b|2 + 2 ( ∑ i,j,k,m,α hαijh α kmrmijk + ∑ i,j,k,m,α hαijh α jmrmkik ) + ∑ i,j,k,α,β hαijh β jkrαiβk − ∑ i,j,k,α,β hαijh β jkrαkβi + ∑ i,j,k,α,β hαijh β ijrαkβk − ∑ i,j,k,α,β hαijh β kkrαiβj + n ∑ i,j hn+1ij hij − nh ∑ i,j,m,α hαijh α mih n+1 mj + ∑ α,β [ tr ( hαhβ )]2 + 3 2 ∑ α,β n ( hαhβ − hβhα ) , (3.25) where n(a) = tr(aat), for all matrix a = (aij). proof. note that 1 2 ∆s = ∑ i,j,α hαij∆h α ij + ∑ i,j,k,α ( hαijk )2 . using the definition of ∆hαij = ∑ k h α ijkk and the fact that |∇b| 2 = ∑ i,j,k(h α ijk) 2 we have 1 2 ∆s = ∑ i,j,k,α hαijh α ijkk + |∇b| 2. using the codazzi equation (2.9) and the fact that hαij = h α ji we get 1 2 ∆s = ∑ i,j,k,α hαijrαijkk + ∑ i,j,k,α hαijh α kijk + |∇b| 2. linear weingarten spacelike submanifolds 65 from (2.10) we obtain 1 2 ∆s = |∇b|2 + ∑ i,j,k,α hαijrαijkk + ∑ i,j,k,α hαijh α kikj + ∑ i,j,k,m,α hαijh α kmrmijk+ + ∑ i,j,k,m,α hαijh α mirmkjk. thence, 1 2 ∆s = |∇b|2 + ∑ i,j,k,α hαijrαijkk + ∑ i,j,k,α hαijh α kkij + ∑ i,j,k,α hαijrαkikj+ + ∑ i,j,k,m,α hαijh α kmrmijk + ∑ i,j,k,m,α hαijh α mirmkjk. using the gauss equation (2.5) we get∑ i,j,k,m,α hαijh α kmrmijk = ∑ i,j,k,m,α hαijh α kmrmijk − ∑ i,j,k,m,α,β hαijh α kmh β mjh β ik+ + ∑ i,j,k,m,α,β hαijh α kmh β mkh β ij and ∑ i,j,k,m,α hαijh α mirmkjk = ∑ i,j,k,m,α hαijh α mirmkjk − n ∑ i,j,m,α,β hαijh α mih β mjh β+ + ∑ i,j,k,m,α,β hαijh α mih β mkh β kj. since we can choose a local orthonormal frame {e1, . . . , en+p} such that en+1 = h h , we have that hn+1 = 1 n tr(hn+1) = h and hα = 1 n tr(hα) = 0, for α ≥ n + 2. thus, we get∑ i,j,k,m,α hαijh α mirmkjk = ∑ i,j,k,m,α hαijh α mirmkjk − n ∑ i,j,m,α hαijh α mih n+1 mj h+ + ∑ i,j,k,m,α,β hαijh α mih β mkh β kj. 66 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez therefore, ∑ i,j,k,m,α hαijh α kmrmijk + ∑ i,j,k,m,α hαijh α mirmkjk = ∑ i,j,k,m,α hαijh α kmrmijk − ∑ i,j,k,m,α,β hαijh α kmh β mjh β ik + ∑ i,j,k,m,α,β hαijh α kmh β mkh β ij + ∑ i,j,k,m,α hαijh α mirmkjk − n ∑ i,j,m,α hαijh α mih n+1 mj h + ∑ i,j,k,m,α,β hαijh α mih β mkh β kj. from (2.11) we have∑ i,j,k,α hαijrαijkk = ∑ i,j,k,α hαijrαijk;k + ∑ i,j,k,α,β hαijh β ikrαβjk + ∑ i,j,k,α,β hαijh β jkrαiβk + ∑ i,j,k,α,β hαijh β kkrαijβ + ∑ i,j,k,m,α hαijh α kmrmijk. using the ricci equation (2.7) , we conclude that∑ i,j,k,α,β hαijh β ikrαβjk = ∑ i,j,k,m,α,β hαijh β ikh α jmh β mk − ∑ i,j,k,m,α,β hαijh β ikh α mkh β jm. thence,∑ i,j,k,α hαijrαijkk = ∑ i,j,k,α hαijrαijk;k + ∑ i,j,k,m,α,β hαijh β ikh α jmh β mk − ∑ i,j,k,m,α,β hαijh β ikh α mkh β jm + ∑ i,j,k,α,β hαijh β jkrαiβk + ∑ i,j,k,α,β hαijh β kkrαijβ + ∑ i,j,k,m,α hαijh α kmrmijk. on other hand∑ i,j,k,α hαijrαkikj = ∑ i,j,k,α hαijrαkik;j + ∑ i,j,k,α,β hαijh β kjrαβik + ∑ i,j,k,α,β hαijh β ijrαkβk + ∑ i,j,k,α,β hαijh β kjrαkiβ + ∑ i,j,k,m,α hαijh α jmrmkik. linear weingarten spacelike submanifolds 67 thence, ∑ i,j,k,α,β hαijh β kjrαβik = ∑ i,j,k,m,α,β hαijh β kjh α imh β mk − ∑ i,j,k,m,α,β hαijh β kjh α mkh β im. therefore, ∑ i,j,k,α hαijrαkikj = ∑ i,j,k,α hαijrαkik;j + ∑ i,j,k,m,α,β hαijh β kjh α imh β mk − ∑ i,j,k,m,α,β hαijh β kjh α mkh β im + ∑ i,j,k,α,β hαijh β ijrαkβk + ∑ i,j,k,α,β hαijh β kjrαkiβ + ∑ i,j,k,m,α hαijh α jmrmkik. hence, ∑ i,j,k,α hαij(rαijkk + rαkikj) = ∑ i,j,k,α hαij(rαijk;k + rαkik;j) + ∑ i,j,k,m,α,β hαijh β ikh α jmh β mk − ∑ i,j,k,m,α,β hαijh β ikh α mkh β jm + ∑ i,j,k,α,β hαijh β jkrαiβk + ∑ i,j,k,α,β hαijh β kkrαijβ + ∑ i,j,k,m,α hαijh α kmrmijk + ∑ i,j,k,m,α,β hαijh β kjh α imh β mk − ∑ i,j,k,m,α,β hαijh β kjh α mkh β im + ∑ i,j,k,α,β hαijh β ijrαkβk + ∑ i,j,k,α,β hαijh β kjrαkiβ + ∑ i,j,k,m,α hαijh α jmrmkik. since l n+p p is locally symmetric, we have that ∑ i,j,k,α hαij(rαijk;k + rαkik;j) = 0. 68 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez thus, ∑ i,j,k,α hαij(rαijkk + rαkikj) = ∑ i,j,k,m,α,β hαijh β ikh α jmh β mk − ∑ i,j,k,m,α,β hαijh β ikh α mkh β jm + ∑ i,j,k,α,β hαijh β jkrαiβk + ∑ i,j,k,α,β hαijh β kkrαijβ + ∑ i,j,k,m,α hαijh α kmrmijk + ∑ i,j,k,m,α,β hαijh β kjh α imh β mk − ∑ i,j,k,m,α,β hαijh β kjh α mkh β im + ∑ i,j,k,α,β hαijh β ijrαkβk + ∑ i,j,k,α,β hαijh β kjrαkiβ + ∑ i,j,k,m,α hαijh α jmrmkik. now, observe that ∑ i,j,k,α hαijh α kkij = n ∑ i,j,α hαijh α ij. using the fact that hkj = h n+1 kl and h α kj = 0, for α > n + 1 we have ∑ i,j,k,α hαijh α kkij = n ∑ i,j hn+1ij hij. finally, we conclude that 1 2 ∆s = |∇b|2 + ∑ i,j,k,m,α,β hαijh β ikh α jmh β mk − ∑ i,j,k,m,α,β hαijh β ikh α mkh β jm + ∑ i,j,k,α,β hαijh β jkrαiβk + nh ∑ i,j,k,α hαijrαijn+1 + ∑ i,j,k,m,α hαijh α kmrmijk + ∑ i,j,k,m,α,β hαijh β kjh α imh β mk − ∑ i,j,k,m,α,β hαijh β kjh α mkh β im + ∑ i,j,k,α,β hαijh β ijrαkβk + linear weingarten spacelike submanifolds 69 + ∑ i,j,k,α,β hαijh β kjrαkiβ + ∑ i,j,k,m,α hαijh α jmrmkik + n ∑ i,j hn+1ij hij + ∑ i,j,k,m,α hαijh α kmrmijk − ∑ i,j,k,m,α,β hαijh α kmh β mjh β ik + ∑ i,j,k,m,α,β hαijh α kmh β mkh β ij + ∑ i,j,k,m,α hαijh α mirmkjk − nh ∑ i,j,m,α hαijh α mih n+1 mj + ∑ i,j,k,m,α,β hαijh α mih β mkh β kj. (3.26) note that ∑ i,j,k,m,α,β hαijh β ikh α jmh β mk − ∑ i,j,k,m,α,β hαijh β ikh α mkh β jm = ∑ α,β tr ( hαhβhβhα ) − ∑ α,β tr ( hαhβ )2 , (3.27) ∑ i,j,k,m,α,β hαijh α kmh β mkh β ij − ∑ i,j,k,m,α,β hαijh α kmh β mjh β ik = ∑ α,β [ tr ( hαhβ )]2 − ∑ α,β tr ( hαhβ )2 , (3.28) ∑ i,j,k,m,α,β hαijh β kjh α imh β mk = ∑ α,β tr ( hαhβhβhα ) (3.29) and ∑ i,j,k,m,α,β hαijh α mih β mkh β kj − ∑ i,j,k,m,α,β hαijh β kjh α mkh β im = 1 2 ∑ α,β n ( hαhβ − hβhα ) . (3.30) therefore, inserting (3.27), (3.28), (3.29) and (3.30) into (3.26) we complete the proof. in order to study linear weingarten submanifolds, we will consider, for each a ∈ r, an appropriated cheng-yau’s modified operator, which is given by l = � + n − 1 2 a∆, (3.31) 70 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez where, according to [7], the square operator is defined by �f = ∑ i,j ( nhδij − nhn+1 ) fij, (3.32) for each f ∈ c∞(m), and the normal vector field en+1 is taken in the direction of the mean curvature vector field, that is, en+1 = h h . the next lemma guarantees us the existence of an omori-type sequence related to the operator l. lemma 3. let mn be a complete linear weingarten spacelike in a locally symmetric semi-riemannian space l n+p p (c) satisfying conditions (3.1), (3.3) and (3.4), such that r = ah + b, with a ≥ 0 and (n − 1)a2 + 4n ( r − b ) ≥ 0. if h is bounded on mn, then there is a sequence of points {qk}k∈n ⊂ mn such that lim k nh(qk) = sup m nh, lim k |∇nh(qk)| = 0 and lim sup k l(nh(qk)) ≤ 0. proof. let us choose a local orthonormal frame {e1, . . . , en} on mn such that hn+1ij = λ n+1 i δij. from (3.31) we have that l(nh) = n ∑ i ( nh + n − 1 2 a − λn+1i ) hii. thus, for all i = 1, . . . , n and since that l n+p p satisfies the conditions (3.1) and (3.4) then from (2.6) and with straightforward computation we get (λn+1i ) 2 ≤ s = n2h2 + n(n − 1) ( ah + b − r ) = ( nh + n − 1 2 a )2 − n − 1 4 [ (n − 1)a2 + 4n ( r − b )] ≤ ( nh + n − 1 2 a )2 , where we have used our assumption that (n−1)a2 +4n ( r − b ) ≥ 0 to obtain the last inequality. consequently, for all i = 1, . . . , n, we have |λn+1i | ≤ ∣∣∣∣nh + n − 12 a ∣∣∣∣ . (3.33) linear weingarten spacelike submanifolds 71 thus, from (2.6) we obtain rijij = rijij − ∑ α hαiih α jj + ∑ α (hαij) 2 ≥ rijij − ∑ α hαiih α jj. since s ≤ ( nh + n − 1 2 a )2 , we get that (hαij) 2 ≤ ( nh + n − 1 2 a )2 , for every α, i, j and, hence, from (3.33) we have hαiih α jj ≤ |h α ii||h α jj| ≤ ( nh + n − 1 2 a )2 . therefore, since we are supposing that h is bounded on mn and l n+p p satisfies the condition (3.3), this is, rijij ≥ c2, it follows that the sectional curvatures of mn are bounded from below. thus, we may apply the well known generalized maximum principle of omori [17] to the function nh, obtaining a sequence of points {qk}k∈n in mn such that lim k nh(qk) = sup nh, lim k |∇nh(qk)| = 0, and lim sup k ∑ i nhii(qk) ≤ 0. (3.34) since supm h > 0, taking subsequences if necessary, we can arrive to a sequence {qk)k∈n in mn which satisfies (3.34) and such that h(qk) ≥ 0. hence, since a ≥ 0, we have 0 ≤ nh(qk) + n − 1 2 a − |λn+1i (qk)| ≤ nh(qk) + n − 1 2 a − λn+1i (qk) ≤ nh(qk) + n − 1 2 a + |λn+1i (qk)| ≤ 2nh(qk) + (n − 1)a. 72 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez this previous estimate shows that function nh(qk) + n−1 2 a − λn+1i (qk) is nonnegative and bounded on mn, for all k ∈ n. therefore, taking into account (3.34), we obtain lim sup k (l(nh)(qk)) ≤ n ∑ i lim sup k [( nh + n − 1 2 a − λn+1i ) (qk)hii(qk) ] ≤ 0. we close this section with the following algebraic lemma, whose proof can be found in [18]. lemma 4. let a, b : rn −→ rn be symmetric linear maps such that ab − ba = 0 and tr(a) = tr(b) = 0. then ∣∣tr(a2b)∣∣ ≤ n − 2√ n(n − 1) n(a) √ n(b), where n(a) = tr(aat), for all matrix a = (aij). moreover, the equality holds if and only if (n − 1) of the eigenvalues xi of b and corresponding eigenvalues yi of a satisfy |xi| = √ n(b) n(n − 1) , xiyi ≥ 0 and yi = √ n(a) n(n − 1) ( resp. − √ n(a) n(n − 1) ) . 4. main results as before, the normal vector field en+1 is taken in the direction of the mean curvature vector field, that is, en+1 = h h . in this setting, we will consider the following symmetric tensor φ = ∑ i,j,α φαijωi ⊗ ωjeα, where φn+1ij = h n+1 ij − hδij and φ α ij = h α ij, n + 2 ≤ α ≤ n + p. let |φ|2 = ∑ i,j,α(φ α ij) 2 be the square of the length of φ. remark 1. since the normalized mean curvature vector of mn is parallel, we have ωn+1α = 0, for α > n+1. thus, from of the structure equations of the normal bundle of mn, it follows that rn+1βij = 0, for all α, i, j. hence, from ricci equation, we have that hn+1hα−hαhn+1 = 0, for all α. this implies that linear weingarten spacelike submanifolds 73 the matrix hn+1 commutes with all the matrix hα. thus, being φα = (φαij), we have that φα = hα − hα and, hence φn+1 = hn+1 − hn+1 and φα = hα, for α > n + 1. these form, φn+1 commutes with all the matrix φα. since the matrix φα is traceless and symmetric, once the matrix hα are symmetric, we can use lemma 4 for the matrix φα and φn+1 in order to obtain ∣∣tr((φα)2φn+1)∣∣ ≤ n − 2√ n(n − 1) n(φα) √ n(φn+1). (4.1) summing (4.1) in α, we have ∑ α ∣∣tr((φα)2φn+1)∣∣ ≤ n − 2√ n(n − 1) ∑ α n(φα) √ n(φn+1). in order to prove our characterization results, it will be essential the following lower boundedness for the laplacian operator acting on the square of the length of the second fundamental form. if l n+p p is a space form then from [8] follows that r⊥ = 0 if and only if there exists an orthogonal basis for tm that diagonalizes simultaneously all bξ, ξ ∈ tm⊥. proposition 1. let mn be a linear weingarten spacelike submanifold in a semi-riemannian locally symmetric space l n+p p satisfying conditions (3.1), (3.2), (3.3) and (3.4), with parallel normalized mean curvature vector field and flat normal bundle. suppose that there exists an orthogonal basis for tm that diagonalizes simultaneously all bξ, ξ ∈ tm⊥. if mn is such that r = ah + b, with (n − 1)a2 + 4n ( r − b ) ≥ 0 and c = c1 n + 2c2, then l(nh) ≥ |φ|2 ( |φ|2 p − n(n − 2)√ n(n − 1) h|φ| − n(h2 − c) ) . proof. let us consider {e1, . . . , en} a local orthonormal frame on mn such that hαij = λ α i δij, for all α ∈ {n + 1, . . . , n + p}. from (3.25), we get 2 ( ∑ i,j,k,m,α hαijh α kmrmijk + ∑ i,j,k,m,α hαijh α jmrmkik ) = 2 ∑ i,k,α ( (λαi ) 2rikik + λ α i λ α k rkiik ) = ∑ i,k,α rikik(λ α i − λ α k ) 2. 74 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez since that l n+p p satisfies the condition (3.3) we have 2 ( ∑ i,j,k,m,α hαijh α kmrmijk + ∑ i,j,k,m,α hαijh α jmrmkik ) ≥ c2 ∑ i,k,α ( λαi − λ α k )2 = 2nc2|φ|2. (4.2) now, for each α, consider hα the symmetric matrix (hαij), and sαβ = ∑ i,j hαijh β ij. then the (p × p)-matrix (sαβ) is symmetric and we can see that is diagonalizable for a choose of en+1, . . . , en+p. thence, sα = sαα = ∑ i,j hαijh α ij, and we have that s = ∑ α sα. since that l n+p p satisfies the condition (3.2) we obtain∑ i,j,k,α,β hαijh β jkrαiβk − ∑ i,j,k,α,β hαijh β jkrαkβi + ∑ i,j,k,α,β hαijh β ijrαkβk − ∑ i,j,k,α,β hαijh β kkrαiβj = ∑ i,k,α (λαi ) 2rαkαk − nh2c1. since that l n+p p satisfies the condition (3.1) we conclude that∑ i,j,k,α,β hαijh β jkrαiβk − ∑ i,j,k,α,β hαijh β jkrαkβi + ∑ i,j,k,α,β hαijh β ijrαkβk − ∑ i,j,k,α,β hαijh β kkrαiβj = c1|φ| 2. (4.3) finally note that ∑ α,β n ( hαhβ − hβhα ) ≥ 0. (4.4) linear weingarten spacelike submanifolds 75 therefore, from (3.25) and using (4.2), (4.3) and (4.4) we conclude that 1 2 ∆s ≥ |∇b|2 + cn|φ|2 + n ∑ i,j hn+1ij hij − nh ∑ i,j,m,α hαijh α mih n+1 mj + ∑ α,β [ tr ( hαhβ )]2 . (4.5) from (3.31) we have l(nh) = �(nh) + n − 1 2 a∆(nh) = ∑ i,j ( nhδij − hn+1ij ) (nh)ij + n − 1 2 a∆(nh) = n2h ∑ i hii − n ∑ i,j hn+1ij hij + n − 1 2 a∆(nh) = n2h∆h − n ∑ i,j hn+1ij hij + n − 1 2 a∆(nh). note that ∆h2 = 2h∆h + 2|∇h|2. thus, l(nh) = 1 2 ∆ ( n2h2 ) − n2|∇h|2 − n ∑ i,j hn+1ij hij + n − 1 2 a∆(nh). since that r = ah + b and l n+p p satisfies the conditions (3.1) and (3.3) we have that r is constant then from (2.6) we get 1 2 n(n − 1)∆(ah) + 1 2 ∆ ( n2h2 ) = 1 2 ∆s. therefore, using the inequality (4.5) and lemma 1 we conclude that l(nh) = 1 2 ∆s − n2|∇h|2 − n ∑ i,j hn+1ij hij ≥ |∇b|2 − n2|∇h|2 + cn|φ|2 − nh ∑ i,j,m,α hαijh α mih n+1 mj + ∑ α,β [ tr ( hαhβ )]2 ≥ cn|φ|2 − nh ∑ i,j,m,α hαijh α mih n+1 mj + ∑ α,β [ tr ( hαhβ )]2 . 76 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez on the other hand, with a straightforward computation we guarantee that − nh ∑ α tr [ hn+1 ( hα )2] + ∑ α,β [ tr ( hαhβ )]2 = −nh ∑ α tr [ φn+1 ( φα )2] − nh2|φ|2 + ∑ α,β [ tr ( φαφβ )]2 ≥ −n(n − 2)√ n(n − 1) h|φ|3 − nh2|φ|2 + |φ|4 p . (4.6) therefore, l(nh) ≥ cn|φ|2 − n(n − 2)√ n(n − 1) h|φ|3 − nh2|φ|2 + |φ|4 p = |φ|2ph,p,c(|φ|), (4.7) where ph,p,c(x) = x2 p − n(n − 2)√ n(n − 1) hx − n ( h2 − c ) . when c > 0, if h2 ≥ 4(n−1)c q(p) , where q(p) = (n − 2)2p + 4(n − 1), then the polynomial ph,p,c defined by ph,p,c(x) = x2 p − n(n − 2)√ n(n − 1) hx − n(h2 − c) has (at least) a positive real root given by c(n, p, h) = √ n 2 √ n − 1 ( p(n − 2)h + √ pq(p)h2 − 4p(n − 1)c ) . on the other hand, in the case that c ≤ 0, the same occurs without any restriction on the values of the mean curvature function h. now, we are in position to present our first theorem. theorem 1. let mn be a complete linear weingarten spacelike submanifold in locally symmetric l n+p p satisfying conditions (3.1), (3.2), (3.3) and (3.4), with parallel normalized mean curvature vector and flat normal bundle, such that r = ah + b with a ≥ 0 and (n − 1)a2 + 4n ( r − b ) ≥ 0. suppose that there exists an orthogonal basis for tm that diagonalizes simultaneously linear weingarten spacelike submanifolds 77 all bξ, ξ ∈ tm⊥. when c > 0, assume in addition that h2 ≥ 4(n−1)c q(p) . if h is bounded on mn and |φ| ≥ c(n, p, sup h), then p = 1 and mn is an isoparametric hypersurface with two distinct principal curvatures one of which is simple. proof. since we are assuming that a ≥ 0 and that inequality (3.11) holds, we can apply lemma 3 to the function nh in order to obtain a sequence of points {qk}k∈n ⊂ mn such that lim k nh(qk) = sup m nh, and lim sup k l(nh)(qk) ≤ 0. (4.8) thus, from (4.7) and (4.8) we have 0 ≥ lim sup k l(nh)(qk) ≥ sup m |φ|2psup h,p,c ( sup m |φ| ) . (4.9) on the other hand, our hypothesis imposed on |φ| guarantees us that supm |φ| > 0. therefore, from (4.9) we conclude that psup h,p,c ( sup m |φ| ) ≤ 0. (4.10) suppose, initially, the case c > 0. from our restrictions on h and |φ|, we have that ph,p,c(|φ|) ≥ 0, with ph,p,c(|φ|) = 0 if, and only if, |φ| = c(n, p, h). consequently, from (4.10) we get sup m |φ| = c(n, p, sup h). taking into account once more our restriction on |φ|, we have that |φ| is constant on mn. thus, since mn is a linear weingarten submanifold, from (3.11) we have that h is also constant on mn. hence, from (4.7) we obtain 0 = l(nh) ≥ |φ|2ph,p,c(|φ|) ≥ 0. since |φ| > 0, we must have ph,p,c(|φ|) = 0. thus, all inequalities obtained along the proof of proposition 1 are, in fact, equalities. in particular, from inequality (4.6) we conclude that tr(φn+1) = |φ|2. so, from (2.6) we get tr(φn+1)2 = |φ|2 = s − nh2. (4.11) 78 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez on the other hand, we also have that tr ( φn+1 )2 = s − ∑ α>n+1 ∑ i,j ( hαij )2 − nh2. (4.12) thus, from (4.11) and (4.12) we conclude that ∑ α>n+1 ∑ i,j(h α ij) 2 = 0. but, from inequality (4.6) we also have that |φ|4 = p ∑ α [ n ( φα )]2 = pn ( φn+1 )2 = p|φ|4. (4.13) hence, since |φ| > 0, we must have that p = 1. in this setting, from (3.12) and (4.13) we get∑ i,j,k ( hn+1ijk )2 = |∇b|2 = n2|∇h|2 = 0, that is, hn+1ijk = 0 for all i, j. hence, we obtain that m n is an isoparametric hypersurface of l n+p p . when c ≤ 0, we proceed as before until reach equation (4.10) and, from |φ| ≥ c(n, p, sup h), we have that ph,p,c(|φ|) ≥ 0. at this point, we can reason as in the previous case to obtain that h is constant, p = 1 and, consequently, we also conclude that mn is an isoparametric hypersurface of l n+p p . hence, since the equality occurs in (4.1), we have that also occurs the equality in lemma 4. consequently, mn has at most two distinct constant principal curvatures. in particular, when the immersed submanifold has constant scalar curvature, from theorem 1 we obtain the following corollary 1. let mn be a complete spacelike submanifold in locally symmetric semi-riemannian l n+p p satisfying conditions (3.1), (3.2), (3.3) and (3.4), with parallel normalized mean curvature vector, flat normal bundle and constant normalized scalar curvature r satisfying r ≤ c. suppose that there exists an orthogonal basis for tm that diagonalizes simultaneously all bξ, ξ ∈ tm⊥. when c > 0, assume in addition that h2 ≥ 4(n−1)c q(p) . if h is bounded on mn and |φ| ≥ c(n, p, sup h), then p = 1 and mn is an isoparametric hypersurface with two distinct principal curvatures one of which is simple. linear weingarten spacelike submanifolds 79 in order to establish our next theorem, we will need of the following lemma obtained by caminha, which can be regarded as an extension of hopf’s maximum principle for complete riemannian manifolds (cf. proposition 2.1 of [5]). in what follows, let l1(m) denote the space of lebesgue integrable functions on mn. lemma 5. let x be a smooth vector field on the n-dimensional complete noncompact oriented riemannian manifold mn, such that divmx does not change sign on mn. if |x| ∈ l1(m), then divmx = 0. we close our paper stating and proving our second characterization theorem. theorem 2. let mn be a complete linear weingarten spacelike submanifold in locally symmetric einstein semi-riemannian l n+p p satisfying conditions (3.1), (3.2), (3.3) and (3.4), with parallel normalized mean curvature vector, flat normal bundle such that r = ah +b, with (n−1)a2 +4n(r−b) ≥ 0. suppose that there exists an orthogonal basis for tm that diagonalizes simultaneously all bξ, ξ ∈ tm⊥. when c > 0, assume in addition that h2 ≥ 4(n−1)c q(p) . if h is bounded on mn, |φ| ≥ c(n, p, h) and |∇h| ∈ l1(m), then p = 1 and mn is a isoparametric hypersurface with two distinct principal curvatures one of which is simple. proof. since the ambient space l n+p p is supposed to be einstein, reasoning as in the first part of the proof of theorem 1.1 in [10], from (3.31) and (3.32) it is not difficult to verify that l(nh) = divm(p(∇h)), (4.14) where p = ( n2h + n(n − 1) 2 a ) i − nhn+1. (4.15) on the other hand, since r = ah + b and h is bounded on mn, from equation (2.6) we have that b is bounded on mn. consequently, from (4.15) we conclude that the operator p is bounded, that is, there exists c1 such that |p | ≤ c1. since we are also assuming that |∇h| ∈ l1(m), we obtain that |p(∇h)| ≤ |p ||∇h| ≤ c1|∇h| ∈ l1(m). (4.16) 80 j.g. araújo, h.f. de lima, f.r. dos santos, m.a.l. velásquez so, from lemma 5 and (4.14) we obtain that l(nh) = 0 on mn. thus, 0 = l(nh) ≥ |φ|2ph,p,c(|φ|) ≥ 0 (4.17) and, consequently, we have that all inequalities are, in fact, equalities. in particular, from (3.11) we obtain |∇b|2 = n2|∇h|2. (4.18) hence, lemma 1 guarantees that h is constant. at this point, we can proceed as in the last part of the proof of theorem 1 to conclude our result. acknowledgements the first author is partially supported by capes, brazil. the second author is partially supported by cnpq, brazil, grant 303977/20159. the fourth author is partially supported by cnpq, brazil, grant 308757/2015-7. the authors would like to thank the referee for reading the manuscript in great detail and for his/her valuable suggestions and useful comments. references [1] j.o. baek, q.m. cheng, y.j. suh, complete spacelike hypersurface in locally symmetric lorentz spaces, j. geom. phys. 49 (2) (2004), 231 – 247. [2] e.r. barbosa, k.o. araújo, on complete submanifolds with bounded mean curvature, j. geom. phys. 61 (10) (2011), 1957 – 1964. [3] a. brasil jr., r.m.b. chaves, a.g. colares, rigidity results for submanifolds with parallel mean curvature vector in the de sitter space, glasg. math. j. 48 (1) (2006), 1 – 10. [4] e. calabi, examples of bernstein problems for some nonlinear equations, proc. sympos. pure math. 15 (1970), 223 – 230. [5] a. caminha, the geometry of closed conformal vector fields on riemannian spaces, bull. braz. math. soc. (n.s.) 42 (2) (2011), 277 – 300. [6] s.y. cheng, s.t. yau, maximal spacelike hypersurfaces in the lorentzminkowski space, ann. of math. 104 (3) (1976), 407 – 419. [7] s.y. cheng , s.t. yau, hypersurfaces with constant scalar curvature, math. ann. 225 (3) (1977), 195 – 204. [8] m. dajczer, “submanifolds and isometric immersions”, mathematics lecture series 13, publish or perish inc., houston, 1990. [9] x. guo, h.li, submanifolds with constant scalar curvature in a unit sphere, tohoku math. j. 65 (3) (2013), 331 – 339. linear weingarten spacelike submanifolds 81 [10] h.f. de lima, j.r. de lima, characterizations of linear weingarten spacelike hypersurfaces in einstein spacetimes, glasg. math. j. 55 (3) (2013), 567 – 579. 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[19] d. yang, z.h. hou, linear weingarten spacelike submanifolds in de sitter space, j. geom. 103 (1) (2012), 177 – 190. e extracta mathematicae vol. 31, núm. 2, 145 – 167 (2016) ascent and essential ascent spectrum of linear relations ezzeddine chafai, maher mnif département de mathématiques, faculté des sciences de sfax, université de sfax, route de soukra km 3.5, b. p. 1171, 3000, sfax, tunisia ezzeddine.chafai@ipeis.rnu.tn maher.mnif@gmail.com presented by manuel gonzález received february 28, 2016 abstract: in the present paper, we study the ascent of a linear relation everywhere defined on a banach space x and the related essential ascent spectrum. some properties and characterization of such spectra are given. in particular, we show that a banach space x is finite dimensional if and only if the ascent and the essential ascent of every closed linear relation in x is finite. as an application, we focus on the stability of the ascent and the essential ascent spectrum under perturbations. we prove that an operator f in x has some finite rank power, if and only if, σeasc(t + f) = σ e asc(t), for every closed linear relation t commuting with f . key words: ascent, essential ascent, perturbation, spectrum, linear relations. ams subject class. (2010): 47a06, 47a53, 47a10. 1. introduction let x denote a linear space over k = r or c. a multivalued linear operator in x or simply a linear relation in x is a mapping from a subspace d(t) ⊂ x, called the domain of t , into the collection of nonempty subsets of x such that t(α1x1 + α2x2) = α1tx1 + α2tx2, for all nonzero scalars α1, α2 ∈ k and x1, x2 ∈ d(t). we use the convention that the domain of t is d(t) := {x ∈ x : tx ̸= ∅}. then we have tx = ∅, for all x ∈ x\d(t). the class of such linear relations t is denoted by lr(x). the subspace t(0) is called the multivalued part of t , and we say that t is a single valued linear operator or simply an operator if t(0) = {0}, that is equivalent to t maps the points of its domain to singletons. a linear relation t in x is uniquely determined by its graph, g(t), which is defined by g(t) := {(x, y) ∈ x ×x : x ∈ d(t), y ∈ tx}, so that we can identify t with g(t). we say that t ∈ lr(x) is closed if its graph g(t) is a closed subspace of x × x. we designate by cr(x) the class of all closed linear relations in x. given a subset a ⊂ x, the image of a is defined by t(a) := ∪{ta : a ∈ a ∩ d(t)}, while r(t) := t(d(t)) is called the range of t . the linear 145 146 e. chafai, m. mnif relation t is said to be surjective whenever its range r(t) coincides with x. the inverse of t is the linear relation t −1 given by g(t −1) := {(y, x) : (x, y) ∈ g(t)}. let ∅ ̸= b ⊂ x, then the inverse image of b under t is defined to be the set t −1(b) := {x ∈ d(t) : b ∩ tx ̸= ∅}. the kernel of t is the subspace n(t) := t −1(0) = {x ∈ d(t) : 0 ∈ tx}, and t is called injective if n(t) = {0}. when t is injective and surjective we say that t is bijective. the quantities α(t) := dim n(t) and β(t) := dim x/r(t) are called the nullity and the conullity of t , respectively, and the index of t is defined by ind(t) := α(t) − β(t) provided α(t) and β(t) are not both infinite. if α(t) and β(t) are both infinite, then t is said to have no index. let m be a subspace of x such that m ∩ d(t) ̸= ∅. then the restriction of t to m, denoted by t/m, is given by g(t/m) := {(m, y) ∈ g(t) : m ∈ m ∩ d(t)}. for linear relations s and t such that d(t) ∩ d(s) ̸= ∅ and λ ∈ c, the linear relations s +t and λt are given by g(s +t) := {(x, y +z) : (x, y) ∈ g(s), (x, z) ∈ g(t)} and g(λt) := {(x, λy) : (x, y) ∈ g(t)}. t − λ stands for t − λi, where i is the identity operator on x. let s, t ∈ lr(x) such that r(t) ∩ d(s) ̸= ∅. the product st is defined as the relation g(st) := {(x, z) : (x, y) ∈ g(t), (y, z) ∈ g(s) for some y ∈ x}. the product of linear relations is clearly associative. hence for t ∈ lr(x) and n ∈ z, t n is defined as usual with t 0 = i and t 1 = t . it is easily seen that (t −1)n = (t n)−1, n ∈ z. the singular chain manifold rc(t) of t is defined by rc(t) := ( +∞∪ n=1 t n(0) ) ∩ ( +∞∪ n=1 n(t n) ) and we say that the linear space rc(t) is trivial if rc(t) = {0}. for a given closed subspace m of a normed space x, let qm denote the natural quotient map from x onto x/m. if t ∈ lr(x), then we shall denote q t(0) by qt . clearly qt t is single valued, and t is closed, if and only if, qt t and t(0) are both closed (see [12, ii.5.3]). for x ∈ x, we define ∥tx∥ := ∥qt tx∥ and thus ∥t∥ = sup{∥tx∥ : ∥x∥ ≤ 1} = ∥qt t∥. the quantity ∥t∥ is referred as the norm of t , though we note that it is in fact a pseudonorm, since ∥t∥ = 0 does not imply t = 0. a linear relation t is said to be continuous if for each open subset v in r(t), t −1(v ) is an open subset in d(t), equivalently ∥t∥ < ∞, open if its inverse t −1 is continuous and bounded below if t is injective and open. continuous everywhere defined linear relations are referred to as bounded relations. it is very well known (see [22, lemmas 3.4 and 3.5]) that (n(t n))n∈n is an increasing sequence and if n(t m) = n(t m+1), for some nonnegative integer ascent and essential ascent spectrum of linear relations 147 m, then n(t m) = n(t n), for all n ≥ m. similarly, (r(t n))n∈n is a decreasing sequence and if r(t m) = r(t m+1), for some m ∈ n, then r(t m) = r(t n) for all n ≥ m. these statements lead to the introduction of the ascent and the descent of a linear relation t in x by a(t) := min{r ∈ n : n(t r) = n(t r+1)}; d(t) := min{s ∈ n : r(t s) = r(t s+1)}; respectively, whenever these minima exist. if no such numbers exist the ascent and the descent of t are defined to be ∞. likewise, the statements of lemma 2.5 below lead to the introduction of the essential ascent of a linear relation t , which are due to chafai and mnif [7], by ae(t) := min{n ∈ n : αn := dim n(t n+1)/n(t n) < ∞}, where the minimum over the empty set is taken to be infinite. if ae(t) is finite, we denote p(t) := min{p ∈ n : αn(t) = αp(t), ∀n ≥ p}. in the sequel, x will be a complex banach space and t ∈ cr(x). we say that t is upper semi-fredholm, usually denoted t ∈ φ+(x), if r(t) is closed and α(t) is finite. clearly, every upper semi-fredholm linear relation has a finite essential ascent precisely we have ae(t) = 0. the such class of linear relations contains every linear relation with finite ascent. the resolvent set of t is the set ρ(t) := {λ ∈ c : t − λ is bijective }; and the spectrum of t is defined as the set σ(t) : c\ρ(t). it is shown (see [12, vi.1.3]), that ρ(t) is an open set and hence σ(t) is closed. the descent resolvent, the ascent resolvent and the essential ascent resolvent sets of t are defined by ρdes(t) := {λ ∈ c : d(t − λ) < ∞}; ρasc(t) := {λ ∈ c : a(t − λ) < ∞ and r((t − λ)a(t−λ)+1) is closed}; ρeasc(t) := {λ ∈ c : ae(t − λ) < ∞ and r((t − λ) ae(t−λ)+1) is closed}; 148 e. chafai, m. mnif respectively. the descent spectrum σdes(t), the ascent spectrum σasc(t) and the essential ascent spectrum σeasc(t) are defined by σdes(t) := c\ρdes(t); σasc(t) := c\ρasc(t); σeasc(t) := c\ρ e asc(t); respectively. linear relations were introduced into functional analysis by j. von neumann [24], motivated by the need to consider adjoints of non-densely defined linear differential operators which are considered by coddington [9], coddington and dijksma [10], dikjsma, sabbah and de snoo [13], among others. one main reason why linear relations are more convenient than operators is that one can define the inverse, the closure and the completion for a linear relation. interesting works on multivalued linear operators include the treatise on partial differential relations by gromov [18], the application of multivalued methods to solution of differential equations by favini and yagi [15], the development of fixed point theory for linear relations to the existence of mild solutions of quasi-linear differential inclusions of evolution and also to many problems of fuzzy theory (see, for example [1]) and several papers on semifredholm linear relations and other classes related to them (see, for examples [5] and [4]). for an operator in a linear space, the notion of ascent and essential ascent was studied in several articles, for instance, we cite [6], [8], [16], [17], [19], [20] and [23]. later, these concepts are extended to the multivalued case. in particular, some well known results concerning the ascent and the essential ascent for the case of linear operators remain valid in the context of linear relations. sometimes, an additional condition is needed which is the linear relation having a trivial singular chain manifold. in [6], the authors study the ascent and the essential ascent spectrum of an operator acting on a banach space. they show that a banach space x has a finite dimension, if and only if, the essential ascent of every operator on x is finite. the aim of this paper is to find conditions under which results of the type mentioned above will still be true in the most general setting of multivalued linear operators between banach spaces. the structure of this work is as follows. throughout section 2, we give some auxiliary results, sometimes purely algebraic, which are used to prove the main results. section 3 is devoted to the study of the ascent spectrum and the essential ascent spectrum of a closed linear relation acting on a banach space. we show that they are closed subsets of the spectrum, and that σeasc(t) is empty precisely when σasc(t) is empty. we shall also prove that x has a finite ascent and essential ascent spectrum of linear relations 149 dimension, if and only if, the essential ascent of every closed linear relation is finite. finally, in section 4, we are concerned with the stability of the essential ascent spectrum under finite rank perturbations. we prove that f k has a finite dimensional range, for some k ∈ n, if and only if, σeasc(t +f) = σeasc(t) (equivalently, σasc(t + f) = σasc(t)) for every closed linear relation t in the commutant of f. 2. preliminary and auxiliary results in this section we collect some algebraic results of the theory of multivalued linear operators which will be needed in the following sections. firstly, we recall the next elementary lemma. lemma 2.1. ([12, i.3.1]) let x be a linear space and t ∈ lr(x). then (i) tt −1(m) = m ∩ r(t) + t(0), for all m ⊂ x. (ii) t −1t(m) = m ∩ d(t) + n(t), for all m ⊂ x. (iii) t(m + n) = t(m) + t(n), for all m ⊂ x and n ⊂ d(t). a proof of the next lemma can be found in [22]. lemma 2.2. ([22, lemmas 4.1 and 4.4]) let t be a linear relation in a linear space x and let n, m ∈ n. then (i) d(t m)/(r(t n) + n(t m)) ∩ d(t m) ≃ r(t m)/r(t m+n). (ii) if, moreover, rc(t) = {0}, then n(t m+n)/n(t n) ≃ n(t m) ∩ r(t n). as an immediate consequence of lemma 2.2, we mention the following useful result which is valid for every linear relation everywhere defined in a linear space and having a trivial singular chain manifold. n(t) ∩ r(t p) = n(t) ∩ r(t p+n), for all n ∈ n, where p := p(t). the inverse image of a closed linear space m of a banach space x under a linear relation t in x is not, in general, a closed subspace of x. in the following lemma, we give conditions for which t −1(m) remains closed. lemma 2.3. let t be an everywhere defined linear relation in a banach space x and let m be a closed subspace of x such that t(0) ⊂ m. then t −1(m) is closed. 150 e. chafai, m. mnif proof. according to [12, iii.4.2] and, since t is closed and everywhere defined, one can deduce that t is bounded. so that, qt t is a bounded operator. on the other hand, since m and t(0) are both closed, it follows, from [3, lemma 13]), that qt (m) = (m + t(0))/t(0) = m/t(0) is also closed. therefore (qt t) −1qt (m) is closed. but clearly, (qt t) −1qt (m) = t −1(m + n(qt )) = t −1(m + t(0)) = t −1(m). the next lemma is used to prove lemma 3.2 below. lemma 2.4. let x be a banach space, t ∈ cr(x) be everywhere defined and let m be a closed subspace of x such that t(0) ∩ m = {0} or t(0) ⊂ m. suppose that m +r(t) and m ∩r(t) are closed. if either t(0) or m ∩r(t) has a finite dimension then, r(t) is closed. proof. write for short n = (m + t(0)) ∩ r(t) = m ∩ r(t) + t(0) if t(0) ∩ m = {0} and n = m ∩ r(t) if t(0) ⊂ m. clearly n is a closed subspace of x and hence, using lemma 2.3, it follows that t −1(n) is also closed . now, let us consider the linear relation t̂ : (x/t −1(n)) ⊕ (m/n) → (r(t) + m)/n defined canonically by t̂(x + m) := {y + m : y ∈ tx}. it is easy to check that t̂ is correctly defined. moreover, for y ∈ t(0), we have y = 0 (as t(0) ⊂ n), so that t̂(0) = 0. which implies that t̂ is an operator. on the other hand, if x ∈ x and m ∈ m such that t̂(x + m) = 0, then tx + m ⊂ n and hence tx ⊂ (m + n) ∩ r(t) = n. it follows that m ∈ n and x ∈ t −1(n), so that x + m = 0. this implies that t̂ is injective and, obviously, t̂ is surjective. thus t̂ is bijective. now, since tt −1(n) = n, then for all x ∈ x, m ∈ m and y ∈ tx we have ∥t̂(x + m)∥ = ∥y + m∥ = d(y + m, n) ≤ d(y, n) + d(m, n) = d(y, tt −1(n)) + d(m, n) = inf x′∈t −1(n) d(y, tx′) + d(m, n). ascent and essential ascent spectrum of linear relations 151 which implies that ∥t̂(x + m) ≤ inf x′∈t −1(n) d(tx, tx′) + d(m, n) = inf x′∈t −1(n) ∥tx − tx′∥ + d(m, n) ≤ ∥t∥ inf x′∈t −1(n) ∥x − x′∥ + d(m, n) = ∥t∥d(x, t −1(n)) + d(m, n) ≤ (1 + ∥t∥)[d(x, t −1(n)) + d(m, n)] = (1 + ∥t∥)∥x + m∥. thus t̂ is bounded and, since t̂ is bijective, then, by the open mapping theorem of linear operators, t̂(x/(t −1(n))) is closed. now let us consider p : r(t) + m → (r(t) + m)/n the canonical projection. then r(t) = p −1(r(t)/n) = p −1(t̂(x/t −1(n))). consequently, r(t) is closed. in order to introduce the ascent and the essential ascent of linear relations we will need the next result. lemma 2.5. let t be a linear relation in a linear space x such that rc(t) = {0}. then, for n ≥ 1, (i) dim n(t n+1)/n(t n) ≤ dim n(t n)/n(t n−1). (ii) if there exists n ∈ n such that dim n(t n+1)/n(t n) is finite then dim n(t m+1)/n(t m) is finite, for all m ≥ n. (iii) dim n(t n+1)/n(t n) < ∞ if and only if dim n(t n+k)/n(t n) < ∞, for all k ≥ 1. proof. (i) we note, by lemma 2.2, that n(t n+1)/n(t n) ≃ r(t n) ∩ n(t) ⊂ r(t n−1) ∩ n(t) ≃ n(t n)/n(t n−1). thus, obviously, dim n(t n+1)/n(t n) ≤ dim n(t n)/n(t n−1). (ii) follows immediately from the part (i). (iii) suppose that dim n(t n+1)/n(t n) < ∞ then, for any k ≥ 1, dim n(t n+k)/n(t n) = k−1∑ i=0 dim n(t n+i+1)/n(t n+i) < ∞. the reverse implication is trivial. 152 e. chafai, m. mnif we close this section with the next lemma which is sometimes useful. lemma 2.6. let t be a linear relation in a banach space x. then (i) rc(t) = {0} if and only if rc(t − λ) = {0} for every λ ∈ c. (ii) if ρ(t) ̸= ∅ then rc(t) = {0}. proof. (i) see [22, lemma 7.1]. (ii) follows immediately from [21, lemma 6.1]. 3. ascent and essential ascent spectrum of linear relations throughout this section, x will denote a complex banach space. the regular spectrum (see definition 3.1 below) of t is defined as those complex numbers λ for which t − λ is not regular. in this section, our interest concentrates on proving that, if 0 /∈ σeasc(t), then either t is regular or 0 is an isolated point of its regular spectrum. this extend the result of theorem 2.3, described in [6], to the multivalued case. the proof requires the following technical lemmas. lemma 3.1. let t ∈ lr(x) be everywhere defined. then (i) d(t) is finite if and only if r(t) + n(t d) = x for some d ∈ n. if, moreover, rc(t) = {0}, then (ii) a(t) is finite if and only if n(t) ∩ r(t p) = {0} for some p ∈ n. (iii) ae(t) is finite if and only if n(t) ∩ r(t p) has a finite dimension in x for some p ∈ n. proof. follows immediately from lemma 2.2. lemma 3.2. let t ∈ cr(x) be everywhere defined such that ρ(t) ̸= ∅ and ae(t) is finite. if r(t n) is closed, for some n > ae(t), then r(t n) is closed, for all n ≥ ae(t). proof. the use of [14, lemma 3.1] proves that t n is closed and everywhere defined, for all n ∈ n. moreover, from lemma 2.6, we get rc(t) = {0}. now, suppose that r(t n) is closed, for some n > ae(t). we need only to show that r(t n−1) and r(t n+1) are both closed. since t is continuous and t(0) ⊂ r(t n), then, by lemma 2.3, one can deduce that t −1(r(t n)) = ascent and essential ascent spectrum of linear relations 153 r(t n−1) + n(t) is closed. now, using lemma 3.1, we get r(t n−1) ∩ n(t) is finite dimensional and hence it is closed. according to lemma 2.4, it follows that r(t n−1) is closed. let t0 be the restriction of t to the banach space r(t n−1) onto the banach space r(t n). evidently, t0 is surjective and closed, which implies that t0 is open. write for short m := r(t n) + n(t0) = r(t n) + n(t) ∩ r(t n−1). clearly, since m and t0 are both closed, t1 := t0/m is also closed . according to [12, ii.6.1] and the fact that t0 is open, it follows that t1 is open. consequently, from [12, ii.5.3], we deduce that r(t1) = t(m) = r(t n+1) + t(n(t) ∩ r(t n−1) = r(t n+1) is closed. this completes the proof. remark 3.1. since ae(t) ≤ a(t), we obviously see, by lemma 3.2, for t ∈ cr(x) everywhere defined such that rc(t) = {0}, that σeasc(t) ⊆ σasc(t) ⊆ σ(t). the following purely algebraic lemma helps to read definition 3.1 below. there exhibits some useful connections between the kernels and the ranges of the iterates t n of a linear relation t in x. lemma 3.3. ([2, lemma 3.7]) let t ∈ lr(x). then the following statements are equivalent. (i) n(t) ⊂ r(t n) for each n ∈ n. (ii) n(t m) ⊂ r(t) for each m ∈ n. (iii) n(t m) ⊂ r(t n) for each m ∈ n and n ∈ n. definition 3.1. we say that a linear relation t ∈ lr(x) is regular if r(t) is closed and t verifies one of the equivalent conditions of lemma 3.3. trivial examples of regular linear relations are surjective multivalued operators as well as injective multivalued operators with closed range. the next perturbation results are shown in [2] and used in the sequel. lemma 3.4. ([2, theorems 23, 25 and 27]) let t ∈ cr(x). (i) if t is regular then there exists γ > 0 such that t − λ is regular for all |λ| < γ. (ii) if t ∈ φ+(x) then there exists γ > 0 such that t − λ ∈ φ+(x) and α(t − λ) is constant in the annulus 0 < |λ| < γ. moreover t is regular if and only if α(t − λ) = α(t) for all 0 < |λ| < γ. 154 e. chafai, m. mnif we shall make frequent use of the following result which is the multivalued version of the corresponding result for operators. lemma 3.5. let t ∈ lr(x) be regular with finite-dimensional kernel and such that rc(t) = {0}. then α(t n) = n α(t). proof. since t is regular then n(t n−1) ⊂ r(t). this implies, by lemma 2.2, that dim n(t n−1) = dim(n(t n−1) ∩ r(t)) = dim(n(t n)/n(t)). thus dim n(t n) = dim n(t n−1) + dim n(t). by induction, we get dim n(t n) = n dim n(t) for all n ∈ n. remark 3.2. as a consequence of [11, theorem 3.1], if t is an everywhere defined linear relation in x with an index and such that rc(t) = {0}, then ind(t n) = n ind(t). now, we are ready to give our first main result of this section. theorem 3.1. let t ∈ cr(x) be everywhere defined and such that ρ(t) ̸= ∅, ae(t) < ∞ and r(t ae(t)+1) is closed. let p := p(t). then there exists γ > 0 such that for each 0 < |λ| < γ the following assertions hold: (i) t − λ is regular. (ii) dim(n(t − λ)n) = n dim(n(t p+1)/n(t p)) for each n ∈ n. (iii) codim r(t − λ)n = n dim(r(t p)/r(t p+1)) for each n ∈ n. proof. according to [12, iii.4.2(a)] and, since t is closed and everywhere defined, we deduce that t is bounded and t(0) is closed. on the other hand, from lemma 3.2, we infer that r(t p+1) and r(t p) are closed. let t0 := t/r(t p), be the restriction of t to r(t p), then t0 is closed (as t and r(t p) are both closed). however, n(t0) = n(t)∩r(t p) = n(t)∩r(t p+n), for all n ∈ n. it follows, from lemma 3.1(iii), that n(t0) is finite dimensional and that n(t0) ⊂ r(t p+n) = r(t n0 ). therefore t0 is both regular and upper semi-fredholm. according to lemma 3.4, there exists γ > 0 such that t0 − λ is both regular and upper semi-fredholm with α(t0) = α(t0 − λ), for each 0 < |λ| < γ. furthermore, rc(t) = {0} (since ρ(t) ̸= ∅) and hence ascent and essential ascent spectrum of linear relations 155 rc(t0) ⊂ rc(t) = {0}. which implies that rc(t0 − λ) = {0} (by lemma 2.6). it follows that dim(n(t − λ)n) = dim(n(t − λ)n ∩ r(t p)) ([22, lemma 7.2]) = dim(n(t0 − λ)n) = n dim n(t0 − λ) (lemma 3.5) = n dim n(t0) (lemma 3.4) = n dim(n(t) ∩ r(t p)) = dim(n(t p+1)/n(t p)) (lemma 2.2). now, for n ≥ 1 and λ ̸= 0, we let us consider the polynomials p and q defined by p(z) = (z − λ)n and q(z) = zp, for all z ∈ c. clearly p and q have no common divisors. then there exist two polynomials u and v such that 1 = p(z)u(z)+q(z)v(z) for all z ∈ c. it follows that x = r(t −λ)n +r(t p), and therefore codim r(t − λ)n = dim x/r(t − λ)n = dim[(r(t p) + r(t − λ)n)/r(t − λ)n] = dim[r(t p)/r(t p) ∩ r(t − λ)n] ([22, lemma 2.3]) = codim r(t0 − λ)n = dim n(t0 − λ)n − ind(t0 − λ)n = n[dim n(t0 − λ) − ind(t0 − λ)] (lemma 3.5 and = n[dim n(t0) − ind(t0)] remark 3.2) = n codim r(t0) = n dim r(t p)/r(t p+1). finally, n(t −λ) = n(t −λ)∩r(t p) = n(t0−λ) ⊆ r(t0−λ)n ⊆ r(t −λ)n. which means that t − λ is regular. the next corollary is an immediate consequence of theorem 3.1. corollary 3.1. let t ∈ cr(x) be everywhere defined such that a(t) < ∞, ρ(t) ̸= ∅ and r(t a(t)+1) is closed. then there exists γ > 0 such that, for each 0 < |λ| < γ, the following assertions holds. (i) t − λ is regular. (ii) t − λ is bounded below. (iii) codim r(t − λ)n = n dim(r(t a(t))/r(t a(t)+1)). 156 e. chafai, m. mnif corollary 3.2. let t ∈ cr(x) be everywhere defined such that ρ(t) ̸= ∅. then σasc(t) and σeasc(t) are two closed subsets of σ(t). moreover σasc(t)\σeasc(t) is an open set. proof. the closedness of σeasc(t) and σasc(t) are two immediate consequences of theorem 3.1 and corollary 3.1, respectively. for the last assertion, let λ ∈ σasc(t)\σeasc(t) and let p := p(t − λ). then by theorem 3.1 there exists a deleted open neighborhood u of λ such that u ∩ σeasc(t) = ∅ and, for all α ∈ u and n ∈ n, dim n(t − α)n ≥ n dim(n(t − λ)p+1/n(t − λ)p). but, since t −λ has an infinite ascent, dim(n(t −λ)p+1/n(t −λ)p) is nonzero, and consequently the sequence (dim n(t −α)n)n is strictly increasing, for each α ∈ u. thus u ⊂ σasc(t), which completes the proof. in the following we denote e(t) := ρasc(t) ∩ ρdes(t) ∩ σ(t). corollary 3.3. let t ∈ cr(x) be everywhere defined such that ρ(t) ̸= ∅. if λ ∈ e(t) then λ is an isolated point of the boundary of σ(t). proof. let λ ∈ e(t). then t −λ has a finite descent and ascent, moreover r((t −λ)a(t−λ)+1) is closed. on the other hand, since t is closed and rc(t) = {0}, one can deduce, by virtue of lemma 2.6 and [12, ii.5.16], that t − λ is closed and rc(t − λ) = {0}. furthermore, applying corollary 3.1 to t − λ instead to t , we conclude that there exists γ > 0 such that t − α is injective and surjective, for all 0 < |α − λ| < γ. this implies that d(λ, γ)\{λ} ⊂ ρ(t). thus λ is isolated and it is in the boundary of the spectrum of t . the ascent and the essential ascent spectrum of a linear relation t can be empty. as a consequence of the following theorem, we show that this occurs precisely when the boundary of the spectrum of t is a subset of the essential ascent resolvent. theorem 3.2. let t ∈ cr(x) be everywhere defined such that ρ(t) ̸= ∅. then ρeasc(t) ∩ ∂σ(t) = ρasc(t) ∩ ∂σ(t) = e(t). (3.1) proof. from corollary 3.3, we have the obviously inclusions e(t) ⊂ ρasc(t) ∩ ∂σ(t) ⊂ ρeasc(t) ∩ ∂σ(t). for the reverse inclusions, it suffices ascent and essential ascent spectrum of linear relations 157 to show that ρeasc(t) ∩ ∂σ(t) ⊂ e(t). let λ be an element of the boundary of σ(t) such that t − λ has a finite essential ascent and r(t ae(t−λ)+1) is closed. then r(t a(t−λ)+1) is closed (see lemma 3.2), t − λ is closed (by [3, lemma 14]) and rc(t − λ) = {0} (by [22, lemma 7.1]). moreover according to theorem 3.1, there exists a punctured neighborhood v of λ such that dim n(t − α) = dim(n(t − λ)p)/(n(t − λ)p) and codim r(t − α) = dim(r(t − λ)p/r(t − λ)p+1), for some p ∈ n and all α ∈ v . since t − α is closed, for all α (see [3, lemma 14] ) and λ ∈ ∂σ(t), then there exists α0 ∈ v \σ(t) ̸= ∅. hence 0 = dim n(t − α0) = codim r(t − α0) = dim(n(t − λ)p+1)/n((t − λ)p) = dim(r((t − λ)p)/r((t − λ)p+1)). it follows that t −λ is of finite ascent and descent and r(t a(t−λ)+1) is closed. this means that λ ∈ e(t). corollary 3.4. let t ∈ cr(x) be everywhere defined such that ρ(t) ̸= ∅. then the following assertions are equivalent. (i) σasc(t) = ∅. (ii) σeasc(t) = ∅. (iii) ∂σ(t) ⊆ ρasc(t). (iv) ∂σ(t) ⊆ ρeasc(t). proof. all the implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) follow immediately from (3.1). for the implication (iv) ⇒ (i), assume that ∂σ(t) ⊆ ρeasc(t) then ∂σ(t) = e(t). according to corollary 3.3, it follows that all points of ∂σ(t) are isolated and hence ∂σ(t) = σ(t). now we get, by (3.1), σ(t) ⊂ ρdes(t) ∩ ρasc(t). which means that c = σ(t) ∪ ρ(t) ⊂ ρdes(t) ∩ ρasc(t). hence ρasc(t) = c, and consequently σasc(t) = ∅. corollary 3.5. let k(x) := {t ∈ cr(x) : d(t) = x and ρ(t) ̸= ∅}. the following assertions are equivalent. (i) x has a finite dimension. (ii) every t ∈ k(x) has a finite ascent and r(t a(t)+1) is closed. (iii) every t ∈ k(x) has a finite essential ascent and r(t ae(t)+1) is closed. 158 e. chafai, m. mnif (iv) σasc(t) is empty, for every t ∈ k(x). (v) σeasc(t) is empty, for every t ∈ k(x). proof. first observe that t ∈ k(x), if and only if, t − λ ∈ k(x), for every λ ∈ c. however, all the implications (i) ⇒ (ii) ⇒ (iii) and (iv) ⇒ (v) are obvious. from corollary 3.4, it follows immediately that (iii) ⇒ (iv). now, suppose that σeasc(t) is empty for every t ∈ k(x), then σeasc(t) is empty, for every bounded linear operator on x. it follows, from [6, corollary 2.8], that x has a finite dimension. thus (v) ⇒ (i). theorem 3.3. let t ∈ cr(x) be everywhere defined such that ρ(t) ̸= ∅. if ω be a connected component of ρeasc(t) then ω ⊂ σ(t) or ω\e(t) ⊂ ρ(t). proof. let ωr := {λ ∈ ω : t − λ is both regular and upper semifredholm}. from theorem 3.2, clearly that ω1 := ω\ωr is at most countable and therefore ωr is connected. suppose that ω ∩ ρ(t) is non-empty, so that is ωr ∩ ρ(t). let λ ∈ ω ∩ ρ(t). the use of lemma 3.4 and [3, corollary 17] leads to dim n(t − λ) = 0 and, by continuity of the index (see [3, theorem 15]), codim r(t − λ) = 0. this implies that ωr ⊂ ρ(t). thus ω\ωr consists of isolated points of the spectrum with finite essential ascent, so that ω\ωr ⊂ ρeasc(t) ∩ ∂σ(t) = e(t). consequently ω\e(t) ⊂ ωr ⊂ ρ(t). corollary 3.6. let t ∈ cr(x) be everywhere defined such that ρ(t) ̸= ∅. the following assertions are equivalent. (i) σ(t) is at most countable. (ii) σasc(t) is at most countable. (iii) σeasc(t) is at most countable. in this case, σasc(t) = σ e asc(t) and σ(t) = σasc(t) ∪ e(t). proof. all the implications are obvious except (iii) ⇒ (i). to show this, assume that σeasc(t) is at most countable, then ρ e asc(t) is connected. the use of theorem 3.3 leads to ρeasc(t)\e(t) ⊂ ρ(t). consequently, σ(t) = σeasc(t) ∪ e(t) is at most countable. ascent and essential ascent spectrum of linear relations 159 for the last assertion, suppose that σ(t) is at most countable, then σasc(t)\σeasc(t) is at most countable and open (see corollary 3.2). consequently it is empty, which means that σeasc(t) = σasc(t). 4. ascent, essential ascent spectrum and perturbations in this section we are interested to investigate the stability of the ascent spectrum and the essential ascent spectrum of a linear relation, everywhere defined in a complex banach space, under commuting finite rank perturbations. we start this section by some technical lemmas which are used in the sequel. lemma 4.1. let a and b be two linear relations in a linear space x and let c be an operator in x. assume that d(a) = d(c) = x, ab = ba, ac = ca and a(0) ⊂ b(0). then an(b + c) = (b + c)an = anb + anc, for all n ∈ n. (4.1) proof. first consider the case n = 1. from [12, i.4.2 (e)], we have an(b + c) = anb + anc, for all ∈ n, (4.2) and, by using [12, i.4.3 (c)], it follows that (b + c)a ⊂ a(b + c). (4.3) now, let y ∈ g(ab + ac). then y ∈ abx + acx = bax + cax, for some x ∈ x. which implies that y ∈ by1 + cy2, for some y1, y2 ∈ ax, so that y ∈ by1 + cy1 + c(y2 − y1) = by1 + cy1 = (b + c)y1 ⊂ (b + c)ax, because c(y2 − y1) ∈ ca(0) = ac(0) = a(0) ⊂ b(0). therefore ab + ac ⊂ (b + c)a. (4.4) the use of (4.2), (4.3) and (4.4) leads to a(b+c) = (b+c)a holds. assume now that (4.1) holds, for some positive integer n. then an+1(b + c) = aan(b + c) = a(b + b)an = (b + c)aan = (b + c)an+1. thus (4.1) holds, for all n ∈ n. lemma 4.2. let t be an everywhere defined linear relation in a linear space x and let f be an operator in x such that d(f) = x and tf = ft . then, 160 e. chafai, m. mnif (i) (t + f)n = n∑ i=0 (ni )t n−if i, for all n ∈ n. (ii) t n − f n = ( n−1∑ i=0 t n−1−if i ) (t − f), for all n ≥ 1. proof. (i) for n = 0 and n = 1 the result is trivial. suppose that (t + f)n = n∑ i=0 (ni )t n−if i, for some n ≥ 1. it follows that (t + f)n+1 = (t + f)(t + f)n = (t + f) ( n∑ i=0 (ni )t n−if i ) = n∑ i=0 (ni )(t + f)t n−if i ([12, i.4.2 (e)]) = n∑ i=0 [(ni )t n−i(t + f)f i] (lemma 4.1) = n∑ i=0 [(ni )t n−if i(t + f)] (lemma 4.1) = n∑ i=0 (ni )t n−if it + n∑ i=0 (ni )t n−if if ([12, i.4.2 (e)] = n∑ i=0 (ni )t n−i+1f i + n∑ i=0 (ni )t n−if i+1 (as tf = ft) = n∑ i=0 (ni )t n−i+1f i + n+1∑ i=1 ( ni−1)t n−i+1f i = t n+1 + n∑ i=1 (n+1i )t n−i+1f i + f n+1 = n+1∑ i=0 (n+1i )t n+1−if i. therefore the required equality holds, for all n ∈ n. (ii) we can easily seen that t n + t kf j − t kf j = t n, for all 0 ≤ k ≤ n and ascent and essential ascent spectrum of linear relations 161 j ∈ n. applying this together with lemma 4.1 and [12, i.4.2 (e)] we obtain ( n−1∑ i=0 t n−1−if i ) (t − f) = (t n−1 + t n−2f + · · · + f n−1)t − (t n−1 + t n−2f + · · · + f n−1)f. we shall now to show, by induction on n, that (t n−1 + t n−2f + · · · + f n−1)t = t(t n−1 + t n−2f + · · · + f n−1). (4.5) the case n = 1 is evident. assume that (4.5) holds for some positive integer n and take a = t, b = t(t n−1 + t n−2f + · · · + f n−1) and c = f n. then, d(a) = d(c) = x, a(0) = t(0) ⊂ b(0) = t n(0), ac = tf n = f nt = ca (as tf = ft) and ab = t(t(t n−1 + t n−2f + · · · + f n−1)) = t(t n−1 + t n−2f + · · · + f n−1)t = ba. now, this fact together with lemma 4.1 and [12, i.4.2 (e)]), lead to (t n + t n−1f + · · · + f n)t = [t(t n−1 + t n−2f + · · · + f n−1) + f n]t = (b + c)a = a(b + c) = t [t(t n−1 + t n−2f + · · · + f n−1) + f n] = t(t n + t n−2f + · · · + f n). hence (4.5) holds for all n ≥ 1. we prove, arguing in a similar way as in preceding, that (t n−1 + t n−2f + · · · + f n−1)f = f(t n−1 + t n−2f + · · · + f n−1). (4.6) combining (4.5) and (4.6) we get ( n−1∑ i=0 t n−1−if i ) (t − f) = t(t n−1 + t n−2f + · · · + f n−1) − f(t n−1 + t n−2f + · · · + f n−1) = (t n + t n−1f + · · · + tf n−1) − (t n−1f + t n−2f 2 + · · · + f n) = t n + (t n−1f − t n−1f + · · · + tf n−1 − tf n−1) − f n = t n − f n. 162 e. chafai, m. mnif in the rest of this section, x, unless otherwise stated, will be a complex banach space. lemma 4.3. let t ∈ cr(x). (i) let f ∈ lr(x) be single valued with finite rank. if t ∈ φ+(x) then t + f ∈ φ+(x). (ii) let s ∈ lr(x) be bounded. if st ∈ φ+(x), then t ∈ φ+(x). proof. follows immediately from [12, v.3.2 and v.2.16]. lemma 4.4. let f be a bounded operator in x such that f k is of finite rank, for some k ≥ 1, and let t ∈ cr(x) be everywhere defined. suppose that ft = tf . if t is upper semi-fredholm then t +f is upper semi-fredholm. proof. according to lemma 4.2, we can write t k −f k = s(t −f) where s := k−1∑ i=0 t k−1−if i. furthermore, since t is closed and upper semi-fredholm, then so is t k (see [2, proposition 24]). from this together with lemma 4.3, it follows that t k − f k is also upper semi-fredholm. on the other hand, since t is closed and everywhere defined and f is bounded, then s is bounded. this means, using lemma 4.3, that t − f is upper semi-fredholm. now, by interchanging f and −f, we obtain t + f is upper semi-fredholm. lemma 4.5. let x be a linear spaces and let t ∈ lr(x) be injective. then dim d(t) ≤ dim(r(t)). proof. we have, by [12, i.6.4], dim d(t)+dim t(0) = dim r(t)+dim n(t). therefore, since n(t) = {0}, we have obviously dim d(t) ≤ dim(r(t)). remark 4.1. as a direct consequence of the above lemma we have, for t ∈ lr(x), dim d(t)/n(t) ≤ dim r(t). lemma 4.6. let f be a bounded operator in x such that f k is of finite rank, for some nonnegative integer k, and let t ∈ cr(x) be everywhere defined. assume that ft = tf . then dim ( n(t n)/n((t + f)n+k−1) ∩ n(t n) ) < dim ( r(f k) ) < ∞, for all n ≥ 1. ascent and essential ascent spectrum of linear relations 163 proof. first, observe, since tf = ft , that t m(0) = (tf)m(0) = (ft)m(0) = f m(t m(0)) ⊂ r(f m), for all m ∈ n. let n ≥ 1 and let m be a subspace of n(t n) such that n(t n) = [ n((t + f)n+k−1) ∩ n(t n) ] ⊕ m. it follows, by lemma 4.2, that (t + f)n+k−1 maps n(t n) into t n+k−1(0) + r(f k) = r(f k). now, since (t +f)n+k−1 is injective on m, and according to lemma 4.5, we get dim(m) ≤ dim(t +f)n+k−1(n(t n)) ≤ dim(r(f k)) < ∞. this proves the lemma. now, we are in the position to give the main theorem of this section. theorem 4.1. let f be a bounded operator on x such that f k is of finite rank, for some nonnegative integer k, and let t ∈ cr(x) be everywhere defined such that ρ(t) ̸= ∅ and ρ(t + f) ̸= ∅. suppose that tf = ft . then (i) a(t) is finite if and only if a(t + f) is finite. (ii) ae(t) is finite if and only if ae(t + f) is finite. in this case r(t ae(t)+1) is closed if and only if r((t +f)ae(t+f)+1) is closed. proof. clearly, since f is a bounded operator and using lemma 4.3, it suffices to show only one direction. (i) let a := a(t) and, for n ≥ a, we let us consider the sequences (an(t))n∈n and (bn(t))n∈n defined as an(t) := dim ( n(t n)/n(t + f)n+k−1 ∩ n(t n) ) = dim ( n(t a)/n(t + f)n+k−1 ∩ n(t a) ) ; bn(t) := dim ( n(t + f)n/n(t + f)n ∩ n(t n+k−1) ) = dim ( n(t + f)n/n(t + f)n ∩ n(t a) ) ; respectively. clearly (an(t))n is a decreasing sequence, which implies that there exists p ≥ a such that an(t) = ap(t), for all n ≥ p. it follows that n(t + f)n+k−1 ∩ n(t a) = n(t + f)p+k−1 ∩ n(t a), for all n ≥ p. furthermore (bn(t))n is an increasing sequence and, by interchanging t and t + f in lemma 4.6, we may infer that bn(t) ≤ dim(r(f k)) < ∞. so, there 164 e. chafai, m. mnif exists q ≥ a such that bn(t) = bq(t), for all n ≥ q. hence, for n ≥ q ≥ p+k−1, dim ( n(t + f)q/n(t + f)q ∩ n(t a) ) = dim ( n(t + f)n+k−1/n(t + f)n+k−1 ∩ n(t a) ) = dim ( n(t + f)n+k−1/n(t + f)p+k−1 ∩ n(t a) ) = dim ( n(t + f)n+k−1/n(t + f)q ∩ n(t a) ) . this implies that n(t + f)q = n(t + f)n+k−1, for all n ≥ q. thus a(t + f) ≤ q. (ii) suppose that t has a finite essential ascent and let p := p(t). given n ≥ k + p, it follows, by lemma 4.6, that dim ( n(t n)/n((t + f)n+k−1) ∩ n(t n) ) < ∞. (4.7) by interchanging t and t + f in (4.7), we obtain dim ( n(t + f)n/n(t n+k−1) ∩ n(t + f)n ) < ∞. on the other hand, since ae(t) < ∞, then dim n(t n+k−1)/n(t p) < ∞, which means that dim ( n((t + f)n)/n(t + f)n ∩ n(t p)) ) < ∞. (4.8) furthermore, n(f k) ∩ n(t p) ⊂ n(t + f)n ∩ n(t p) ⊂ n(t p) and, since f k is finite dimensional range, it follows that dim ( n(t p)/(n(f k) ∩ n(t p)) ) < ∞. (4.9) hence dim ( n(t p)/(n(t + f)n ∩ n(t p)) ) < ∞ (4.10) now, the use of (4.8) combined with (4.9), leads to dim ( n(t + f)n/n(f k) ∩ n(t p) ) = dim ( n(t + f)n/n(t + f)n ∩ n(t p) ) + dim ( n(t + f)n ∩ n(t p)/n(f k) ∩ n(t p) ) ≤ dim ( n(t + f)n/n(t + f)n ∩ n(t p) ) + dim ( n(t p)/n(f k) ∩ n(t p) ) < ∞. ascent and essential ascent spectrum of linear relations 165 therefore dim(n(t + f)n+1/n(t + f)n) = dim(n(t + f)n+1/n(f k) ∩ n(t p)) − dim(n(t + f)n/n(f k) ∩ n(t p)) < ∞. thus ae(t + f) ≤ p + k. now, assume that r(t ae(t)+1) is closed and let n ≥ k + q, where q := max(p(t), p(t + f)). denote by t0 and f0 the restrictions of t and f, respectively, to r(t q). since r(t q) is closed (by lemma 3.2) and t is closed, then t0 is closed. on the other hand, n(t0) = n(t) ∩ r(t q) is finite dimensional (by lemma 3.1), which means that t0 is upper semi-fredholm. moreover t0 + f0 is closed (as t0 is closed and f0 is a bounded operator). furthermore, clearly t0(0) = t(0) ⊂ n(f) ∩ r(t q) = n(f0), and hence, by lemma 4.4, t0 +f0 is upper semi fredholm. it follows, from [2, proposition 24], that (t0 + f0) n is also upper semi-fredholm. this means that t q(r(t + f)n) = r(t0 − f0)n is closed. therefore, since t q is everywhere defined and closed and t q(0) ⊂ t q(r(t + f)n), we infer that t −qt q(r(t +f)n) is closed. thus r((t +f)n)+n(t q) = t −qt q(r(t +f)n) is closed. on the other hand, using (4.10), we get dim ( r(t + f)n ∩ n(t q)/r(t + f)n ∩ n(t + f)n ∩ n(t q) ) < ∞. however, since ae(t + f) < ∞ and using lemma 2.2, we deduce that n(t + f)n ∩ r(t + f)n has a finite dimension. this implies that dim r(t + f)n ∩ n(t q) < ∞, in particular r(t + f)n ∩ n(t q) is closed. by the hypothesis rc(t) = {0}, we infer that (t + f)n(0) ∩ n(t q) = t n(0) ∩ n(t q) = {0}. further, since t +f is closed and ρ(t +f) ̸= ∅, then (t +f)n is closed. the use of [14, lemma 3.2] leads to r(t +f)n is closed. hence, applying lemma 3.2, it follows that r(t + f)ae(t+f)+1 is closed. for the reverse implication it suffices to interchange t and t + f. as applications of theorem 4.1 we give the following corollaries. corollary 4.1. let f be a bounded operator in x and let kf := { t ∈ cr(x) : d(t) = x, tf = ft, ρ(t) ̸= ∅ and ρ(t + f) ̸= ∅ } . then the following assertions are equivalent. (i) f k has a finite rank, for some k ≥ 1. (ii) σasc(t + f) = σasc(t), for every t ∈ kf . (iii) σeasc(t + f) = σ e asc(t), for every t ∈ kf . the proof of this corollary requires the following lemma. 166 e. chafai, m. mnif lemma 4.7. ([6, theorem 3.2]) let f be a bounded operator in x. the following conditions are equivalent: (i) there exists a positive integer n such that f n has a finite rank. (ii) σeasc(t +f) = σ e asc(t), for all bounded operator t ∈ lr(x) commuting with f. (iii) σasc(t +f) = σasc(t), for all bounded operator t ∈ lr(x) commuting with f. proof of corollary 4.1. the implications (i) ⇒ (ii) and (i) ⇒ (iii) follow immediately from theorem 4.1. now, since all bounded operators commuting with f on x belong to kf and using lemma 4.7, we conclude that (ii) ⇒ (i) and (iii) ⇒ (i). corollary 4.2. let t ∈ cr(x) be everywhere defined such that ρ(t) ̸= ∅. then σeasc(t) ⊂ ∩ f∈ft (x) σasc(t + f) where ft (x) denotes the set of all bounded finite-rank operators f on x commuting with t and such that ρ(t + f) ̸= ∅. references [1] r. p. agarwal, m. meehan, d. o’regan, “fixed point theory and applications”, cambridge tracts in mathematics, 141, cambridge university press, cambridge, 2001. [2] t. álvarez, on regular linear relations, acta math. sin. (english series) 28 (1) (2012), 183 – 194. [3] t. álvarez, small perturbation of normally solvable relations, publ. math. debrecen 80 (1-2) (2012), 155 – 168. 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[16] s. grabiner, uniform ascent and descent of bounded operators, j. math. soc. japan 34 (2) (1982), 317 – 337. [17] s. grabiner and j. zemánek, ascent, descent and ergodic properties of linear operators, j. operator theory 48 (1) (2002), 69 – 81. [18] m. gromov, “partial differential relations”, springer-verlag, berlin, 1986. [19] m. a. kaashoek, d. c. lay, ascent, descent and commuting perturbation, trans. amer. math. soc. 169 (1972), 35 – 47. [20] m. mbekhta, ascente, descente et spectre essentiel quasi-fredholm, rend. circ. mat. palermo (2) 46 (2) (1997), 175 – 196. [21] a. sandovici, h. de snoo, an index formula for the product of linear relations, linear algebra appl. 431 (11) (2009), 2160 – 2171. [22] a. sandovici, h. de snoo, h. winkler, ascent, descent, nullity, defect, and related notions for linear relations in linear spaces, linear algebra appl. 423 (2-3) (2007), 456 – 497. 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[24] j. von neumann, functional operators ii, the geometry of orthogonal spaces, annals of math. studies, 22, princeton university press, princeton n. j., 1950. e extracta mathematicae vol. 32, núm. 2, 173 – 208 (2017) the differences between birkhoff and isosceles orthogonalities in radon planes hiroyasu mizuguchi student affairs department-shinnarashino educational affairs section, chiba institute of technology, narashino, japan hiroyasu.mizuguchi@p.chibakoudai.jp presented by javier alonso received may 22, 2017 abstract: the notion of orthogonality for vectors in inner product spaces is simple, interesting and fruitful. when moving to normed spaces, we have many possibilities to extend this notion. we consider birkhoff orthogonality and isosceles orthogonality. recently the constants which measure the difference between these orthogonalities have been investigated. the usual orthognality in inner product spaces and isosceles orthogonality in normed spaces are symmetric. however, birkhoff orthogonality in normed spaces is not symmetric in general. a two-dimensional normed space in which birkhoff orthogonality is symmetric is called a radon plane. in this paper, we consider the difference between birkhoff and isosceles orthogonalities in radon planes. key words: birkhoff orthogonality, isosceles orthogonality, minkowski plane, minkowski geometry, radon plane. ams subject class. (2010): 46b20, 51b20, 52a21, 26d20. 1.. introduction we denote by x a real normed space with the norm ∥ · ∥, the unit ball bx and the unit sphere sx. throughout this paper, we assume that the dimension of x is at least two. in case of that x is an inner product space, an element x ∈ x is said to be orthogonal to y ∈ x (denoted by x ⊥ y) if the inner product ⟨x,y⟩ is zero. in the general setting of normed spaces, many notions of orthogonality have been introduced by means of equivalent propositions to the usual orthogonality in inner product spaces. for example, roberts [20] introduced roberts orthogonality: for any x,y ∈ x, x is said to be roberts orthogonal to y (denoted by x ⊥r y) if ∥x + ty∥ = ∥x − ty∥ for all t ∈ r. birkhoff [4] introduced birkhoff orthogonality: x is said to be birkhoff orthogonal to y (denoted by x ⊥b y) if ∥x + ty∥ ≥ ∥x∥ for all t ∈ r. 173 174 h. mizuguchi james [4] introduced isosceles orthogonality: x is said to be isosceles orthogonal to y (denoted by x ⊥i y) if ∥x + y∥ = ∥x − y∥. these generalized orthogonality types have been studied in a lot of papers ([1], [2], [8] and so on). recently, quantitative studies of the difference between two orthogonality types have been performed: d(x) = inf { inf λ∈r ∥x + λy∥ : x,y ∈ sx,x ⊥i y } , d′(x) = sup{∥x + y∥ − ∥x − y∥ : x,y ∈ sx,x ⊥b y}, br(x) = sup α>0 { ∥x + αy∥ − ∥x − αy∥ α : x,y ∈ sx,x ⊥b y } = sup { ∥x + y∥ − ∥x − y∥ ∥y∥ : x,y ∈ x,x,y ̸= 0,x ⊥b y } , bi(x) = sup { ∥x + y∥ − ∥x − y∥ ∥x∥ : x,y ∈ x,x,y ̸= 0,x ⊥b y } , ib(x) = inf { infλ∈r ∥x + λy∥ ∥x∥ : x,y ∈ x,x,y ̸= 0,x ⊥i y } . (see [10], [14], [19]). an orthogonality notion“⊥ ” is called symmetric if x ⊥ y implies y ⊥ x. the usual orthogonality in inner product spaces is, of course symmetric. by the definition, isosceles orthogonality in normed spaces is symmetric, too. however birkhoff orthogonality is not symmetric in general. birkhoff [4] proved that if birkhoff orthogonality is symmetric in a strictly convex normed space whose dimension is at least three, then the space is an inner product space. day [6] and james [9] showed that the assumption of strict convexity in birkhoff’s result can be released. theorem 1.1. ([2], [6], [9] ) a normed space x whose dimension is at least three is an inner product space if and only if birkhoff orthogonality is symmetric in x. the assumption of the dimension of the space in the above theorem cannot be omitted. a two-dimensional normed space in which birkhoff orthogonality is symmetric is called a radon plane. difference between orthogonality in radon planes 175 in this paper, we consider the constant ib(x) in radon planes. the inequality 1/2 ≤ ib(x) ≤ 1 holds for any normed space x. under the assumption that the space x is a radon plane, an inequality 8/9 ≤ ib(x) ≤ 1 is proved, and the radon plane in which ib(x) = 8/9 is characterized. on the other hand, a radon plane is made by connecting the unit sphere of a two-dimensional normed space and its dual ([6], [12], [13]). a collection of normed spaces in which ib(x) < 8/9 holds and that constant of the induced radon plane is equal to 8/9 is obtained. 2.. the difference between two orthogonality types in radon planes to consider the difference between birkhoff and isosceles orthogonalities, the results obtained by james in [7] are important. proposition 2.1. ([7]) (i) if x (̸= 0) and y are isosceles orthogonal elements in a normed space, then ∥x + ky∥ > 1 2 ∥x∥ for all k. (ii) if x (̸= 0) and y are isosceles orthogonal elements in a normed space, and ∥y∥ ≤ ∥x∥, then ∥x + ky∥ ≥ 2( √ 2 − 1)∥x∥ for all k. from this, one can has 1/2 ≤ ib(x) ≤ 1 and 2( √ 2 − 1) ≤ d(x) ≤ 1 for any normed space. for two elements x,y in the unit sphere in a normed space x, the sine function s(x,y) is defined by s(x,y) = inf t∈r ∥x + ty∥ ([22]). v. balestro, h. martini, and r. teixeira [3] showed the following proposition 2.2. ([3]) a two dimensional normed space x is a radon plane if and only if its associated sine function is symmetric. thus for elements x,y in the unit sphere in a radon plane x with x ⊥i y we have infλ∈r ∥x+λy∥ = infµ∈r ∥y +µx∥. hence the inequality 2( √ 2 − 1) ≤ ib(x) ≤ 1 holds for a radon plane x. using proposition 2.2 again, we start to consider the lower bound of ib(x) in a radon plane. 176 h. mizuguchi proposition 2.3. let x be a radon plane, an element x ∈ sx be isosceles orthogonality to αy for another element y ∈ sx and a real number α ∈ r. take numbers k,l ∈ r such that ∥x + ky∥ = minλ∈r ∥x + λy∥ = minµ∈r ∥y + µx∥ = ∥y + lx∥. then, in the estimation of the constant ib(x), we may only consider the situation 0 ≤ α ≤ 1, 0 ≤ k and 0 ≤ l. in this case, k ≤ min{1/2,α} and l ≤ 1/2 hold. proof. since x ⊥i αy implies x ⊥i −αy and y ⊥i x/α, we can suppose 0 ≤ α ≤ 1. from the assumption ∥x + ky∥ = minλ∈r ∥x + λy∥ = minµ∈r ∥y + µx∥ = ∥y + lx∥, we can also suppose 0 ≤ k and 0 ≤ l. then it follows from x ⊥i αy and ∥x + ky∥ = minλ∈r ∥x + λy∥ that k ≤ α. the assumption ∥x+ky∥ = minλ∈r ∥x+λy∥ implies that x+ky is birkhoff orthogonal to y. from the symmetry of birkhoff orthogonality in a radon plane, y is birkhoff orthogonal to x + ky. using this fact, one has α + k ≤ ∥x + ky − (α + k)y∥ = ∥x − αy∥ = ∥x + αy∥ = ∥x + ky + (α − k)y∥ ≤ ∥x + ky∥ + α − k and hence 2k ≤ ∥x + ky∥ = minλ∈r ∥x + λy∥ ≤ 1. in a similar way, from the fact that x is birkhoff orthogonal to y + lx, we have 2l ≤ ∥y + lx∥ ≤ 1. proposition 2.4. let x be a radon plane, an element x ∈ sx be isosceles orthogonality to αy for another element y ∈ sx and a number α ∈ [0,1]. take numbers k ∈ [0,min{1/2,α}] and l ∈ [0,1/2] such that ∥x + ky∥ = minλ∈r ∥x + λy∥ = minµ∈r ∥y + µx∥ = ∥y + lx∥. then ∥x + ky∥ ≥ max { (α + k)(1 − kl) (α + k)(1 − kl) + k(1 − l)(α − k) , (1 + αl)(1 − kl) (1 + αl)(1 − kl) + l(1 − k)(1 − αl) } . proof. it follows from x = α(x + ky) + k(x − αy) α + k difference between orthogonality in radon planes 177 and x ⊥i αy that α + k ≤ α∥x + ky∥ + k∥x − αy∥ = α∥x + ky∥ + k∥x + αy∥. for c = α − k 1 + α − k − αl and d = 1 − kl 1 + α − k − αl , the equality d(x + αy) = (1 − c)(x + ky) + c(y + lx) holds, and hence one has ∥x + αy∥ ≤ ∥x + ky∥ d = 1 + α − k − αl 1 − kl ∥x + ky∥. thus, we obtain α + k ≤ ( α + k · 1 + α − k − αl 1 − kl ) ∥x + ky∥ = (α + k)(1 − kl) + k(α − k − αl + kl) 1 − kl ∥x + ky∥ = (α + k)(1 − kl) + k(1 − l)(α − k) 1 − kl ∥x + ky∥. meanwhile, from the equality y = l(−x + αy) + y + lx 1 + αl , we obtain 1 + αl ≤ ∥y + lx∥ + l∥ − x + αy∥ = ∥x + ky∥ + l∥x + αy∥ ≤ ( 1 + l · 1 + α − k − αl 1 − kl ) ∥x + ky∥ = (1 + αl)(1 − kl) + l(1 − k)(1 − αl) 1 − kl ∥x + ky∥. let f(α,k,l) = k(1 − l)(α − k) (α + k)(1 − kl) and g(α,k,l) = l(1 − k)(1 − αl) (1 + αl)(1 − kl) . 178 h. mizuguchi from the above proposition, the inequality ∥x + ky∥−1 ≤ 1 + min { f(α,k,l), g(α,k,l) } (2.1) holds. it follows from 1 − l 1 − kl = 1 k + 1 − k k(kl − 1) that the function f(α,k,l) is decreasing on l in the interval [0,1]. in a similar way, g(α,k,l) is decreasing on k in the interval [0,1]. let us consider the upper bound of min{f(α,k,l), g(α,k,l)}. lemma 2.5. let 0 ≤ α ≤ 1, 0 ≤ k ≤ min{α,1/2} and k ≤ l ≤ 1/2. then min { f(α,k,l), g(α,k,l) } = f(α,k,l) ≤ k(1 − k) (1 + k)2 . proof. let 0 ≤ α ≤ 1, 0 ≤ k ≤ min{α,1/2} and k ≤ l ≤ 1/2. for the function h(α,k,l) := ( g(α,k,l) − f(α,k,l) ) (1 − kl), we have h(α,k,l) = l(1 − k) 1 − αl 1 + αl − k(1 − l) α − k α + k and hence ∂h ∂α = l(1 − k) ∂ ∂α ( 1 − αl 1 + αl ) − k(1 − l) ∂ ∂α ( α − k α + k ) = − 2l2(1 − k) (1 + αl)2 − 2k2(1 − l) (α + k)2 ≤ 0. this implies that h is decreasing on α. thus we obtain the inequality h(α,k,l) ≥ h(1,k, l) = l(1 − k) 1 − l 1 + l − k(1 − l) 1 − k 1 + k = (1 − k)(1 − l)(l − k) (1 + k)(1 + l) ≥ 0, and so f(α,k,l) ≤ g(α,k,l) holds. difference between orthogonality in radon planes 179 using the fact that f(α,k,l) is a decreasing function on l, min { f(α,k,l), g(α,k,l) } = f(α,k,l) ≤ f(α,k,k) = k(α − k) (1 + k)(α + k) . from the fact that the function (α − k)/(α + k) is increasing on α, it follows that k(α − k) (1 + k)(α + k) ≤ k(1 − k) (1 + k)2 , which completes the proof. lemma 2.6. let 0 ≤ α ≤ 1, 0 ≤ k ≤ min{α,1/3} and 0 ≤ l < k. then min { f(α,k,l), g(α,k,l) } ≤ k(1 − k) (1 + k)2 . proof. let 0 ≤ α ≤ 1, 0 ≤ k ≤ min{α,1/3} and 0 ≤ l < k. then min { f(α,k,l), g(α,k,l) } ≤ (1 − k)lf(α,k,l) + (1 − l)k g(α,k,l) (1 − k)l + (1 − l)k = 2α(1 − k)k(1 − l)l (α + k)(1 + αl) ( (1 − k)l + (1 − l)k ) = (∗). we have that (∗) ≤ k(1−k) (1+k)2 if and only if the function f(α,k,l) := 2α(1 + k)2(1 − l)l − (α + k)(1 + αl) ( (1 − k)l + (1 − l)k ) is negative. one can has f(α,k,l) = ( (2 + α)l − 1 ) k2 + ( α(1 − l)(4l − 1 − αl) − l(1 + αl)(1 − α) ) k + αl ( 1 − (2 + α)l ) and hence ∂f ∂k = 2ak + b, where a = (2 + α)l − 1 and b = α(1 − l)(4l − 1 − αl) − l(1 + αl)(1 − α). from the fact l < k ≤ 1/3, we obtain a ≤ (3l − 1) ≤ 0 and b ≤ α(1 − l)(4l − 1 − αl) − l(1 − α) ≤ α(1 − l)(l − αl) − l(1 − α) = l(1 − α) ( α(1 − l) − 1 ) ≤ 0. 180 h. mizuguchi thus the function f is decreasing with respect to k and hence f(α,k,l) ≤ f(α,l, l) = 2α(1 + l)2(1 − l)l − 2(α + l)(1 + αl)(1 − l)l = 2(1 − l)l ( α(1 + l)2 − (α + l)(1 + αl) ) = −2(1 − l)l2(1 − α)2 ≤ 0. this completes the proof. under the assumption 1/3 < k and l < k, we consider the upper bound of (∗), too. lemma 2.7. let 0 ≤ α ≤ 1, 1/3 < k ≤ min{α,1/2} and 0 ≤ l < k. then min { f(α,k,l), g(α,k,l) } ≤ 2k(1 − k) (√ 2(1 − k) − √ k )2 (1 + k) (√ 2(1 − 2k) + √ k(1 − k) )2 . proof. as in the above lemma, min{f(α,k,l), g(α,k,l)} is less than 2α(1 − k)k(1 − l)l (α + k)(1 + αl) ( (1 − k)l + (1 − l)k ) = (∗). the inequality (∗) ≤ 2k(1 − k) (√ 2(1 − k) − √ k )2 (1 + k) (√ 2(1 − 2k) + √ k(1 − k) )2 is equivalent to g(α,k,l) : = α(1 − l)l (α + k)(1 + αl) ( (1 − k)l + (1 − l)k ) ≤ (√ 2(1 − k) − √ k )2 (1 + k) (√ 2(1 − 2k) + √ k(1 − k) )2 . on this function g, one can see ∂g ∂α = (1 − l)l l(1 − k) + k(1 − l) × ∂ ∂α ( α (α + k)(1 + αl) ) = (1 − l)l l(1 − k) + k(1 − l) × k − α2l (α + k)2(1 + αl)2 . difference between orthogonality in radon planes 181 from the assumption k > l, the function g is increasing on α and so g(α,k,l) ≤ g(1,k, l) = (1 − l)l (1 + k)(1 + l) ( l(1 − k) + k(1 − l) ). we have that g(1,k, l) ≤ (√ 2(1 − k) − √ k )2 (1 + k) (√ 2(1 − 2k) + √ k(1 − k) )2 if and only if pk(l) := (√ 2(1 − 2k) + √ k(1 − k) )2 (1 − l)l − (√ 2(1 − k) − √ k )2 (1 + l) ( l(1 − k) + k(1 − l) ) ≤ 0. letting lk = k k + √ 2k(1 − k) , we have lk(1 − k) + k(1 − lk) = k + (1 − 2k)lk = 1 − k + √ 2k(1 − k) k + √ 2k(1 − k) k, and hence (1 + lk) ( k + (1 − 2k)lk ) (1 − lk)lk = ( 2k + √ 2k(1 − k) ) {1 − k + √ 2k(1 − k)}√ 2k(1 − k) = 2 √ 2k(1 − k) + 1 + k = (√ 1 − k + √ 2k )2 . meanwhile one can easily check(√ 1 − k + √ 2k )(√ 2(1 − k) − √ k ) = √ 2(1 − 2k) + √ k(1 − k). thus we obtain (1 + lk) ( lk(1 − k) + k(1 − lk) ) (1 − lk)lk = (√ 1 − k + √ 2k )2 = (√ 2(1 − 2k) + √ k(1 − k) )2(√ 2(1 − k) − √ k )2 , 182 h. mizuguchi which implies pk(lk) = 0. we consider the derivation p ′k(l) = (√ 2(1 − 2k) + √ k(1 − k) )2 (1 − 2l) − (√ 2(1 − k) − √ k )2( (1 − k) + 2(1 − 2k)l ) , too. for lk, we have 1 − k + 2(1 − 2k)lk = 1 − k + 2(1 − 2k)k k + √ 2k(1 − k) = 3k − 5k2 + (1 − k) √ 2k(1 − k) k + √ 2k(1 − k) , and hence 1 − k + 2(1 − 2k)lk 1 − 2lk = 3k − 5k2 + (1 − k) √ 2k(1 − k) −k + √ 2k(1 − k) . on the other hand, a equality(√ 1 − k + √ 2k )2( − k + √2k(1 − k)) = ( 1 + k + 2 √ 2k(1 − k) )( − k + √ 2k(1 − k) ) = 3k − 5k2 + (1 − k) √ 2k(1 − k) holds. thus we have 1 − k + 2(1 − 2k)lk 1 − 2lk = (√ 1 − k + √ 2k )2 = (√ 2(1 − 2k) + √ k(1 − k) )2(√ 2(1 − k) − √ k )2 . this implies p ′k(lk) = 0. combining the fact pk(0) = −k (√ 2(1 − k) − √ k )2 ≤ 0 with pk(lk) = 0 and p ′k(lk) = 0, one can see that pk(l) ≤ 0 for any real number l. therefore the inequality min { f(α,k,l), g(α,k,l) } ≤ 2k(1 − k) (√ 2(1 − k) − √ k )2 (1 + k) (√ 2(1 − 2k) + √ k(1 − k) )2 holds. difference between orthogonality in radon planes 183 a fundamental derivation implies that the function k(1−k)/(1+k)2 takes maximum 1/8 at k = 1/3. now we let h(k) = k(1 − k)( √ 2(1 − k) − √ k)2 (1 + k) (√ 2(1 − 2k) + √ k(1 − k) )2 = k (√ 2(1 − k) − √ k(1 − k) )2 (1 + k) (√ 2(1 − 2k) + √ k(1 − k) )2 and consider the maximum of h(k). lemma 2.8. the function h(k) in the interval [0,1/2] takes maximum 1/16 at k = 1/3. proof. we can consider the derivation h′(k) as follows: (1 + k)2 (√ 2(1 − 2k) + √ k(1 − k) )4 h′(k) = [(√ 2(1 − k) − √ k(1 − k) )2 + 2k (√ 2(1 − k) − √ k(1 − k) )( − √ 2 − 1 − 2k 2 √ k(1 − k) )] × (1 + k) (√ 2(1 − 2k) + √ k(1 − k) )2 − k(√2(1 − k) − √k(1 − k))2 × [(√ 2(1 − 2k) + √ k(1 − k) )2 + 2(1 + k) (√ 2(1 − 2k) + √ k(1 − k) )( − 2 √ 2 + 1 − 2k 2 √ k(1 − k) )] . thus we obtain√ k(1 − k)(1 + k)2 (√ 2(1 − k) + √ k(1 − k) )−1 × (√ 2(1 − 2k) + √ k(1 − k) )3 h′(k) = ( (1 − k)(2 − 5k) + √ 2k √ k(1 − k) )√ k(1 − k) + k(1 + k) ( 4 √ k(1 − k) − √ 2(2 − k) ) = (9k2 − 3k + 2) √ k(1 − k) − 2 √ 2k 184 h. mizuguchi and hence √ 1 − k(1 + k)2 (√ 2(1 − k) + √ k(1 − k) )−1(√ 2(1 − 2k) + √ k(1 − k) )3 h′(k) = (9k2 − 3k + 2) √ 1 − k − 2 √ 2k. we note that (9k2 − 3k + 2) √ 1 − k − 2 √ 2k is positive if and only if (9k2 − 3k + 2)2(1 − k) − 8k is so. meanwhile, one has (9k2 − 3k + 2)2(1 − k) − 8k = ( 3k(3k − 1) + 2 )2 (1 − k) − 8k = 9k2(3k − 1)2(1 − k) + 12k(3k − 1)(1 − k) + 4(1 − k) − 8k = (3k − 1) ( 9k2(3k − 1)(1 − k) + 12k(3k − 1)(1 − k) − 4 ) = −(3k − 1)(1 + 3k2)(2 − 3k)2. therefore we obtain that the function h(k) takes maximum at k = 1/3. one can easily have h(1/3) = 1/16, which completes the proof. from the inequality (2.1) and the above lemmas we have theorem 2.9. let x be a radon plane. then 8/9 ≤ ib(x) ≤ 1. in addition, we are able to characterize a radon plane x satisfying ib(x) = 8/9. for simplicity, we use the notation ẑ as z/∥z∥ for any nonzero z ∈ x. theorem 2.10. let x be a radon plane. then ib(x) = 8/9 if and only if its unit sphere is an affine regular hexagon. proof. suppose that x is a radon plane and the equality ib(x) = 8/9 holds. then there exist elements x,y ∈ sx and a real number α such that ∥x + αy∥ = ∥x − αy∥ and minλ∈r ∥x + λy∥ = minµ∈r ∥y + µx∥ = 8/9. for k and l in the above lemmas, all inequalities in the proofs have to turn into equalities and hence k = l = 1/3. as one of them, the inequality α + k ≤ α∥x + ky∥ + k∥x − αy∥ = α∥x + ky∥ + k∥x + αy∥ also becomes an equality for α = 1 and k = 1/3. this implies 4 3 = ∥∥∥x + 1 3 y ∥∥∥ + 1 3 ∥x − y∥ = 8 9 + 1 3 ∥x − y∥ difference between orthogonality in radon planes 185 and hence ∥x + y∥ = ∥x − y∥ = 4/3. using these facts, one has x̂ + y = 3 4 (x + y) = 9 16 (( x + 1 3 y ) + ( y + 1 3 x )) = 1 2 ( ̂ x + 1 3 y + ̂ y + 1 3 x ) . this implies ∥∥∥∥∥12 ( ̂ x + 1 3 y + ̂ y + 1 3 x )∥∥∥∥∥ = ∥x̂ + y∥ = 1. on the other hand, for x = 3 4 ( x + 1 3 y ) + 1 4 (x − y), from ∥∥∥x + 1 3 y ∥∥∥ = 8 9 and ∥x − y∥ = 4 3 we have x = 2 3 ( ̂ x + 1 3 y ) + 1 3 (x̂ − y) and hence ∥∥∥∥23 ( ̂ x + 1 3 y ) + 1 3 (x̂ − y) ∥∥∥∥ = ∥x∥ = 1. in a similar way, the equality∥∥∥∥23 ( ̂ y + 1 3 x ) + 1 3 (−̂x + y) ∥∥∥∥ = ∥y∥ = 1 holds. thus the three segments[ x̂ − y, ̂ x + 1 3 y ] , [ ̂ x + 1 3 y, ̂ y + 1 3 x ] and [ ̂ y + 1 3 x, −̂x + y ] are contained in the unit sphere sx. moreover we obtain (x̂ − y) + ( ̂ y + 1 3 x ) = 3 4 (x − y) + 9 8 ( y + 1 3 x ) = 9 8 ( x + 1 3 y ) = ̂ x + 1 3 y. 186 h. mizuguchi therefore, the unit sphere sx is an affine regular hexagon. conversely, suppose that sx is an affine regular hexagon (and therefore x is a radon plane). then there exist u,v ∈ sx such that ±u, ±v and ±(u+v) are the vertices of sx. letting x = u + 1 3 v and y = − 1 3 u − v, we have x + y = 2 3 (u − v) and x − y = 4 3 (u + v) . thus ∥x + y∥ = 4/3 = ∥x − y∥ and hence x ⊥i y. meanwhile, one has x + 1 3 y = u + 1 3 v + 1 3 ( − 1 3 u − v ) = 8 9 u. therefore, the inequality ib(x) = inf { infλ∈r ∥x + λy∥ ∥x∥ : x,y ∈ x, x,y ̸= 0, x ⊥i y } ≤ 8 9 holds. this implies ib(x) = 8/9. 3.. practical radon planes and a calculation a radon plane is made by connecting the unit sphere of a normed plane and its dual ([6]). hereafter, we make a collection of the space x in which the unit sphere sx is a hexagon, the constant ib(x) is less than 8/9 and that of the induced radon plane coincides with 8/9. a norm ∥ · ∥ on r2 is said to be absolute if ∥(a,b)∥ = ∥(|a|, |b|)∥ for any (a,b) ∈ r2, and normalized if ∥(1,0)∥ = ∥(0,1)∥ = 1. let an2 denote the family of all absolute normalized norm on r2, and ψ2 denote the family of all continuous convex function ψ on [0,1] such that max{1 − t, t} ≤ ψ(t) ≤ 1 for all t ∈ [0,1]. as in [5, 21], it is well known that an2 and ψ2 are in a one-to-one correspondence under the equation ψ(t) = ∥(1 − t,t)∥ for t ∈ [0,1] and ∥(a,b)∥ψ =   (|a| + |b|)ψ ( |b| |a| + |b| ) if (a,b) ̸= (0,0), 0 if (a,b) = (0,0). let ∥ · ∥ψ denote an absolute normalized norm associated with a convex function ψ ∈ ψ2. difference between orthogonality in radon planes 187 for ψ ∈ ψ2, the dual function ψ∗ on [0,1] is defined by ψ∗(s) = sup { (1 − t)(1 − s) + ts ψ(t) : t ∈ [0,1] } for s ∈ [0,1]. it is known that ψ∗ ∈ ψ2 and that ∥·∥ψ∗ ∈ an2 is the dual norm of ∥ · ∥ψ, that is, (r2,∥ · ∥ψ)∗ is identified with (r2,∥ · ∥ψ∗) (cf. [16, 17, 18]). meanwhile, for ψ ∈ ψ2, the function ψ̃ ∈ ψ2 is defined by ψ̃(t) = ψ(1 − t) for any t ∈ [0,1]. one can easily check (̃ψ∗) = ( ψ̃ )∗ . so we write it ψ̃∗. according to [6], [12] and [13], for any ψ ∈ ψ2, the day-james space ℓψ-ℓψ̃∗ becomes a radon plane. for any c ∈ [0,1], let ψc(t) = { −ct + 1 if 0 ≤ t ≤ (1 + c)−1, t if (1 + c)−1 ≤ t ≤ 1. then the norm of (a,b) ∈ r2 is computed by ∥(a,b)∥ψc = { |a| + (1 − c)|b| if |a| ≥ c|b|, |b| if |a| ≤ c|b|. the dual function is calculated as follows: proposition 3.1. let c ∈ [0,1]. then ψ∗c(s) =   1 − s if 0 ≤ s ≤ 1 − c 2 − c , (1 − c)s + c if 1 − c 2 − c ≤ s ≤ 1. proof. fix s ∈ [0,1]. we define the function fc,s(t) from [0,1] into r by fc,s(t) = (1 − t)(1 − s) + ts ψc(t) . we note that ψ∗c(s) = max{fc,s(t) : 0 ≤ t ≤ 1} and calculate the maximum of fc,s on [0,1]. by the definition of ψc, we have fc,s(t) =   1 − s + (2s − 1)t −ct + 1 if 0 ≤ t ≤ (1 + c)−1, 2s − 1 + 1 − s t if (1 + c)−1 ≤ t ≤ 1. 188 h. mizuguchi the function 2s − 1 + (1 − s)/t is clearly decreasing on t. if 0 ≤ s ≤ (1 − c)/(2 − c), then the function fc,s(t) is decreasing on [0,(1 + c)−1]. hence we have ψ∗c(s) = fc,s(0) = 1 − s. suppose that (1−c)/(2−c) ≤ s ≤ 1. then the function fc,s(t) is increasing on [0,(1 + c)−1]. thus we have ψ∗c(s) = fc,s ( 1 1 + c ) = (1 − c)s + c. therefore we obtain this proposition. from this result, one has proposition 3.2. let c ∈ [0,1]. then ∥(a,b)∥ψ∗c = { |a| if |b| ≤ (1 − c)|a|, c|a| + |b| if (1 − c)|a| ≤ |b|. thus the radon plane ℓψc-ℓψ̃∗c induced by ψc is the space r2 with the norm ∥(a,b)∥ ψc,ψ̃∗c =   |a| + (1 − c)|b| if c|b| ≤ |a| and ab ≥ 0, |b| if − (1 − c)|b| ≤ a ≤ c|b| and b ≥ 0, |b| if − (1 − c)|b| ≤ −a ≤ c|b| and b ≤ 0, |a| + c|b| if (1 − c)|b| ≤ |a| and ab ≤ 0. therefore the unit sphere of this space is an affine regular hexagon with the vertices ±(1,0), ±(1 − c,1), ±(−c,1) and hence the constant ib(ℓψc-ℓψ̃∗c ) coincide with 8/9 by the theorem 2.10. next, we calculate the constants ib((r2,∥ · ∥ψc)) and ib((r 2,∥ · ∥ ψ̃∗c )). then we obtain that the values are smaller than ib(ℓψc-ℓψ̃∗c ) = 8/9 and equal to 8/9 only when c = 1/2. we note that ψ̃∗c = ψ1−c and it is enough to calculate ib((r2,∥ · ∥ψc)) for c ∈ [0,1]. to do this, we need to recall the dunkl-williams constant defined in [11]: dw(x) = sup { ∥x∥ + ∥y∥ ∥x − y∥ ∥∥∥∥ x∥x∥ − y∥y∥ ∥∥∥∥ : x,y ∈ x, x,y ̸= 0, x ̸= y } = sup { ∥u + v∥ ∥(1 − t)u + tv∥ : u,v ∈ sx, 0 ≤ t ≤ 1 } . difference between orthogonality in radon planes 189 the unit sphere of (r2,∥ · ∥ψc) and (r 2,∥ · ∥ ψ̃∗c ). the unit sphere of radon plane ℓψc-ℓψ̃∗c . for any normed space, the equality 2 ≤ dw(x) ≤ 4 holds. in [14], it is shown that the equality ib(x)dw(x) = 2 holds for any normed space x. one can find a formula to calculate this constant in the paper [15]. for each x ∈ sx and for each y ∈ x with x ⊥b y, we put m(x,y) = sup {∥∥∥∥x + λ + µ2 y ∥∥∥∥ : λ ≤ 0 ≤ µ, ∥x + λy∥ = ∥x + µy∥ } . we define the positive number m(x) by m(x) = sup { m(x,y) : x ⊥b y } . using these notions, the dunkl-williams constant can be calculated as dw(x) = 2 sup { m(x) : x ∈ sx } = 2 sup { m(x) : x ∈ fr(bx) } , 190 h. mizuguchi where fr(bx) is the frame of unit ball. an element x ∈ sx is called an extreme point of bx if y,z ∈ sx and x = (y + z)/2 implies x = y = z. the set of all extreme points of bx is denoted by ext(bx). suppose that the space x has two-dimension. then the above calculation method is turned into dw(x) = 2 sup { m(x) : x ∈ ext(bx) } . here, we reduce the amount of calculation a little more. as in section 2, we use the notation ẑ. proposition 3.3. let x be a two-dimensional normed space. then dw(x) = sup { ∥u + v∥ ∥(1 − t)u + tv∥ : u ∈ ext(bx), v ∈ sx, 0 ≤ t ≤ 1 } . proof. take arbitrary elements u,v ∈ sx \ ext(bx). if the segment [u,v] belongs to the unit sphere sx, then ∥u + v∥ ∥(1 − t)u + tv∥ = 2 for any t ∈ [0,1]. so we may assume [u,v] ̸⊂ sx. then we have t0 ∈ [0,1] such that min 0≤t≤1 ∥(1 − t)u + tv∥ = ∥(1 − t0)u + t0v∥. letting x = ̂(1 − t0)u + t0v and y = û − v, we have four elements u1, u2, v1, v2 ∈ sx such that at least two elements among them belong to ext(bx) and satisfying u ∈ [u1,u2] ⊂ sx, v ∈ [v1,v2] ⊂ sx and û1 − v1 = y = û2 − v2. for these elements, from the fact that three vectors u − v, u1 − v1 and u2 −v2 are parallel each other, we can take numbers s0 ∈ (0,1) satisfying u = (1−s0)u1+s0u2, v = (1−s0)v1+s0v2. meanwhile there exist t1, t2 ∈ (0,1) such that min 0≤t≤1 ∥(1 − t)u1 + tv1∥ = ∥(1 − t1)u1 + t1v1∥, min 0≤t≤1 ∥(1 − t)u2 + tv2∥ = ∥(1 − t2)u2 + t2v2∥. it follow from x ⊥b y and û1 − v1 = y = û2 − v2 that ̂(1 − t2)u2 + t2v2 = x and ̂(1 − t1)u1 + t1v1 = ±x. in case of ̂(1 − t1)u1 + t1v1 = −x, the element −u1 belongs to the arc between v1 and x. letting v3 = −u1, we difference between orthogonality in radon planes 191 can take element u3 satisfying û3 − v3 = y, again. hence we may consider ̂(1 − t1)u1 + t1v1 = x. then the equalities (1 − t0)u + t0v = (1 − s0) ( (1 − t1)u1 + t1v1 ) + s0 ( (1 − t2)u2 + t2v2 ) and ∥(1 − t0)u + t0v∥ = (1 − s0)∥(1 − t1)u1 + t1v1∥ + s0∥(1 − t2)u2 + t2v2∥ holds. thus, using triangle inequality and the fact that an inequality (1 − α)a + αb (1 − α)c + αd ≤ max { a c , b d } holds for α ∈ [0,1] and positive numbers a,b,c,d, we obtain ∥u + v∥ ∥(1 − t0)u + t0v∥ = ∥(1 − s0)u1 + s0u2 + (1 − s0)v1 + s0v2∥ (1 − s0)∥(1 − t1)u1 + t1v1∥ + s0∥(1 − t2)u2 + t2v2∥ ≤ (1 − s0)∥u1 + v1∥ + s0∥u2 + v2∥ (1 − s0)∥(1 − t1)u1 + t1v1∥ + s0∥(1 − t2)u2 + t2v2∥ ≤ max { ∥u1 + v1∥ ∥(1 − t1)u1 + t1v1∥ , ∥u2 + v2∥ ∥(1 − t2)u2 + t2v2∥ } ≤ sup { ∥u + v∥ ∥(1 − t)u + tv∥ : u ∈ ext(bx), v ∈ sx, 0 ≤ t ≤ 1 } . this completes the proof. thus, to obtain the value of the dunkl-williams constant, in the above calculation method, for x ∈ ext(bx) and y ∈ x with x ⊥b y, the value m(x,y) can be computed as m(x,y) = sup {∥∥∥∥x + λ + µ2 y ∥∥∥∥ : λ ≤ 0 ≤ µ, ∥x + λy∥ = ∥x + µy∥, x̂ + λy ∈ ext(bx) } . 192 h. mizuguchi 4.. the constant ib(x) in hexagonal planes now, we start to compute dw((r2,∥ · ∥ψc)) and ib((r 2,∥ · ∥ψc)) for c ∈ [0,1]. for simplicity we write xc and ∥ · ∥ for (r2,∥ · ∥ψc) and ∥ · ∥ψc, respectively. first we suppose 1/2 ≤ c. let e1 = (1,0), u = (c,1). then, by [15, proposition 2.5], dw(xc) = 2 max{m(e1),m(u)}. putting vt = (−t,1) and wt = (1 − t)(−e1) + t(−c,1) = (−1 + t − ct,t) for t ∈ [0,1], we have e1 ⊥b vt for t ∈ [0,1 − c], u ⊥b vt for t ∈ [1 − c,c] and u ⊥b wt for t ∈ [0,1]. by [15, theorem 2.9 and corollary 2.10], one has m(e1) = sup { m(e1,vt) : t ∈ (0,1 − c) } and m(u) = max { sup{m(u,vt) : t ∈ (1 − c,c)}, sup{m(u,wt) : t ∈ (0,1) \ {1/2}} } . lemma 4.1. let c ∈ [1/2,1]. then, in xc, m(e1) = 1 + 1 − c (1 + √ 2c)2 . proof. let t ∈ (0,1 − c). then the norm of e1 + λvt is computed as ∥e1 + λvt∥ =   −λ if λ ≤ −(c − t)−1, 1 − (1 − c + t)λ if − (c − t)−1 ≤ λ ≤ 0, 1 + (1 − c − t)λ if 0 ≤ λ ≤ (c + t)−1, λ if (c + t)−1 ≤ λ. from the inequality∥∥∥∥e1 + 1c + tvt ∥∥∥∥ = 1 + 1 − c − tc + t < 1 + 1 − c + tc − t = ∥∥∥∥e1 − 1c − tvt ∥∥∥∥ , we can find real numbers pt ∈ (−(c − t)−1,0) and qt more than (c + t)−1 such that ∥e1 + ptvt∥ = ∥∥∥∥e1 + 1c + tvt ∥∥∥∥ and ∥e1 + qtvt∥ = ∥∥∥∥e1 − 1c − tvt ∥∥∥∥ , difference between orthogonality in radon planes 193 respectively. to obtain m(e1,vt), it is enough to consider∥∥∥∥e1 + 12 ( pt + 1 c + t ) vt ∥∥∥∥ and ∥∥∥∥e1 + 12 ( − 1 c + t + qt ) vt ∥∥∥∥ . since the equality qt = ∥e1 + qtvt∥ = ∥∥∥∥e1 − 1c − tvt ∥∥∥∥ = 1 + 1 − c + tc − t = 1c − t holds, one has ( −(c − t)−1 + qt ) /2 = 0. on the other hand, from the equality 1 − (1 − c + t)pt = ∥e1 + ptvt∥ = ∥∥∥∥e1 + 1c + tvt ∥∥∥∥ = 1 + 1 − c − tc + t , we have pt = − 1 − c − t (1 − c + t)(c + t) and hence 1 2 ( pt + 1 c + t ) = t (1 − c + t)(c + t) . it follows from 0 < t (1 − c + t)(c + t) = 1 2 ( pt + 1 c + t ) < 1 c + t that ∥∥∥∥e1 + 12 ( pt + 1 c + t ) vt ∥∥∥∥ = 1 + (1 − c − t)t(1 − c + t)(c + t). this implies that m(e1,vt) = 1 + (1 − c − t)t (1 − c + t)(c + t) . letting fc(t) = (1 − c − t)t (1 − c + t)(c + t) , one can figure out (1 − c + t)2(c + t)2f ′c(t) = (−2t + 1 − c)(1 − c + t)(c + t) − (2t + 1)(1 − c − t)t = −(2 − c)t2 − 2c(1 − c)t + c(1 − c)2. 194 h. mizuguchi let t0 be the larger solution of the equation −(2−c)t2−2c(1−c)t+c(1−c)2 = 0. then t0 = c(1 − c) √ 2c + c ∈ (0,1 − c) and fc takes maximum at t0. this t0 satisfies the equality (−2t0 + 1 − c)(1 − c + t0)(c + t0) = (1 − c − t0)t0(2t0 + 1), too. thus we obtain m(e1) = 1 + (1 − c − t0)t0 (1 − c + t0)(c + t0) = 1 + −2t0 + 1 − c 2t0 + 1 = 1 + −2c(1 − c) + ( √ 2c + c)(1 − c) 2c(1 − c) + √ 2c + c = 1 + 1 − c( 1 + √ 2c )2 . lemma 4.2. let c ∈ [1/2,1]. then, in xc, sup { m(u,vt) : t ∈ (1 − c,c) } = 2c. proof. let t ∈ (1 − c,c). then the norm of u + λvt is calculated by ∥u + λvt∥ =   −(1 + λ) if λ ≤ −2c/(c − t), 2c − 1 − {t + (1 − c)}λ if − 2c/(c − t) ≤ λ ≤ −1, 1 − {t − (1 − c)}λ if − 1 ≤ λ ≤ 0, 1 + λ if 0 ≤ λ. there exist two real numbers αt, βt satisfying 0 < αt < βt, ∥u + αtvt∥ = ∥u − vt∥ and ∥u + βtvt∥ = ∥∥∥∥u − 2cc − tvt ∥∥∥∥ . it is enough to consider ∥u + 1 2 (−1 + αt)vt∥ and∥∥∥∥u + 12 ( − 2c c − t + βt ) vt ∥∥∥∥ . difference between orthogonality in radon planes 195 from the equality 1 + αt = ∥u + αtvt∥ = ∥u − vt∥ = 1 + ( t − (1 − c) ) , we have αt = t−(1−c) and hence (−1+αt)/2 = − ( 2−(t+c) ) /2. meanwhile, it follows from 1 + βt = ∥u + βtvt∥ = ∥∥∥∥u − 2cc − tvt ∥∥∥∥ = − ( 1 − 2c c − t ) that 1 2 ( − 2c c − t + βt ) = −1. by the inequality 1 2 ( − 2c c − t + βt ) = −1 < − ( 2 − (t + c) ) /2 = (−1 + αt)/2 < 0, we obtain m(u,vt) = ∥u−vt∥ = t+c and hence sup{m(u,vt) : t ∈ (1−c,c)} = 2c. next, for t ∈ (0,1), the norm of u + λwt is calculated by ∥u + λwt∥ =   2c − 1 − λ if λ ≤ −1/t, 1 − {1 − 2(1 − c)t}λ if − 1/t ≤ λ ≤ 0, 1 + tλ if 0 ≤ λ ≤ 2c/(1 − t), −(2c − 1) + λ if 2c/(1 − t) ≤ λ. in particular we have ∥∥∥∥u + 2c1 − twt ∥∥∥∥ = 1 + 2c1 − tt,∥∥∥∥u − 1twt ∥∥∥∥ = 1 + 1 − 2(1 − c)tt , and hence∥∥∥∥u − 1twt ∥∥∥∥ − ∥∥∥∥u + 2c1 − twt ∥∥∥∥ = (1 − t) ( 1 − 2(1 − c)t ) − 2ct2 t(1 − t) = (1 − 2t) ( 1 + (2c − 1)t ) t(1 − t) . 196 h. mizuguchi from this equality, we obtain that if t ∈ (0,1/2) then∥∥∥∥u + 2c1 − twt ∥∥∥∥ < ∥∥∥∥u − 1twt ∥∥∥∥ and that if t ∈ (1/2,1) then∥∥∥∥u − 1twt ∥∥∥∥ < ∥∥∥∥u + 2c1 − twt ∥∥∥∥ . lemma 4.3. let c ∈ [1/2,(1 + √ 5)/4]. then, in xc, sup { m(u,wt) : t ∈ (0,1/2) } = max { 1 2 + c, 1 + c {1 + √ 2(1 − c)}2 } . proof. let t ∈ (0,1/2). then there exist two numbers γt ∈ (−1/t,0) and δt greater than 2c/(1 − t) satisfying ∥u + γtwt∥ = ∥∥∥∥u + 2c1 − twt ∥∥∥∥ and ∥u + δtwt∥ = ∥∥∥∥u − 1twt ∥∥∥∥ , respectively. to obtain m(u,wt) it is enough to consider∥∥∥∥u + 12 ( γt + 2c 1 − t ) wt ∥∥∥∥ and ∥∥∥∥u + 12 ( − 1 t + δt ) wt ∥∥∥∥ . from the equality −(2c − 1) + δt = ∥u + δtwt∥ = ∥∥∥∥u − 1twt ∥∥∥∥ = 2c − 1 − ( − 1 t ) , one has 1 2 ( − 1 t + δt ) = 2c − 1. it is easy to check 2c − 1 < 2c/(1 − t) and hence we obtain∥∥∥∥u + 12 ( − 1 t + δt ) wt ∥∥∥∥ = 1 + (2c − 1)t. under the assumption t ∈ (0,1/2), this function takes the supremum 1/2 + c. meanwhile, it follows from 1 − ( 1 − 2(1 − c)t ) γt = ∥u + γtwt∥ = ∥∥∥∥u + 2c1 − twt ∥∥∥∥ = 1 + 2ct1 − t difference between orthogonality in radon planes 197 that γt = − 2ct( 1 − 2(1 − c)t ) (1 − t) . hence we have 1 2 ( γt + 2c 1 − t ) = c ( 1 − (3 − 2c)t ) (1 − t) ( 1 − 2(1 − c)t ) and ∥∥∥∥u + 12 ( γt + 2c 1 − t ) wt ∥∥∥∥ = 1 + ct ( 1 − (3 − 2c)t ) (1 − t) ( 1 − 2(1 − c)t ). we note that 1 − (3 − 2c)t > 1 − (3 − 2c)/2 = c − 1/2 > 0. letting gc(t) = t ( 1 − (3 − 2c)t ) (1 − t) ( 1 − 2(1 − c)t ) in the interval [0,1/2], we have( 1 − 2(1 − c)t )2 (1 − t)2g′c(t) = ( − 2(3 − 2c)t + 1 ) (1 − t) ( 1 − 2(1 − c)t ) − ( 4(1 − c)t − (3 − 2c) ) t ( 1 − (3 − 2c)t ) = ( (3 − 2c)2 − 2(1 − c) ) t2 − 2(3 − 2c)t + 1 we note that (3−2c)2 −2(1−c) = 4c2 −10c+7 = 4(c−5/4)2 +3/4 > 0. let t1 be the smaller solution of equality ( (3−2c)2 −2(1−c) ) t2 −2(3−2c)t+1 = 0, i.e., t1 = 1 3 − 2c + √ 2(1 − c) . if c < (1 + √ 5)/4, then this t1 belongs to the interval (0,1/2). thus gc(t) takes the maximum gc(t1) = 1( 1 + √ 2(1 − c) )2 . this implies that ∥∥∥∥u + 12 ( γ + 2c 1 − t ) wt ∥∥∥∥ takes maximum 1 + c( 1 + √ 2(1 − c) )2 . 198 h. mizuguchi in case of (1 + √ 5)/4 ≤ c, the solution t1 is more than 1/2 and hence the function gc(t) takes the maximum at t = 1/2. one can follow the above proof except for this part, and obtain the following: lemma 4.4. let c ∈ [(1 + √ 5)/4,1]. then, in xc, sup { m(u,wt) : t ∈ (0,1/2) } = 1/2 + c. next we consider sup{m(u,wt) : t ∈ (1/2,1)}. lemma 4.5. let c ∈ [1/2,1]. then, in xc, sup { m(u,wt) : t ∈ (1/2,1) } =   9 8 if 1 2 ≤ c ≤ 9 16 , 2c if 9 16 ≤ c ≤ 1. proof. let t ∈ (1/2,1). then from∥∥∥∥u − 1twt ∥∥∥∥ < ∥∥∥∥u + 2c1 − twt ∥∥∥∥ one can take γt less than −1/t and δt ∈ (0,2c/(1 − t)) satisfying ∥u + γtwt∥ = ∥∥∥∥u + 2c1 − twt ∥∥∥∥ and ∥u + δtwt∥ = ∥∥∥∥u − 1twt ∥∥∥∥ , respectively. from the equality 2c − 1 − γt = ∥u + γtwt∥ = ∥∥∥∥u + 2c1 − twt ∥∥∥∥ = −(2c − 1) + 2c1 − t, we have 1 2 ( γt + 2c 1 − t ) = 2c − 1. the fact 0 ≤ 2c − 1 = 1 2 ( γt + 2c 1 − t ) < 2c 1 − t implies that∥∥∥∥u + 12 ( γt + 2c 1 − t ) wt ∥∥∥∥ = ∥u + (2c − 1)wt∥ = 1 + (2c − 1)t. difference between orthogonality in radon planes 199 it is clear that the function 1 + (2c − 1)t takes the supremum 2c. on the other hand, it follows from 1 + tδt = ∥u + δtwt∥ = ∥∥∥∥u − 1twt ∥∥∥∥ = 1 + 1 − 2(1 − c)tt that δt = 1 − 2(1 − c)t t2 and hence 1 2 ( − 1 t + δt ) = 1 − (3 − 2c)t 2t2 . under the assumption c ∈ [1/2,1], one can easily check (3 − 2c)−1 ∈ (1/2,1). in case of t ∈ (1/2,(3 − 2c)−1], from the inequality 0 ≤ 1 − (3 − 2c)t 2t2 = 1 2 ( − 1 t + δt ) < δt < 2c 1 − t , we have ∥∥∥∥u + 12 ( − 1 t + δt ) wt ∥∥∥∥ = 1 + 1 − (3 − 2c)t2t . it is easy to check that this function takes the supremum 1/2 + c. suppose that t ∈ ((3 − 2c)−1,1). then, from the inequality − 1 t < 1 2 ( − 1 t + δt ) = − (3 − 2c)t − 1 2t2 < 0, one has ∥∥∥∥u + 12 ( − 1 t + δt ) wt ∥∥∥∥ = 1 + ( 1 − 2(1 − c)t )( (3 − 2c)t − 1 ) 2t2 . considering the function hc(t) in the interval [1/2,1] defined by hc(t) = ( 1 − 2(1 − c)t )( (3 − 2c)t − 1 ) t2 , we figure out t4h′c(t) = ( − 4(1 − c)(3 − 2c)t + 5 − 4c ) t2 − 2 ( 1 − 2(1 − c)t )( (3 − 2c)t − 1 ) t 200 h. mizuguchi and hence t3h′c(t) = ( − 4(1 − c)(3 − 2c)t + 5 − 4c ) t − 2 ( 1 − 2(1 − c)t )( (3 − 2c)t − 1 ) = −(5 − 4c)t + 2. since the function −(5 − 4c)t + 2 is decreasing, we have the following: if c < 3/4, then one has 2/(5 − 4c) ∈ ((3 − 2c)−1,1) and hence max { hc(t) : t ∈ ((3 − 2c)−1,1) } = hc ( 2 5 − 4c ) = ( (5 − 4c) − 4(1 − c) )( 2(3 − 2c) − (5 − 4c) ) 4 = 1 4 . this implies that ∥∥∥∥u + 12 ( − 1 t + δt ) wt ∥∥∥∥ takes the maximum 9/8 at t = 2/(5 − 4c). meanwhile, 9/8 is greater than 2c only when c < 9/16. in case of 3/4 ≤ c, from 1 ≤ 2/(5 − 4c) one has that hc(t) is increasing. hence we have max { hc(t) : t ∈ ((3 − 2c)−1,1) } = h(1) = 2(1 − c)(2c − 1) this implies that ∥∥∥∥u + 12 ( − 1 t + δt ) wt ∥∥∥∥ takes the supremum 1 + (1 − c)(2c − 1). we note that 1 + (1 − c)(2c − 1) < 1 + (2c − 1) = 2c. therefore we obtain the following proposition. difference between orthogonality in radon planes 201 proposition 4.6. let c ∈ [1/2,1]. then ib(xc)−1 = dw(xc)/2 coincide with  max { 1 + 1 − c {1 + √ 2c}2 , 1 + c {1 + √ 2(1 − c)}2 , 9 8 } if 1 2 ≤ c ≤ 9 16 , max { 1 + 1 − c {1 + √ 2c}2 , 1 + c {1 + √ 2(1 − c)}2 , 2c } if 9 16 < c < 1 + √ 5 4 , max { 1 + 1 − c {1 + √ 2c}2 , 2c } if 1 + √ 5 4 ≤ c ≤ 1. hereafter we suppose c < 1/2. similarly to the above paragraph, dw(xc) = 2 max{m(e1),m(u)} holds. on the other hand, for vt and wt, birkhoff orthogonality relations differ from the above paragraph. we have e1 ⊥b vt for t ∈ [0,c], e1 ⊥b wt for t ∈ [1/2(1 − c),1] and u ⊥b wt for t ∈ [0,1/2(1 − c)]. by [15, theorem 2.9 and corollary 2.10], one figure out m(e1) = max { sup{m(e1,vt) : t ∈ (0,c)}, sup{m(e1,wt) : t ∈ (1/2(1 − c),1)} } and m(u) = sup { m(u,wt) : t ∈ (0,1/2(1 − c)) \ {1/2} } . lemma 4.7. let c ∈ [0,(3 − √ 5)/4]. then in xc, sup { m(e1,vt) : t ∈ (0,c) } = 3 2 − c. proof. let t ∈ (0,c). then in a similar way to the proof of lemma 4.1, we have m(e1,vt) = 1 + (1 − c − t)t (1 − c + t)(c + t) . moreover letting fc(t) = (1 − c − t)t (1 − c + t)(c + t) , we also have (1 − c + t)2(c + t)2f ′c(t) = −(2 − c)t 2 − 2c(1 − c)t + c(1 − c)2 202 h. mizuguchi again. from c ∈ [0,(3 − √ 5)/4], it is more than −(2 − c)c2 − 2c(1 − c)c + c(1 − c)2 = c(4c2 − 6c + 1) ≥ 0. from this fact, fc(t) increases and hence sup { m(e1,vt) : t ∈ (0,c) } = 1 + fc(c) = 3 2 − c. suppose that c ∈ ((3 − √ 5)/4,1/2). then for t0 defined in the same formula to the proof of lemma 4.1, we have t0 ∈ (0,c) and f ′c(t0) = 0. hence we obtain lemma 4.8. let c ∈ ((3 − √ 5)/4,1/2). then in xc, sup { m(e1,vt) : t ∈ (0,c) } = 1 + 1 − c( 1 + √ 2c )2 . lemma 4.9. let c ∈ [0,1/2). then in xc, sup { m(e1,wt) : t ∈ (1/2(1 − c),1) } = 2(1 − c). proof. let t ∈ (1/2(1 − c),1). then the norm of e1 + λwt is calculated as ∥e1 + λwt∥ =   1 − λ if λ ≤ 0, 1 + ( 2(1 − c)t − 1 ) λ if 0 ≤ λ ≤ ( 1 − (1 − 2c)t )−1 , tλ if ( 1 − (1 − 2c)t )−1 ≤ λ ≤ (1 − t)−1, −1 + λ if (1 − t)−1 ≤ λ. one can take two real numbers st,rt satisfying st < rt < 0, ∥e1 + rtwt∥ =∥∥e1 +(1 − (1 − 2c)t)−1wt∥∥ and ∥e1 + stwt∥ = ∥e1 + (1 − t)−1wt∥. it is enough to consider ∥∥e1 + 12(rt +(1 − (1 − 2c)t)−1)wt∥∥ and ∥∥e1 + 12(st + (1 −t)−1)wt∥∥. from the equality 1 − rt = ∥e1 + rtwt∥ = ∥∥e1 + (1 − (1 − 2c)t)−1wt∥∥ = 1 + 2(1 − c)t − 1 1 − (1 − 2c)t , one has rt = − 2(1 − c)t − 1 1 − (1 − 2c)t difference between orthogonality in radon planes 203 and hence 1 2 ( rt + 1 1 − (1 − 2c)t ) = 1 − (1 − c)t 1 − (1 − 2c)t . it follows from 1 − st = ∥e1 + stwt∥ = ∥e1 + (1 − t)−1wt∥ = −1 + 1 1 − t that 1 2 (st + (1 − t)−1) = 1. since the inequality 0 < 1 − (1 − c)t 1 − (1 − 2c)t < 1 < 1 1 − (1 − 2c)t holds, we obtain m(e1,wt) = ∥∥∥∥e1 + ( st + 1 1 − t ) wt ∥∥∥∥ = 1 + (2(1 − c)t − 1) × 1 = 2(1 − c)t. this implies sup { m(e1,wt) : t ∈ (1/2(1 − c),1) } = 2(1 − c). for t ∈ (0,1/2(1−c)) the norm of u+λwt is calculated in a similar way to the case of c ∈ [1/2,1]. now we suppose c ∈ [0,1/2) and so 1/2 ≤ 1/2(1−c) < 1 holds. thus we have to consider the following two cases again: if t ∈ (0,1/2) then ∥∥∥∥u + 2c1 − twt ∥∥∥∥ < ∥∥∥∥u − 1twt ∥∥∥∥ . if t ∈ (1/2,1) then ∥∥∥∥u − 1twt ∥∥∥∥ < ∥∥∥∥u + 2c1 − twt ∥∥∥∥ . lemma 4.10. let c ∈ [0,1/2). then, in xc sup { m(u,wt) : t ∈ (0,1/2) } = max { 2(1 − c),1 + c( 1 + √ 2(1 − c) )2 } . 204 h. mizuguchi proof. let t ∈ (0,1/2). in a similar way to lemma 4.3, one can take δt and figure out that this constant satisfy ∥u + 1 2 (−1/t + δt)wt∥ = 1 + (1 − 2c) ( 1−2(1−c)t ) and that this function of t takes the supremum 2(1−c). we also have γt and that 1 2 ( γt + 2c 1 − t ) = c ( 1 − (3 − 2c)t ) (1 − t) ( 1 − 2(1 − c)t ). now we are considering the case of c ∈ [0,1/2) and so 1/(3 − 2c) ∈ (0,1/2). if t ∈ (0,1/(3 − 2c)), then we have 0 < 1 2 ( γt + 2c 1 − t ) < 2c 1 − t and hence ∥∥∥∥u + 12 ( γt + 2c 1 − t ) wt ∥∥∥∥ = 1 + ct ( 1 − (3 − 2c)t ) (1 − t) ( 1 − 2(1 − c)t ). for t1 defined by same formula to lemma 4.3, we have t1 ∈ (0,1/(3 − 2c)) and that the function ∥∥∥∥u + 12 ( γt + 2c 1 − t ) wt ∥∥∥∥ takes maximum 1 + c/ ( 1 + √ 2(1 − c) )2 at t1. assume that t ∈ (1/(3 − 2c),1/2). then from the inequality − 1 t < γt < 1 2 ( γt + 2c 1 − t ) = − c ( (3 − 2c)t − 1 ) (1 − t) ( 1 − 2(1 − c)t ) < 0, we obtain ∥∥∥∥u + 12 ( γt + 2c 1 − t ) wt ∥∥∥∥ = 1 + c ( (3 − 2c)t − 1 ) 1 − t . this function of t is increasing and hence less than 1 + c ( (3 − 2c)/2 − 1 ) 1 − 1/2 = (1 − c)(1 + 2c) < 2(1 − c). thus we obtain ∥∥∥∥u + 12 ( γt + 2c 1 − t ) wt ∥∥∥∥ < 2(1 − c), which completes the proof. difference between orthogonality in radon planes 205 lemma 4.11. let c ∈ [0,1/2). then, in xc, sup { m(u,wt) : t ∈ (1/2,1/(2−c)) } =   (1 − c)(1 + 2c) if 0 < c ≤ 1 4 , max { (1 − c)(1 + 2c), 9 8 } if 1 4 < c < 1 2 . proof. let t ∈ (1/2,1/(2 − c)). in a similar way to lemma 4.5, we take γt less than −1/t and δt ∈ (0,2c/(1 − t)). then we have 1 2 ( γt + 2c 1 − t ) = −(1 − 2c) it follows from −1/t < −2(1 − c) < −(1 − 2c) < 0 that∥∥∥∥u + 12 ( γt + 2c 1 − t ) wt ∥∥∥∥ = 1 + (1 − 2c)(1 − 2(1 − c)t) in the situation t ∈ (1/2,1/(2 − c)), it takes supremum 1 + c(1 − 2c) = (1 − c)(1 + 2c). in addition, we have − 1 t < 1 2 ( − 1 t + δt ) = − (3 − 2c)t − 1 2t2 < 0. hence the equality∥∥∥∥u + 12 ( − 1 t + δt ) wt ∥∥∥∥ = 1 + ( 1 − 2(1 − c)t )( (3 − 2c)t − 1 ) 2t2 holds. as in lemma 4.5, one can consider the following two cases: if 0 < c < 1/4, then the above function is decreasing and hence takes the supremum 1 + ( 1 − 2(1 − c)/2 )( (3 − 2c)/2 − 1 ) 2(1/2)2 = 1 + c(1 − 2c) = (1 − c)(1 + 2c) when 1/4 ≤ c < 1/2, we have that the above function takes maximum 9/8 at t = 2/(5 − 4c). indeed, (1−c)(1+2c) is less than 2(1−c) for any c ∈ [0,1/2). meanwhile, it is easy to see that 2(1 − c) < 9/8 only if c > 7/16. therefore we have 206 h. mizuguchi proposition 4.12. let c ∈ [0,1/2]. then ib(xc)−1 = dw(xc)/2 coincide with  max { 2(1 − c), 1 + c( 1 + √ 2(1 − c) )2 } if 0 ≤ c ≤ 3 − √ 5 4 , max { 2(1 − c), 1 + c( 1 + √ 2(1 − c) )2 , 1 + 1 − c( 1 + √ 2c )2 } if 3 − √ 5 4 < c < 7 16 , max { 1 + c( 1 + √ 2(1 − c) )2 , 1 + 1 − c( 1 + √ 2c )2 , 98 } if 7 16 ≤ c ≤ 1 2 . considering the symmetry of the functions c( 1 + √ 2(1 − c) )2 and 1 − c( 1 + √ 2c )2 and that these function takes value 1/8 at t = 1/2, we finally obtain theorem 4.13. let c ∈ [0,1] and put d = max{c,1 − c}. then both dw(xc) and dw(x ∗ c ) coincide with 2 max { 2d, 1 + d( 1 + √ 2(1 − d) )2 } . this is more than dw(ℓψc-ℓψ̃∗c ) = 9/4 and the equality holds only when c = 1/2. theorem 4.14. let c ∈ [0,1] and put d = max{c,1 − c}. then both ib(xc) and ib(x ∗ c ) coincide with min { 1 2d , ( 1 + √ 2(1 − d) )2 d + ( 1 + √ 2(1 − d) )2 } . this is less than ib(ℓψc-ℓψ̃∗c ) = 8/9 and the equality holds only when c = 1/2. difference between orthogonality in radon planes 207 the graphs of the functions y = 2x, y = 2(1 − x), y = 1 + x( 1 + √ 2(1 − x) )2 , y = 1 + 1 − x( 1 + √ 2x )2 and y = 98 on the interval [0,1]. acknowledgements the author would like to thank the anonymous referee for the careful reading, 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[22] t. szostok, on a generalization of the sine function, glas. mat. ser. iii 38(58) (1) (2003), 29 – 44. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 36, num. 1 (2021), 81 – 98 doi:10.17398/2605-5686.36.1.81 available online december 10, 2020 cone asymptotes of convex sets v. soltan department of mathematical sciences, george mason university 4400 university drive, fairfax, va 22030, usa vsoltan@gmu.edu received november 18, 2020 presented by horst martini accepted november 28, 2020 abstract: based on the notion of plane asymptote, we introduce the new concept of cone asymptote of a set in the n-dimensional euclidean space. we discuss the existence and describe some families of cone asymptotes. key words: plane asymptote, cone asymptote, convex set. msc (2020): 52a20, 90c25. 1. introduction originated in geometry, a widely used definition of an asymptote says that it is a line (or a halfline) that continually approaches a curve or a surface but does not meet it. in convex geometry, this definition was generalized by gale and klee [2], who defined an asymptote of a nonempty closed subset x of the euclidean space rn as a halfline h which lies in rn \x and satisfies the condition δ(x,h) = 0, where the inf -distance δ(x,y ) between nonempty sets x and y in rn is given by δ(x,y ) = inf{‖x−y‖ : x ∈ x, y ∈ y}. later, klee [4] introduced the concept of j-asymptote of a closed convex set k ⊂ rn, as a plane l ⊂ rn of dimension j, 1 ≤ j ≤ n − 1, satisfying the conditions k ∩l = ∅ and δ(k,l) = 0. in particular, a 1-asymptote is a line asymptote (see figure 1). plane asymptotes appeared to be a useful tool in the study of various properties of convex sets. for instance, closed convex sets in rn without plane asymptotes are precisely those whose affine images are closed (see [1, 5, 9]). certain classes of convex sets (like continuous convex sets, m-decomposable sets, and m-polyhedral sets) have suitable geometric properties due to the issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.36.1.81 mailto:vsoltan@gmu.edu https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 82 v. soltan k l figure 1: a plane asymptote l of a convex set k. absence of plane asymptotes (see [2, 4, 7, 8, 11]). the range of dimensions of plane asymptotes of a given closed convex set in rn was investigated in [3, 5]. in the study of asymptotic behavior of convex sets, it is often desirable to describe their line or halfline asymptotes. this task is often difficult (if possible at all) partly due to the fact that the family of plane asymptotes is not hereditary (see, for instance, example 3.5 below). our goal here is to show that plane asymptotes of a given convex set k ⊂ rn always contain closed cones which are asymptotes of k (see theorem 3.3 and theorem 4.1 below). also, we study some geometric properties of cone asymptotes. for terminological convenience, we will say that a nonempty closed set x ⊂ rn is an asymptote of a nonempty closed set y ⊂ rn provided x∩y = ∅ and δ(x,y ) = 0. furthermore, we use the expression cone asymptote to distinguish it from another established concept, asymptotic cone cx of a nonempty set x ⊂ rn, which is a cone with apex at the origin o, defined by cx = ⋂ ( cl((0,ε]x) : ε > 0 ) . we conclude this section with necessary definitions, notation, and results on convex sets in the n-dimensional euclidean space rn (see, e.g., [10] for details). the elements of rn are called vectors, or points. in what follows, o stands for the zero vector of rn. we denote by [u,v], and [u,v〉, respectively, the closed segment with endpoints u,v ∈ rn and the closed halfline through v with endpoint u: [u,v] = {(1 −λ)u + λv : 0 ≤ λ ≤ 1}, [u,v〉 = {(1 −λ)u + λv : λ ≥ 0}. also, u ·v will mean the dot product of u and v, and ‖u‖ denotes the norm of u. given a pair of subspaces t ⊂ s of rn, a subspace f ⊂ rn will be called complementary to t in s provided f ∩ t = {o} and f + t = s. by an r-dimensional plane l in rn, where 0 ≤ r ≤ n, we mean a translate of cone asymptotes of convex sets 83 a suitable r-dimensional subspace sl of rn: l = u + sl, where u ∈ rn. the subspace sl is uniquely determined by l and equals l−v for any choice of v ∈ l. a closed halfspace v of rn is defined by v = {x ∈ rn : x·e ≤ γ}, where e is a nonzero vector and γ is a scalar. if l ⊂ rn is a plane of positive dimension, then by a closed halfplane of l we will mean any set of the form l∩v , where v is a closed halfspace of rn satisfying the condition ∅ 6= l∩v 6= l. in what follows, k means a nonempty convex set in rn. the relative interior and relative boundary of k are denoted rint k and rbd k, respectively. a convex set k ⊂ rn is called line-free if it contains no lines. for a nonempty set x ⊂ rn, the notations cl x and conv x stand, respectively, for the closure and the convex hull of x. if x is bounded, then the set y = conv(cl x) is compact. we recall that a nonempty set c in rn is a cone with apex v ∈ rn if v + λ(x − v) ∈ c whenever λ ≥ 0 and x ∈ c. (obviously, this definition implies that v ∈ c, although a stronger condition λ > 0 can be beneficial; see, e.g., [6].) a cone c with apex v is called convex if it is a convex set, and is called nontrivial if c 6= {v}. the polar cone c◦ of a convex cone c ⊂ rn with apex o is defined by c◦ = {x ∈ rn : x·e ≤ 0 ∀e ∈ c}. it is known that c◦ is a closed convex cone with apex o. furthermore, c◦ is n-dimensional provided c is line-free. for a convex set k ⊂ rn and a point v ∈ rn, the generated cone cv(k) is defined by cv(k) = {v + λ(x−v) : x ∈ k, λ ≥ 0}. both sets cv(k) and cl cv(k) are convex cones with apex v. the recession cone of a convex set k ⊂ rn is defined by rec k = {e ∈ rn : x + λe ∈ k whenever x ∈ k and λ ≥ 0}, and the lineality space of k is the subspace defined by lin k = rec k ∩ (−rec k). we will use the following properties of recession cones and lineality spaces of closed convex sets. (p1) rec k is a closed convex cone with apex o; it is distinct from {o} if and only if k is unbounded. 84 v. soltan (p2) for any given point u ∈ k, the cone rec k is the largest among all closed convex cones c ⊂ rn with apex o satisfying the inclusion u + c ⊂ k. (p3) one has k + lin k = k. (p4) if c ⊂ rn is a closed convex cone with apex o, then lin c is the largest subspace contained in c. 2. halfplane asymptotes klee [4] (also see [3]) observed that a line asymptote of a closed convex set k ⊂ rn contains a halfline asymptote. our first result partly generalizes this assertion to the case of halfplanes of a plane asymptote. theorem 2.1. if k ⊂ rn is a closed convex set and l ⊂ rn is a plane asymptote of k, then the following assertions hold. (a) for any bounded set x ⊂ l, there is a closed halfplane of l, disjoint from x, which is an asymptote of k. (b) for any bounded set z ⊂ l, there is a closed halfplane of l, containing z, which is an asymptote of k. proof. (a) consider the sets mr = {x ∈ l : δ(k,x) ≤ 1/r}, r ≥ 1. we state that every set mr is nonempty, closed, and convex. indeed, since δ(k,l) = 0, there are points p ∈ k and q ∈ l such that ‖p − q‖ ≤ 1/r. obviously, δ(k,q) ≤ ‖p − q‖ ≤ 1/r, which gives the inclusion q ∈ mr. if a sequence of points q1,q2, . . . ∈ mr converges to a point q, then q ∈ l due to the closedness of l. because the function δ(k,x) continuously depends on x ∈ rn (see, e.g., [10, theorem 8.21]), one has δ(k,q) = lim i→∞ δ(k,qi) ≤ 1/r, implying the inclusion q ∈ mr. so, the set mr is closed. for the convexity of mr, choose any points q,q ′ ∈ mr and a scalar ε > 0. there are points p,p′ ∈ k such that ‖p−q‖≤ δ(k,q) + ε ≤ 1/r + ε, ‖p′ −q′‖≤ δ(k,q′) + ε ≤ 1/r + ε. cone asymptotes of convex sets 85 given a scalar λ ∈ [0, 1], put p0 = (1 −λ)p + λp′ and q0 = (1 −λ)q + λq′. then p0 ∈ k by the convexity of k, and q0 ∈ l because (1 −µ)q + µq′ ∈ l whenever µ ∈ r (see [10, theorem 2.38]). furthermore, δ(k,q0) ≤‖p0 −q0‖ ≤ (1 −λ)‖p−q‖ + λ‖p′ −q′‖≤ 1/r + ε for each ε > 0. thus δ(k,q0) ≤ 1/r, which gives q0 ∈ mr. so, the set mr is convex. next, we observe that m1 ∩m2 ∩·· · = ∅. indeed, assuming the existence of a point u ∈ m1 ∩m2 ∩·· · ⊂ l, we would have δ(k,u) ≤ inf{1/r : r ≥ 1} = 0, which gives u ∈ k. the latter contradicts the assumption k ∩l = ∅. let x be a bounded subset of l. then the convex set y = conv(cl x) is compact. by the above argument, y ∩m1∩m2∩·· · = ∅. since the sequence of closed sets m1,m2, . . . is decreasing, the compactness of y implies the existence of an integer t such that y ∩ mr = ∅ for all r ≥ t. furthermore, δ(y,mt) > 0. hence there is a hyperplane h ⊂ rn separating y and mt such that these sets are contained in the opposite open halfspaces determined by h (see, e.g., [10, theorem 10.12]). denote by v the closed halfspace determined by h and containing mt, and let p = l∩v . clearly, p is a closed halfplane of l containing mt and disjoint from y . furthermore, δ(k,p) ≤ inf{δ(k,mr) : r ≥ t}≤ inf{1/r : r ≥ t} = 0. summing up, p is an asymptote of k. (b) let z be a bounded subset of l, and let p be a closed halfplane of l which is an asymptote of k (the existence of p is proved above). by a compactness argument, one can choose a translate v + p of p which lies in l and contains p ∪z. obviously, the halfplane v + p is an asymptote of k. we observe that the method of proof of theorem 2.1 gives a limited choice of halfplane asymptotes of k, as illustrated by the next example. example 2.2. consider k = {(x,y,z) : x > 0,y ≥ 1/x}, closed convex set in r3. it is easy to see that the coordinate xz-plane of r3, say l, is an asymptote of k. the sets mr ⊂ l defined in theorem 2.1 are vertical closed 86 v. soltan halfplanes of l of the form {(x, 0,z) : x ≥ αr}, r ≥ 1. consequently, the halfplane asymptote p of k described in theorem 2.1 should be of a similar form. on the other hand, it is easy to see that the only closed halfplanes of l, which are not asymptotes of k, have the form {(x, 0,z) : x ≤ λ}. 3. cone asymptotes we are going to refine the argument of theorem 2.1 to describe a wider family of asymptotes which are subsets of a given plane asymptote. for a nontrivial closed convex cone c ⊂ rn with apex o, consider its conic εneighborhood dε(c), ε > 0, defined as the union of all closed halflines h ⊂ rn with apex o which form with c an angle of size at most ε. clearly, dε(c) is a closed cone (not necessarily convex) with apex o. for the following lemma, we recall that sl denotes the subspace which is a translate of a plane l ⊂ rn. lemma 3.1. if l ⊂ rn is a plane asymptote of a closed convex set k ⊂ rn, then {o} 6= rec k ∩sl 6⊂ lin k. furthermore, rec k ∩sl 6= sl. proof. the assertion {o} 6= rec k ∩ sl 6⊂ lin k is proved in [9]. next, assume for a moment that rec k ∩ sl = sl. then sl ⊂ rec k, and a combination of (p1) and (p4) would imply the inclusion sl ⊂ lin k, contrary to the first assertion. we will need one more technical lemma. lemma 3.2. let k ⊂ rn be a unbounded closed convex set, and u ∈ k, v ∈ rn be points. any unbounded sequence of points x1,x2, . . . ∈ k \{u,v} contains a subsequence, say x′1,x ′ 2, . . . , such that the unit vectors ei = x′i −u ‖x′i −u‖ and ci = x′i −v ‖x′i −v‖ , i ≥ 1, (1) converge to the same unit vector e ∈ rec k. proof. since all vectors (xi−u)/‖xi−u‖, i ≥ 1, belong to the unit sphere of rn, a compactness argument implies the existence of a subsequence, say x′1,x ′ 2, . . . , of x1,x2, . . . such that the vectors e1,e2, . . . given by (1) converge to a unit vector e. the inequalities ‖x′i‖−‖u‖≤‖x ′ i −u‖≤‖x ′ i‖ + ‖u‖, ‖x′i‖−‖v‖≤‖x ′ i −v‖≤‖x ′ i‖ + ‖v‖ cone asymptotes of convex sets 87 give lim i→∞ ‖x′i −u‖ = lim i→∞ ‖x′i −v‖ = ∞ and lim i→∞ ‖x′i −u‖ ‖x′i −v‖ = 1. therefore, lim i→∞ ci = lim i→∞ ( x′i −u ‖x′i −v‖ + u−v ‖x′i −v‖ ) = lim i→∞ ( ‖x′i −u‖ ‖x′i −v‖ x′i −u ‖x′i −u‖ + u−v ‖x′i −v‖ ) = 1e + o = e. next, we assert that the closed halfline [u,u + e〉 lies in k. indeed, choose any point y in [u,u + e〉. then y = (1 −λ)u + λ(u + e) = u + λe, λ ≥ 0. clearly, ‖y − u‖ = λ. choose an integer i0 such that ‖x′i − u‖ ≥ max{1,λ} for all i ≥ i0. then u + ei = ( 1 − 1 ‖x′i −u‖ ) u + 1 ‖x′i −u‖ x′i ∈ [u,x ′ i], i ≥ i0. with yi = u + λei, one has yi = (1 −λ)u + λ(u + ei) ∈ [u,u + ei〉 = [u,x′i〉. now, the inequality ‖yi −u‖ = λ ≤‖x′i −u‖, i ≥ i0, gives yi ∈ [u,x′i] ⊂ k for all i ≥ i0. finally, y = u + λe = lim i→∞ (u + λei) = lim i→∞ yi ∈ k. summing up, [u,u + e〉⊂ k. finally, (p2) shows that e ∈ rec k. theorem 3.3. let k ⊂ rn be a closed convex set, and let l ⊂ rn be a plane asymptote of k. with c = rec k ∩sl, the following assertions hold. (a) for any point v ∈ l and a scalar ε > 0, the closed cone v + dε(c)∩sl lies in l and is an asymptote of k. furthermore, there is a scalar ε0 > 0 such that v + dε(c) ∩sl is a proper subset of l for all 0 < ε < ε0. 88 v. soltan (b) if a closed subset m of l is an asymptote of k, then, for any point v ∈ l and a scalar ε > 0, the closed set m ∩ (v + dε(c) ∩ sl) is an asymptote of k. proof. (a) since rec k is a closed convex cone with apex o, so is the set c. by lemma 3.1, {o} 6= c 6= sl. then the polar cone c◦l of c within the space sl is nontrivial. if e is a nonzero vector in c ◦ l, then c lies in the closed halfplane p = {x ∈ sl : x·e ≤ 0} of sl. a simple geometric argument shows the existence of a scalar ε0 > 0 such that the closed cone dε(c) ∩ sl is a proper subset of sl for all 0 < ε < ε0. obviously, v + dε(c) ∩sl is a closed cone with apex v. furthermore, v + dε(c) ∩sl ⊂ v + sl = l. by the above argument, v + dε(c) ∩ sl is a proper subset of l for all 0 < ε < ε0. because l is an asymptote of k, there are sequences of points p1,p2, · · · ∈ k and q1,q2, . . . ∈ l such that limi→∞‖pi − qi‖ = 0. clearly, both sets {p1,p2, . . .} and {q1,q2, . . .} are unbounded. indeed, otherwise, one could choose respective subsequences converging to the same point in k∩l, contrary to the assumption k ∩l = ∅. choose a point u ∈ k. by lemma 3.2, there is an unbounded subsequence pi1,pi2, . . . such that the unit vectors eij = pij −u ‖pij −u‖ and cij = pij −v ‖pij −v‖ , j ≥ 1, converge to the same unit vector e ∈ rec k. let c′ij = qij −v ‖qij −v‖ , j ≥ 1. since the subsequence qi1,qi2, . . . is unbounded and ‖pij −qij‖→ 0 as j →∞, the inequalities ‖qij −v‖−‖pij −qij‖≤‖pij −v‖≤‖qij −v‖ + ‖pij −qij‖ give lim j→∞ ‖pij −v‖ ‖qij −v‖ = 1. cone asymptotes of convex sets 89 therefore, lim j→∞ c′ij = limj→∞ ( pij −v ‖qij −v‖ − pij −qij ‖qij −v‖ ) = lim j→∞ ( ‖pij −v‖ ‖qij −v‖ pij −v ‖pij −v‖ − pij −qij ‖qij −v‖ ) = 1e + o = e. because c′ij ∈ sl for all j ≥ 1, the limit vector e also belongs to sl. thus e ∈ rec k ∩sl = c. clearly, there is an index r ≥ 1 such that every vector eij , j ≥ r, forms with e an angle of size at most ε. equivalently, every halfline [o,qij − v〉 forms with the halfline h = [o,e〉 ⊂ c an angle of size at most ε. thus qij ∈ [v,qij〉 = v + [o,qij −v〉⊂ v + dε(h) ⊂ v + dε(c) ∩sl, j ≥ r. finally, because limj→∞‖pij − qij‖ = 0, the cone v + dε(c) ∩ sl is an asymptote of k. (b) suppose that a closed subset m of l is an asymptote of k. then there are sequences of points p1,p2, . . . ∈ k and q1,q2, . . . ∈ m such that limi→∞‖pi−qi‖ = 0. an obvious modification of the argument from part (a) shows the existence of unbounded subsequences pi1,pi2, . . . ∈ k and qi1,qi2, . . . ∈ m ∩ (v + dε(c) ∩sl) such that limj→∞‖pij − qij‖ = 0. so, the set m ∩ (v + dε(c) ∩ sl) is an asymptote of k. we observe that, generally, theorem 2.1 does not follow from theorem 3.3. indeed, if k is the convex set from example 2.2, then its recession cone is the polyhedron {(x,y,z) : x ≥ 0, y ≥ 0}. in terms of theorem 3.3, sl = l, c is the halfplane {(x, 0,z) : x ≥ 0} of l, and v + dε(c)∩sl is the nonconvex closed cone in l of angle size π + 2ε. clearly, this cone is larger than any halfplane asymptote of k contained in v + dε(c)∩sl, and there is no halfplane asymptote of k containing this cone. analysis of the proof of theorem 3.3 results in the following corollary, which shows the existence (but not a constructive description) of arbitrarily sharp pencil cone asymptotes. corollary 3.4. let k ⊂ rn be a closed convex set, and l ⊂ rn be a plane asymptote of k. there is a closed halfline h ⊂ rec k ∩ sl with endpoint o such that for any point v ∈ l and scalar ε > 0, the pencil cone (v + dε(h)) ∩l is an asymptote of k. 90 v. soltan the next example shows that in theorem 3.3, the set dε(c) cannot be replaced with c and even with the closed metric ε-neighborhood of c, defined by nε(c) = {x ∈ rn : ρ(x,c) ≤ ε}. consequently, no translate of the cone v + c may be an asymptote of k. example 3.5. the set k ⊂ r3 given by the conditions k = {(x,y,z) : x > 0, y ≥ 1/x, z ≥ (x + y)2} is closed and convex. furthermore, the following properties of k hold: (a) the coordinate xz-plane, say l, is an asymptote of k; (b) no closed slab between a pair of parallel lines in l is an asymptote of k; (c) rec k is the vertical halfline h = {(0, 0,z) : z ≥ 0}, and no metric neighborhood v + nε(h) ∩ l of the halfline v + h, with v ∈ l, is an asymptote of k; (d) for any choice of scalars α ≥ 0, and β,γ ∈ r, the planar cone d(α,β,γ) = {(x, 0,z) : z ≥ α|x−β| + γ}⊂ l (2) is an asymptote of k. indeed, the set k is closed and convex as the intersection of closed and convex sets k1 = {(x,y,z) : x > 0, y ≥ 1/x} and k2 = {(x,y,z) : z ≥ (x + y)2}. (a) clearly, k ∩l = ∅, and the sequences of points pi = ( i, 1/i, [(i2 + 1)/i]2 ) ∈ k, qi = ( i, 0, [(i2 + 1)/i]2 ) ∈ l, i ≥ 1, satisfy the condition ‖pi − qi‖ = 1/i. so, δ(k,l) = 0, which shows that l is an asymptote of k. (b) let q be any closed slab between a pair of parallel lines in l. assume first that q is vertical. then it can be described as q = {(x, 0,z) : α ≤ x ≤ β}, where α ≤ β. cone asymptotes of convex sets 91 since k1∩q = ∅ and k1 is a both-way unbounded vertical cylinder, δ(k1,q) equals the distance between the branch of hyperbola {(x,y, 0) : x > 0,xy = 1} ⊂ bd k1 and the point (α, 0, 0) ∈ q, which is positive. therefore, the inclusion k ⊂ k1 gives δ(k,q) ≥ δ(k1,q) > 0. hence no vertical slab in l is an asymptote of k. suppose now that q is slant. then q = {(x, 0,z) : αx + β ≤ z ≤ αx + β′} for suitable scalars α and β ≤ β′. assume for a moment that δ(k,q) = 0. then one can choose sequences of points pi ∈ k and qi ∈ q, i ≥ 1, such that ‖pi − qi‖→ 0 as i →∞. denote by p′i the point at which the segment [pi,qi] meets bd k, i ≥ 1. let p′i = (xi,yi,zi) and qi = (ui, 0,vi), αui + β ≤ vi ≤ αui + β ′, i ≥ 1. by a convexity argument, xiyi = 1 and ‖p′i−qi‖≤‖pi−qi‖. hence ‖p ′ i−qi‖→ 0 as i →∞, or, equivalently, lim i→∞ ( (xi −ui)2 + 1/x2i + (zi −vi) 2 ) = 0. (3) thus xi →∞ and ui = xi + εi, where εi → 0 as i →∞. consequently, there is an index i0 ≥ 0 and a scalar γ > 0 such that |vi| ≤ max { |αui + β|, |αui + β′| } ≤ |αxi| + |αεi| + max{|β|, |β′|}≤ |αxi| + γ whenever i ≥ i0. therefore, |zi −vi| ≥ |zi|− |vi| ≥ ( (xi + 1/xi) 2 −|αxi|−γ ) →∞ as i →∞, in contradiction with (3). summing up, q cannot be an asymptote of k. (c) given the point p = (1, 1, 4) ∈ k, the halfline p+h = {(1, 1,z) : z ≥ 4} is included in k. this argument and (p2) imply that h ⊂ rec k. for the opposite inclusion, choose any vector e = (u,v,w) ∈ rec k. then p + λe = (1 + λu, 1 + λv, 4 + λw) ∈ k for all λ > 0. equivalently, for all λ > 0, one has 1 + λu > 0, 1 + λv ≥ 1/(1 + λu), 4 + λw ≥ (2 + λ(u + v))2. 92 v. soltan the first condition implies that u ≥ 0. this inequality and the second condition give v ≥ 0, while the third condition results in u+v = 0. hence u = v = 0 and w ≥ 0, implying the inclusion e ∈ h. summing up, rec k = h. it is easy to see that a metric neighborhood v + nε(h) ∩ l of v + h, with v ∈ l, is a subset of a suitable vertical slab q ⊂ l. according to part (b) above, q is not an asymptote of k. hence v + nε(h) ∩l is not an asymptote of k. (d) let sl = l and v = (β, 0,γ). if α > 0, then the planar cone d(α,β,γ) from (2) can be expressed as v + dε(h) ∩sl, where ε = cot−1(α). if α = 0, then d(0,β,γ) = v + dπ/2(h) ∩sl. in either case, corollary 3.4 shows that d(α,β,γ) is an asymptote of k. 4. line-free cone asymptotes we describe below a family of plane asymptotes m contained in a plane asymptote l of k such that the cone dε(rec k ∩m) ∩sm becomes line-free provided ε is sufficiently small. theorem 4.1. let k ⊂ rn be a closed convex set, l ⊂ rn be a plane asymptote of k, and f be a subspace complementary to lin k ∩ sl in sl. the following assertions hold. (a) for any point v ∈ l, the plane m = v+f lies in l and is an asymptote of k. (b) for any point v ∈ l and any ε > 0, the closed cone v+dε(rec k∩f)∩f is a line-free asymptote of k. (c) there is an n-dimensional line-free closed convex cone c′ ⊂ rn with apex o such that c′ ⊂ dε(rec k ∩ f) and the cone v + c′ ∩ f is an asymptote of k. the proof of theorem 4.1 is divided into lemmas 4.2 – 4.4. lemma 4.2. let k ⊂ rn be a closed convex set, l ⊂ rn be a plane asymptote of k, and v be a point in l. for any subspace f ⊂ rn complementary to lin k ∩ sl in sl, the plane m = v + f lies in l and is an asymptote of k. proof. since lin k ⊂ rec k, lemma 3.1 shows that the subspace t = lin k ∩ sl is distinct from sl (possibly, t = {o}). by (p3), k = k + t. cone asymptotes of convex sets 93 choose a subspace f ⊂ rn complementary to t in sl and consider the plane m = v + f. since m = v + f ⊂ v + sl = l and k ∩ l = ∅, it remains to show that δ(k,m) = 0. indeed, choose sequences of points p1,p2, . . . ∈ k and q1,q2, . . . ∈ l such that limi→∞‖pi − qi‖ = 0. clearly, qi − v ∈ sl, i ≥ 1. denote by q′i the projection of qi − v on f in the direction of t, and let p′i = pi + (v + q ′ i −qi), i ≥ 1. because v + q′i −qi = q ′ i − (qi −v) ∈ t, one has p′i = pi + (v + q ′ i −qi) ∈ k + t = k, i ≥ 1. furthermore, v + q′i ∈ v + f = m, i ≥ 1, and δ(k,m) ≤ lim i→∞ ‖p′i − (v + q ′ i)‖ = lim i→∞ ‖(pi + v + q′i −qi) − (v + q ′ i)‖ = lim i→∞ ‖pi −qi‖ = 0. summing up, m is an asymptote of k. lemma 4.3. let k ⊂ rn be a closed convex set, and l ⊂ rn be a plane asymptote of k. given a subspace f complementary to lin k∩sl in sl, the closed convex cone c = rec k ∩f with apex o is nontrivial and line-free. proof. by lemma 4.2, every translate of the form v+f, where v ∈ l, is an asymptote of k. consequently, lemma 3.1 shows that the closed convex cone c = rec k ∩ f is distinct from {o}. it remains to prove that c is line-free. assume for a moment that c contains a line l. we consider separately the following two cases. (i) let o ∈ l. by (p4), l is a 1-dimensional subspace contained in lin k. consequently, l ⊂ lin k ∩f, contrary to the choice of f. (ii) let o /∈ l. because o is an apex of rec k, every closed halfline [o,x〉, where x ∈ l, is contained in rec k. clearly, the union of all such halflines coincides with {o}∪ p , where p is the open halfplane of the 2-dimensional subspace, span l, bounded by the 1-dimensional subspace l′ = l − l. then l′ ⊂ cl p ⊂ rec k, and, as above, l′ ∈ lin k ∩f, contrary to the choice of f . summing up, c should be line-free. 94 v. soltan lemma 4.4. if c ⊂ rn is a closed line-free convex cone with apex o, then there is a scalar ε > 0 such that the closed cone dε(c) is line-free. furthermore, there is an n-dimensional closed convex cone c′ ⊂ rn with apex o and a scalar ε′ ∈ (0,ε) such that dε′(c) ⊂ c′ ⊂ dε(c). proof. since c is line-free, the polar cone c◦ is n-dimensional (see, e.g., [10, theorem 8.4]). choose a nonzero vector e ∈ (− int c◦). then c \{o} is contained in the open halfspace w = {x ∈ rn : x·e > 0} (see [10, theorem 8.6]). denote by s the unit sphere of rn, and let e = c ∩ s. clearly, c = {λx : λ ≥ 0, x ∈ e}. since e is a compact subset of w , there is an ε > 0 such that the closed set vρ(e) = {x ∈ s : δ(x,e) ≤ ρ}, ρ = 2 sin(ε/2), is contained in w . we observe that a closed halfline h = [o,u〉, with u ∈ s, is contained in dε(c) if and only if there is a closed halfline h ′ = [o,u′〉, with u′ ∈ e, such that the angle between h and h′ is of size at most ε. considering the isosceles triangle ∆(o,u,u′), we deduce that the angle between h and h′ is of size at most ε if and only if ‖u−u′‖≤ 2 sin(ε/2). this argument implies that dε(c) = {λx : λ ≥ 0, x ∈ vρ(e)}. it is easy to see that every closed halfline [o,x〉, where x ∈ vρ(e), is contained in {o}∪w . hence dε(c) ⊂{o}∪w . an argument similar to that of lemma 4.3 shows that the cone dε(c) is line-free. finally, consider the hyperplane h = {x ∈ rn : x·e = 1}. then both sets a = c ∩ h and b = dε(c) ∩ h are compact (see [10, theorem 8.15]), and a is convex. furthermore, both sets a and b have dimension n− 1 and rint a ⊂ rint b (see [10, theorem 8.14]). given any point a ∈ rint a, the set a′ = a+µ(a−a), with µ > 1, is convex and contains a in its relative interior. a compactness argument implies that a′ ⊂ b provided µ is sufficiently close to 1. for this value of µ, let c′ = co(a ′) = {λx : λ ≥ 0, x ∈ a′}. then c′ is a closed convex cone satisfying the inclusions c ⊂{o}∪ int c′ and c′ ⊂ dε(c), as desired. the following example shows that the cone dε(c) in lemma 4.4 may be nonconvex even if the cone c is convex and line-free. cone asymptotes of convex sets 95 example 4.5. let c = {(x,y, 0) : y ≥ |x|/10} be the planar cone in r3. clearly, c is closed, convex, and line-free. let ε = sin−1(0.1) ≈ 0.1. the point u = (10, 1, 1) belongs to dε(c) because u ′ = (10, 1, 0) ∈ c and the angle between the halflines [o,u〉 and [o,u′〉 equals sin−1(1/ √ 102) < ε. similarly, the point v = (−10, 1, 1) belongs to dε(c). on the other hand, the midpoint w = (0, 1, 1) of the segment [u,v] does not belong to dε(c). indeed, the angle between [o,w〉 and c is achieved on the pair of halflines [o,w〉 and [o,w′〉, where w = (0, 1, 0), and this angle equals sin−1(1/ √ 2) = π/4 > ε. so, w /∈ dε(c), implying that the cone dε(c) is not convex. in view of theorem 4.1, one may ask about the existence of minimal (under inclusion) cone asymptotes of k. the following theorem shows that such asymptotes (if any) should be halflines. theorem 4.6. let x ⊂ rn be a nonempty closed set, and c ⊂ rn be a closed convex cone with apex v, which is an asymptote of x. if dim c ≥ 2, then there is a convex cone c′ with apex v which is an asymptote of k and a proper subset of c. proof. choose a point u ∈ rint c\{v}, and consider a hyperplane h ⊂ rn containing {u,v} such that c meets the interiors of both closed halfspaces, say v1 and v2, determined by h. the sets c1 = c ∩v1 and c2 = c ∩v2 are closed convex cones with common apex v, whose union is c, and each of these cones is a proper subset of c. because c is an asymptote of k, there is an unbounded sequence of points x1,x2, . . . ∈ c such that δ(xi,x) → 0 as i →∞. clearly, one of the cones c1 and c2, say c1, contains an infinite subsequence x′1,x ′ 2, . . . of x1,x2, . . . . since c1 ∩ x ⊂ c ∩ x = ∅ and δ(x ′ i,x) → 0 as i →∞, the cone c1 is an asymptote of x. 5. properties of cone asymptotes the next two theorems show that some properties of plane asymptotes can be generalized to the case of cone asymptotes (see [4] and [9] for the original statements). theorem 5.1. let c ⊂ rn be a closed convex cone with apex o. if a translate of c is an asymptote of a closed convex set k ⊂ rn, then rec k ∩c 6= {o}. furthermore, for any nonzero vector e ∈ rec k ∩c, there is a translate of the closed halfline h = [o,e〉 which either lies in bd k or is an asymptote of k. 96 v. soltan proof. let a translate d = v + c of c be an asymptote of k. then k ∩d = ∅ and there are sequences of points p1,p2, . . . ∈ k and q1,q2, . . . ∈ d such that limi→∞‖pi − qi‖ = 0. we observe that both sequences are unbounded. indeed, otherwise one could choose respective subsequences converging to the same point in k ∩d, contrary to the assumption k ∩d = ∅. as shown in the proof of theorem 3.3, there are subsequences pi1,pi2, . . . and qi1,qi2, . . . such that the unit vectors eij = pij −u ‖pij −u‖ an d c′ij = qij −v ‖qij −v‖ , j ≥ 1, converge to the same unit vector e ∈ rec k. since qij − v ∈ c for all j ≥ 1 and the cone c is closed, we obtain that e ∈ c. consider the halfline h = [o,e〉. then h ⊂ c and, by (p2), the halfline u + h lies in k. consider the family of parallel halflines h(λ) = ((1 −λ)v + λu) + h, 0 ≤ λ ≤ 1. by the above, h(0) ⊂ d and h(1) ⊂ k. consequently, k ∩h(0) = ∅. let λ′ = sup{λ ∈ [0, 1] : k ∩h(λ) = ∅}. it is easy to see that int k∩h(λ′) = ∅ and δ(k,h(λ′)) = 0. if k∩h(λ′) = ∅, then h(λ′) is an asymptote of k. suppose that k ∩h(λ′) 6= ∅ and choose a point w ∈ k ∩h(λ). by (p2), the halfline w + h lies in k, and the inclusion w + h ⊂ h(λ′) implies that w + h ⊂ bd k, as desired. the following example shows that in theorem 5.1 (unlike lemma 3.1) the inclusion rec k ∩c ⊂ lin k is possible. example 5.2. let the closed convex sets k ⊂ r3 be given by k = {(x,y,z) : x ≥ 0, y ≥ 0, x + y ≥ 1}. clearly, rec k = {(x,y,z) : x ≥ 0, y ≥ 0} and lin k is the z-axis. let c be the closed convex cone with apex o generated by the circular disk d = {(x,y, 1) : x2 + (y + 1)2 ≤ 1}. the cone c is an asymptote of k because k ∩ c = ∅ and δ(c,l) = 0, where l ⊂ k is the vertical line {(1, 0,z) : z ∈ r}. at the same time, rec k ∩c = {(0, 0,z) : z ≥ 0} is a subset of lin k. cone asymptotes of convex sets 97 theorem 5.3. let x ⊂ rn be a nonempty closed set and c ⊂ rn be a closed convex cone with apex o. there is a translate of c which is an asymptote of x if and only if the set x −c is not closed. proof. let a translate d = v + c of c be an asymptote of x. we first observe that v /∈ x−c. indeed, assume for a moment that v ∈ x−c. then v = x−u, where x ∈ x and u ∈ c. consequently, x = v + u ∈ v + c = d, contrary to the assumption d ∩ x = ∅. since d is an asymptote of x, there are sequences of points x1,x2, . . . ∈ x and y1,y2, . . . ∈ d such that limi→∞‖xi−yi‖ = 0. expressing yi as yi = v + zi for a suitable zi ∈ c, i ≥ 1, one has xi −zi ∈ x −c and lim i→∞ ‖v − (xi −zi)‖ = lim i→∞ ‖yi −xi‖ = 0. so, v ∈ cl(x −c) \ (x −c). thus the set x −c is not closed. conversely, suppose that the set x −c is not closed. choose a point v ∈ cl(x −c) \ (x −c) and consider the cone d = v +c. we assert that d∩x = ∅. indeed, assume the existence of a point z ∈ d ∩x. then z = v + u = x, where u ∈ c and x ∈ x. consequently, v = x−u ∈ x −c, which is impossible by the choice of v. next, the inclusion v ∈ cl(x − c) implies the existence of a sequence u1,u2, . . . of points in x − c converging to v. with ui = xi − zi, where xi ∈ x and zi ∈ c, and with wi = v + zi ∈ d, i ≥ 1, we obtain that lim i→∞ ‖xi −wi‖ = lim i→∞ ‖(ui + zi) − (v + zi)‖ = lim i→∞ ‖ui −v‖ = 0. hence δ(x,d) = 0. summing up, d is an asymptote of x. acknowledgements the author is thankful to the referee for the careful reading and considered suggestions leading to a better presented paper. 98 v. soltan references [1] a. auslender, m. teboulle, “asymptotic cones and functions in optimization and variational inequalities”, springer-verlag, new york, 2003. [2] d. gale, v. klee, continuous convex sets, math. scand. 7 (1959), 379 – 391. [3] p. goossens, hyperbolic sets and asymptotes, j. math. anal. appl. 116 (1986), 604 – 618. [4] v. klee, asymptotes and projections of convex sets, math. scand. 8 (1960), 356 – 362. [5] v.l. klee, asymptotes of convex bodies, math. scand. 20 (1967), 89 – 90. [6] j. lawrence, v. soltan, on unions and intersections of nested families of cones, beitr. algebra geom. 57 (2016), 655 – 665. [7] j.e. mart́ınez-legaz, d. noll, w. sosa, minimization of quadratic functions on convex sets without asymptotes, j. convex anal. 25 (2018), 623 – 641. [8] j.e. mart́ınez-legaz, d. noll, w. sosa, non-polyhedral extensions of the frank and wolfe theorem, in “splitting algorithms, modern operator theory, and applications” (h. bauschke, r. burachik, d. luke editors), springer, cham, 2019, 309 – 329. [9] v. soltan, asymptotic planes and closedness conditions for linear images and vector sums of sets, j. convex anal. 25 (2018), 1183 – 1196. [10] v. soltan, “lectures on convex sets”, second edition, world scientific, hackensack, nj, 2020. [11] v. soltan, on m-decomposable sets, j. math. anal. appl. 485 (2020), 123816, 15 pp. introduction halfplane asymptotes cone asymptotes line-free cone asymptotes properties of cone asymptotes e extracta mathematicae vol. 31, núm. 1, 69 – 88 (2016) sharp upper estimates for the first eigenvalue of a jacobi type operator h.f. de lima1,∗, a.f. de sousa2, f.r. dos santos1, m.a.l. velásquez1 1 departamento de matemática, universidade federal de campina grande 58.429-970 campina grande, paráıba, brasil 2 departamento de matemática, universidade federal de pernambuco 50.740-560 recife, pernambuco, brazil henrique@dme.ufcg.edu.br, tonysousa3@gmail.com fabio@dme.ufcg.edu.br, marco.velasquez@pq.cnpq.br presented by manuel de león received november 3, 2015 abstract: our purpose in this article is to obtain sharp upper estimates for the first positive eigenvalue of a jacobi type operator, which is a suitable extension of the linearized operators of the higher order mean curvatures of a closed hypersurface immersed either in the euclidean space or in the euclidean sphere. key words: euclidean space, euclidean sphere, closed hypersurfaces, r-th mean curvatures, jacobi operator, reilly type inequalities. ams subject class. (2010): primary 53c42; secondary 53b30. 1. introduction in the last decades has been increasing the study of the first positive eigenvalue of certain elliptic operators defined on riemannian manifolds. this study was initiated in 1977 when reilly [13] established some inequalities estimates for the first positive eigenvalue λ1 of the laplacian operator ∆ of a closed hypersurface mn immersed in the euclidean space rn+1. for instance, he obtained the following sharp estimate λ1 (∫ m hr dm )2 ≤ n vol(m) ∫ m h2r+1 dm , for every 0 ≤ r ≤ n − 1, where hr stands for the r-th mean curvature of mn, and the equality holds precisely when mn is a round sphere of rn+1. ∗ corresponding author. 69 70 h.f. de lima, a.f. de sousa, f.r. dos santos, m.a.l. velázquez several authors presented generalizations and extensions of the previous reilly’s inequality to some other ambient spaces (we refer, for instance, the works [1], [6], [7], [8], [9], [10], [11] and [16]). also in this setting, we note that aĺıas and malacarne [3] extended techniques due to takahashi [15] and veeravalli [16] in order to derive sharp upper bounds for the first positive eigenvalue of the linearized operator lr of the r-th mean curvature hr of a closed hypersurface immersed either in the euclidean space rn+1 or in the euclidean sphere sn+1. our aim in this work is study the first positive eigenvalue λ lr,s 1 of the jacobi type (or simply, jacobi) operator lr,s, which is defined as follows: fixed integer numbers r, s such that 0 ≤ r ≤ s ≤ n − 1, lr,s : c∞(m) → c∞(m) is given by (1.1) lr,s(f) = s∑ j=r (j + 1)ajlj(f) , where lj are the linearized operators of the j-th mean curvatures hj, aj are nonnegative real numbers (with at least one nonzero) for all j ∈ {r, . . . , s} and f is a smooth function on the hypersurface mn which is supposed immersed either in rn+1 or in sn+1. we point out that the authors in [17] established the notion of (r, s)stability concerning closed hypersurfaces with higher order mean curvatures linearly related in a space form. in this setting, they obtained a suitable characterization of the (r, s)-stability through of the analysis of the first positive eigenvalue λ lr,s 1 of the jacobi operator lr,s, which is associated to the corresponding variational problem (cf. [17, theorem 5.3]). our purpose in this work, is exactly obtain sharp upper estimates for λ lr,s 1 . consequently, the results that we will present along this paper are naturally attached with the study of (r, s)-stable closed hypersurfaces in a space form. this manuscript is organized in the following way: in section 2, we recall some basic facts concerning r-th mean curvatures hr and their corresponding linearized operators lr. afterwards, in section 3 we obtain a version of the classical result of takahashi [15] (cf. proposition 1) for the jacobi operator lr,s defined in (1.1) and we apply it to obtain a reilly type inequality for λ lr,s 1 (cf. lemma 3). next, in section 4 we apply our previous reilly type inequality in order to prove sharp upper bound for λ lr,s 1 (cf. theorem 1, theorem 2, theorem 3 and corollary1). finally, in section5 we consider the case when the ambient space is sn+1 (cf. theorem 4). sharp estimates for the first eigenvalue 71 2. preliminaries given a connected and orientable hypersurface x : mn → mn+1(c) into a riemannian space form of constant sectional curvature c, one can choose a globally defined unit normal vector field n on mn. let a denote the shape operator with respect to n, so that, at each p ∈ mn, a restricts to a selfadjoint linear map ap : tpm → tpm. associated to the shape operator a of mn one has n algebraic invariants, namely, the elementary symmetric functions sr of the principal curvatures κ1, . . . , κn of a, given by sr = σr (κ1, . . . , κn) = ∑ i1<··· 0), and x : mn → m n+1 (c) be a closed hypersurface. if hr+1 > 0 then (a) each operator lj is elliptic, (b) each j-th mean curvature hj is positive, for all j ∈ {1, . . . , r}. when m n+1 (c) is the euclidian space, [3, corollary 3] also gives the following another sufficient criteria of ellipticity. lemma 2. let x : mn → rn+1 be a closed hypersurface with positive ricci curvature (hence, necessarily embedded). then (a) each operator lj is elliptic, (b) each j-th mean curvature hj is positive, for all j ∈ {1, . . . , r}. 3. a reilly-type inequality in the euclidean space given a closed hypersurface x : mn → rn+1, its center of gravity c is defined by (3.1) c = 1 vol(m) ∫ m x dm ∈ rn+1, sharp estimates for the first eigenvalue 73 where vol(m) denotes the n-dimensional volume of mn. in this setting, let us consider on mn the support functions la = ⟨x − c, a⟩ and fa = ⟨n, a⟩ with respect to a fixed nonzero vector a ∈ rn+1. it is not difficult to verify that the gradient of function la is given by ∇la = a⊤, where a⊤ = a − fan ∈ x(m). thus, for x ∈ x(m) we have that (3.2) ∇x∇la = faax . from (2.1) and (3.2), for each j ∈ {r, . . . , s}, we get (3.3) lj(la) = bjhj+1fa . consequently, considering the jacobi operator lr,s defined in (1.1), from (3.3) we obtain (3.4) lr,s(la) = ( s∑ j=r (j + 1)ajbjhj+1 ) fa . thus, denoting by {e1, . . . , en+1} the canonical orthonormal basis of rn+1, from (3.4) we can write (3.5) lr,s(x − c) = ( s∑ j=r (j + 1)ajbjhj+1 ) n. now, we are in position to present a version of a classical result due to takahashi [15]. proposition 1. let x : mn → rn+1 be an orientable closed connected hypersurface and c its center of gravity. if lr,s is the jacobi operator defined in (1.1), then (3.6) lr,s(x − c) + λ(x − c) = 0 , for some real number λ ̸= 0 if, and only if, x(m) is a round sphere of rn+1 centered at c. proof. suppose that (3.6) is true for some λ ̸= 0. from expression (3.5) we have (3.7) ( s∑ j=r (j + 1)ajbjhj+1 ) n + λ(x − c) = 0 . 74 h.f. de lima, a.f. de sousa, f.r. dos santos, m.a.l. velázquez taking the covariant derivative in (3.7) we obtain (3.8) x ( s∑ j=r (j + 1)ajbjhj+1 ) n − ( s∑ j=r (j + 1)ajbjhj+1 ) ax + λx = 0 , for all x ∈ x(m). consequently, taking into account that λ ̸= 0, from (3.8) we conclude that ∑s j=r(j + 1)ajbjhj+1 is a nonzero constant. thus, returning to (3.8), we obtain a = ( s∑ j=r (j + 1)ajbjhj+1 )−1 · λi, an, hence, x(m) is a totally umbilical hypersurface of rn+1. therefore, unless of translations and homotheties, x(m) is a round sphere of rn+1 centered at c. reciprocally, for a the round sphere of rn+1 centered at c and of radius ρ > 0, let us consider n = −1 ρ (x − c), and thus its j-th mean curvature is hj+1 = 1 ρj+1 . then, from (3.5) we have that (3.6) is satisfied for λ = s∑ j=r (j + 1)ajbj ρj+2 ̸= 0 , since at least on of aj are supposed be nonzero. remark 1. we note that the first positive eigenvalue of the operator lr,s on a round sphere sn(ρ) ⊂ rn+1 of radius ρ > 0 is given by λ lr,s 1 = s∑ j=r (j + 1)ajbjhj+2 . indeed, since sn(ρ) is totally umbilical with a = 1 ρ i, the j-th newton transformation is given by pj = bj nρj , where bj = (j + 1) ( n j+1 ) . then ljf = bj ρj ∆f for each f ∈ c∞(m) . sharp estimates for the first eigenvalue 75 hence, for integers r, s such that 0 ≤ r ≤ s ≤ n − 1 and nonnegative real numbers aj (with at least one nonzero) for all 1 ≤ j ≤ n, we have lr,s = s∑ j=r (j + 1)ajlj = s∑ j=r (j + 1)ajbj nρj ∆ . since the first positive eigenvalue for the laplacian operator ∆ in sn(ρ) is given by λ∆1 = n ρ2 , we conclude that λ lr,s 1 = s∑ j=r (j + 1)ajbj ρj+2 = s∑ j=r (j + 1)ajbjhj+2 . let us consider (x − c)⊤ = (x − c) − ⟨x − c, n⟩n ∈ x(m), where (x − c)⊤ denotes the component tangent of x − c along mn. for every j ∈ {r, . . . , s}, using (2.1), it is not difficult to verify that divpj(x − c)⊤ = bj ( hj + ⟨x − c, n⟩hj+1 ) . consequently, (3.9) s∑ j=r (j + 1)aj [ divpj(x − c)⊤ ] = s∑ j=r (j + 1)ajbj ( hj + ⟨x − c, n⟩hj+1 ) , where bj = (j + 1) ( n j+1 ) = (n − j) ( n j ) and aj are nonnegative real numbers (with at least one nonzero) for all j ∈ {r, . . . , s}. at this point, we will assume that the hypersurface mn is closed. so, from (3.9) we obtain the following minkowski type integral formula (3.10) s∑ j=r (j + 1)ajbj ∫ m ( hj + ⟨x − c, n⟩hj+1 ) dm = 0 . in the next result, motivated by remark 1, we apply proposition 1 to obtain a reilly type inequality for the jacobi operator lr,s. lemma 3. let x : mn → rn+1 be an orientable closed connected hypersurface and let c be its center of gravity. if either hs+1 > 0, for some integer number s ∈ {1, . . . , n − 1}, or the ricci curvature of mn is positive (hence, necessarily embedded), then (3.11) λ lr,s 1 ∫ m |x − c|2 dm ≤ s∑ j=r (j + 1)ajbj ∫ m hj dm , 76 h.f. de lima, a.f. de sousa, f.r. dos santos, m.a.l. velázquez for all r ∈ {0, . . . , s − 1}, where λlr,s1 is the first positive eigenvalue of jacobi operator lr,s defined in (1.1), aj are nonnegative real numbers (with at least one nonzero) for all j ∈ {r, . . . , s} and bj = (j + 1) ( n j+1 ) . in particular, the equality occurs in (3.11) if and only if x(m) is a round sphere of rn+1 centered at c. proof. since either hs+1 > 0 or the ricci curvature of m n is positive, lemma 1 and lemma 2 guarantee that lj is elliptic for j ∈ {1, . . . , s} and, hence, lr,s is elliptic. thus, it holds the following characterization of λl1 (3.12) λ lr,s 1 = inf { − ∫ m flr,s(f) dm∫ m f2 dm : ∫ m f dm = 0 } . let {e1, . . . , en+1} be the canonical orthonormal basis of rn+1. for every 1 ≤ k ≤ n + 1, we consider the k-th coordinate function fk = ⟨x − c, ek⟩. thus, for every 1 ≤ k ≤ n + 1, from (3.1) we have that ∫ m fk dm = 0. so, from (3.12) we get (3.13) λ lr,s 1 ∫ m f2k dm ≤ − ∫ m fklr,s(fk) dm . furthermore, from (3.4) we obtain (3.14) λ lr,s 1 ∫ m f2k dm ≤ − s∑ j=r (j + 1)ajbj ∫ m fk⟨n, ek⟩hj+1 dm . now, summing on k of 1 until n + 1 in (3.14) and taking into account that n+1∑ k=1 f2k = |x − c| 2 and n+1∑ k=1 fk⟨n, ek⟩ = ⟨n, x − c⟩ , we get (3.15) λ lr,s 1 ∫ m |x − c|2 dm ≤ − s∑ j=r (j + 1)ajbj ∫ m ⟨n, x − c⟩hj+1 dm . hence, from (3.15) and (3.10) we have λ lr,s 1 ∫ m |x − c|2 dm ≤ s∑ j=r (j + 1)ajbj ∫ m hj dm . sharp estimates for the first eigenvalue 77 if occurs the equality in (3.11), all of the above inequalities are, in fact, equalities and, in particular, from (3.13) we get lr,s(fk) + λ lr,s 1 fk = 0 , for every k = 1, . . . , n + 1, which happens if and only if lr,s(x − c) + λ lr,s 1 (x − c) = 0. in this case, proposition 1 assures that x(m) is a round sphere centered at c. 4. upper estimates for λ lr,s 1 in r n+1 in [3, theorem 9], aĺıas and malacarne obtained the following sharp estimate for the first positive eigenvalue λlr1 of linearized operator lr concerning a closed hypersurface immersed in the euclidean space rn+1 λlr1 (∫ m hs dm )2 ≤ br ∫ m hr dm ∫ m h2s+1 dm , 0 ≤ s ≤ n − 1 , occurring the equality if and only if mn is a round sphere of rn+1. in our next result, we extend the ideas of aĺıas and malacarne [3] in order to get a sharp estimate for the first positive eigenvalue of the jacobi operator lr,s which was defined in (1.1). theorem 1. let x : mn → rn+1 be an orientable closed connected hypersurface and let c be its center of gravity. if either hs+1 > 0, for some integer number s ∈ {1, . . . , n − 1}, or the ricci curvature of mn is positive (hence, necessarily embedded), then λ lr,s 1 (∫ m s∑ i=r (i + 1)ãihi dm )2 ≤ ( s∑ j=r (j + 1)ajbj ∫ m hj dm )∫ m ( s∑ i=r (i + 1)ãihi+1 )2 dm ,(4.1) for all r ∈ {0, . . . , s − 1}, where λlr,s1 is the first positive eigenvalue of jacobi operator lr,s defined in (1.1), aj and ãi are nonnegative real numbers (with at least one nonzero) for all i, j ∈ {r, . . . , s} and bj = (j + 1) ( n j+1 ) . in particular, the equality in (4.1) holds if and only if x(m) is a round sphere of rn+1 centered at c. 78 h.f. de lima, a.f. de sousa, f.r. dos santos, m.a.l. velázquez proof. let c the center of gravity of m defined in (3.1). if we multiply both sides of (3.11) by ∫ m (∑s i=r(i + 1)ãihi+1 )2 dm, we obtain λ lr,s 1 ∫ m |x − c|2 dm ∫ m ( s∑ i=r (i + 1)ãihi+1 )2 dm ≤ s∑ j=r (j + 1)ajbj ∫ m hj dm ∫ m ( s∑ i=r (i + 1)ãihi+1 )2 dm . using cauchy-schwarz inequality, the left side can be developed as follows λ lr,s 1 ∫ m |x − c|2 dm ∫ m ( s∑ i=r (i + 1)ãihi+1 )2 dm ≥ λlr,s1 (∫ m |x − c| ∣∣∣∣∣ s∑ i=r (i + 1)ãihi+1 ∣∣∣∣∣dm )2 ≥ λlr,s1 ( s∑ i=r (i + 1)ãi ∫ m ⟨x − c, n⟩hi+1 dm )2 = λ lr,s 1 ( s∑ i=r (i + 1)ãi ∫ m hi dm )2 , where in the last equality, it was used the minkowski type integral formula (3.10). hence, λ lr,s 1 (∫ m s∑ i=r (i + 1)ãihi dm )2 ≤ s∑ j=r (j + 1)ajbj ∫ m hj dm ∫ m ( s∑ i=r (i + 1)ãihi+1 )2 dm . now if the equality occurs in (4.1), then the equality occurs also in (3.11), implying that m is a round sphere centered at c. proceeding, we also get the following result. theorem 2. let x : mn → rn+1 be an orientable closed connected hypersurface and let c be its center of gravity. assume that, either hs+1 > 0, sharp estimates for the first eigenvalue 79 for some integer number s ∈ {1, . . . , n − 1}, or the ricci curvature of mn is positive (hence, necessarily embedded). if hk+1 is constant for some k ∈ {r, . . . , s} then (4.2) λ lr,s 1 ≤ 1 vol(m) ( hk+1 ) 2 k+1 ( s∑ j=r (j + 1)ajbj ∫ m hj dm ) . where λ lr,s 1 is the first positive eigenvalue of jacobi operator lr,s defined in (1.1), aj are nonnegative real numbers (with at least one nonzero) for all j ∈ {r, . . . , s} and bj = (j + 1) ( n j+1 ) . in particular, the equality in (4.2) holds if and only if x(m) is a round sphere of rn+1 centered at c. proof. taking ãi = { 0 , for i ̸= k ∈ {r, . . . , s}, 1 k+1 , for i = k ∈ {r, . . . , s}, in theorem 1 and supposing hk+1 constant, for some k ∈ {r, . . . , s}, we obtain (4.3) λ lr,s 1 (∫ m hk dm )2 ≤ vol(m)h2k+1 ( s∑ j=r (j + 1)ajbj ∫ m hj dm ) . since hs+1 > 0, we have that h 1 k+1 k+1 ≤ h 1 k k (cf. [5, proposition 2.3]). hence, h k k+1 k+1 ≤ hk and consequently, from (4.3) we get inequality (4.2). moreover, if equality occurs in (4.2), then in (4.1) we also have an equality and hence x(m) is a round sphere of rn+1 centered at c. as a consequence of theorem 2 we have the following corollary 1. let x : mn → rn+1 be an orientable closed connected hypersurface and let c be its center of gravity. assume that, either hs+1 > 0, for some integer number s ∈ {1, . . . , n − 1}, or the ricci curvature of mn is positive (hence, necessarily embedded). if hk+1 is constant for some k ∈ {r, . . . , s} then (4.4) λ lr,s 1 ≤ 1 vol(m) inf m (hm) 2 m ( s∑ j=r (j + 1)ajbj ∫ m hj dm ) , 80 h.f. de lima, a.f. de sousa, f.r. dos santos, m.a.l. velázquez for any m ∈ {2, . . . , k+1}, where λlr,s1 is the first positive eigenvalue of jacobi operator lr,s defined in (1.1), aj are nonnegative real numbers (with at least one nonzero) for all j ∈ {r, . . . , s} and bj = (j + 1) ( n j+1 ) . in particular, the equality in (4.4) holds if and only if x(m) is a round sphere of rn+1 centered at c. proof. since h 1 k+1 k+1 ≤ h 1 m m , for all m ∈ {2, . . . , k + 1} (cf. [5, proposition 2.3]), then from (4.2) we have λ lr,s 1 ≤ 1 vol(m) inf m ( hk+1 ) 2 k+1 ( s∑ j=r (j + 1)ajbj ∫ m hj dm ) ≤ 1 vol(m) inf m ( hm ) 2 m ( s∑ j=r (j + 1)ajbj ∫ m hj dm ) , for any m ∈ {2, . . . , k + 1}. when equality occurs in (4.4), the same happens in (4.2) and in this case x(m) is a round sphere of rn+1 centered at c. we close this section with the following theorem 3. let x : mn → rn+1 be an orientable closed connected hypersurface and let c be its center of gravity. if either hs+1 > 0, for some integer number s ∈ {1, . . . , n − 1}, or the ricci curvature of mn is positive (hence, necessarily embedded), then (4.5) λ lr,s 1 (∫ m ⟨x − c, n⟩ dm )2 ≤ vol(m) s∑ j=r (j + 1)ajbj ∫ m hj dm , for all r ∈ {0, . . . , s − 1}, where λlr,s1 is the first positive eigenvalue of jacobi operator lr,s defined in (1.1), aj are nonnegative real numbers (with at least one nonzero) for all j ∈ {r, . . . , s} and bj = (j + 1) ( n j+1 ) . in particular, the equality occurs in (4.5) if and only if x(m) is a round sphere of rn+1 centered at c. moreover, if mn embedded in rn+1, then (4.6) λ lr,s 1 ≤ vol(m) (n + 1)2vol(ω)2 s∑ j=r (j + 1)ajbj ∫ m hj dm , with equality if and only if x(m) is a round sphere in rn+1 centered at c. here ω is the compact domain in rn+1 bounded by mn and vol(ω) denotes its (n + 1)-dimensional volume. sharp estimates for the first eigenvalue 81 proof. if we multiply both sides of (3.11) by ∫ m 12 dm, and use cauchyschwarz inequality, we obtain vol(m) s∑ j=r (j + 1)ajbj ∫ m hj dm ≥ λ lr,s 1 ∫ m |x − c|2 dm ∫ m 12 dm ≥ λlr,s1 (∫ m |x − c| dm )2 ≥ λlr,s1 (∫ m ⟨x − c, n⟩ dm )2 , showing that (4.5) holds. now, if the equality occurs in (4.5), then the equality also occurs in (3.11) and, hence, x(m) is a round sphere in rn+1 centered at c. moreover, in the case in that mn is embedded in rn+1, let ω be a compact domain in rn+1 bounded by mn so that m = ∂ω. according to the proof of [3, theorem 10], let us consider the vector field y (p) = p − c defined on ω, as div(y ) = (n + 1). so, it follows from divergence theorem that (n + 1)vol(ω) = ∫ m div(y ) dω = ∫ m ⟨x − c, n⟩ dm . therefore, from (4.5) we get λ lr,s 1 ≤ vol(m) (n + 1)2vol(ω)2 s∑ j=r (j + 1)ajbj ∫ m hj dm . 5. upper estimates for λ lr,s 1 in s n+1 in this last section, we will consider orientable closed connected hypersurface hypersurfaces x : mn → sn+1 immersed into the euclidean sphere sn+1 ↪→ rn+2. according to [3], we defined a center of gravity of mn as a critical point of the smooth function e : sn+1 → r given by e(p) = ∫ m ⟨x, p⟩ dm , p ∈ sn+1. in this way, a point c ∈ sn+1 is a center of gravity of mn if, and only if, dec(v) = ∫ m ⟨x, v⟩ dm = ⟨∫ m x dm, v ⟩ = 0 , 82 h.f. de lima, a.f. de sousa, f.r. dos santos, m.a.l. velázquez for every v ∈ tcsn+1 = c⊥ = { p ∈ rn+2 : ⟨p, c⟩ = 0 } . hence a center of gravity of mn is given by c = 1 | ∫ m x dm| ∫ m x dm ∈ sn+1, whenever ∫ m x dm ̸= 0 ∈ rn+2. for a fixed nonzero vector a ∈ rn+2, let us the smooth function ⟨x, a⟩ defined on mn. then, the gradient of the function ⟨x, a⟩ is given by ∇⟨x, a⟩ = a⊤ = a − ⟨n, a⟩n − ⟨x, a⟩x ∈ x(m) , where n is the orientation of x : mn → sn+1. moreover, ∇x∇⟨x, a⟩ = ⟨n, a⟩ax − ⟨x, a⟩x, for all x ∈ x(m) and,hence, from (2.1) lr,s(⟨x, a⟩) = s∑ j=r (j + 1)ajlj(⟨x, a⟩) = s∑ j=r (j + 1)aj tr ( pj ◦ hess(⟨x, a⟩) ) = s∑ j=r (j + 1)aj ( ⟨n, a⟩tr(a ◦ pj) − ⟨x, a⟩tr(pj) ) (5.1) = s∑ j=r (j + 1)ajbj ( ⟨n, a⟩hj+1 − ⟨x, a⟩hj ) , where bj = (j + 1) ( n j+1 ) = (n − j) ( n j ) . proceeding with the above notation, in what follows we are able to establish an extension of lemma 3 for the case that mn is a hypersurface immersed in sn+1. lemma 4. let x : mn → sn+1 be an orientable closed connected hypersurface, which lies in an open hemisphere of sn+1, and let c be its center of gravity. if hs+1 > 0, for some integer number s ∈ {1, . . . , n − 1}, then (5.2) λ lr,s 1 ∫ m ( 1 − ⟨x, c⟩2 ) dm ≤ s∑ j=r (j + 1)ajbj ∫ m hj dm , sharp estimates for the first eigenvalue 83 for all r ∈ {0, . . . , s − 1}, where λlr,s1 is the first positive eigenvalue of jacobi operator lr,s defined in (1.1), aj are nonnegative real numbers (with at least one nonzero) for all j ∈ {r, . . . , s} and bj = (j + 1) ( n j+1 ) . in particular, the equality occurs in (5.2) if and only if x(m) is an geodesic sphere in sn+1 centered at c. proof. since hs+1 > 0, lemma 1 guarantees that lr,s is elliptic and, hence, it holds the characterization of its first positive eigenvalue given in (3.12). we consider the canonical basis {e1 . . . , en+1} ⊂ rn+2 of tcsn+1 = c⊥ ={ v ∈ rn+2 : ⟨v, c⟩ = 0 } and for every 1 ≤ k ≤ n + 1, let us fk = ⟨x, ek⟩. then, as before, ∫ m fk dm = 0, for every 1 ≤ k ≤ n + 1, and from (5.1) (5.3) lr,s(fk) = s∑ j=r (j + 1)ajbj ( ⟨n, ek⟩hj+1 − ⟨x, ek⟩hj ) . hence, from (3.12) we have λ lr,s 1 ∫ m f2k dm ≤ − ∫ m fklr,s(fk) dm = s∑ j=r (j + 1)ajbj ∫ m ( f2k hj − fk⟨n, ek⟩hj+1 ) dm .(5.4) on the one hand, x = n+1∑ k=1 fkek + ⟨x, c⟩c and n = n+1∑ k=1 ⟨n, ek⟩ek + ⟨c, n⟩c , so that (5.5) n+1∑ k=1 fk⟨n, ek⟩ = −⟨c, n⟩⟨x, c⟩ and 1 − ⟨x, c⟩2 = n+1∑ k=1 f2k . summing on k of 1 until n + 1 in (5.4) and using relations (5.5), we obtain λ lr,s 1 ∫ m (1 − ⟨x, c⟩2) dm ≤ s∑ j=r (j + 1)ajbj (∫ m (1 − ⟨x, c⟩2)hj dm + ∫ m ⟨c, n⟩⟨x, c⟩hj+1 dm ) . (5.6) 84 h.f. de lima, a.f. de sousa, f.r. dos santos, m.a.l. velázquez now, taking a = c in (5.1) (5.7) lr,s(⟨x, c⟩) = s∑ j=r (j + 1)ajbj ( ⟨c, n⟩hj+1 − ⟨x, c⟩hj ) , multiply both sides of (5.7) by ⟨x, c⟩, we obtain (5.8) ⟨x, c⟩lr,s(⟨x, c⟩) = s∑ j=r (j + 1)ajbj ( ⟨x, c⟩⟨c, n⟩hj+1 − ⟨x, c⟩2hj ) . replacing (5.8) in (5.6), we get λ lr,s 1 ∫ m (1−⟨x, c⟩2) dm ≤ s∑ j=r (j + 1)ajbj (∫ m hj dm + ∫ m ⟨x, c⟩lr,s(⟨x, c⟩)hj+1 dm ) .(5.9) with a straightforward computation, we see that lr,s ( ⟨x, c⟩2 ) = s∑ j=r (j + 1)aj ⟨ ∇⟨x, c⟩, pj (∇⟨x, c⟩) ⟩ + ⟨x, c⟩lr,s(⟨x, c⟩) . integrating over mn and using divergence theorem we obtain (5.10) ∫ m ⟨x, c⟩lr,s (⟨x, c⟩) dm = − s∑ j=r (j + 1)aj ∫ m ⟨ c⊤, pj(c ⊤) ⟩ dm , where c⊤ = ∇⟨x, c⟩. from (5.9) and (5.10), we get λ lr,s 1 ∫ m ( 1 − ⟨x, c⟩2 ) dm ≤ s∑ j=r (j + 1)ajbj ∫ m hj dm − s∑ j=r (j + 1)aj ∫ m ⟨ c⊤, pj(c ⊤) ⟩ dm .(5.11) since each operator lj is elliptic, for r ≤ j ≤ s, from lemma 1 we have that the operator p̃ = ∑s j=r(j +1)ajpj is positive. consequently, from (5.11) we get λ lr,s 1 ∫ m ( 1 − ⟨x, c⟩2 ) dm ≤ s∑ j=r (j + 1)ajbj ∫ m hj dm , with the equality occurs if and only if c⊤ = ∇⟨x, c⟩ = 0, that is, if and only if x(m) is a geodesic sphere sn+1 centered at the point c. sharp estimates for the first eigenvalue 85 before to present our last result, we observe that integrating (5.1) over mn and using divergence theorem we obtain the following minkowski type formula for hypersurfaces immersed in sn+1 (5.12) s∑ j=r (j + 1)ajbj ∫ m ( ⟨n, a⟩hj+1 dm − ⟨x, a⟩hj ) dm = 0 , where a ∈ rn+2 is arbitrary. as an application of lemma 4, we derive the following reilly type inequality for the first positive eigenvalue of the jacobi operator lr,s of a closed hypersurface in sphere, which extend [3, theorem 16]. theorem 4. let x : mn → sn+1 orientable closed connected hypersurface, which lies in an open hemisphere of sn+1, and let c be its center of gravity. if hs+1 > 0, for some integer number s ∈ {1, . . . , n − 1}, then we have following inequalities λ lr,s 1 ( s∑ i=r (i + 1)ãi ∫ m hi⟨x, c⟩ dm )2 ≤ s∑ j=r (j + 1)ajbj ∫ m hj dm ∫ m ( s∑ i=r (i + 1)ãihi+1 )2 dm(5.13) and (5.14) λ lr,s 1 (∫ m ⟨c, n⟩ dm )2 ≤ vol(m) s∑ j=r (j + 1)ajbj ∫ m hj dm , for all r ∈ {0, . . . , s − 1}, where λlr,s1 is the first positive eigenvalue of jacobi operator lr,s defined in (1.1), aj and ãi are nonnegative real numbers (with at least one nonzero) for all i, j ∈ {r, . . . , s}, bj = (j + 1) ( n j+1 ) and vol(m) denotes the n-dimensional volume of mn. in particular, if m is embedded in sn+1 then (5.13) results in (5.15) λ lr,s 1 (∫ ω ⟨c, p⟩ dω(p) )2 ≤ vol(m) (n + 1)2 s∑ j=r (j + 1)ajbj ∫ m hj dm , where ω is any one of the two compact domains of sn+1 bounded by mn. moreover, the equality occurs in one of these three inequalities if and only if x(m) is a geodesic sphere in sn+1 centered at c. 86 h.f. de lima, a.f. de sousa, f.r. dos santos, m.a.l. velázquez proof. multiply both sides of (5.2) by ∫ m ( ∑s i=r(i + 1)ãihi+1) 2 dm, we have λ lr,s 1 ∫ m (1−⟨x, c⟩2) dm ∫ m ( s∑ i=r (i + 1)ãihi+1 )2 dm ≤ s∑ j=r (j + 1)ajbj ∫ m hj dm ∫ m ( s∑ i=r (i + 1)ãihi+1 )2 dm . using cauchy-schwarz inequality, the side left can be developed as in following way λ lr,s 1 ∫ m (1 − ⟨x, c⟩2) dm ∫ m ( s∑ i=r (i + 1)ãihi+1 )2 dm ≥ λlr,s1 (∫ m √ 1 − ⟨x, c⟩2 ∣∣∣∣∣ s∑ i=r (i + 1)ãihi+1 ∣∣∣∣∣dm )2 .(5.16) on the other hand, c = c⊤ + ⟨c, n⟩n + ⟨x, c⟩x, so that 1 − ⟨x, c⟩2 = ∣∣c⊤∣∣2 + ⟨c, n⟩2 ≥ ⟨c, n⟩2, which implies (5.17) √ 1 − ⟨x, c⟩2 ≥ |⟨c, n⟩| . occurring equality if and only if ∇⟨x, c⟩ = c⊤ = 0, that is, if and only if x(m) is a geodesic sphere in sn+1 centered at c. replacing (5.17) in (5.16) and using the minkowski type formula (5.12) with a = c, we obtain λ lr,s 1 (∫ m √ 1 − ⟨x, c⟩2 ∣∣∣∣∣ s∑ i=r (i + 1)ãihi+1 ∣∣∣∣∣dm )2 ≥ λlr,s1 (∫ m |⟨c, n⟩| ∣∣∣∣∣ s∑ i=r (i + 1)ãihi+1 ∣∣∣∣∣dm )2 ≥ λlr,s1 ( s∑ i=r (i + 1)ãi ∫ m ⟨c, n⟩hi+1 dm )2 = λ lr,s 1 ( s∑ i=r (i + 1)ãi ∫ m ⟨x, c⟩hi dm )2 , sharp estimates for the first eigenvalue 87 which proves (5.13). for proof the of (5.14), we multiply both sides of (5.2) by vol(m) =∫ m 12 dm, using cauchy-schwarz inequality in (5.17), we have vol(m) s∑ j=r (j + 1)ajbj ∫ m hj dm ≥ λ lr,s 1 ∫ m (1 − ⟨x, c⟩2) dm ∫ m 12 dm ≥ λlr,s1 (∫ m √ 1 − ⟨x, c⟩2 dm )2 ≥ λlr,s1 (∫ m ⟨c, n⟩ dm )2 , which shows (5.14). moreover, if occurs the equality either in (5.13) or in (5.14), then occurs in (5.2), and x(m) is a geodesic sphere in sn+1 centered at c. now, if mn is embedded in sn+1, following the same steps of [3, theorem 16], let us consider the vector field y on sn+1 defined by y (p) = c − ⟨c, p⟩p, p ∈ sn+1. observe that y is a conformal vector field on sn+1 with singularities in c and −c, and with spherical divergence given by divy = −(n + 1)⟨c, p⟩ . moreover, if ω denotes one of the two compact domains in sn+1 bounded by mn so that ∂ω = m, then (5.18) (n + 1)2 (∫ ω ⟨c, p⟩ dω(p) )2 = (∫ m ⟨c, n⟩ dm )2 . therefore, replacing (5.18) in (5.13), we obtain (5.15). acknowledgements the first author is partially supported by cnpq, brazil, grant 303977/2015-9. the fourth author is partially supported by cnpq, brazil, grant 308757/2015-7. references [1] h. alencar, m. de carmo, f. marques, upper bounds for the first eigenvalue of the operator lr and some applications, illinois j. math. 45 (2001), 851 – 863. 88 h.f. de lima, a.f. de sousa, f.r. dos santos, m.a.l. velázquez [2] h. alencar, m. de carmo, h. rosemberg, on the first eigenvalue of the linearized operator of the r-th mean curvature of a hypersurface, ann. global anal. geom. 11 (1993), 387 – 395. 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[17] m.a. velásquez, a.f. de sousa, h.f. de lima, on the stability of hypersurfaces in space forms, j. math. anal. appl. 406 (2013), 134–146. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 2 (2019), 285 – 301 doi:10.17398/2605-5686.34.2.285 available online july 2, 2019 fractional ostrowski type inequalities for functions whose derivatives are s-preinvex b. meftah 1,2, m. merad 2, a. souahi 3 1 laboratoire des télécommunications, faculté des sciences et de la technologie university of 8 may 1945 guelma, p.o. box 401, 24000 guelma, algeria 2 département des mathématiques, faculté des mathématiques, de l’informatique et des sciences de la matière, université 8 mai 1945 guelma, algeria 3 laboratory of advanced materials, university badji mokhtar annaba, algeria badrimeftah@yahoo.fr , mrad.meriem@gmail.com , arsouahi@yahoo.fr received january 16, 2019 presented by manuel maestre accepted may 25, 2019 abstract: in this paper, we establish a new integral identity, and then we derive some new fractional ostrowski type inequalities for functions whose derivatives are s-preinvex. key words: integral inequality, -preinvex functions, hölder inequality, power mean inequality. ams subject class. (2010): 26d15, 26d20, 26a51. 1. introduction in 1938, a.m. ostrowski proved an interesting integral inequality, estimating the absolute value of the derivative of a differentiable function by its integral mean as follows theorem 1. ([21]) let i ⊆ r be an interval. let f : i → r, be a differentiable mapping in the interior i◦of i, and a,b ∈ i◦ with a < b. if |f ′(x)| ≤ m for all x ∈ [a,b], then∣∣∣∣f(x) − 1b−a ∫ b a f(t)dt ∣∣∣∣ ≤ m (b−a) [ 1 4 + (x−a+b2 ) 2 (b−a)2 ] . (1.1) the above inequality has attracted many researchers, various generalizations, refinements, extensions and variants have appeared in the literature see for instance [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 24, 25], and references therein. issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.2.285 mailto:badrimeftah@yahoo.fr mailto:mrad.meriem@gmail.com mailto:arsouahi@yahoo.fr https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 286 b. meftah, m. merad, a. souahi set [25], gave the following fractional ostrwski inequality for differentiable s-convex functions∣∣∣((x−a)α+(b−x)αb−a )f (x) − γ(α+1)b−a ((iαx−f) (a) + (iαx+f) (b))∣∣∣ ≤ m b−a ( 1 + γ(α+1)γ(s+1) γ(α+s+1) )( (x−a)α+1+(b−x)α+1 α+s+1 ) and ∣∣∣((x−a)α+(b−x)αb−a )f (x) − γ(α+1)b−a ((iαx−f) (a) + (iαx+f) (b))∣∣∣ ≤ m (1+pα) 1 p ( 2 s+1 )1 q ( (x−a)α+(b−x)α b−a ) . kirmaci et al. [4], presented some results connected with inequality (1.1)∣∣∣∣ 1b−a ∫ b a f(x)dx−f ( a+b 2 )∣∣∣∣ ≤ b−a8 (∣∣f ′ (a)∣∣ + ∣∣f ′ (b)∣∣). zhu et al. [27], established the following fractional midpoint inequality∣∣∣ γ(α+1)2(b−a)α[ (jαa+f) (b) + (jαb−f) (a) ]−f (a+b2 )∣∣∣ ≤ b−a 4(1+α) (∣∣f ′ (a)∣∣ + ∣∣f ′ (b)∣∣)(α + 3 − 1 2α−1 ) . motivated by the above results, in this paper, we establish a new integral identity, and then we derive some new fractional ostrowski type inequalities for functions whose derivatives are s-preinvex. 2. preliminaries in this sections we recall some definitions and lemmas definition 1. ([23]) a function f : i → r is said to be convex, if f (tx + (1 − t) y) ≤ tf (x) + (1 − t) f(y) holds for all x,y ∈ i and all t ∈ [0, 1]. definition 2. ([1]) a nonnegative function f : i ⊂ [0,∞) → r is said to be s-convex in the second sense for some fixed s ∈ (0, 1], if f(tx + (1 − t)y) ≤ tsf(x) + (1 − t)sf(y) holds for all x,y ∈ i and t ∈ [0, 1]. fractional ostrowski type inequalities 287 definition 3. ([18]) a set k ⊆ rn is said to be an invex with respect to the bifunction η : k ×k → rn, if for all x,y ∈ k, we have x + tη (y,x) ∈ k. in what follows we assume that k ⊆ r be an invex set with respect to the bifunction η : k ×k → r. definition 4. ([26]) a function f : k → r is said to be preinvex with respect to η, if f (x + tη (y,x)) ≤ (1 − t) f (x) + tf(y) holds for all x,y ∈ k and all t ∈ [0, 1]. definition 5. ([5]) a nonnegative function f : k ⊂ [0,∞) → r is said to be s-preinvex in the second sense with respect to η for some fixed s ∈ (0, 1], if f (x + tη (y,x)) ≤ (1 − t)sf(x) + tsf(y) holds for all x,y ∈ k and t ∈ [0, 1]. definition 6. ([2, 3, 17]) let f ∈ l1[a,b]. the riemann-liouville fractional integrals iα a+ f and iα b− f of order α > 0 with a ≥ 0 are defined by iαa+f(x) = 1 γ (α) ∫ x a (x− t)α−1 f(t)dt, x > a, iαb−f(x) = 1 γ (α) ∫ b x (t−x)α−1 f(t)dt, b > x, respectively, where γ(α) = ∫∞ 0 e−ttα−1dt, is the gamma function and i0 a+ f(x) = i0 b− f(x) = f(x). we also recall that the beta function for any complex numbers and nonpositive integers ρ,τ such that re(ρ) > 0 and re(τ) > 0 is defined by b (ρ,τ) = ∫ 1 0 θρ−1 (1 −θ)τ−1 dθ = γ (ρ) γ (τ) γ (ρ + τ) . the incomplete beta function is given by bt (ρ,τ) = ∫ t 0 θρ−1 (1 −θ)τ−1 dθ , 0 < t < 1 . lemma 1. ([22]) for any 0 ≤ a < b in r and fixed p ≥ 1, we have (b−a)p ≤ bp −ap. 288 b. meftah, m. merad, a. souahi 3. main results in what follows, in order to simplify and lighten the writing, we note in all the proofs η(b,a) by c. lemma 2. let f : [a,a + η(b,a)] → r be a differentiable mapping on (a,a + η(b,a)) with η(b,a) > 0, and assume that f ′ ∈ l ( [a,a + η(b,a)] ) . then the following equality holds f (x) − γ(α+1) 2ηα(b,a) (( iαa+f ) (a + η (b,a)) + ( iα (a+η(b,a))− f ) (a) ) = η(b,a) 2 (∫ 1 0 kf ′ (a + tη (b,a)) dt + ∫ 1 0 (tα − (1 − t)α) f ′ (a + tη (b,a)) dt ) , (3.1) where k =   1 if 0 ≤ t < x−a η(b,a) , −1 if x−a η(b,a) ≤ t < 1 . (3.2) proof. let c = η (b,a). and let i = ∫ 1 0 kf ′(a + tc)dt + ∫ 1 0 (tα − (1 − t)α)f ′(a + tc)dt = i1 + i2 , (3.3) where i1 = ∫ 1 0 kf ′(a + tc)dt, (3.4) i2 = ∫ 1 0 (tα − (1 − t)α)f ′(a + tc)dt, (3.5) and k is defined by (3.2). clearly, i1 = ∫ 1 0 kf ′(a + tc)dt = ∫ x−a c 0 f ′(a + tc)dt− ∫ 1 x−a c f ′(a + tc)dt = 2 c f(x) − 1 c [f(a) + f(a + c)]. (3.6) fractional ostrowski type inequalities 289 now, by integration by parts, i2 gives i2 = 1 c f (a + c) + 1 c f (a) − α c (∫ 1 0 tα−1f (a + tc) dt + ∫ 1 0 (1 − t)α−1 f (a + tc) dt ) = 1 c f (a + c) + 1 c f (a) (3.7) − α cα+1 (∫ a+c a (u−a)α−1 f (u) du + ∫ a+c a (c + a−u)α−1 f (u) du ) = 1 c f (a + c) + 1 c f (a) − γ(α+1) cα+1 ( (iαa+f) (a + c) + ( iα (a+c)− f ) (a) ) . substituting (3.6) and (3.7) in (3.3). multiplying the resulting equality by c 2 , and then replacing c by η(a,b), we get the desired result. theorem 2. let f : [a,a + η(b,a)] → r be a positive function on [a,b] with η(b,a) > 0 and f ∈ l[a,b]. if |f ′| is s-preinvex function with s ∈ (0, 1], then the following fractional inequality holds∣∣∣∣f (x) − γ(α+1)2ηα(b,a) ((iαa+f) (a + η (b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ ≤ η(b,a) 2 (∣∣f ′(a)∣∣ + ∣∣f ′ (b)∣∣) (3.8) × ( 1 s+1 + 1−( 12 ) α+s α+s+1 + b1 2 (s + 1,α + 1) −b1 2 (α + 1,s + 1) ) . where b1 2 (·, ·) is the incomplete beta function. proof. let c = η(a,b). from lemma 2, and properties of modulus, we have∣∣∣∣f (x) − γ(α+1)2cα ((iαa+f) (a + c) + (iα(a+c)−f) (a)) ∣∣∣∣ ≤ c 2 [∫ x−a c 0 ∣∣f ′ (a + tc)∣∣dt + ∫ 1 x−a c ∣∣f ′ (a + tc)∣∣dt + ∫ 1 2 0 ((1 − t)α − tα) ∣∣f ′ (a + tc)∣∣dt +∫ 1 1 2 (tα − (1 − t)α) ∣∣f ′ (a + tc)∣∣dt ] . 290 b. meftah, m. merad, a. souahi using the s-preinvexity of |f ′|, the above inequality gives∣∣∣∣f(x) − γ(α+1)2cα ((iαa+f) (a + c) + (iα(a+c)−f) (a)) ∣∣∣∣ ≤ c 2 (∫ x−a c 0 ( (1 − t)s|f ′(a)| + ts|f ′(b)| ) dt + ∫ 1 x−a c ( (1 − t)s|f ′(a)| + ts|f ′(b)| ) dt + ∫ 1 2 0 ( (1 − t)α − tα )( (1 − t)s|f ′(a)| + ts|f ′(b)| ) dt + ∫ 1 1 2 ( (1 − t)α − tα )( (1 − t)s|f ′(a)| + ts|f ′(b)| ) dt ) = c 2 [(( 1−(1−x−ac ) s+1 ) |f′(a)|+( x−ac ) s+1 |f′(b)| s+1 ) + ( (1−x−ac ) s+1 |f′(a)|+ ( 1−( x−ac ) s+1 ) |f′(b)| s+1 ) + ∣∣f ′(a)∣∣ (∫ 1 2 0 ((1 − t)α+s − tα(1 − t)s)dt + ∫ 1 1 2 ( tα(1 − t)s − (1 − t)α+s ) dt ) + ∣∣f ′ (b)∣∣ (∫ 1 2 0 (ts (1 − t)α − tα+s)dt + ∫ 1 1 2 (tα+s − ts(1 − t)α)dt )] = c 2 (∣∣f ′(a)∣∣ + ∣∣f ′ (b)∣∣) × ( 1 s+1 + ∫ 1 2 0 ((1 − t)α+s − tα(1 − t)s) dt + ∫ 1 2 0 (ts (1 − t)α − tα+s)dt ) = c 2 (∣∣f ′(a)∣∣ + ∣∣f ′ (b)∣∣) × ( 1 s+1 + 1−( 12 ) α+s α+s+1 + b1 2 (s + 1,α + 1) −b1 2 (α + 1,s + 1) ) . replacing c by η (b,a) in the above inequality, we get the desired result. the proof is completed. fractional ostrowski type inequalities 291 corollary 1. in theorem 2 if we choose x = 2a+η(b,a) 2 , we obtain the following fractional midpoint inequality∣∣∣∣f (2a+η(b,a)2 )− γ(α+1)2ηα(b,a) ((iαa+f) (a + η (b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ ≤ η(b,a) 2 ( ∣∣f ′(a)∣∣ + ∣∣f ′ (b)∣∣) × ( 1 s+1 + 1−( 12 ) α+s α+s+1 + b1 2 (s + 1,α + 1) −b1 2 (α + 1,s + 1) ) . moreover if we take η(b,a) = b−a, we obtain∣∣∣∣f (a+b2 )− γ(α+1)2(b−a)α ((iαa+f) (b) + (iαb−f) (a)) ∣∣∣∣ ≤ b−a 2 ( ∣∣f ′(a)∣∣ + ∣∣f ′ (b)∣∣) × ( 1 s+1 + 1−( 12 ) α+s α+s+1 + b1 2 (s + 1,α + 1) −b1 2 (α + 1,s + 1) ) . corollary 2. in theorem 2 if we put s = 1, we obtain∣∣∣∣f (x) − γ(α+1)2ηα(b,a) ((iαa+f) (a + η (b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ ≤ η(b,a) 4(α+1) ( ∣∣f ′(a)∣∣ + ∣∣f ′ (b)∣∣) (α + 3 −(1 2 )α−1) . moreover if we choose x = 2a+η(b,a) 2 , we obtain the following fractional midpoint inequality∣∣∣∣f (2a+η(b,a)2 )− γ(α+1)2ηα(b,a) ((iαa+f) (a + η (b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ ≤ η(b,a) 4(α+1) ( ∣∣f ′(a)∣∣ + ∣∣f ′ (b)∣∣) 1 2(α+1) ( α + 3 − ( 1 2 )α−1) . remark 1. theorem 2 will be reduced to theorem 2.3 from [27], if we choose η(b,a) = b−a, x = a+b 2 and s = 1. theorem 3. let f : [a,b] ⊂ [0,∞) → r be a positive function on [a,b] with a < b and f ∈ l[a,b]. if |f ′|q is s-preinvex function, where s ∈ (0, 1] and 292 b. meftah, m. merad, a. souahi q ≥ 1, then the following fractional inequality holds∣∣∣∣f(x) − γ(α+1)2ηα(b,a) ((iαa+f) (a + η (b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ ≤ η(b,a) 2  ( ( 1− ( 1− x−a η(b,a) )s+1) |f′(a)|q+ ( x−a η(b,a) )s+1 |f′(b)|qd s+1 )1 q + (( 1− x−a η(b,a) )s+1 |f′(a)|q+ ( 1− ( x−a η(b,a) )s+1) |f′(b)|qd s+1 )1 q + ( 2α−1 (α+1)2α )1−1 q (( µ1α,s ∣∣f ′(a)∣∣q + µ2α,s ∣∣f ′(b)∣∣q)1q + ( µ2α,s ∣∣f ′(a)∣∣q + µ1α,s ∣∣f ′(b)∣∣q)1q )   , (3.9) where µ1α,s = 1−( 12 ) α+s+1 α+s+1 −b1 2 (α + 1,s + 1) (3.10) and µ2α,s = b1 2 (s + 1,α + 1) − ( 1 2 ) α+s+1 α+s+1 . (3.11) proof. let c = η (b,a). from lemma 2, properties of modulus, and power mean inequality, we have∣∣∣∣f(x) − γ(α+1)2cα ((iαa+f) (a + c) + (iα(a+c)−f) (a)) ∣∣∣∣ ≤ c 2  (∫ x−ac 0 ∣∣f ′ (a + tc)∣∣q dt )1 q + (∫ 1 x−a c ∣∣f ′ (a + tc)∣∣q dt )1 q + (∫ 1 2 0 ((1 − t)α − tα) dt )1−1 q (∫ 1 2 0 ((1 − t)α − tα) ∣∣f ′(a + tc)∣∣q dt )1 q + (∫ 1 1 2 (tα − (1 − t)α) dt )1−1 q (∫ 1 1 2 (tα − (1 − t)α) ∣∣f ′ (a + tc)∣∣q dt )1 q   . fractional ostrowski type inequalities 293 since |f ′|q is s-preinvex function, we deduce∣∣∣∣f(x) − γ(α+1)2cα ((iαa+f) (a + c) + (iα(a+c)−f) (a)) ∣∣∣∣ ≤ c 2  (∫ x−ac 0 (1 − t)s ∣∣f ′(a)∣∣q + ts ∣∣f ′ (b)∣∣q dt )1 q + (∫ 1 x−a c (1 − t)s ∣∣f ′(a)∣∣q + ts ∣∣f ′ (b)∣∣q dt )1 q + ( 1−( 12 ) α α+1 )1−1 q   (∫ 1 2 0 ((1 − t)α − tα) ((1 − t)s ∣∣f ′(a)∣∣q + ts ∣∣f ′ (b)∣∣q)dt )1 q + (∫ 1 2 0 ((1 − t)α − tα) (ts ∣∣f ′(a)∣∣q + (1 − t)s ∣∣f ′(b)∣∣q)dt )1 q     = c 2  ((1−(1−x−ac )s+1)|f′(a)|q+( x−ac )s+1|f′(b)|q s+1 )1 q + ( (1−x−ac ) s+1 |f′(a)|q+ ( 1−( x−ac ) s+1 ) |f′(b)|q s+1 )1 q + ( 1−( 12 ) α α+1 )1−1 q ({( 1−( 12 ) α+s+1 α+s+1 −b1 2 (α + 1,s + 1) )∣∣f ′(a)∣∣q + ( b1 2 (s + 1,α + 1) − ( 1 2 ) α+s+1 α+s+1 )∣∣f ′ (b)∣∣q}1q + {( b1 2 (s + 1,α + 1) − ( 1 2 ) α+s+1 α+s+1 )∣∣f ′(a)∣∣q + ( 1−( 12 ) α+s+1 α+s+1 −b1 2 (α + 1,s + 1) )∣∣f ′ (b)∣∣q}1q ) 294 b. meftah, m. merad, a. souahi = c 2 [(( 1−(1−x−ac ) s+1 ) |f′(a)|q+( x−ac ) s+1 |f′(b)|q s+1 )1 q + ( −(1−x−ac ) s+1 |f′(a)|q+ ( 1−( x−ac ) s+1 ) |f′(b)|q s+1 )1 q + ( 2α−1 (α+1)2α )1−1 q {( µ1α,s ∣∣f ′(a)∣∣q + µ2α,s ∣∣f ′ (b)∣∣q )1q + ( µ2α,s ∣∣f ′(a)∣∣q + µ1α,s ∣∣f ′ (b)∣∣q )1q} ] , where µ1α,s and µ 2 α,s are defined as in (3.10) and (3.11) respectively. by replacing c by η (b,a) in the above inequality, we get the desired result. the prove is completed. corollary 3. in theorem 3 if we choose x = 2a+η(b,a) 2 , we obtain the following fractional midpoint inequality ∣∣∣∣f (2a+η(b,a)2 )− γ(α+1)2ηα(b,a) ((iαa+f) (a + η(b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ ≤ η(b,a) 2  ((1−( 12 )s+1)|f′(a)|q+( 12 )s+1|f′(b)|qd s+1 )1 q + ( (( 12 )) s+1 |f′(a)|q+ ( 1−( 12 ) s+1 ) |f′(b)|qd s+1 )1 q + ( 2α−1 (α+1)2α )1−1 q {( µ1α,s ∣∣f ′(a)∣∣q + µ2α,s ∣∣f ′ (b)∣∣q )1q + ( µ2α,s ∣∣f ′(a)∣∣q + µ1α,s ∣∣f ′ (b)∣∣q )1q}   . moreover if we take η (b,a) = b−a, we obtain fractional ostrowski type inequalities 295 ∣∣∣∣f (a+b2 )− γ(α+1)2(b−a)α ((iαa+f) (b) + (iαb−f) (a)) ∣∣∣∣ ≤ b−a 2  ((1−( 12 )s+1)|f′(a)|q+( 12 )s+1|f′(b)|qd s+1 )1 q + ( (( 12 )) s+1 |f′(a)|q+ ( 1−( 12 ) s+1 ) |f′(b)|qd s+1 )1 q + ( 2α−1 (α+1)2α )1−1 q {( µ1α,s ∣∣f ′(a)∣∣q + µ2α,s ∣∣f ′ (b)∣∣q )1q + ( µ2α,s ∣∣f ′(a)∣∣q + µ1α,s ∣∣f ′ (b)∣∣q )1q}   . corollary 4. in theorem 3 if we put s = 1, we obtain∣∣∣∣f (x) − γ(α+1)2ηα(b,a) ((iαa+f) (a + η (b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ ≤ η(b,a) 2  ( ( 1− ( 1− x−a η(b,a) )2) |f′(a)|q+ ( x−a η(b,a) )2 |f′(b)|qd 2 )1 q + (( 1− x−a η(b,a) )2 |f′(a)|q+ ( 1− ( x−a η(b,a) )2) |f′(b)|qd 2 )1 q + ( 2α−1 (α+1)2α )1−1 q {( µ1α,s ∣∣f ′(a)∣∣q + µ2α,s ∣∣f ′ (b)∣∣q )1q + ( µ2α,s ∣∣f ′(a)∣∣q + µ1α,s ∣∣f ′ (b)∣∣q )1q}   , where µ1α,1 = 1 α+2 − 1 α+1 ( 1 2 )α+1 (3.12) and µ2α,1 = 1 (α+1)(α+2) − 1 α+1 ( 1 2 )α+1 . (3.13) 296 b. meftah, m. merad, a. souahi moreover if we choose x = 2a+η(b,a) 2 , we obtain the following fractional midpoint inequality∣∣∣∣f (2a+η(b,a)2 )− γ(α+1)2ηα(b,a) ((iαa+f) (a + η (b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ ≤ η(b,a) 2 [( 3|f′(a)|q+|f′(b)|qd 8 )1 q + ( |f′(a)|q+3|f′(b)|qd 8 )1 q + ( 2α−1 (α+1)2α )1−1 q {( µ1α,1 ∣∣f ′(a)∣∣q + µ2α,1 ∣∣f ′ (b)∣∣q )1q + ( µ2α,1 ∣∣f ′(a)∣∣q + µ1α,1 ∣∣f ′ (b)∣∣q )1q} ] . additionally if we take η (b,a) = b−a, we obtain∣∣∣∣f (a+b2 )− γ(α+1)2(b−a)α ((iαa+f) (b) + (iαb−f) (a)) ∣∣∣∣ ≤ b−a 2 [( 3|f′(a)|q+|f′(b)|qd 8 )1 q + ( |f′(a)|q+3|f′(b)|qd 8 )1 q + ( 2α−1 (α+1)2α )1−1 q {( µ1α,1 ∣∣f ′(a)∣∣q + µ2α,1 ∣∣f ′ (b)∣∣q )1q + ( µ2α,1 ∣∣f ′(a)∣∣q + µ1α,1 ∣∣f ′ (b)∣∣q )1q} ] , where µ1α,1 and µ 2 α,1 are defined as in (3.12) and (3.13) respectively. theorem 4. let f : [a,b] ⊂ [0,∞) → r be a positive function on [a,b] with a < b and f ∈ l[a,b]. if |f ′|q is s-preinvex function, where s ∈ (0, 1], and q > 1 with 1 p + 1 q = 1, then the following fractional inequality holds∣∣∣∣f (x) − γ(α+1)2ηα(b,a) ((iαa+f) (a + η (b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ ≤ η(b,a) 2(s+1) 1 q [(( 1 − ( 1 − x−a η(b,a) )s+1)∣∣f ′(a)∣∣q + ( x−a η(b,a) )s+1 ∣∣f ′(b)∣∣q)1q + (( 1 − x−a η(b,a) )s+1 ∣∣f ′(a)∣∣q + (1 −( x−a η(b,a) )s+1)∣∣f ′(b)∣∣q)1q fractional ostrowski type inequalities 297 + ( 1−( 12 ) αp αp+1 )1 p × (( (2s+1−1)|f′(a)|q+|f′(b)|q 2s+1 )1 q + ( |f′(a)|q+(2s+1−1)|f′(b)|q 2s+1 )1 q )] . proof. let c = η (b,a). from lemma 2, properties of modulus, hölder’s inequality, and lemma 1, we have∣∣∣∣f(x) − γ(α+1)2cα ((iαa+f) (a + c) + (iα(a+c)−f) (a)) ∣∣∣∣ ≤ c 2  (∫ x−ac 0 ∣∣f ′ (a + tc)∣∣q dt )1 q + (∫ 1 x−a c ∣∣f ′(a + tc)∣∣q dt )1 q + (∫ 1 2 0 ((1 − t)α − tα)p dt )1 p (∫ 1 2 0 ∣∣f ′ (a + tc)∣∣q dt )1 q + (∫ 1 1 2 (tα − (1 − t)α)p dt )1 p (∫ 1 1 2 |f ′(a + tc)|qdt )1 q   ≤ c 2  (∫ x−ac 0 ∣∣f ′ (a + tc)∣∣q dt )1 q + (∫ 1 x−a c |f ′(a + tc)|qdt )1 q + (∫ 1 2 0 ((1 − t)αp − tαp) dt )1 p (∫ 1 2 0 |f ′(a + tc)|qdt )1 q + (∫ 1 1 2 (tαp − (1 − t)αp) dt )1 p (∫ 1 1 2 |f ′(a + tc)|qdt )1 q   = c 2  (∫ x−ac 0 |f ′(a + tc)|qdt )1 q + (∫ 1 x−a c |f ′(a + tc)|qdt )1 q + ( 1−( 12 ) αp αp+1 )1 p ×  (∫ 12 0 |f ′(a + tc)|qdt )1 q + (∫ 1 1 2 |f ′(a + tc)|qdt )1 q     . 298 b. meftah, m. merad, a. souahi since |f ′|q is s-preinvex function, we have ∣∣∣∣f(x) − γ(α+1)2cα ((iαa+f) (a + c) + (iα(a+ct)−f) (a)) ∣∣∣∣ ≤ c 2  (∫ x−ac 0 ((1 − t)s ∣∣f ′(a)∣∣q + ts|f ′(b)|q)dt )1 q + (∫ 1 x−a c ((1 − t)s|f ′(a)|q + ts|f ′(b)|q)dt )1 q + ( 1−( 12 ) αp αp+1 )1 p   (∫ 1 2 0 ((1 − t)s ∣∣f ′(a)∣∣q + ts ∣∣f ′(b)∣∣q)dt )1 q + (∫ 1 1 2 ((1 − t)s ∣∣f ′(a)∣∣q + ts ∣∣f ′(b)∣∣q)dt )1 q     = c 2(s+1) 1 q [(( 1 − ( 1 − x−a c )s+1)∣∣f ′(a)∣∣q + (x−a c )s+1 ∣∣f ′(b)∣∣q )1 q + (( 1 − x−a c )s+1 ∣∣f ′(a)∣∣q + (1 −(x−a c )s+1)∣∣f ′ (b)∣∣q )1 q + ( 1−( 12 ) αp αp+1 )1 p × (( (2s+1−1)|f′(a)|q+|f′(b)|q 2s+1 )1 q + ( |f′(a)|q+(2s+1−1)|f′(b)|q 2s+1 )1 q )] . replacing c by η(b,a) in the above inequality, we get the desired result. corollary 5. in theorem 4 if we choose x = 2a+η(b,a) 2 , we obtain the following fractional midpoint inequality ∣∣∣∣f (2a+η(b,a)2 )− γ(α+1)2ηα(b,a) ((iαa+f) (a + η (b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ fractional ostrowski type inequalities 299 ≤ η(b,a) 2(s+1) 1 q ( 1 + ( 1−( 12 ) αp αp+1 )1 p ) × (( (2s+1−1)|f′(a)|q+|f′(b)|q 2s+1 )1 q + ( |f′(a)|q+(2s+1−1)|f′(b)|q 2s+1 )1 q ) . moreover if we take η (b,a) = b−a, we obtain∣∣∣∣f (a+b2 )− γ(α+1)2(b−a)α ((iαa+f) (b) + (iαb−f) (a)) ∣∣∣∣ ≤ b−a 2(s+1) 1 q ( 1 + ( 1−( 12 ) αp αp+1 )1 p ) × (( (2s+1−1)|f′(a)1q+|f′(b)|q 2s+1 )1 q + ( |f′(a)|q+(2s+1−1)|f′(b)|q 2s+1 )1 q ) . corollary 6. in theorem 4 if we put s = 1, we obtain∣∣∣∣f (x) − γ(α+1)2ηα(b,a) ((iαa+f) (a + η (b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ ≤ η(b,a) 2 1+ 1q [(( 1 − ( 1 − x−a η(b,a) )2)∣∣f ′(a)∣∣q + ( x−a η(b,a) )2 ∣∣f ′(b)∣∣q)1q + (( 1 − x−a η(b,a) )2 ∣∣f ′(a)∣∣q + (1 −( x−a η(b,a) )2)∣∣f ′(b)∣∣q)1q + ( 1−( 12 ) αp αp+1 )1 p (( 3|f′(a)|q+|f′(b)|q 4 )1 q + ( |f′(a)|q+3|f′(b)|q 4 )1 q )] . moreover if we choose x = 2a+η(b,a) 2 , we obtain the following fractional midpoint inequality∣∣∣∣f (2a+η(b,a)2 )− γ(α+1)2ηα(b,a) ((iαa+f) (a + η(b,a)) + (iα(a+η(b,a))−f) (a)) ∣∣∣∣ ≤ η(b,a) 2 1+ 1q ( 1 + ( 1−( 12 ) αp αp+1 )1 p )(( 3|f′(a)|q+|f′(b)|q 4 )1 q + ( |f′(a)|q+3|f′(b)|q 4 )1 q ) . 300 b. meftah, m. merad, a. souahi additionally if we take η (b,a) = b−a, we obtain∣∣∣∣f (a+b2 )− γ(α+1)2(b−a)α ((iαa+f) (b) + (iαb−f) (a)) ∣∣∣∣ ≤ b−a 2 1+ 1q ( 1 + ( 1−( 12 ) αp αp+1 )1 p )(( 3|f′(a)|q+|f′(b)|q 4 )1 q + ( |f′(a)|q+3|f′(b)|q 4 )1 q ) . references [1] w.w. breckner, stetigkeitsaussagen für eine klasse verallgemeinerter konvexer funktionen in topologischen linearen räumen (german), publ. inst. math. 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[27] c. zhu, m. fečkan, j. wang, fractional integral inequalities for differentiable convex mappings and applications to special means and a midpoint formula, j. appl. math. stat. inf. 8 (2) (2012), 21 – 28. introduction preliminaries main results � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 2 (2019), 269 – 283 doi:10.17398/2605-5686.34.2.269 available online september 3, 2019 aff(1|1)-trivial deformations of aff(2|1)-modules of weighted densities on the superspace r1|2 ismail laraiedh département de mathématiques, faculté des sciences de sfax, bp 802, 3038 sfax, tunisie departement of mathematics, college of sciences and humanities-kowaiyia shaqra university, kingdom of saudi arabia ismail.laraiedh@gmail.com , ismail.laraiedh@su.edu.sa received march 15, 2019 presented by rosa navarro accepted june 14, 2019 abstract: over the (1|2)-dimensional real superspace, we study aff(1|1)-trivial deformations of the action of the affine lie superalgebra aff(2|1) on the direct sum of the superspaces of weighted densities. we compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this action and we prove that any formal deformation is equivalent to its infinitisemal part. key words: relative cohomology, trivial deformation, lie superalgebra, symbol. ams subject class. (2010): 17b56, 53d55, 58h15. 1. introduction let vect(r) be the lie algebra of polynomial vector fields on the real space. consider the 1-parameter deformation of vect(r)-action on c∞(r) lλ x d dx (f) = xf ′ + λx′f, where x, f ∈ c∞(r) and x′ := dx dx . denote by fλ the vect(r)-module structure on c∞(r) defined by lλ for a fixed λ. geometrically, fλ = { fdxλ : f ∈ c∞(r)} is the space of polynomial weighted densities of weight λ ∈ r. the space fλ coincides with the space of vector fields, functions and differential 1-forms for λ = −1 , 0 and 1, respectively. the superspace dλ,µ := homdiff (fλ,fµ) the linear differential operators with the natural vect(r)-action denoted lλ,µx (a) = l µ x ◦a−a◦l λ x. each module dλ,µ has a natural filtration by the order of differential operators; the graded module sλ,µ := gr dλ,µ is the space of symbols. the quotient-module dkλ,µ / d k−1 λ,µ is isomorphic to the module of weighted densities fµ−λ−k; the issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.2.269 mailto:ismail.laraiedh@gmail.com mailto:ismail.laraiedh@su.edu.sa https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 270 i. laraiedh isomorphism is defined by the principal symbol map σpr defined by a = k∑ i=0 ai(x) ( ∂ ∂x )i 7−→ σpr(a) = ak(x)(dx)µ−λ−k, (see, e.g.,[15]). therefore, as a vect(r)-module, the space sλ,µ depends on the difference β = µ−λ, so that sλ,µ be written as sβ, and sβ = ∞⊕ k=0 fβ−k as vect(r)-modules. the space of symbols of order ≤ n is snβ := n⊕ k=0 fβ−k. the space dλ,µ cannot be isomorphic as a vect(r)-module to the space of symbols, but is a deformation of this space in the sense of richardson and neijenhuis [19]. deformation theory plays a crucial role in all branches of physics. in physics the mathematical theory of deformations has been proved to be a powerful tool in modeling physical reality. the concepts symmetry and deformations are considered to be two fundamental guiding principles for developing the physical theory further. the notion of deformation was applied to lie algebras by nijenhuis and richardson [19, 18]. this theory is developed by ovsienko and by other authors [3, 9, 19]. we consider the superspace r1|2 endowed with its standard contact structure defined by the 1-form α2, and k(2) of contact vector fields on r1|2. we introduce the k(2)-module f2λ of λ-densities on r 1|2 and the k(2)-module of linear differential operators, d2λ,µ := homdiff (f 2 λ,f 2 µ), which are super analogs of the spaces fλ and dλ,µ, respectively. the lie superalgebra aff(2|1), a super analog of aff(1), is a subalgebra of k(2). we classify the aff(1|1)-trivial deformations of the structure of the aff(2|1)-module s2µ−λ = ∞⊕ k=0 f2 µ−λ−k 2 , which is super analog of the space sβ. we prove that any formal deformation is equivalent to its infinitesimal part and we give an example of deformation with one parameter. aff(1|1)-trivial deformations of aff(2|1)-modules 271 2. definitions and notations we briefly give in this section the basics definitions of geometrical objects on r1|2 that will be needed for our purpose, for more details, see [7, 11, 6, 16, 15, 17]. 2.1. the lie superalgebra of contact vector fields on r1|2. let r1|2 be the superspace with coordinates (x, θ1, θ2), where θ1 and θ2 are odd indeterminates: θiθj = −θjθi. we consider the superspace c∞(r1|2) of polynomial functions. any element of c∞(r1|2) has the form f = f0 + f1θ1 + f2θ2 + f12θ1θ2, where f0,f1,f2,f12 ∈ c∞(r). even elements in c∞(r1|2) are the functions f(x,θ) = f0(x) + f12(x)θ1θ2, the functions f(x,θ) = f1(x)θ1 + f2(x)θ2 are odd elements. we denote by |f| the parity of a function f. let vect(r1|2) be the space of polynomial vector fields on r1|2: vect(r1|2) = { f0∂x + f1∂1 + f2∂2 : fi ∈ c∞(r1|2) } , where ∂i and ∂x stand for ∂ ∂θi and ∂ ∂x . the superbracket of two vector fields is bilinear and defined for two homogeneous vector fields by [x,y ] = x ◦y − (−1)|x||y |y ◦x. the supespace r1|2 is equipped with the contact structure given by the 1-form α2 = dx + θ1dθ1 + θ2dθ2. this contact structure is equivalently defined by the kernel of α2, spanned by the odd vector fields ηi = ∂i −θi∂x. we consider the superspace k(2) of contact vector fields on r1|2. that is, k(2) = { x ∈ vect(r1|2) : ∃ f ∈ c∞(r1|2) such that lx(α2) = fα2 } , where lx is the lie derivative of a vector field, acting on the space of functions, forms, vector fields, . . . . any contact vector field on r1|2 can be expressed as xf = f∂x − 1 2 (−1)|f| 2∑ i=1 ηi(f)ηi , where f ∈ c ∞(r1|2). 272 i. laraiedh of course, k(2) is a subalgebra of vect(r1|2), and k(2) acts on c∞(r1|2) through lxf (g) = fg ′ − 1 2 (−1)|f | 2∑ i=1 ηi(f) ·ηi(g), (2.1) where g ∈ c∞(r1|2). the contact bracket is defined by [xf , xg] = x{f,g}. the space c∞(r1|2) is thus equipped with a lie superalgebra structure isomorphic to k(2). the explicit formula can be easily calculated: {f,g} = fg′ −f ′g− 1 2 (−1)|f| 2∑ i=1 ηi(f) ·ηi(g). (2.2) 2.2. the superalgebra aff(2|1). recall that the lie algebra aff(1) can be realized as a subalgebra of vect(r): aff(1) = span (x1, xx) , and the affine lie superalgebra aff(1|1) is realized as a subalgebra of k(1): aff(1|1) = span(x1, xx, xθ). the space aff(1|1)0 is isomorphic to aff(1), while (aff(1|1))1̄ = span(xθ). similarly, the affine lie superalgebra aff(2|1) can be realized as a subalgebra of k(2): aff(2|1) = span(x1, xx, xθ1, xθ2, xθ1θ2 ), where (aff(2|1))0̄ = span(x1, xx, xθ1θ2 ), (aff(2|1))1̄ = span(xθ1, xθ2 ). we easily see that aff(1|1) is subalgebra of the lie superalgebra aff(2|1). aff(1|1)-trivial deformations of aff(2|1)-modules 273 2.3. the space of weighted densities on r1|2. we introduce a oneparameter family of modules over the lie superalgebra k(2). as vector spaces all these modules are isomorphic to c∞(r1|2), but not as k(2)-modules. for every contact vector field xf , define a one-parameter family of firstorder differential operators on c∞(r1|2): lλxf = xf + λf ′, λ ∈ r. (2.3) we easily check that [lλxf ,l λ xg ] = lλx{f,g}. (2.4) we thus obtain a one-parameter family of k(2)-modules on c∞(r1|2) that we denote f2λ, the space of all weighted densities on c ∞(r1|2) of weight λ with respect to α2: f2λ = { fαλ2 : f ∈ c ∞(r1|2) } . (2.5) in particular, we have f0λ = fλ. obviously the adjoint k(2)-module is isomorphic to the space of weighted densities on r1|2 of weight −1. 2.4. differential operators on weighted densities. a differential operator on r1|2 is an operator on c∞(r1|2) of the form: a = m∑ j=0 aj∂ j x + 2∑ i=1 ni∑ k=0 bk,i∂ k x∂i + n∑ `=0 c`∂ ` x∂1∂2, (2.6) where aj, bk,i, c` ∈ c∞(r1|2). any differential operator defines a linear mapping fαλ2 7→ (af)α µ 2 from f 2 λ to f 2 µ for any λ, µ ∈ k; thus, the space of differential operators becomes a family of osp(2|2)-modules d2λ,µ for the natural action: xf ·a = l µ xf ◦a− (−1)|a||f|a◦lλxf . (2.7) proposition 2.1. every differential operator a ∈ d2λ,µ can be expressed in the form a(fαλ2 ) = ∑ `,m a`,m(x,θ)η ` 1η m 2 (f)α µ 2, (2.8) where a`,m(x,θ) are arbitrary functions. proof. since −η2i = ∂x, and ∂i = ηi − θiη 2 i , every differential operator a given by (2.6) is a polynomial expression in η1 and η2. 274 i. laraiedh proposition 2.2. as a aff(1|1)-module, we have d2λ,µ ' d 1 λ,µ ⊕d 1 λ+ 1 2 ,µ+ 1 2 ⊕ π ( d1 λ,µ+ 1 2 ⊕d1 λ+ 1 2 ,µ ) . (2.9) proof. any element f ∈ c∞(r1|2) can be uniquely written as follows: f = f1 + f2θ2, where ∂2f1 = ∂2f2 = 0. therefore, for any xh ∈ aff(1|1), we easily chek that lλxh (f) = l λ xh (f1) + l λ+ 1 2 xh (f2)θ2. thus, the following map is an aff(1|1)-isomorphism: φλ : f 2 λ −→ f 1 λ ⊕ π(f 1 λ+ 1 2 ) fαλ2 7−→ ( f1α λ 1 , π(f2α λ+ 1 2 1 ) ) . (2.10) so, we deduce an aff(1|1)-isomorphism: ψλ,µ : d 1 λ,µ ⊕d 1 λ+ 1 2 ,µ+ 1 2 ⊕ π ( d1 λ,µ+ 1 2 ⊕d1 λ+ 1 2 ,µ ) −→ d2λ,µ (2.11) a 7−→ φ−1µ ◦a◦ φλ. we identify the aff(1|1)-modules the following isomorphisms: π ( d1 λ,µ+ 1 2 ) −→ homdiff ( f1λ, π(f 1 µ+ 1 2 ) ) , π(a) 7−→ π ◦a, π ( d1 λ+ 1 2 ,µ ) −→ homdiff ( f1 λ+ 1 2 , π(f1µ) ) , π(a) 7−→ a◦ π, d1 λ+ 1 2 ,µ+ 1 2 −→ homdiff ( π(f1 λ+ 1 2 ), π(f1 µ+ 1 2 ) ) , π(a) 7−→ π ◦a◦ π. 3. aff(1|1)-trivial deformation of aff(2|1)-modules deformation theory of lie algebra was first considered with one-parameter of deformation [13, 19, 12, 21, 4, 5]. recently, deformations of lie (super)algebras with multi-parameters were intensively studied (see, e.g., [1, 2, 3, 8, 20]). 3.1. infinitesimal deformations and the first cohomology. let ρ0 : g → end(v ) be an action of a lie superalgebra g on a vector superspace v and let h be a subagebra of g (if h is omitted it assumed to be {0}). when aff(1|1)-trivial deformations of aff(2|1)-modules 275 studying h-trivial deformations of the g-action ρ0, one usually starts with infinitesimal deformations ρ = ρ0 + t υ, (3.1) where υ : g → end(v ) is a linear map vanishing on h and t is a formal parameter with p(t) = p(υ). the homomorphism condition [ρ(x),ρ(y)] = ρ([x,y]), (3.2) where x, y ∈ g, is satisfied in order 1 in t if and only if υ is a h-relative 1-cocycle. that is, the map υ satisfies (−1)|x||υ|[ρ0(x), υ(y)] − (−1)|y|(|x|+|υ|)[ρ0(y), υ(x)] − υ([x, y]) = 0. moreover, two h-trivial infinitesimal deformations ρ = ρ0 + t υ1, and ρ = ρ0 + t υ2, are equivalents if and only if υ1 − υ2 is h-relative coboundary: (υ1 − υ2)(x) = (−1)|x||a|[ρ0(x),a] := δa(x), where a ∈ end(v )h and δ stands for differential of cochains on g with values in end(v ) (see, e.g., [14, 19]). so, the space h1(g,h; end(v )) determines and classifies infinitesimal deformations up to equivalence. if dim h1(g,h; end(v )) = m, then choose 1-cocycles υ1, . . . , υm representing a basis of h 1(g,h; end(v )) and consider the infinitesimal deformation ρ = ρ0 + m∑ i=1 ti υi, (3.3) where t1, . . . , tm are independent parameters with |ti| = |υi|. since we are interested in the aff(1|1)-trivial deformations of the aff(2|1)module structure on the space s 2,m β = 2m⊕ k=0 f2 β−k 2 , where m ∈ 1 2 n. (3.4) the first differential cohomology spaces h1diff (aff(2|1),aff(1|1),d 2 λ,µ) was computed in [10]. the result is as follows: dim(h1(aff(2|1),aff(1|1),d2λ,µ)) =   1 if µ = λ, 2 if µ−λ = k, k ∈{1, 2, . . .}, 0 otherwise. 276 i. laraiedh the following 1-cocycles span the corresponding cohomology spaces: υλ,λ(xg) = η1η2(g), γ1λ,λ+k(xg) = η1η2(g)∂ k x, γ2λ,λ+k(xg) = η1η2(g)η1η2∂ k−1 x . we consider the space h1(aff(2|1), aff(1|1), end(s2,mβ )) spanned by the classes υiλ,λ and γ i λ,λ+k, i = 1, 2, where k ∈ {1, . . . , [m]}, [m] denoting the integer part of m, and 2(β − λ) ∈ {2k, . . . , 2m} for a generic β. any infinitesimal aff(1|1)-trivial deformation of the aff(2|1)-module structure on s2,mβ is then of the form l̃xf = lxf + l (1) xf , (3.5) where lxf is the lie derivate of s 2,m β along the vector xf defined by (2.3), and l (1) xf = ∑ λ tλ,λυλ,λ(xf ) + ∑ λ [m]∑ k=1 2∑ i=1 tiλ,λ+kγ i λ,λ+k(xf ), (3.6) where tλ,λ and t i λ,λ+k are independent parameters with |tλ,λ| = |υλ,λ| and |tiλ,λ+k| = |γ i λ,λ+k|. 3.2. integrability conditions and deformations over supercommutative algebras. consider the superalgebra with unity c[[t1, . . . , tm]] and consider the problem of integrability of infinitesimal deformations. starting with the infinitesimal deformation (3.3), we look for a formal series ρ = ρ0 + m∑ i=1 ti υi + ∑ i,j titj ρ (2) ij + · · · , (3.7) where the higher order terms ρ (2) ij , ρ (3) ijk, . . . are linear maps from g to end(v ) with |ρ(2)ij | = |titj|, |ρ (3) ijk| = |titjtk|, . . . such that the map ρ : g → c[[t1, . . . , tm]] ⊗ end(v ) (3.8) satisfies the homomorphism condition (3.2). quite often the above problem has no solution. following [13] and [2], we will impose extra algebraic relations on the parameters t1, . . . , tm. let r be an ideal in c[[t1, . . . , tm]] generated by some set of relations, and we can aff(1|1)-trivial deformations of aff(2|1)-modules 277 speak about deformations with base a = c[[t1, . . . , tm]]/r, (for details, see [13]). the map (3.8) sends g to a⊗ end(v ). setting ϕt = ρ−ρ0, ρ(1) = ∑ ti υi, ρ (2) = ∑ titj ρ (2) ij , . . . , we can rewrite the relation (3.2) in the following way: [ϕt(x),ρ0(y)] + [ρ0(x),ϕt(y)] −ϕt([x,y]) + ∑ i,j>0 [ρ(i)(x),ρ(j)(y)] = 0. (3.9) the first three terms are (δϕt)(x,y). for arbitrary linear maps γ1, γ2 : g → end(v ), consider the standard cup-product : [[γ1,γ2]] : g⊗g → end(v ) defined by: [[γ1,γ2]](x,y) = (−1)|γ2|(|γ1|+|x|)[γ1(x),γ2(y)] + (−1)|γ1||x|[γ2(x),γ1(y)]. (3.10) the relation (3.9) becomes now equivalent to δϕt + 1 2 [[ϕt,ϕt]] = 0. (3.11) expanding (3.11) in power series in t1, . . . , tm, we obtain the following equation for ρ(k): δρ(k) + 1 2 ∑ i+j=k [[ρ(i),ρ(j)]] = 0. (3.12) the first non-trivial relation δρ(2) + 1 2 [[ρ(1),ρ(1)]] = 0 gives the first obstruction to integration of an infinitesimal deformation. thus, considering the coefficient of ti tj, we get δρ (2) ij + 1 2 [[υi, υj]] = 0. (3.13) it is easy to check that for any two 1-cocycles γ1 and γ2 ∈ z1(g,h; end(v )), the bilinear map [[γ1,γ2]] is a h-relative 2-cocycle. the relation (3.13) is precisely the condition for this cocycle to be a coboundary. moreover, if one of the cocycles γ1 or γ2 is a h-relative coboundary, then [[γ1,γ2]] is a h-relative 2coboundary. therefore, we naturally deduce that the operation (3.10) defines a bilinear map: h1(g,h; end(v )) ⊗ h1(g,h; end(v )) −→ h2(g,h; end(v )). (3.14) all the obstructions lie in h2(g,h; end(v )) and they are in the image of h1(g,h; end(v )) under the cup-product. 278 i. laraiedh 3.3. equivalence. two deformations ρ and ρ′ of a g-module v over a are said to be equivalent (see [13]) if there exists an inner automorphism ψ of the associative superalgebra a⊗ end(v ) such that ψ ◦ρ = ρ′ and ψ(i) = i, where i is the unity of the superalgebra a⊗ end(v ). the following notion of miniversal deformation is fundamental. it assigns to a g-module v a canonical commutative associative algebra a and a canonical deformation over a. a deformation (3.7) over a is said to be miniversal if (i) for any other deformation ρ′ with base (local) a′, there exists a homomorphism ψ : a′ →a satisfying ψ(1) = 1, such that ρ = (ψ ⊗ id) ◦ρ′; (ii) under notation of (i), if ρ is infinitesimal, then ψ is unique. if ρ satisfies only the condition (i), then it is called versal. this definition does not depend on the choice 1-cocycles υ1, . . . , υm representing a basis of h1(g,h; end(v )). the miniversal deformation corresponds to the smallest ideal r. we refer to [13] for a construction of miniversal deformations of lie algebras and to [2] for miniversal deformations of g-modules. superization of these results is immediate by the sign rule. 3.4. integrability conditions. in this subsection, we obtain the integrability conditions for the infinitesimal deformation (3.5). theorem 3.1. (i) the following conditions are necessary and sufficient for integrability of the infinitesimal deformation (3.5): t1λ,λ+ktλ,λ − tλ+k,λ+k t 1 λ,λ+k = 0 t2λ,λ+ktλ,λ − tλ+k,λ+k t 2 λ,λ+k = 0 } for 2(β −λ) ∈{2k,. . . , 2m} . (3.15) (ii) any formal aff(1|1)-trivial deformation of the aff(2|1)-module s 2,m β is equivalent to a deformation of order 1, that is, to a deformation given by (3.5). aff(1|1)-trivial deformations of aff(2|1)-modules 279 the super-commutative algebra defined by relations (3.15) corresponds to the miniversal deformation of the lie derivative lx. note that the supercommutative algebra defined in theorem 3.1 is infinite-dimensional. the proof of theorem 3.1 consists in two steps. first, we compute explicitly the obstructions for existence of the second-order term, this will prove that relations (3.15) are necessary. second we show that under relations (3.15) the highest-order terms of the deformation can be chosen identically zero, so that relations (3.15) are indeed sufficient. proof. assume that the infinitesimal deformation (3.5) can be integrated to a formal deformation l̃x = lx + l (1) x + l (2) x + · · · , where l (1) x is given by (3.6) and l (2) x is a quadratic polynomial in t with coefficients in s 2,m β . considering the homomorphism condition, we compute the second order term l(2) which is a solution of the maurer-cartan equation: ∂(l(2)) = − 1 2 [[l(1), l(1)]]. (3.16) for arbitrary λ, the right hand side of (3.16) yields the following aff(1|1)relative 2-cocycles: bλ,λ = [[υλ,λ, υλ,λ]] : aff(2|1) ⊗aff(2|1) −→ d2λ,λ, b̃iλ,λ+k = [[γ i λ,λ+k, υλ,λ]] : aff(2|1) ⊗aff(2|1) −→ d 2 λ,λ+k, i ∈{1, 2}, b i λ,λ+k = [[υλ+k,λ+k, γ i λ,λ+k]] : aff(2|1) ⊗aff(2|1) −→ d 2 λ,λ+k, i ∈{1, 2}. by a straightforward computation, we check that bλ,λ = 0, b̃ 1 λ,λ+k = −b 1 λ,λ+k and b̃ 2 λ,λ+k = −b 2 λ,λ+k, with b̃1λ,λ+k(xf ,xg) = ( η1η2(f)g ′ −f ′η1η2(g) ) ∂kx, b̃2λ,λ+k(xf ,xg) = ( η1η2(f)g ′ −f ′η1η2(g) ) η1η2∂ k−1 x . we will need the following: 280 i. laraiedh proposition 3.2. each of the bilinear map b̃1λ,λ+k, b̃ 2 λ,λ+k define generically nontrivial cohomology class. moreover, these cohomology classes are linearly independent. proof. each map b̃iλ,λ+k, i = 1, 2, is a aff(1|1)-relative 2-cocycle on aff(2|1) since it is the kolmogorov-alexander product of two aff(1|1)-relative 1-cocycles. assume that, for some differential 1-cochain bλ,λ+k on aff(2|1) with coefficients in d2λ,λ+k, we have b̃ i λ,λ+k = ∂(bλ,λ+k). the general form of such a cochain is bλ,λ+k(xf ) = ∑ a`1`2m1m2 (x,θ1,θ2)η `1 1 η `2 2 (f)η m1 1 η m2 2 , where the coefficients a`1`2m1m2 (x,θ1,θ2) are arbitrary functions. to complete the proof of the proposition we will need the following: lemma 3.3. the condition b̃iλ,λ+k = ∂(bλ,λ+k) implies that the coefficients a`1`2m1m2 (x,θ1,θ2) are functions of θi, not depending on x. proof. the condition b̃iλ,λ+k = ∂(bλ,λ+k) reads b̃iλ,λ+k(xf ,xg) = xf · bλ,λ+k(xg) − (−1) |g||f |xg · bλ,λ+k(xf ) − bλ,λ+k([xf , xg]). (3.17) we choose a constant function f = 1, and prove that x1 ·bλ,λ+k = 0. indeed, we have (x1 · bλ,λ+k)(xg) = x1 · bλ,λ+k(xg) − bλ,λ+k([x1, xg]). since `1 + `2 ≥ 1 in the expression of bλ,λ+k, it follows that bλ,λ+k(x1) = 0, and thus the last equality gives (x1 · bλ,λ+k)(xg) = x1 · bλ,λ+k(xg) −xg · bλ,λ+k(x1) − bλ,λ+k([x1, xg]) = ∂(bλ,λ+k)(x1,xg). by assumption, b̃iλ,λ+k = ∂(bλ,λ+k), from the explicit formula for b̃ i λ,λ+k, we obtain b̃iλ,λ+k(x1,xg) = 0 for all xg ∈ aff(2|1). therefore, x1 · bλ,λ+k = 0. lemma 3.3 is proved. aff(1|1)-trivial deformations of aff(2|1)-modules 281 now, by direct computation, we check that equation (3.17) has no solution. this is contradiction with the assumption, b̃iλ,λ+k = ∂(bλ,λ+k). thus, the maps b̃iλ,λ+k are nontrivial aff(1|1)-relative 2-cocycles. moreover, by a direct computation, we can check that, for some differential 1-cochain bλ,λ+k on aff(2|1) with coefficients in d2λ,λ+k, the system( b̃1λ,λ+k, b̃ 2 λ,λ+k, ∂(bλ,λ+k) ) is linearly independent. thus, the cohomology classes of b̃iλ,λ+k are linearly independent. this completes the proof of proposition 3.2. proposition 3.2 implies that equation (3.16) has solutions if and only if the quadratic polynomials given by (3.15) vanish simultaneously. we thus proved that conditions (3.15) are, indeed, necessary. to prove that the conditions (3.15) are sufficient, we will find explicitly a deformation of lxf , whenever the conditions (3.15) are satisfied. the solution l(2) of (3.16) can be chosen identically zero. choosing the hightest-order terms l(m) with m ≥ 3, also identically zero, one obviously obtains a deformation which is of order 1 in t. theorem 3.1, part (i) is proved. the solution l(2) of (3.16) is defined up to a aff(1|1)-relative 1-cocycle and it has been shown in [13, 2] that different choices of solutions of the maurercartan equation correspond to equivalent deformations. thus, we can always reduce l(2) to zero by equivalence. then, by recurrence, the hightest-order terms l(m) satisfy the equation ∂(l(m)) = 0 and can also be reduced to the identically zero map. this completes the proof of part (ii). example 3.1. for m ∈ 1 2 n and for arbitrary generic λ ∈ r, the following example is a 1-parameter aff(1|1)-trivial deformation of the aff(2|1)-module s 2,m λ+m l̃xf = lxf + t 2m∑ `=0 [ 2m−` 2 ]∑ k=1 2∑ i=1 γi λ+ ` 2 ,λ+ ` 2 +k , that is , we put ti λ+ ` 2 ,λ+ ` 2 +k = t and tλ,λ = 0. of course it is easy to give many other examples of true deformations with one parameter or with several parameters. acknowledgements we would like to thank mabrouk ben ammar, nizar ben fraj, hafedh khalfoun and alice fialowski for their interest in this work. 282 i. laraiedh references [1] m. abdaoui, h. khalfoun, i. laraiedh, deformation of modules of weighted densities on the superspace r1|n , acta. math. hungar. 145 (2015), 104 – 123. 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[14] d.b. fuchs, “cohomology of infinite-dimensional lie algebras”, springer, new york, 1986. arxiv:math.ph/0807.4811 arxiv:math.rt/1004.1700 arxiv:math.rt/0702664 http://dx.doi.org/10.1063/1.4789539 aff(1|1)-trivial deformations of aff(2|1)-modules 283 [15] h. gargoubi, n. mellouli, v. ovsienko, differential operators on supercircle: conformally equivariant quantization and symbol calculus, lett. math. phys. 79 (2007), 51 – 65. [16] f. gieres, s. theisen, superconformally covariant operators and super-walgebras, j. math. phys. 34 (1993), 5964 – 5985. [17] p. grozman, d. leites, i. shchepochkina, lie superalgebras of string theories, acta math. vietnam. 26 (1) (2001), 27 – 63. arxiv:hep-th/9702120 [18] a. nijenhuis, r. richardson, cohomology and deformations in graded lie algebras, bull. amer. math. soc. 72, (1966) 1 – 29. [19] a. nijenhuis, r.w.jr. richardson , deformations of homomorphisms of lie groups and lie algebras, bull. amer. math. soc. 73 (1967), 175 – 179. [20] v. ovsienko, c. roger, deforming the lie algebra of vector fields on s1 inside the lie algebra of pseudodifferential symbols on s1, in “ differential topology, infinite-dimensional lie algebras, and applications ”, d.b. fuchs’ 60th anniversary collection, edited by alexander astashkevich and serge tabachnikov, american mathematical society translations, series 2, 194. adv. math. sci., 44. amer. math. soc., providence, ri, 1999, 211 – 226, [21] r.w. richardson, deformations of subalgebras of lie algebras, j. diff. geom. 3 (1969), 289 – 308. arxiv:hep-th/9702120 introduction definitions and notations the lie superalgebra of contact vector fields on r1|2. the superalgebra aff(2|1). the space of weighted densities on r1|2. differential operators on weighted densities. aff(1|1)-trivial deformation of aff(2|1)-modules infinitesimal deformations and the first cohomology. integrability conditions and deformations over supercommutative algebras. equivalence. integrability conditions. e extracta mathematicae vol. 32, núm. 2, 125 – 161 (2017) essential g-ascent and g-descent of a closed linear relation in hilbert spaces ∗ zied garbouj, häıkel skhiri faculté des sciences de monastir, département de mathématiques, avenue de l’environnement, 5019 monastir, tunisia zied.garbouj.fsm@gmail.com haikel.skhiri@gmail.com haikel.skhiri@fsm.rnu.tn presented by jesús m. f. castillo received january 26, 2017 abstract: we define and discuss for a closed linear relation in a hilbert space the notions of essential g-ascent (resp. g-descent) and g-ascent (resp. g-descent) spectrums. we improve in the hilbert space case some results given by e. chafai in a banach space [acta mathematica sinica, 34 b, 1212-1224, 2014] and several results related to the ascent (resp. essential ascent) spectrum for a bounded linear operator on a banach space [studia math, 187, 59-73, 2008] are extended to closed linear relations on hilbert spaces. we prove also a decomposition theorem for closed linear relations with finite essential g-ascent or g-descent. key words: range subspace, closed linear relation, spectrum, ascent, essential ascent, descent, essential descent, semi-fredholm relation. ams subject class. (2010): 47a06, 47a05, 47a10, 47a55 1. introduction and terminology let h be a complex hilbert space. a multivalued linear operator t : h −→ h or simply a linear relation is a mapping from a subspace d(t) ⊆ h, called the domain of t, into the collection of nonempty subsets of h such that t(λx+µy) = λt(x)+µt(y) for all nonzero scalars λ, µ and x, y ∈ d(t). we denote by lr(h) the class of linear relations on h. if t maps the points of its domain to singletons, then t is said to be a single valued linear operator or simply an operator. the graph g(t) of t ∈ lr(h) is defined by : g(t) = {(x, y) ∈ h × h : x ∈ d(t), y ∈ tx}. we say that t is closed if its graph is a closed subspace of h × h. the class of such linear relations will be denoted by cr(h). a linear relation t ∈ lr(h) is said to be continuous if for each open set ω ⊆ im(t), t −1(ω) is an open ∗this work is supported by the higher education and scientific research in tunisia, ur11es52 : analyse, géométrie et applications 125 126 z. garbouj, h. skhiri set in d(t). continuous everywhere defined linear relations are referred to as bounded relations. the kernel of a linear relation t is the subspace ker(t) := t −1(0). the subspace im(t) := t(d(t)) is called the range of t. the nullity and the defect of a linear relation t ∈ lr(h) are defined by α(t) = dim ker(t) and β(t) = dim h/im(t), respectively. recall that t ∈ cr(h) is said to be upper semi-fredholm if t has closed range and α(t) < +∞, and t is said to be lower semi-fredholm if β(t) < +∞. if t is upper or lower semi-fredholm we say that t is semi-fredholm, and we denote by φ±(h) the class of all semi-fredholm relations. for t ∈ φ±(h) we define the index of t by ind(t) = α(t) − β(t). a linear relation is fredholm if max{α(t), β(t)} < +∞. we denote by φ(h) (respectively, φ+(h), φ−(h)) the class of all fredholm (respectively, upper semi-fredholm, lower semi-fredholm) relations. the linear relation t ∈ cr(h) is called regular if im(t) is closed and ker(t) ⊆ im(t n), for every n ∈ n (see [1]). recall that the resolvent set of t ∈ lr(h) is defined (see, [4, chapter vi]) by ϱ(t) = {λ ∈ c : λi − t is injective, open and has dense range} and the spectrum of t is the set σ(t) = c\ϱ(t). it is clear from the closed graph theorem for a linear relation that if t is a closed linear relation then ϱ(t) = {λ ∈ c : λi − t is bijective}. we say that t ∈ lr(h) has a trivial singular chain manifold if rc(t) = {0} where rc(t) = [ ∞∪ i=1 ker(t i) ] ∩ [ ∞∪ i=1 t i(0) ] . let λ ∈ c, by [10, lemma 7.1], we know that rc(t) = {0} if and only if rc(λi − t) = {0}. it is easy to see that rc(t) = {0} when ϱ(t) ̸= ∅. for two subspaces m and n of h, we recall that dim m/m ∩ n = dim(m + n)/n and (n + w) ∩ m = n ∩ m + w whenever w is a subspace of m. following [3], the ascent and the descent of t ∈ lr(h) are respectively defined by a(t) = inf{k ∈ n : ker(t k+1) = ker(t k)}, d(t) = inf{k ∈ n : im(t k+1) = im(t k)}, essential g-ascent and g-descent of a closed relation 127 whenever these minima exist. if no such numbers exist the ascent and descent of t are defined to be +∞. for t ∈ lr(h) and n ∈ n, we define the following quantities : αn(t) = dim ker(t n+1)/ker(t n), βn(t) = dim im(t n)/im(t n+1). let us recall from [9, lemma 3.2] and [10, lemma 4.1], the following properties αn(t) = dim[im(t n) ∩ ker(t)]/[t n(0) ∩ ker(t)] (1.1) and βn(t) = dim d(t n)/[im(t) + ker(t n)] ∩ d(t n) = dim[im(t) + d(t n)]/[im(t) + ker(t n)]. (1.2) in [6], we show that (βn(t))n≥0 and (αn(t))n≥0 are decreasing sequences. recall that for t ∈ lr(h), the essential ascent, ae(t), and the essential descent, de(t), are defined by ([3]) ae(t) = inf{n ∈ n : αn(t) < +∞}, de(t) = inf{n ∈ n : βn(t) < +∞}, where the infimum over the empty set is taken to be infinite. for t ∈ lr(h) we consider the two decreasing sequences α̃n(t) = dim im(t n) ∩ ker(t), β̃n(t) = dim h/[im(t) + ker(t n)], n ∈ n. remark 1.1. from the equalities (1.1) and (1.2), we see that αn(t) ≤ α̃n(t) and βn(t) ≤ β̃n(t), for all n ∈ n. observe that ker(t) ∩ t j(0) ⊆ rc(t), for all j ∈ n. thus, by equality (1.1) (resp. (1.2)), it follows that if rc(t) = {0} (resp. d(t i) + im(t) = h, for all i ∈ n), so αn(t) = α̃n(t) (resp. βn(t) = β̃n(t)), for all n ∈ n. the above remark leads to the introduction of a new concept of g-ascent (resp. essential g-ascent, g-descent, essential g-descent) for a linear relation. definition 1.2. let t ∈ lr(h). (i) the g-ascent, ã(t), of t is defined by ã(t) = inf{n ∈ n : α̃n(t) = 0}. 128 z. garbouj, h. skhiri (ii) the essential g-ascent, ãe(t), of t is defined as ãe(t) = inf{n ∈ n : α̃n(t) < +∞}. (iii) the g-descent, d̃(t), of t is defined by d̃(t) = inf{n ∈ n : β̃n(t) = 0}. (iv) the essential g-descent, d̃e(t), of t is defined by d̃e(t) = inf{n ∈ n : β̃n(t) < +∞}, where as usual the infimum over the empty set is taken to be +∞. it is clear that a(t) ≤ ã(t) and ae(t) ≤ ãe(t) (resp. d(t) ≤ d̃(t) and de(t) ≤ d̃e(t)), and equality holds when rc(t) = {0} (resp. d(t i)+im(t) = h, for all i ∈ n). the notion of ascent (resp. descent, essential ascent, essential descent) of a linear operator was studied in several papers (see for examples [2, 5]). in recent years some work has been devoted to extend these concepts to the case of linear relations, (see [3, 6, 10]). in [3], many basic results related to the ascent (resp. descent, essential ascent, essential descent) spectrum of linear operators have been extended to linear relations (usually with additional conditions). in this context, we prove that the results in [3] related to the spectral mapping theorem of ascent and essential ascent (resp. descent and essential descent) spectrums of a closed linear relation everywhere defined such that ϱ(t) ̸= ∅ (resp. dim t(0) < +∞) remain valid when t ∈ υ(h) (see page 142) (resp. t ∈ lr(h)) and without the assumption that ϱ(t) ̸= ∅ (resp. dim t(0) < +∞) and d(t) = h. in [2], the ascent spectrum and the essential ascent spectrum of a bounded operator acting in banach spaces are introduced and studied. in this paper, we extend in the hilbert space case these notions to multivalued linear operators. however, the techniques used in this work are different from those used in [2, 3, 5]. our approach here is based in the concept of range subspaces of hilbert spaces (see, [7]). the paper is organized as follows. in the next section, we first established some algebraic lemmas that will be used throughout this work. in sections 3, 4 and 5, we are interested in the spectral theory of closed linear relations in hilbert spaces having a finite essential g-ascent or finite essential g-descent. for example, in theorem 3.4, we show that a closed linear relation with finite essential g-ascent and g-descent of a closed relation 129 essential g-ascent is stable under perturbations of the form λi, where λ ∈ c. in theorem 3.8, we study the spectrum boundary points of a closed linear relation with finite essential g-ascent. in theorem 4.12 and theorem 5.4, we prove, under some conditions, that the essential g-ascent spectrum and essential g-descent spectrum satisfy the polynomial version of the spectral mapping theorem for a closed linear relation. finally, in section 6, we prove that if t ∈ cr(h), then there exists n ∈ n such that ãe(t) ≤ n and im(t) + ker(t n) is closed (resp. d̃e(t) ≤ n and im(t n) ∩ ker(t) is closed) if and only if there exist d ∈ n and two closed subspaces m and n such that : (i) h = m u n; (ii) im(t d) ⊆ m, t(m) ⊆ m, n ⊆ ker(t d) and, if d > 0, n ̸⊂ ker(t d−1); (iii) g(t) = [g(t) ∩ (m × m)] u [g(t) ∩ (n × n)]; (iv) the restriction of t to m is both upper semi-fredholm (resp. lower semi-fredholm) and regular relation; (v) if a ∈ lr(n) such that its graph is the subspace g(t) ∩ (n × n), then a is a bounded operator everywhere defined and g(ad) = n × {0}. 2. algebraic preliminaries throughout this paper the symbol u denotes the topological direct sum of closed subspaces in h, i.e., x0 = x1 u x2 if the linear space x0 = x1 + x2 is closed and x1 ∩ x2 = {0}. next we give an example of quantities introduced below. example 2.1. (i) let m be a subspace of h and define the linear relation t in h by : d(t) = m and t(x) = m, ∀ x ∈ m. clearly we have ker(t n) = im(t n) = m, ∀ n ≥ 1. (1) • case 1 : if dim m = +∞, from (1), we have a(t) = ae(t) = 1 < ã(t) = ãe(t) = +∞. 130 z. garbouj, h. skhiri • case 2 : if 0 < dim m < +∞, by (1), we deduce that ae(t) = ãe(t) = 0 < a(t) = 1 < ã(t) = +∞. • case 3 : if 0 < dim h/m < +∞, from (1), we obtain d̃e(t) = de(t) = 0 < d(t) = 1 < d̃(t) = +∞. • case 4 : if dim h/m = +∞, it follows from (1) that d(t) = de(t) = 1 < d̃e(t) = d̃(t) = +∞. (ii) let m and n be a pair of closed subspaces of h such that h = m u n and {0} n h. let p be the linear projection with domain h, range n and kernel m and let l = p −1, then d(l) = n and l(x) = x + m, ∀ x ∈ n. since (i − l)x = m, with x ∈ n, it follows that ker[(i − l)n] = n and im[(i − l)n] = m, for all n ≥ 1. thus ãe(i − l) = ae(i − l) ≤ a(i − l) = ã(i − l) = 1, d̃e(i − l) = de(i − l) ≤ d(i − l) = d̃(i − l) = 1. in particular, if dim n < +∞, then ãe(i −l) = ae(i −l) = d̃e(i −l) = de(i − l) = 0. in this section, we prove some algebraic results of the theory of linear relations which are used to prove the main results in this work. for t ∈ lr(h), we consider the sequence sn(t) = dim[im(t n) ∩ ker(t)]/[im(t n+1) ∩ ker(t)], n ∈ n. from [10, lemma 4.2], we get sn(t) = dim[im(t) + ker(t n+1)]/[im(t) + ker(t n)], ∀ n ∈ n. the degree of stable iteration, p(t), of t is defined by p(t) = inf{n ∈ n : sm(t) = 0, ∀ m ≥ n}, where the infimum over the empty set is taken to be infinite. the following lemma helps to characterize the relationship between the degree of stable iteration and both the finite essential ascent and essential g-ascent of linear relations. essential g-ascent and g-descent of a closed relation 131 lemma 2.2. let t ∈ lr(h). (i) if ae(t) < +∞, then p(t) ≤ inf{n ∈ n : αn(t) = αi(t), ∀ i ≥ n} < +∞. (ii) if ãe(t) < +∞ (resp. ã(t) < +∞), then p(t) = inf{n ∈ n : α̃n(t) = α̃m(t), ∀ m ≥ n} < +∞ and ãe(t) ≤ p(t) (resp. ã(t) = p(t)). proof. (i) if ae(t) < +∞, then αn(t) < +∞, for every n ≥ ae(t). so, there exists m ≥ ae(t) such that αn(t) = αm(t), for all n ≥ m. now by [6, equality (16)], we get sn(t) = 0, for all n ≥ m, and this proves that p(t) ≤ m. (ii) by [6, equality (18)], we can prove this assertion similarly as in (i), which completes the proof. remark 2.3. let t ∈ lr(h) such that ae(t) < +∞ (resp. a(t) < +∞). we note that ae(t) ≤ p(t) (resp. a(t) = p(t)) in general is not true. indeed, let t be defined as in case 1 of example 2.1, then p(t) = 0 < ae(t) = a(t) = 1. the next lemma exhibits some useful entirely algebraic properties of the degree of stable iteration of linear relations. lemma 2.4. let t ∈ lr(h). (i) if de(t) < +∞, then p(t) ≤ m = inf{n ∈ n : βn(t) = βi(t), ∀ i ≥ n} < +∞. (ii) if d̃e(t) < +∞ (resp. d̃(t) < +∞), then p(t) = inf{n ∈ n : β̃n(t) = β̃m(t), ∀ m ≥ n} < +∞ and d̃e(t) ≤ p(t) (resp. d̃(t) = p(t)). 132 z. garbouj, h. skhiri proof. since ker(t n) + im(t) ⊆ ker(t n+1) + im(t) ⊆ d(t n+1) + im(t) ⊆ d(t n) + im(t), we deduce that βn(t) = dim[d(t n) + im(t)]/[d(t n+1) + im(t)] + βn+1(t) + sn(t). (1) but, since im(t) + ker(t n) ⊆ im(t) + ker(t n+1) ⊆ h, it follows that β̃n(t) = sn(t) + β̃n+1(t). (2) finally, the assertions (i) and (ii) follow from (1) and (2). the proof is therefore complete. remark 2.5. let t ∈ lr(h) such that de(t) < +∞ (resp. d(t) < +∞). we note that de(t) ≤ p(t) (resp. d(t) = p(t)) in general is not true. indeed, let t be defined as in case 4 of example 2.1, then p(t) = 0 < de(t) = d(t) = 1. 3. essential g-ascent spectrum and g-ascent spectrum of a closed relation this section contains the main results of this work, in which we generalize some results of [2, section 2] and our results of [5, section 3] to the case of a closed linear relation in a hilbert space h. throughout the remainder of the paper, for t ∈ lr(h) and λ ∈ c, we denote by tλ the relation λi − t. the ascent resolvent set of t ∈ cr(h) is the set ϱasc(t) = {λ ∈ c : a(tλ) < +∞ and im(tλ) + ker[(tλ)a(tλ)] is closed} and its ascent spectrum σasc(t) = c\ϱasc(t). the essential ascent resolvent and the essential ascent spectrum of t ∈ cr(h) are defined respectively by ϱeasc(t) = {λ ∈ c : ae(tλ) < +∞ and im(tλ) + ker[(tλ) ae(tλ)] is closed} essential g-ascent and g-descent of a closed relation 133 and σeasc(t) = c\ϱ e asc(t). the g-ascent resolvent set of t ∈ cr(h) is the set ϱgasc(t) = {λ ∈ c : ã(tλ) < +∞ and im(tλ) + ker[(tλ) ã(tλ)] is closed} and its g-ascent spectrum σgasc(t) = c\ϱ g asc(t). the essential g-ascent resolvent and the essential g-ascent spectrum of t ∈ cr(h) are defined respectively by ϱe, gasc(t) = {λ ∈ c : ãe(tλ) < +∞ and im(tλ) + ker[(tλ) ãe(tλ)] is closed} and σe, gasc(t) = c\ϱ e, g asc(t). from [6, lemma 2.9], we deduce easily the following ϱ(t) ⊆ ϱgasc(t) ⊆ ϱasc(t) ⊆ ϱ e asc(t), ϱ(t) ⊆ ϱgasc(t) ⊆ ϱ e, g asc(t) ⊆ ϱ e asc(t). let us recall the following definition. definition 3.1. ([7], definition 3.3, definition 4.1) (i) a subspace m of a hilbert space h is said to be a range subspace of h if there exist a hilbert space n and a bounded operator t from n to h such that m = im(t). in particular, a closed subspace of a hilbert space h is a range subspace of h. (ii) an operator or a relation t ∈ lr(h) is said to be a range space operator or range space relation if its graph g(t) is a range subspace of h × h. it is clear that a closed relation in a hilbert space h is a range space relation in h. our approach here is based in the concept of range subspaces of hilbert spaces (see, [7]). we have the following lemma, which will be needed in the sequel. 134 z. garbouj, h. skhiri lemma 3.2. let t ∈ cr(h). (i) if ae(t) < +∞, t k(0) and im(t) + ker(t ae(t)) are closed for all k ∈ n, then a) im(t n)+ker(t m) is closed, for all m+n ≥ p(t). in particular im(t n) is closed for all n ≥ p(t). b) im(t n) + ker(t m) is closed, for all n ∈ n and m ≥ ae(t). (ii) if ãe(t) < +∞ and im(t) + ker(t ãe(t)) is closed, then a) t n(0) is closed, for all n ∈ n, and so from [6, lemma 2.9] and (i), im(t n) + ker(t m) is closed, for all m + n ≥ p(t) (resp. n ∈ n and m ≥ ae(t)), b) ker(t n) and im(t ae(t)+n) are closed, for all n ∈ n. proof. (i) from [6, equality (17)], we have im(t n) + ker(t m) is closed for all n, m ∈ n such that m + n ≥ p(t). in particular im(t n) + ker(t p(t)+ae(t)) is closed for every n ∈ n, so by [6, lemma 2.9], im(t n) + ker(t m) is closed, for all n ∈ n and m ≥ ae(t). (ii) from [6, lemma 2.10], it follows that t n(0) and im(t ae(t)+n) are closed, for all n ∈ n. let s = t|im(t p(t )) be the restriction of t to im(t p(t)). now, let n ∈ n, by lemma 2.2, we have ãe(t) ≤ p(t), this implies that dim ker(t n) ∩ im(t p(t)) = α(sn) ≤ n α(s) = n α̃p(t)(t) < +∞ (see [9, lemma 5.1]) and as im(t p(t))+ker(t n) is closed, so by [7, propositions 3.9, 3.10, 4.8, lemma 4.2], ker(t n) is closed, for all n ∈ n. this completes the proof of the lemma. for t ∈ lr(h) and k ∈ n, t̃k denotes the following linear relation : t̃k : d(t)/ker(t k) ⊆ h/ker(t k) −→ h/ker(t k) x 7−→ tx := {z : z ∈ tx}. we will prove first that the linear relation t̃k is well-defined. to do this, let us choose x1, x2 ∈ d(t) such that x1 −x2 ∈ ker(t k). so, 0 ∈ t k(x1 −x2), and therefore there exists x ∈ t(x1 − x2) = tx1 − tx2, such that 0 ∈ t k−1(x). essential g-ascent and g-descent of a closed relation 135 from this, we get x ∈ ker(t k−1) and 0 ∈ tx1 − tx2. let y ∈ t(x1), since t(x1) = y + t(0) then 0 ∈ y + t(0) − tx2 = y − tx2. hence, t(x1) ⊆ t(x2) and by interchanging x1 and x2, we deduce that t(x1) = t(x2). let m be a closed subspace of h, then h/m is a hilbert space with the inner product < · , · >m : h/m × h/m −→ r (x , y) 7−→ < p(x) , p(y) >, where p is the orthogonal projection onto m⊥ and < · , · > is the standard inner product on h. note, the hilbert space topology of (h/m, < · , · >m) coincides with the quotient topology of h/m : ∥x∥ = √ < x, x >m = √ < p(x), p(x) > = dist(x, m), where dist(x, m) denotes, as usual, the distance of x to m. now, let t ∈ cr(h) such that ãe(t) < +∞ and im(t) + ker(t ãe(t)) is closed. from lemma 3.2, it follows easily that h/ker(t n) is a hilbert space, for all n ∈ n. in [2, lemma 2.1], it was established that if t is a bounded operator and t admits a finite essential g-ascent such that im(t) + ker(t ãe(t)) is closed, then t̃j is both regular and upper semi-fredholm operator, for all j ≥ p(t), where t̃j is the operator induced by t on h/ker(t j). in [5, lemma 3.4], this result was extended to the case of unbounded closed operators, and in the following lemma, we prove that this result remains valid even in the context of closed linear relations. lemma 3.3. let t ∈ cr(h) such that ãe(t) < +∞ and j ≥ p(t). if im(t) + ker(t ãe(t)) is closed, then t̃ : d(t)/ker(t j) ⊆ h/ker(t j) −→ h/ker(t j) x 7−→ tx := {y : y ∈ tx} is both regular and upper semi-fredholm relation. proof. first, recall that from lemma 3.2, we have ker(t n) is closed for all n ∈ n, and in particular, h/ker(t j) is a hilbert space. now, we will 136 z. garbouj, h. skhiri show that ker(t̃) = ker(t j+1)/ker(t j). to do this let x ∈ ker(t j+1), so y ∈ tx and 0 ∈ t jy for some y ∈ h. since 0h/ker(t j) = y ∈ tx, it follows that ker(t j+1)/ker(t j) ⊆ ker(t̃). in order to prove the converse inclusion, assume that x ∈ ker(t̃). from 0 ∈ tx, we deduce that t(x) ∩ ker(t j) ̸= ∅, which shows that x ∈ t −1(ker(t j)) = ker(t j+1). consequently ker(t̃) ⊆ ker(t j+1)/ker(t j). moreover, it is clear that im(t̃) = [im(t) + ker(t j)]/ker(t j), so from lemma 3.2, we get im(t̃) is closed. let π be the natural quotient map with domain h×h and null space ker(t j)×ker(t j), then g(t̃) = π(g(t)). from [7, corollary 4.9], we know that there exist a hilbert space z and a bounded operator θ from z to h × h such that im(θ) = g(t). this implies that πθ(z) = g(t̃). therefore t̃ is a range space relation. on the other hand, we have α(t̃) = αj(t) < +∞ and im(t̃) is closed. so from [7, lemma 4.6], t̃ is closed, and hence t̃ is upper semi-fredholm. furthermore, by [7, lemma 2.5], it follows that ker(t j+1) ⊆ im(t n)+ker(t j), for all n ∈ n. consequently, ker(t̃) ⊆ im(t̃ n), for all n ∈ n, and the proof is therefore complete. now, we are ready to state our main result of this section, which is an extension of [2, theorem 2.3] and [5, theorem 3.8] to closed multivalued linear operators. theorem 3.4. let t ∈ cr(h) such that ãe(t) < +∞ and im(t) + ker(t ãe(t)) is closed. then there is ε > 0 such that for every 0 < |λ| < ε, we have : (i) tλ is both regular and upper semi-fredholm, (ii) αp(t)(t) ≤ α(tλ) ≤ (p(t) + 1) α̃p(t)(t), (iii) β(tλ) = dim h/[im(t) + ker(t p(t))]. proof. let k = p(t) and let t̃k be the relation induced by t on h/ker(t k). first, from lemma 3.3, t̃k is both regular and upper semi-fredholm relation. using [1, theorems 23, 25], we deduce that there exists ε > 0, such that λi−t̃k is both regular and upper semi-fredholm relation for every 0 < |λ| < ε. furthermore, by [4, corollary v.15.7] and [1, theorem 27], we have α(t̃k) = α(λi − t̃k), β(t̃k) = β(λi − t̃k), ∀ λ ∈ c, 0 < |λ| < ε. (1) we prove first that ker(λi − t̃k) = [ker(λi − t) + ker(t k)]/ker(t k), (2) essential g-ascent and g-descent of a closed relation 137 im[(λi − t̃k)n] = im[(λi − t)n]/ker(t k), ∀ n ∈ n. (3) if x ∈ ker(λi − t̃k), then (λi − t)(0) = (λi − t)(x), and so t(0) + ker(t k) = (λi−t)x+ker(t k). it is clear that x ∈ d(t i), for all i ∈ n and t(0)+t −k(0) = (λi −t)x+t −k(0). from [8, corollary 2.1] and [4, corollary i.2.10], we obtain t k+1(0) + t k(0) = (λi − t)t kx + t k(0). since t k(0) ⊆ t k+1(0) = (λi − t)t k(0) (see [8, theorem 3.6 ]), it follows that (λi −t)t kx = (λi −t)t k(0), which implies that x ∈ ker[(λi − t)t k] and x ∈ ker[(λi − t)t k]/ker(t k). consequently ker(λi − t̃k) ⊆ ker[(λi − t)t k]/ker(t k). to prove the converse inclusion, let x ∈ ker[(λi − t)t k]/ker(t k), so 0 ∈ t k(λi − t)x. this implies that there exists z ∈ h such that z ∈ (λi − t)x and 0 ∈ t kz. hence 0 = z ∈ (λi − t)x, and x ∈ ker(λi − t̃k). now from [8, theorem 3.4], we obtain ker(λi − t̃k) = [ker(λi − t) + ker(t k)]/ker(t k). since ker(t k) ⊆ im[(λi − t)n], it follows that im[(λi − t̃k)n] = [im[(λi − t)n] + ker(t k)]/ker(t k) = im[(λi − t)n]/ker(t k). (i) let s = t|im(t k) be the restriction of t to im(t k). by (1) and (2), we obtain dim ker(λi − t)/ker(λi − t) ∩ ker(t k) = dim[ker(λi − t) + ker(t k)]/ker(t k) =α(t̃k) < +∞ (4) and dim ker(λi − t) ∩ ker(t k) ≤ dim im(t k) ∩ ker(t k) = α(sk) ≤ kα(s) = k α̃k(t) < +∞ (5) (see [9, lemma 5.1]). in particular, this proves that α(λi − t) < +∞. from (3), we infer that im(λi − t) is closed. finally, since λi − t̃k is regular, by (2) and (3), we deduce that λi − t is regular. (ii) from (4) and (5), we get αk(t) = α(t̃k) ≤ α(λi − t) = αk(t) + dim ker(λi − t) ∩ ker(t k) ≤ α̃k(t) + k α̃k(t) ≤ (k + 1) α̃k(t). assertion (iii) follows from (1) and (3), which completes the proof. 138 z. garbouj, h. skhiri the following theorem is a simple consequence of theorem 3.4. theorem 3.5. let t ∈ cr(h) such that ã(t) < +∞ and im(t) + ker(t ã(t)) is closed. then there is ε > 0 such that for every 0 < |λ| < ε, we have (i) tλ is injective with closed range, (ii) β(tλ) = dim h/[im(t) + ker(t p(t))]. corollary 3.6. let t ∈ cr(h), then σ g asc(t) and σ e, g asc(t) are closed. for t ∈ cr(h), we consider the set : e(t) = {λ ∈ σ(t) : λ an isolated point, ã(tλ) < +∞, d̃(tλ) = m < +∞ and im[(tλ)m] is closed}. the following lemma is the key to prove theorem 3.8. lemma 3.7. let t ∈ cr(h), then e(t) = {λ ∈ σ(t) : ã(tλ) < +∞ and d̃(tλ) < +∞}. proof. assume that t has finite g-ascent and g-descent. first, by lemma 2.2 and lemma 2.4, we note that ã(t) = d̃(t) = p(t). let m = d̃(t), s = t|im(t m) be the restriction of t to im(t m) and t̃m be the relation induced by t on h/ker(t m), then by [9, lemma 5.1], dim im(t m) ∩ ker(t k) = α(sk) ≤ k α(s) = kα̃m(t) = 0 and dim h/[im(t k) + ker(t m)] = β(t̃m k ) ≤ k β(t̃m) = k β̃m(t) = 0, for all k ∈ n. therefore im(t m) ∩ ker(t k) = {0} and im(t k) + ker(t m) = h, ∀ k ≥ 0. (1) using now the equality (1) for k = m and [7, propositions 3.10 and 4.8, lemma 4.2], we get im(t m) is closed. from (1) and theorem 3.5, we deduce that there exists ε > 0 such that λi − t is bijective, for every 0 < |λ| < ε. consequently if 0 ∈ σ(t), then 0 is an isolated point of σ(t). this completes the proof of lemma 3.7. essential g-ascent and g-descent of a closed relation 139 the following theorem is an extension of [2, theorem 2.7] and [5, theorem 3.10], to case of closed linear relations. theorem 3.8. let t ∈ cr(h), then ϱe, gasc(t) ∩ ∂σ(t) = ϱ g asc(t) ∩ ∂σ(t) = ϱeasc(t) ∩ ∂σ(t) = ϱasc(t) ∩ ∂σ(t) = e(t). proof. by arguing as in the proof of [5, theorem 3.10], with theorem 3.4 and lemma 3.7, we get the result. as an immediate consequence of theorem 3.8, we have the following result. corollary 3.9. let t ∈ cr(h) such that ϱ(t) ̸= ∅. then the following assertions are equivalent : (i) σasc(t) = ∅; (ii) σeasc(t) = ∅; (iii) ∂σ(t) ⊆ ϱasc(t); (iv) ∂σ(t) ⊆ ϱeasc(t); (v) σ(t) = e(t). remark 3.10. we note that corollary 3.9 in general is not true if ϱ(t) = ∅. indeed, let m ̸= {0} be a closed subspace of h and let t : h −→ h be the linear relation defined by tx = m for all x ∈ h. it is clear that t ∈ cr(h). for each λ ∈ c and for each n ∈ n\{0}, we have (tλ) nx = λnx + m, ∀ x ∈ h. since ker[(tλ) n] = { h if λ = 0 m if λ ̸= 0 ̸= {0} and im[(tλ)n] = { h if λ ̸= 0 m if λ = 0, it follows that a(tλ) = 1 and im(tλ) + ker(tλ) = h, for all λ ∈ c. this implies that ϱ(t) = ∅ and ϱasc(t) = c. hence, ϱeasc(t) = c and t satisfies the conditions (i)-(iv) of corollary 3.9. since e(t) consists of isolated points of σ(t), we deduce that e(t) = ∅, and thus t does not satisfy the condition (v). 140 z. garbouj, h. skhiri 4. spectral mapping theorems of g-ascent and essential g-ascent spectrums we start this section with the following proposition. proposition 4.1. let t ∈ cr(h) be everywhere defined and suppose that t k(0) is closed for all k ∈ n. if n = ae(t) (resp. a(t), ãe(t), ã(t)) is finite, then im(t) + ker(t n) is closed if and only if im(t n+1) is closed. proof. first, from [7, lemma 4.6, corollary 4.9], we get t k ∈ cr(h), for all k ∈ n. put n = ae(t) (resp. a(t), ãe(t), ã(t)). if im(t) + ker(t n) is closed, by [6, lemma 2.9] and lemma 3.2, we get im(t n+1)+ker(t n) is closed and by [9, lemma 3.2], we have dim[im(t n+1) ∩ ker(t n)]/[t n+1(0) ∩ ker(t n)] = dim ker(t 2n+1)/ker(t n+1) = n∑ i=1 αn+i(t) ≤ n αn+1(t) < +∞. then from [7, propositions 3.9, 3.10, 4.8, lemma 4.2], it follows that im(t n+1) is closed. now, suppose that im(t n+1) is closed, from [3, lemma 2.4], we obtain im(t) + ker(t n) = t −n[im(t n+1)] is closed, which completes the proof. corollary 4.2. let t ∈ cr(h) be everywhere defined and suppose that t n(0) is closed for all n ∈ n. then (i) ϱ g asc(t) = {λ ∈ c : ã(tλ) < +∞ and im[(tλ)ã(tλ)+1] is closed}, (ii) ϱ e, g asc(t) = {λ ∈ c : ãe(tλ) < +∞ and im[(tλ)ãe(tλ)+1] is closed}, (iii) ϱasc(t) = {λ ∈ c : a(tλ) < +∞ and im[(tλ)a(tλ)+1] is closed}, (iv) ϱeasc(t) = {λ ∈ c : ae(tλ) < +∞ and im[(tλ)ae(tλ)+1] is closed}. let t ∈ lr(h), for every non-constant polynomial p = n∏ i=1 (λi−x)αi, with coefficients in c, we can associate the linear relation p(t) ∈ lr(h) defined by : p(t) := n∏ i=1 (λii − t)αi. essential g-ascent and g-descent of a closed relation 141 remark 4.3. let t ∈ cr(h) be everywhere defined such that ϱ(t) ̸= ∅. if p is a non-constant complex polynomial, from corollary 4.2 and [3, theorem 4.7], it follows that σasc(p(t)) = p(σasc(t)) and σ e asc(p(t)) = p(σ e asc(t)). for t ∈ lr(h), we remark that d(t −1) = inf{n ∈ n : d(t n) = d(t n+1)}, where as usual the infimum over the empty set is taken to be +∞. hence, if d(t −1) < +∞ then d(t d(t −1)) = d(t d(t −1)+n) ⊆ d(t n), ∀ n ∈ n. example 4.4. let h be a separable hilbert space and let k ∈ lr(h). consider the linear relation t : ⊕∞ i=0 h −→ ⊕∞ i=0 h defined by t(h0 ⊕ h1 ⊕ h2 ⊕ . . .) = k(h1) ⊕ h2 ⊕ h3 ⊕ . . . . clearly, im(t 2) = im(t) and d(t k) = h ⊕ i=k⊕ i=1 d(k) ⊕ ∞⊕ i=k+1 h, ∀k ≥ 1. hence, if d(k) h then d(t −1) = +∞, and, if d(k) = h then d(t −1) = 0. let s = t −1, so if im(k) h then d(s−1) = d(t) = 1, and, d(s−1) = 0 when im(k) = h. let us assume that t is a range space relation (see, definition 3.1) such that q = d(t −1). it is clear that if p = (λ1−x)α1(λ2−x)α2 · · · (λm−x)αm is a complex polynomial then p(t) is a range space relation (see, [7, propositions 4.7, 4.8]) and j = d(p(t)−1) ≤ q according to [8, theorem 3.2]. furthermore, if p is a non-constant polynomial then d([p(t)]j) = d(t q). let us assume that p is a non-constant complex polynomial and t(0) ⊆ d(t q), then for all n ≥ q and m ∈ n, we have im ( [p(t)]n ) ⊆ d(t m). indeed, we prove by induction that t n(0) ⊆ d(t q). the cases n = 0, 1 are obvious. suppose that t n(0) ⊆ d(t q), then t n+1(0) = t ( t n(0) ) ⊆ t ( d(t q+1) ) = tt −1(d(t q)) = d(t q) ∩ im(t) + t(0) ⊆ d(t q). this implies that t n(0) ⊆ d(t q) for all n ∈ n, and consequently t n(0) ⊆ d(t m) for all n, m ∈ n. finally, by [10, lemma 4.1], we get im ( [p(t)]n ) ⊆ d(t q) ⊆ d(t m), for every n ≥ q and m ∈ n. 142 z. garbouj, h. skhiri in the following, we define υ(h) = { t ∈ lr(h) : t is a range space relation, q = d(t −1) < +∞, t(0) ⊆ d(t q), t n(0), d(t n+2) and im(tλ) + d(t q) are closed, ∀ λ ∈ c, ∀ n ∈ n } . clearly, υ(h) ̸= ∅, because t ∈ υ(h), when t is a closed linear relation everywhere defined such that t n(0) is closed for all n ∈ n. for family of vectors (xi)i∈i in h, we denote by vect(xi, i ∈ i), the vector subspace generated by (xi)i∈i. example 4.5. (i) let h be a separable hilbert space and let k ∈ cr(h) such that d(k) h is closed. let h = ⊕3 i=0 h and consider the linear relation t : h −→ h defined by t(h0 ⊕ h1 ⊕ h2 ⊕ h3) = k(h1) ⊕ h2 ⊕ h3 ⊕ h3. clearly, t n(0 ⊕ 0 ⊕ 0 ⊕ 0) = k(0) ⊕ 0 ⊕ 0 ⊕ 0 is closed for all n ≥ 1 and d(t k) =   h ⊕ d(k) ⊕ h ⊕ h if k = 1 h ⊕ d(k) ⊕ d(k) ⊕ h if k = 2 h ⊕ d(k) ⊕ d(k) ⊕ d(k) if k ≥ 3 is closed. hence d(t −1) = 3 and t(0 ⊕ 0 ⊕ 0 ⊕ 0) ⊆ d(t 3). it is not difficult to see that im(tλ) + d(t 3) = { h ⊕ h ⊕ h ⊕ d(k) if λ = 1 h ⊕ h ⊕ h ⊕ h if λ ̸= 1 is closed. since t ∈ cr(h), it follows that t ∈ υ(h). assume that k(0) h and let s = t −1. it is easy to see that t n(h0 ⊕ h1 ⊕ h2 ⊕ h3) = k(h3) ⊕ h3 ⊕ h3 ⊕ h3, ∀h0 ∈ h, ∀ h1, h2, h3 ∈ d(k), for all n ≥ 3. therefore d(s−1) = d(t) ≤ 3. let h ∈ h\k(0) and ξ = h ⊕ 0 ⊕ 0 ⊕ 0, then ξ ∈ s(0) = ker(t) = h ⊕ ker(k) ⊕ {0} ⊕ {0} and ξ ̸∈ d(s3) = im(t 3). consequently, s ̸∈ υ(h). (ii) let h be a separable hilbert space and (en)n∈n be an orthonormal basis of h. define the following operators t and l in h by d(t) = d(l) = vect(en : n ≥ 2), t(en) = en+1 essential g-ascent and g-descent of a closed relation 143 and l(en) = en−1, for all n ≥ 2. it is clear that d(t k) = d(t) and d(lk) = vect(en : n ≥ 1 + k), for all k ≥ 1 and hence d(t −1) = 1 and d(l−1) = +∞ (l ̸∈ υ(h)). since t ∈ cr(h), im(tλ) ⊆ d(t) for all λ ∈ c and d(t) is closed, then t ∈ υ(h). (iii) let t be defined as in (ii) and k ≥ 2. now, we define the following relation s := im(t k) + t (i.e., s(x) = {y + z : y ∈ im(t k), z ∈ t(x)}, for all x ∈ d(t)). since s(0) = im(t k) is closed (because β(t) = 3), t is a closed operator and d(s) = d(t) is closed, then ∥qss(x)∥ = ∥tx∥ ≤ ∥tx∥ ≤ ∥t∥ ∥x∥, for all x ∈ d(s). this proves that qss is closed and by [4, proposition ii.5.3], we get that s ∈ cr(h). it is clear that sj = { im(t k) + t j if j < k im(t k) if j ≥ k. for all j ≥ 1, we have d(sj) = d(t) and sj(0) = im(t k) are closed in h, and from this we get that d(s−1) = 1 and s(0) ⊆ d(s). moreover, for all λ ∈ c, we see that im(sλ) + d(s) = im(tλ) + im(t k) + d(t) = d(t) is closed. now, we can conclude that s ∈ υ(h). (iv) let t be defined as in example 2.1, where m is a closed subspace of h and m h. it is easy to see that g(t) = m × m and t n(0) = d(t n) = im(tλ) = m, ∀ n ≥ 1, λ ∈ c. from this it follows that t ∈ υ(h). (v) let l be defined as in example 2.1. it is not difficult to see that l is a closed relation, d(l−1) = 1, d(ln) = n is closed, d(ln)+im(lλ) = h, ∀n ≥ 1, λ ∈ c. but l(0) = m ̸⊂ d(l), so that l ̸∈ υ(h). 144 z. garbouj, h. skhiri let us show that if t ∈ υ(h) and p = (λ1−x)α1(λ2−x)α2 · · · (λm−x)αm is a non-constant complex polynomial, then a = p(t) ∈ υ(h). put q = d(t −1) and define the following relation t : d(t) ⊆ h/d(t q) −→ h/d(t q) x 7−→ tx. note that the linear relation t is well-defined. indeed, since t(0) ⊆ d(t q), t(d(t q)) = t(d(t q+1)) = tt −1(d(t q)) = d(t q) ∩ im(t) + t(0) ⊆ d(t q), which implies that if x1, x2 ∈ d(t) such that x1 = x2, then tx1 = tx2. we remark also that t(0) = t(0) = 0 and thus t is an unbounded operator. now, let λ ∈ c and x ∈ ker(λi − t), then tλx = tλ(0) = 0, which implies that tλx ⊆ d(t q) and x ∈ d(t q+1) = d(t q). consequently, x = 0, and thus ker(λi − t) = {0}. on the other hand, it is clear that im(λi − t) = ( im(tλ) + d(t q) ) /d(t q) is closed. as in the proof of lemma 3.3, we obtain λi − t is a range space operator. now, applying [7, lemma 4.6], we get λi − t is a closed operator, and thus λi − t ∈ φ+(h/d(t q)). recall that, if s, l ∈ cr(h) are two operators such that l ∈ φ+(h) and im(s) is closed, then ls ∈ cr(h) and im(ls) is closed. since, for every i, j ∈ {1, 2, · · · , m}, λii − t ∈ φ+(h/d(t q)) and im(λji − t) is closed, we deduce that (λii − t)(λji − t) ∈ cr(h/d(t q)) and im[(λii − t)(λji − t)] is closed. consequently, from ker ( (λii − t)(λji − t) ) = {0}, it follows that (λii − t)(λji − t) ∈ φ+(h/d(t q)). therefore im(p(t)) = ( im(p(t)) + d(t q) ) /d(t q) is closed, and finally we obtain im(a) + d(ad(a −1)) = im ( p(t) ) + d(t q) is closed. now, let λ ∈ c and put q = λ −p = a m∏ i=1 (µi −x)βi. arguing in the same way as previous, we can conclude that im(aλ) + d(ad(a −1)) is closed. let k = m∑ i=0 αi and n ∈ n, by [8, theorems 3.2, 3.6], we get an(0) = t k n(0) and d(an+2) = d(t (n+2)k) are closed, and a(0) ⊆ d(t q) = d(ad(a −1)). this proves that, a ∈ υ(h). the aim of this section is to establish the spectral mapping theorem of gascent and essential g-ascent spectrums of a closed linear relation t ∈ υ(h). first, we have the following remark. essential g-ascent and g-descent of a closed relation 145 remark 4.6. (i) if t ∈ υ(h), then p(t) ∈ cr(h), for any complex polynomial p of degree n ≥ min{d(t −1), 2}. indeed, we have d[p(t)] = d(t n) and p(t)(0) = t n(0) are closed, and p(t) is a range space relation. so, by [7, lemma 4.6], it follows that p(t) ∈ cr(h). (ii) let t ∈ υ(h) such that d(t) is closed or t be a closed relation. by (i) and [7, lemma 4.6], we deduce that p(t) ∈ cr(h), for any complex polynomial p. the next lemma is used to prove lemma 4.8. lemma 4.7. let t ∈ lr(h), p = m∑ i=0 aix i = α m∏ i=1 (λi − x) and q = n∑ i=0 bix i = β n∏ i=1 (λi − x) be non-constants complex polynomials. then (i) p(t) = m∑ i=0 ait i, (ii) p(t) + q(t) = (p + q)(t) + t s − t s, where s = max{n, m}. proof. it is easy to see that if ξ ∈ c and i, j ∈ n such that j ≤ i, then ξt j(x) + t i(0) = t j(ξx) + t i(0), ∀ x ∈ d(t j). (1) it follows from this that for all µ ∈ c\{0}, ξt j(x) + µt i(x) = t j(ξx) + µt i(x), ∀ x ∈ d(t i). (2) (i) we will prove that p(t) = m∑ i=0 ait i and am = (−1)mα. by [4, proposition i.4.2], we know that if r, s, l ∈ lr(h) such that d[l(r+s)] = d(lr+ls), then l(r + s) = lr + ls. so by [8, theorem 3.2] and (2), we get (λ1i − t)(λ2i − t)x = (λ1i − t)(λ2x) − (λ1i − t)tx = λ1λ2x − t(λ2x) − t(λ1x) + t 2(x) = λ1λ2x − t [(λ2 + λ1)x] + t 2(x) = λ1λ2x − (λ2 + λ1)t(x) + t 2(x), ∀ x ∈ d(t 2). suppose that α(λ1i − t)(λ2i − t) · · · (λm−1i − t) = m−2∑ i=0 αit i + α (−1)m−1t m−1 146 z. garbouj, h. skhiri and let us show that α(λ1i − t)(λ2i − t) · · · (λmi − t) = m−1∑ i=0 γit i + α (−1)mt m. by [8, corollary 2.1] and (2), we obtain p(t)x = α(λ1i − t)(λ2i − t) · · · (λmi − t)x = (λmi − t)[α(λ1i − t)(λ2i − t) · · · (λm−1i − t)]x = (λmi − t) ( m−2∑ i=0 αit i + α (−1)m−1t m−1 ) x = (λmi − t) ( m−2∑ i=0 t i(αix) + α (−1)m−1t m−1x ) = m−2∑ i=0 t i(λmi − t)(αix) + α (−1)m−1t m−1(λmi − t)x = m−2∑ i=0 t i(λmαix) − m−2∑ i=0 t i+1(αix) + α (−1)m−1t m−1(λmx) + α (−1)mt mx = m−2∑ i=0 t i(λmαix) − m−1∑ i=1 t i(αi−1x) + α (−1)m−1t m−1(λmx) + α (−1)mt mx so, p(t)x = λmα0x + m−2∑ i=1 t i[(λmαi − αi−1)x] + t m−1[(α (−1)m−1λm − αm−2)x] + α (−1)mt mx = λmα0︸︷︷︸ γ0 x + m−2∑ i=1 (λmαi − αi−1)︸︷︷︸ γi t i(x) + (α (−1)m−1λm − αm−2)︸︷︷︸ γm−1 t m−1(x) + α (−1)mt mx = m−1∑ i=0 γit ix + α (−1)mt mx, ∀ x ∈ d(t m). essential g-ascent and g-descent of a closed relation 147 this shows that p = m−1∑ i=0 γix i + α (−1)mxm and hence ai = γi, for all i = 1, · · · , m − 1 and am = α (−1)m. (ii) assume that m ≤ n, from (i) and (1), we see that [p(t) + q(t)](x) = [p(t) + q(t)](x) + [p(t) + q(t)](0) = m∑ i=0 ait ix + n∑ i=0 bit ix + t n(0) = m∑ i=0 (ait ix + bit ix) + n∑ i=m+1 bit ix + t n(0) = m∑ i=0 [t i(aix) + t i(bix)] + n∑ i=m+1 bit i(x) + t n(0) = m∑ i=0 t i[(ai + bi)x] + n∑ i=m+1 bit i(x) + t n(0) = m∑ i=0 (ai + bi)t i(x) + n∑ i=m+1 bit i(x) + t n(0) = n∑ i=0 ωit i(x) + t n(0), ∀ x ∈ d(t n), since p + q = n∑ i=0 ωix i, then p(t) + q(t) = (p + q)(t) + t n − t n. this completes the proof. the following result is an improvement of [5, lemma 4.4] to closed linear relations. lemma 4.8. let t ∈ lr(h) and let p and q be two relatively prime complex polynomials. if a = p(t) and b = q(t), then (i) im(anbn) = im(an) ∩ im(bn), for all n ∈ n, (ii) ker(anbn) = ker(an) + ker(bn), for all n ∈ n, (iii) ker(an) ⊆ im(bm) and ker(bn) ⊆ im(am), for all n, m ∈ n, (iv) ãe(ab) = max{ãe(a), ãe(b)} and ã(ab) = max{ã(a), ã(b)}. in addition, assume that t ∈ υ(h), 148 z. garbouj, h. skhiri (v) if max{ãe(a), ãe(b)} < +∞, then im(a) + ker(aãe(a)) and im(b) + ker(bãe(b)) are both closed if and only if im(ab) + ker[(ab)ãe(ab)] is closed, (vi) if max{ã(a), ã(b)} < +∞, then im(a) + ker(aã(a)) and im(b) + ker(bã(b)) are both closed if and only if im(ab) + ker[(ab)ã(ab)] is closed. proof. the proof is trivial when p or q is a constant polynomial. assume that p and q are non-constants polynomials. so, p = (λ1 − x)α1(λ2 − x)α2 · · · (λm − x)αm and q = (µ1 − x)β1(µ2 − x)β2 · · · (µs − x)βs. first, it is clear that the assertions (i) and (ii) follow immediately from [8, theorems 3.3, 3.4]. (iii) by [10, lemma 7.2], we know that ker[(λii − t)αi] ⊆ im[(µji − t)βj ], so from [8, theorems 3.3, 3.4], ker(an) ⊆ im(bm), for all n, m ∈ n. (iv) let n ∈ n, we have im(anbn) ∩ ker(ab) = im(an) ∩ im(bn) ∩ [ker(a) + ker(b)] = im(an) ∩ [ker(a) + im(bn) ∩ ker(b)] = [im(an) ∩ ker(a)] + [im(bn) ∩ ker(b)]. therefore α̃n(ab) = 0 ⇐⇒ max{α̃n(b), α̃n(a)} = 0, α̃n(ab) < +∞ ⇐⇒ max{α̃n(b), α̃n(a)} < +∞. (v) let n ∈ n\{0}. since p n and qn are relatively prime, we know that there exist two polynomials pn = a p∏ i=1 (νi − x)ri and qn = b r∏ i=1 (ωi − x)ji such that p npn + q nqn = 1. let pn (resp. k) be the degree of pn (resp. p) and α(n) = nk + pn. then, the degree of p npn (resp. q nqn) is α(n) and by [8, theorem 3.2], we get d[anpn(t) + bnqn(t)] = d[anpn(t)] ∩ d[bnqn(t)] = d(t α(n)). now, by lemma 4.7, we obtain anpn(t)x + b nqn(t)x = x + t α(n)(0), ∀ x ∈ d(t α(n)). (1) if n ≥ q = d(t −1), then α(n) ≥ q, and by (1), it is clear that d(t q) = d(t α(n)) ⊆ im(an) + im(bn). essential g-ascent and g-descent of a closed relation 149 since im(an) ⊆ d(t q) and im(bn) ⊆ d(t q), for every j ∈ n, it follows that d(t q) = im(an) + im(bn) = [ker(aj) + im(an)] + [im(bn) + ker(bj)] (2) and ker(ajbj) + im(anbn) = ker(aj) + ker(bj) + im(an) ∩ im(bn) = [ker(aj) + im(an)] ∩ im(bn) + ker(bj) = [ker(aj) + im(an)] ∩ [im(bn) + ker(bj)]. (3) from [7, propositions 3.9, 3.10, 4.8, lemma 4.2] and by using (2) and (3), we get ker(ajbj) + im(anbn) is closed ⇔ { ker(aj) + im(an), ker(bj) + im(bn) are both closed.(4) let n > q = d(t −1) and j = ãe(a) + ãe(b) + ãe(ab) + p(a) + p(b) + p(ab). let us assume that im(ab) + ker(ajbj) is closed. since the degree of pq is greater than or equal to two, then from remark 4.6, ab ∈ cr(h). it follows from [6, lemma 2.9] and lemma 3.2 that im(anbn) + ker(ajbj) is closed. let us assume that n = im(anbn)+ker(ajbj) is closed. as d(bn−1an−1) = d(t q) is closed and im(bn−1an−1) ⊆ d(t q), s : d(t q) −→ d(t q) x 7−→ bn−1an−1x is well-defined. from [7, lemmas 4.6, 4.10, proposition 4.8], it follows that s ∈ cr(d(t q)). because d(s) = d(t q) is closed and s(0) ⊆ n ⊆ d(t q), by [3, lemma 2.4],( im(ab) + ker[(ab)j+n−1] ) ∩ d(t q) = s−1(n) is closed. since (im(ab) + ker[(ab)j+n−1]) + d(t q) = im(ab) + d[(ab)d(a −1b−1)] is closed, from [7, propositions 3.9, 3.10, 4.8, lemma 4.2], it follows that im(ab) + ker[(ab)j+n−1] is closed. now, by [6, lemma 2.9], im(ab) + ker[(ab)j] is closed, and thus im(ab) + ker(ajbj) is closed ⇐⇒ im(anbn) + ker(ajbj) is closed. (5) 150 z. garbouj, h. skhiri arguing in the same way as previous, we can conclude that im(a) + ker(aj) is closed ⇐⇒ im(an) + ker(aj) is closed, im(b) + ker(bj) is closed ⇐⇒ im(bn) + ker(bj) is closed. (6) finally, it follows from (4), (5), (6) and [6, lemma 2.9] that im(a)+ker(aãe(a)) and im(b)+ker(bãe(b)) are both closed if and only if im(ab)+ker[(ab)ãe(ab)] is closed. (vi) finally, by [6, lemma 2.9] and the assertion (v), we see that the following assertions are equivalent : a) im(a) + ker(aã(a)) is closed and im(b) + ker(bã(b)) is closed, b) im(a) + ker(aãe(a)) is closed and im(b) + ker(bãe(b)) is closed, c) im(ab) + ker[(ab)ãe(ab)] is closed, d) im(ab) + ker[(ab)ã(ab)] is closed. the proof is complete. theorem 4.9. let t ∈ υ(h) be a closed linear relation. if a and b are defined as in lemma 4.8, then 0 ∈ ϱe, gasc(ab) ⇐⇒ 0 ∈ ϱ e, g asc(a) ∩ ϱ e, g asc(b) and 0 ∈ ϱgasc(ab) ⇐⇒ 0 ∈ ϱ g asc(a) ∩ ϱ g asc(b). proof. this is an obvious consequence of remark 4.6 and lemma 4.8. theorem 4.10. let t ∈ υ(h) be a closed linear relation and m ∈ n\{0}. then 0 ∈ ϱe, gasc(t) ⇐⇒ 0 ∈ ϱ e, g asc(t m) and 0 ∈ ϱgasc(t) ⇐⇒ 0 ∈ ϱ g asc(t m). proof. first, since t ∈ υ(h) is closed, we obtain t m is closed (see remark 4.6). let n, m ∈ n\{0} and s = t|im(t mn) be the restriction of t to im(t mn). from [9, lemma 5.1], it follows that, α̃nm(t) ≤ α̃n(t m) = α(sm) ≤ mα(s) = mα̃nm(t), essential g-ascent and g-descent of a closed relation 151 and this proves that ãe(t) < +∞ if and only if ãe(t m) < +∞. now, put k = max{p(t m), p(t)} < +∞, then by lemma 2.2, k ≥ max{ãe(t), ãe(t m)}. let n > d(t −1), as in the proof of lemma 4.8 and according to [6, lemma 2.9] and lemma 3.2, we deduce that im(t m) + ker(t m k) is closed =⇒ im(t m n) + ker(t m k) is closed, =⇒ im(t) + ker(t m k) is closed. hence, from [6, lemma 2.9] and lemma 3.2, im(t) + ker(t ãe(t)) is closed ⇐⇒ im(t) + ker(t m k) is closed, ⇐⇒ im(t m) + ker(t m k) is closed, ⇐⇒ im(t m) + ker[(t m)ãe(t m)] is closed. this gives 0 ∈ ϱe, gasc(t) ⇐⇒ 0 ∈ ϱ e, g asc(t m). furthermore, for every m ∈ n\{0}, ã(t) < +∞ if and only if ã(t m) < +∞. so that im(t) + ker(t ã(t)) is closed ⇐⇒ im(t) + ker(t ãe(t)) is closed, ⇐⇒ im(t m) + ker(t m ãe(t m)) is closed, ⇐⇒ im(t m) + ker(t m ã(t m)) is closed. consequently, 0 ∈ ϱgasc(t) ⇐⇒ 0 ∈ ϱ g asc(t m), which completes the proof. corollary 4.11. let t ∈ υ(h) be a closed linear relation and let p = (λ1 − x)m1(λ2 − x)m2 · · · (λn − x)mn be a complex polynomial such that mi ̸= 0 for every i = 1, 2, · · · , n. then 0 ∈ ϱe, gasc(p(t)) ⇐⇒ λi ∈ ϱ e, g asc(t), ∀ 1 ≤ i ≤ n and 0 ∈ ϱgasc(p(t)) ⇐⇒ λi ∈ ϱ g asc(t), ∀ 1 ≤ i ≤ n. proof. first, since t ∈ υ(h) is closed, p(t) is closed (see remark 4.6). now, from theorem 4.9 and theorem 4.10, we deduce 0 ∈ ϱe, gasc(p(t)) ⇐⇒ 0 ∈ ∩ 1≤i≤n ϱ e, g asc[(λii − t)mi], ⇐⇒ 0 ∈ ∩ 1≤i≤n ϱ e, g asc(λii − t), ⇐⇒ λi ∈ ϱ e, g asc(t), ∀ 1 ≤ i ≤ n. 152 z. garbouj, h. skhiri in the same way, we prove that 0 ∈ ϱgasc(p(t)) ⇐⇒ λi ∈ ϱ g asc(t), ∀ 1 ≤ i ≤ n, and the proof is therefore complete. the following extends [3, theorem 4.7], we do not require that the relation t be everywhere defined and ϱ(t) ̸= ∅. theorem 4.12. let t ∈ υ(h) be a closed linear relation and p = (λ1 − x)m1(λ2 − x)m2 · · · (λn − x)mn be a complex polynomial such that mi ̸= 0 for all i = 1, 2, · · · , n. then p(σe, gasc(t)) = σ e, g asc(p(t)) and p(σ g asc(t)) = σ g asc(p(t)). proof. first note, by remark 4.6, p(t) is a closed linear relation. now, from corollary 4.11, it follows that λ ∈ p(σe, gasc(t)) ⇐⇒ λ = p(µ), where µ ∈ σ e, g asc(t), ⇐⇒ λ − p(z) = (µ − z)kq(z), where q(µ) ̸= 0, ⇐⇒ λ ∈ σe, gasc(p(t)). the second equality, can be proved in the same way as the first one. this finishes the proof of the theorem. example 4.13. (i) let t as in (i) of example 4.5, we have t ∈ υ(h) is a closed linear relation and d(t) h. hence, if p is a non-constant complex polynomial, by [3, theorem 4.7], it is not possible to deduce that p(σ g asc(t)) = σ g asc(p(t)) or p(σ e, g asc(t)) = σ e, g asc(p(t)). however, from theorem 4.12, we can do this. (ii) let b = (e1, e2, e3) be a basis of c3. consider the linear relation : t ( 3∑ i=1 αi ei ) = vect(e1, e2) + α3 e3, ∀ α1, α2, α3 ∈ c. clearly, t ∈ υ(h), because t is a closed linear relation everywhere defined and t n(0) is closed for all n ∈ n. for all λ ∈ c, we note that 0 ∈ (λi − t)(e1) = vect(e1, e2), this implies that ϱ(t) = ∅. hence, if p is a non-constant complex polynomial, by [3, theorem 4.7], it is not possible to deduce that p(σ g asc(t)) = σ g asc(p(t)) or p(σ e, g asc(t)) = σ e, g asc(p(t)). however, from theorem 4.12, we can do this. essential g-ascent and g-descent of a closed relation 153 remark 4.14. we note that theorem 4.12 is false in general without the assumption that p is a non-constant polynomial, even if t is a bounded linear operator. for example, if h is a separable hilbert space and (en)n∈n is an orthonormal basis of h, we define the following bounded operator on h, t ( +∞∑ n=0 xnen ) = +∞∑ n=0 xn n + 1 en. since ker(t) = {0} (which gives ae(t) = 0) and im(t) is not closed in h, it follows that 0 ∈ σeasc(t) ⊆ σasc(t). put p = c ∈ c, since σeasc(t) and σasc(t) are non-empty sets, it follows that p(σeasc(t)) = p(σasc(t)) = {c}. however, ϱasc(p(t)) = ϱasc(ci) = c. indeed, c\{c} = ϱ(ci) ⊆ ϱasc(ci), and ci − ci is the zero operator on h with ascent is equal to 1 and kernel is equal to h. hence σeasc(p(t)) = σasc(p(t)) = ∅. 5. a spectral mapping theorems for essential g-descent and g-descent spectrums we start this section with the following definitions. the descent and the essential descent resolvent sets of t ∈ lr(h) are respectively defined by ϱdes(t) = {λ ∈ c : d(tλ) < +∞} and ϱedes(t) = {λ ∈ c : de(tλ) < +∞}. the descent and the essential descent spectrums of t are respectively σdes(t) := c\ϱdes(t) and σedes(t) := c\ϱ e des(t). the g-descent resolvent set and the essential g-descent resolvent set of t ∈ lr(h) are respectively defined by ϱ g des(t) = {λ ∈ c : d̃(tλ) < +∞} and ϱ e, g des(t) = {λ ∈ c : d̃e(tλ) < +∞}. the g-descent and the essential g-descent spectrums of t are respectively σ g des(t) := c\ϱ g des(t) and σ e, g des(t) := c\ϱ e, g des(t). evidently ϱ(t) ⊆ ϱgdes(t) ⊆ ϱdes(t) ⊆ ϱ e des(t), ϱ(t) ⊆ ϱ g des(t) ⊆ ϱ e, g des(t) ⊆ ϱ e des(t). this section will be devoted to study the spectral mapping theorems of the g-descent and the essential g-descent spectrums of linear relations. the following lemmas will be used to prove the main result of this section. 154 z. garbouj, h. skhiri lemma 5.1. let t ∈ lr(h), p and q are relatively prime complex polynomials. let n ∈ n\{0}, a = p(t) and b = q(t). then (i) d̃e(ab) = max{d̃e(a), d̃e(b)} and d̃(ab) = max{d̃(a), d̃(b)}, (ii) t possess a finite g-descent (resp. essential g-descent) if and only if the same holds for t n. proof. (i) let n ∈ n, from lemma 4.8, we have ker(anbn) + im(ab) = ker(an) + ker(bn) + im(a) ∩ im(b), = ( ker(an) + im(a) ) ∩ im(b) + ker(bn), = ( ker(an) + im(a) ) ∩ ( im(b) + ker(bn) ) . therefore ker(anbn) + im(ab) ⊆ ker(an) + im(a) ⊆ h, and consequently, β̃n(ab) = β̃n(a) + dim ( ker(an) + im(a) ) / ( [ker(an) + im(a)] ∩ [im(b) + ker(bn)] ) , max{β̃n(a), β̃n(b)} ≤ β̃n(ab) ≤ β̃n(a) + β̃n(b). (ii) let m ∈ n and s be the relation induced by t on h/ker(t nm). thus from [9, lemma 5.1], we obtain β̃nm(t) = β(s) ≤ β(sn) = β̃m(t n) ≤ n β(s) = n β̃nm(t). this completes the proof. lemma 5.2. let t ∈ lr(h) and m ∈ n\{0}. assume that a and b are defined as in lemma 5.1. then (i) 0 ∈ ϱe, gdes(t) if and only if 0 ∈ ϱ e, g des(t m), (ii) 0 ∈ ϱgdes(t) if and only if 0 ∈ ϱ g des(t m), (iii) 0 ∈ ϱe, gdes(ab) if and only if 0 ∈ ϱ e, g des(a) ∩ ϱ e, g des(b), (iv) 0 ∈ ϱgdes(ab) if and only if 0 ∈ ϱ g des(a) ∩ ϱ g des(b). proof. it is an obvious consequence of lemma 5.1. essential g-ascent and g-descent of a closed relation 155 corollary 5.3. let t ∈ lr(h) and let p = (λ1 − x)m1(λ2 − x)m2 · · · (λn − x)mn be a complex polynomial such that mi ̸= 0 for every i = 1, 2, · · · , n. then 0 ∈ ϱe, gdes(p(t)) ⇐⇒ λi ∈ ϱ e, g des(t), ∀ 1 ≤ i ≤ n and 0 ∈ ϱgdes(p(t)) ⇐⇒ λi ∈ ϱ g des(t), ∀ 1 ≤ i ≤ n. proof. from lemma 5.2, it follows that 0 ∈ ϱe, gdes(p(t)) ⇐⇒ 0 ∈ ∩ 1≤i≤n ϱ e, g des[(λii − t) mi], ⇐⇒ 0 ∈ ∩ 1≤i≤n ϱ e, g des(λii − t), ⇐⇒ λi ∈ ϱ e, g des(t), ∀ 1 ≤ i ≤ n. in the same way, we obtain 0 ∈ ϱgdes(p(t)) ⇐⇒ λi ∈ ϱ g des(t), ∀ 1 ≤ i ≤ n. the proof is complete. first note, that the results of this section are true for banach spaces. hence, the following extends [3, theorem 3.4], we do not require that the relation t be everywhere defined and dim t(0) < +∞. theorem 5.4. let t ∈ lr(h) and p = (λ1 − x)α1 · · · (λm − x)αm be a complex polynomial such that αi ̸= 0 for all i = 1, 2, · · · , m. then p(σ e, g des(t)) = σ e, g des(p(t)) and p(σ g des(t)) = σ g des(p(t)). proof. from corollary 5.3, it follows that λ ∈ p(σe, gdes(t)) ⇐⇒ λ = p(µ), where µ ∈ σ e, g des(t), ⇐⇒ λ − p(z) = (µ − z)kq(z), where q(µ) ̸= 0, ⇐⇒ λ ∈ σe, gdes(p(t)). the second equality, can be proved in the same way as the first one. this finishes the proof of the theorem. 156 z. garbouj, h. skhiri example 5.5. let t and k as in (i) of example 4.5 (resp. suppose that dim k(0) = +∞ and we replace the condition d(k) h by d(k) ⊆ h), we have t ∈ υ(h) is a closed linear relation and d(t) h (resp. dim t(0) = +∞). hence, if p is a non-constant complex polynomial, by [3, theorem 3.4], it is not possible to deduce that p(σ g des(t)) = σ g des(p(t)) (resp. p(σ e, g des(t)) = σ e, g des(p(t))). however, from theorem 5.4, we can do this. remark 5.6. we note that theorem 5.4 is false in general without the assumption that p is a non-constant polynomial, even if t is a bounded linear operator. for example, let t be defined as in remark 4.14. since ker(t) = {0} and im(t) is not closed, then β̃n(t) = β(t) = +∞, for all n ∈ n, and thus 0 ∈ σe, gdes(t) ⊆ σ g des(t). put p = c ∈ c. since σ e, g des(t) and σ g des(t) are non-empty sets, it follows that p(σ e, g des(t)) = p(σ g des(t)) = {c}. however, ϱ g des(p(t)) = ϱ g des(ci) = c. indeed, c\{c} = ϱ(ci) ⊆ ϱ g des(ci), and ci − ci is the zero operator with g-descent is equal to 1. therefore σ g des(p(t)) = σ e, g des(p(t)) = ∅. 6. decomposition theorems first observe that if t ∈ lr(h) is a range space relation such that ãe(t) < +∞ and im(t)+ker(t ãe(t)) is closed in h, then t is a quasi-fredholm relation (see, [7, definition 5.1]). in the following, we prove a decomposition theorem of linear relations with finite essential g-ascent such that im(t) + ker(t ãe(t)) is closed in h. theorem 6.1. let t ∈ cr(h). then there exists n ∈ n such that ãe(t) ≤ n and im(t) + ker(t n) is closed in h if and only if there exist d ∈ n and two closed subspaces m and n such that : (i) h = m u n; (ii) im(t d) ⊆ m, t(m) ⊆ m, n ⊆ ker(t d) and, if d > 0, n ̸⊂ ker(t d−1); (iii) g(t) = [g(t) ∩ (m × m)] u [g(t) ∩ (n × n)]; (iv) the restriction of t to m is both upper semi-fredholm and regular relation; essential g-ascent and g-descent of a closed relation 157 (v) a ∈ lr(n) such that its graph is the subspace g(t) ∩ (n × n), then a is a bounded operator everywhere defined and g(ad) = n × {0}. proof. ” =⇒ ” first, the assertions (i)-(iii) and (v) follow from [7, theorem 5.2], and by the same theorem we know that s = t|m the restriction of t to m is regular. let m = max{d, n}, since s is regular and im(t m) ⊆ m, it follows that ker(t) ∩ im(t m) = ker(s) ∩ im(sm) = ker(s), and hence α(s) = α̃m(t) < +∞. therefore s ∈ φ+(m). ” ⇐= ” let s = t|m be the restriction of t to m and a ∈ lr(n) such that g(a) = g(t) ∩ (n × n), so ker(t) ∩ im(t d) = ker(s) ∩ im(sd) = ker(s) and α̃d(t) < +∞. this implies that ãe(t) ≤ d. by [7, theorem 6.4], im(t) + ker(t d) is closed and from [6, lemma 2.9], it follows that im(t) + ker(t n) is closed for all n ≥ ãe(t), which completes the proof. now from [6, lemma 2.9], we know that if ã(t) = n < +∞ and im(t) + ker(t n) is closed, then ãe(t) ≤ n and im(t) + ker(t n) is closed. so, as a consequence of theorem 6.1, we obtain the following theorem. theorem 6.2. let t ∈ cr(h). then there exists n ∈ n such that ã(t) = n < +∞ and im(t) + ker(t n) is closed if and only if there exist d ∈ n and two closed subspaces m and n such that : (i) h = m u n, (ii) t(m ∩ d(t)) ⊆ m, im(t d) ⊆ m, n ⊆ ker(t d) and, if d > 0, n ̸⊂ ker(t d−1), (iii) g(t) = [g(t) ∩ (m × m)] u [g(t) ∩ (n × n)], (iv) the restriction of t to m is injective with closed range, (v) if a ∈ lr(n) such that its graph is the subspace g(t) ∩ (n × n), then a is a bounded operator everywhere defined and g(ad) = n × {0}. the following lemma will be needed in the proof of theorem 6.4. 158 z. garbouj, h. skhiri lemma 6.3. let t ∈ cr(h) such that de(t) < +∞. the following statements are equivalent : (i) im(t n) ∩ ker(t) is closed for some n ≥ de(t), (ii) im(t n) ∩ ker(t) is closed for all n ≥ de(t). proof. it is clear that only the implication ”(i) =⇒ (ii)” needs to prove. let n0 ≥ de(t) such that im(t n0) ∩ ker(t) is closed. first, we prove that im(t n0+1) ∩ ker(t) is closed. by the equality (1) in the proof of lemma 2.4, we get βn0(t) ≥ sn0(t) = dim ( im(t n0) ∩ ker(t) ) / ( im(t n0+1) ∩ ker(t) ) , and hence from [7, propositions 3.9, 3.10 and 4.8, lemma 4.2], im(t n0+1) ∩ ker(t) is closed. now if n0 > de(t), then n0 − 1 ≥ de(t) and so dim ( im(t n0−1) ∩ ker(t) ) / ( im(t n0) ∩ ker(t) ) < +∞. therefore im(t n0−1) ∩ ker(t) is also closed. this completes the proof. in the following result, we prove a decomposition theorem for t ∈ cr(h), with n = d̃e(t) < +∞ and im(t n) ∩ ker(t) is closed in h. theorem 6.4. let t ∈ cr(h). then d̃e(t) ≤ n and im(t n) ∩ ker(t) is closed for some n ∈ n if and only if there exist d ∈ n and two closed subspaces m and n such that : (i) h = m u n, (ii) t(m ∩ d(t)) ⊆ m, im(t d) ⊆ m, n ⊆ ker(t d) and, if d > 0, n ̸⊂ ker(t d−1), (iii) g(t) = [g(t) ∩ (m × m)] u [g(t) ∩ (n × n)], (iv) the restriction of t to m is both lower semi-fredholm and regular relation, (v) if a ∈ lr(n) such that its graph is the subspace g(t) ∩ (n × n), then a is a bounded operator everywhere defined and g(ad) = n × {0}. proof. ” =⇒ ” first, from lemma 2.4, we have d̃e(t) ≤ p(t) < +∞ and β̃p(t)(t) is finite, which implies that im(t) + ker(t p(t)) is closed in h. moreover, by lemma 6.3, t is a quasi-fredholm relation and so from [7, essential g-ascent and g-descent of a closed relation 159 theorem 5.2], it follows that there exist d ∈ n and two closed subspaces m and n such that h = mun, t(m∩d(t)) ⊆ m, n ⊆ ker(t d) ⊆ d(t), if d > 0, n ̸⊂ ker(t d−1) and the restriction of t to m, s = t|m, is regular. now, let m = max{d, n}, then im(t) + ker(t m) = im(s) u n (see the equality (6.7) in the proof of [7, theorem 6.4]). this implies that dim m/im(s) = dim[m u n]/[im(s) u n] = β̃m(t) < +∞ and consequently s ∈ φ−(m). ” ⇐= ” let s = t|m be the restriction of t to m and a ∈ lr(n) such that g(a) = g(t) ∩ (n × n), so that im(t) + ker(t d) = im(s) u n, ker(t) ∩ im(t d) = ker(s) ∩ im(sd) = ker(s). this implies that β̃d(t) ≤ β(s) < +∞ and ker(t) ∩ im(t d) is closed. this completes the proof of the theorem. now from lemma 6.3, we know that if d̃(t) = n < +∞ and im(t n) ∩ ker(t) is closed, then d̃e(t) ≤ n and im(t n) ∩ ker(t) is closed. therefore, we can prove the following theorem similarly as theorem 6.4. theorem 6.5. let t ∈ cr(h). then d̃(t) ≤ n and im(t n) ∩ ker(t) is closed for some n ∈ n if and only if there exist d ∈ n and two closed subspaces m and n such that : (i) h = m u n, (ii) t(m ∩ d(t)) ⊆ m, im(t d) ⊆ m, n ⊆ ker(t d) and, if d > 0, n ̸⊂ ker(t d−1), (iii) g(t) = [g(t) ∩ (m × m)] u [g(t) ∩ (n × n)], (iv) the restriction of t to m is surjective, (v) if a ∈ lr(n) such that its graph is the subspace g(t) ∩ (n × n), then a is a bounded operator everywhere defined and g(ad) = n × {0}. remark 6.6. let t ∈ cr(h) such that max{ã(t), d̃(t)} ≤ m, for some m ∈ n. it is clear that s = t|im(t m), the restriction of t to im(t m) is bijective, h = im(t m) + ker(t m) and im(t m) ∩ ker(t m) = {0} (see the equality (1) in the proof of lemma 3.7), so from [7, propositions 3.10, 4.8, lemma 4.2], it 160 z. garbouj, h. skhiri follows that im(t m) and ker(t m) are both closed. let a ∈ lr(ker(t m)) such that g(a) = g(t) ∩ (ker(t m) × ker(t m)). first note that a(0) ⊆ ker(t m) ∩ im(t m) = {0} and g(a) is closed, which implies that a is a closed operator. since d(a) = {x ∈ ker(t m) : ax ̸= ∅} = {x ∈ ker(t m) : tx ̸= ∅ in ker(t m)} = {x ∈ ker(t m) : ∃ y ∈ tx and y ∈ ker(t m)} = {x ∈ ker(t m) : ∃ y ∈ tx and 0 ∈ t m(y)} = {x ∈ ker(t m) : 0 ∈ t m+1(x)} = {x ∈ ker(t m) : x ∈ ker(t m+1)} = ker(t m), then a is a bounded operator everywhere defined. but am(ker(t m)) ⊆ ker(t m) ∩ im(t m) = {0}, so g(am) = g(t m) ∩ (ker(t m) × ker(t m)) = ker(t m) × {0}. now, we will show that g(t) = [g(t) ∩ (im(t m) × im(t m))] u [g(t) ∩ (ker(t m) × ker(t m))]. let (x, y) ∈ g(t), then x = x1 + x2 with x1 ∈ im(t m) and x2 ∈ ker(t m). therefore, there exist y1 ∈ t(x1) ⊆ im(t m) and y2 ∈ t(x2) such that y = y1 + y2. clearly, y2 ∈ t(x2) ⊆ t ( ker(t m) ) = t ( ker(t m+1) ) = tt −1 ( ker(t m) ) = ker(t m) ∩ im(t) + t(0) ⊆ ker(t m) + t(0) and hence y2 = y ′ 2 + y ′′ 2, for some y ′ 2 ∈ ker(t m) and y′′2 ∈ t(0). we have, y1 + y ′′ 2 ∈ t(x1) + t(0) = t(x1) ⊆ im(t m) and y′2 = y2 − y ′′ 2 ∈ t(x2) + t(0) = t(x2), so (x, y) = (x1, y1 + y ′′ 2) + (x2, y ′ 2) ∈ [g(t) ∩ (im(t m) × im(t m))] u [g(t) ∩ (ker(t m) × ker(t m))]. this implies that g(t) = [g(t) ∩ (im(t m) × im(t m))] u [g(t) ∩ (ker(t m) × ker(t m))]. finally, if we put m = im(t m) and n = ker(t m), then m and n satisfy the conditions (i)-(v) of theorems 6.2 and 6.5. references [1] t. álvarez, on regular linear relations, acta mathematica sinica, english series 28 (1) (2012), 183 – 194. essential g-ascent and g-descent of a closed relation 161 [2] o. bel hadj fredj, m. burgos, m. oudghiri, ascent spectrum and essential ascent spectrum, studia math. 187 (2008), 59 – 73. [3] e. chafai, m. mnif, spectral mapping theorem for ascent, essential ascent, descent and essential descent spectrum of linear relations, acta mathematica scientia, 34b (4) (2014), 1212 – 1224. [4] r. cross, “multivalued linear operators”, marcel dekker, new york, 1998. [5] z. garbouj, h. skhiri, essential ascent of closed operator and some decomposition theorems, commun. math. anal. 16 (2) (2014), 19 – 47. [6] z. garbouj, h. skhiri, minimum modulus, perturbation for essential ascent and descent of a closed linear relation in hilbert spaces, acta mathematica hungarica 151 (2) (2017), 328 – 360. [7] j.ph. labrousse, a. sandovici, h.s.v. de snoo, h. winkler, the kato decomposition of quasi-fredholm relations, operators and matrices 4 (1) (2010), 1 – 51. [8] a. sandovici, some basic properties of polynomials in a linear relation in linear spaces, in “oper. theory adv. appl. 175”, birkhäuser, basel, (2007), 231 – 240. [9] a. sandovici, h. de snoo, an index formula for the product of linear relations, linear algebra appl. 431 (11) (2009), 2160 – 2171. [10] a. sandovici, h.s.v. de snoo, h. winkler, ascent, descent, nullity, defect, and related notions for linear relations in linear spaces, linear algebra appl. 423 (2-3) (2007), 456 – 497. e extracta mathematicae vol. 31, núm. 2, 199 – 225 (2016) on generalized lie bialgebroids and jacobi groupoids apurba das stat-math unit, indian statistical institute, kolkata 700108, west bengal, india apurbadas348@gmail.com presented by juan c. marrero received november 23, 2016 abstract: generalized lie bialgebroids are generalization of lie bialgebroids and arises naturally from jacobi manifolds. it is known that the base of a generalized lie bialgebroid carries a jacobi structure. in this paper, we introduce a notion of morphism between generalized lie bialgebroids over a same base and prove that the induce jacobi structure on the base is unique up to a morphism. next we give a characterization of generalized lie bialgebroids and use it to show that generalized lie bialgebroids are infinitesimal form of jacobi groupoids. we also introduce coisotropic subgroupoids of a jacobi groupoid and these subgroupoids corresponds to, so called coisotropic subalgebroids of the corresponding generalized lie bialgebroid. key words: jacobi manifolds, coisotropic submanifolds, (generalized) lie bialgebroids, jacobi groupoids. ams subject class. (2010): 17b62, 53c15, 53d17. 1. introduction the notion of lie bialgebroid was introduced by mackenzie and xu [11] as a generalization of lie bialgebra and infinitesimal version of poisson groupoid. roughly, a lie bialgebroid (a,a∗) over m is a lie algebroid a over m such that its dual vector bundle a∗ also carries a lie algebroid structure which is compatible in a certain way with that of a. as an example, if (m,π) is a poisson manifold, then (tm,t∗m) forms a lie bialgebroid over m, where tm is the usual tangent lie algebroid and t∗m is the cotangent lie algebroid of the poisson manifold m. a duality theorem for a lie bialgebroid was shown in [11], that is, if (a,a∗) satisfy the criteria of a lie bialgebroid, then (a∗,a) also satisfies a similar criteria. it was also proved in [11] that the base space of a lie bialgebroid carries a natural poisson structure. in [13], kosmannschwarzbach gave a simple proof of the duality theorem for lie bialgebroid and the fact that the base of a lie bialgebroid has a poisson structure. jacobi manifolds are generalization of poisson manifolds. in [6], iglesias and marrero introduced a notion of generalized lie bialgebroid (genaralization of lie bialgebroid) in such a way that a jacobi manifold associates a 199 200 a. das canonical generalized lie bialgebroid structure. more precisely, a generalized lie bialgebroid ( (a,ϕ0),(a ∗,x0) ) over m is a lie algebroid a over m together with 1-cocycle ϕ0 ∈ γa∗ and such that the dual bundle a∗ also carries a lie algebroid structure with x0 ∈ γa be a 1-cocycle of it and satisfy some compatibility conditions like a lie bialgebroid. given a jacobi manifold (m,λ,e), if we consider the lie algebroid tm × r → m with 1-cocycle (0,1) ∈ γ(t∗m × r) = γ(t∗m) ⊕ c∞(m) and the lie algebroid on the dual 1-jet bundle t∗m × r → m with 1-cocycle (−e,0) ∈ γ(tm × r) = γ(tm) ⊕ c∞(m), then the pair ( (tm × r,(0,1)),(t∗m × r,(−e,0)) ) is a generalized lie bialgebroid over m [6]. moreover it is shown in [6] that if( (a,ϕ0),(a ∗,x0) ) is a generalized lie bialgebroid over m, the base manifold m induces a jacobi structure. we remark that, lie algebroid structures on a vector bundle a → m are in one-to-one correspondence with the linear poisson structures on the dual bundle a∗ [10]. this correspondence has been extended in [5] to the jacobi setup by introducing a notion of linear jacobi structure on a vector bundle. more precisely, they showed that given a lie algebroid a → m with a 1cocycle ϕ ∈ γa∗, the dual bundle a∗ carries a linear jacobi structure and conversely, given a linear jacobi structure on the dual a∗ of a vector bundle a induces a lie algebroid structure on a together with a 1-cocycle ϕ ∈ γa∗. the notion of morphism between lie bialgebroids were introduced in [11]. in this paper, we define a notion of morphism between generalized lie bialgebroids over a same base. if all the cocycles are zero, then it reduces to the morphism between lie bialgebroids. we also showed that, if there is a morphism between two generalized lie bialgebroids over a same base, then the induced jacobi structures on the base coming from two generalized lie bialgebroids are same (section 3). poisson groupoids were introduced by weinstein [15] as a unification of poisson lie groups and symplectic groupoids. in [11], the authors gave an equivalent definition of a poisson groupoid as a lie groupoid g ⇒ m with a poisson structure λ on the total space g such that the bundle map λ♯ : t∗g → tg is a lie groupoid morphism from the cotangent lie groupoid t∗g → a∗g to the tangent lie groupoid tg → tm, where ag → m is the lie algebroid of g. this motivates them to give an equivalent characterization of its infinitesimal object, that is lie bialgebroid. they showed that a pair of lie algebroids (a,a∗) in duality over m is a lie bialgebroid if and only if the bundle map λ ♯ a ◦ ra : t ∗a∗ → t∗a → ta is a lie algebroid morphism from the cotangent lie algebroid t∗a∗ → a∗ of the linear poisson manifold on generalized lie bialgebroids and jacobi groupoids 201 a∗, to the tangent lie algebroid ta → tm of a, where ra : t∗a∗ → t∗a is the canonical isomorphism defined in [11] and λa being the induced linear poisson structure on a. this description is rather complicated but useful to show that lie bialgebroids are infinitesimal of poisson groupoids. jacobi groupoids were introduced by iglesias and marrero [9] as a generalization of both poisson groupois and contact groupoids. more precisely, a jacobi groupoid is a lie groupoid g ⇒ m together with a jacobi structure (λ,e) on g and a multiplicative function σ on g, such that the bundle map (λ,e)♯ : t∗g×r → tg×r defined by (λ,e)♯(ωg,γ) = (λ♯(ωg)+γe(g),−⟨ωg,e(g)⟩), is a lie groupoid morphism between the twisted cotangent lie groupoid t∗g × r ⇒ a∗g to the twisted tangent lie groupoid tg × r ⇒ tm × r (cf. definition 4.7). these twisting has been done by using the multiplicative function σ. thus it is possible to give an equivalent characterization of a generalized lie bialgebroid in terms of some lie algebroid morphism. more precisely, we show that a pair of lie algebroids with 1-cocycles ( (a,ϕ0),(a ∗,x0) ) in duality over m is a generalized lie bialgebroid over m if and only if the map (λa,ea) ♯ ◦ (ra,−id) : t∗a∗ × r → t∗a × r → ta × r is a lie algebroid morphism from the 1-jet lie algebroid t∗a∗ × r → a∗ of the linear jacobi manifold a∗, to the twisted tangent lie algebroid ta × r → tm × r of (a,ϕ0), where (λa,ea) be the linear jacobi structure on a (cf. theorem 4.2). note that an another characterization of a generalized lie bialgebroid in terms of some jacobi algebroid morphism was given in the thesis of d. iglesias ponte [4]. then the theorem 4.2 is being used to show that the infinitesimal form of jacobi groupoids are generalized lie bialgebroids (cf. theorem 4.9) (section 4). coisotropic subgroupoids of a poisson groupoid were introduced in [16] as a generalization of coisotropic subgroups of a poisson lie group. the lie algebroids of coisotropic subgroupoids of a poisson groupoid are coisotropic subalgebroids of the corresponding lie bialgebroid. here we introduce coisotropic subgroupoids of a jacobi groupoid and show that the lie algebroid of a coisotropic subgroupoid appears as some notion of coisotropic subalgebroid of the corresponding generalized lie bialgebroid (section 5). notations. given a lie groupoid g ⇒ m, by α,β : g → m, we denote the source and target maps and ϵ : m → g the unit map. two elements g,h ∈ g are composable if α(g) = β(h), and denote by g(2) ⊂ g × g the set of composable pairs. a morphism between two lie groupoids g1 ⇒ m1 and g2 ⇒ m2 is a smooth map f : g1 → g2 over f : m1 → m2 which commutes with all structure maps. 202 a. das given a vector bundle a → m, there is a cononical isomorphism ra : t∗a∗ → t∗a which is defined as follows. suppose the vector bundle a is locally a ∣∣ u = u × v , where u ⊆ m is open, then the map ra is locally defined by ra(χ,ψ,y ) = (−χ,y,ψ), where χ ∈ t∗m, ψ ∈ v ∗, y ∈ v (see [11] for more details). 2. preliminaries in this section, we recall the definitions and basic facts about jacobi manifolds, lie algebroids and (generalized) lie bialgebroids [3, 5, 6, 10, 11]. • jacobi manifolds definition 2.1. let m be a smooth manifold. a jacobi structure on m is a pair (λ,e), where λ is a 2-vector field and e is a vector field on m satisfying [λ,λ] = 2e ∧ λ, leλ = [e,λ] = 0, where [ , ] is the schouten bracket on the space of multivector fields of m. the manifold m endowed with a jacobi structure is called a jacobi manifold. if (m,λ,e) is a jacobi manifold, then one can define a bilinear, skewsymmetric bracket on the space of smooth functions by the following {f,g} = λ(δf,δg) + fe(g) − ge(f) for all f,g ∈ c∞(m). the bracket { , } satisfies the jacobi identity and the property of a first order differential operator on each arguments, that is, {f,gh} = g{f,h} + h{f,g} − gh{f,1} for all f,g,h ∈ c∞(m). conversely, any bilinear, skew-symmetric bracket on c∞(m) which satisfies jacobi identity and also first order differential operator on each argument defines a jacobi structure on m. note that, if e = 0, then λ defines a poisson structure on m [14]. apart from poisson and symplectic manifolds, contact and locally conformal symplectic (l.c.s) manifolds are also examples of jacobi manifolds [3, 9]. given a jacobi bracket { , } on m and any nowhere zero function a ∈ c∞(m), one can define a new jacobi bracket { , }a by the following {f,g}a = 1 a {af,ag} ∀f,g ∈ c∞(m). on generalized lie bialgebroids and jacobi groupoids 203 a smooth map φ : m → n between two jacobi manifolds is called a jacobi map if {h ◦ φ,h′ ◦ φ}m = {h,h′}n ◦ φ, for any h,h′ ∈ c∞(n). for a nowhere zero function a ∈ c∞(m), the pair (φ,a) is called a conformal jacobi map if φ is a jacobi map between (m,{ , }am) and (n,{ , }n). remark 2.2. (i) given a jacobi manifold (m,λ,e), there is a vector bundle morphism (λ,e)♯ : t∗m × r → tm × r given by (λ,e)♯(ωm,γ) = ( λ♯(ωm) + γe(m),−⟨ωm,e(m)⟩ ) for (ωm,γ) ∈ t∗mm × r, m ∈ m. (ii) let (λ,e) be a jacobi structure on m. then the product manifold m ×r carries a poisson structure λ̃ = e−t ( λ + ∂ ∂t ∧ e ) , where t is the usual coordinate on r. the manifold m × r together with the poisson structure λ̃ is called the poissonization of the jacobi manifold (m,λ,e). the notion of coisotropic submanifolds of a poisson manifold [15] is extended to the context of jacobi manifolds. definition 2.3. ([3]) let (m,λ,e) be a jacobi manifold. then a submanifold s ↪→ m is called a coisotropic submanifold of m if λ♯(txs) 0 ⊆ txs for all x ∈ s, where λ♯ : t∗m → tm is the bundle map induced by the bivector field λ, and (txs) 0 = {α ∈ t∗xm| α(v) = 0,∀v ∈ txs}. similar to the poisson case, one can prove the following result. proposition 2.4. let (m,λ,e) be a jacobi manifold with corresponding jacobi bracket { , }, and c ↪→ m be a closed submanifold. then the followings are equivalent: (i) c is a coisotropic submanifold; (ii) the vanishing ideal i(c) = {f ∈ c∞(m) ∣∣f|c ≡ 0} is a lie subalgebra of (c∞(m),{ , }). 204 a. das • lie algebroids definition 2.5. a lie algebroid (a, [ , ],ρ) over a manifold m is a smooth vector bundle a over m together with a lie bracket [ , ] on the space γa of smooth sections of a and a bundle map ρ : a → tm, called the anchor, such that (i) the induced map ρ : γa → x(m) is a lie algebra homomorphism; (ii) for any f ∈ c∞(m) and x,y ∈ γa, the following condition holds [x,fy ] = f[x,y ] + (ρ(x)f)y. given a lie groupoid g ⇒ m, its lie algebroid consist of a vector bundle ag → m whose fiber at x ∈ m consist of the tangent space tϵ(x)(β−1(x)). then the space of sections γag can be identified with the left invariant vector fields on g. since the space of left invariant vector fields on g is closed under the lie bracket, thus it induces a lie bracket on γag. the anchor is defined to be the differential of α restricted to ag. any lie algebra is a lie algebroid over a point, and the tangent bundle of any smooth manifold is a lie algebroid with the usual lie bracket on vector fields and identity as anchor. given any jacobi manifold, there is a canonical lie algebroid associated to it, given by the following example. example 2.6. ([6, 9]) let (m,λ,e) be a jacobi manifold, then the 1-jet bundle t∗m × r has a lie algebroid structure (t∗m × r, [ , ](λ,e),ρ(λ,e)) over m, where the bracket and anchor are given by [(α,f),(β,g)](λ,e) = ( lλ♯(α)β − lλ♯(β)α − δ(λ(α,β)) + fleβ − gleα − ιe(α ∧ β), λ(β,α) + λ♯(α)(g) − λ♯(β)(f) + fe(g) − ge(f) ) and ρ(λ,e)(α,f) = λ ♯(α) + fe for all (α,f),(β,g) ∈ γ(t∗m × r) = ω1(m) ⊕ c∞(m). when e = 0 ( that is, when λ is a poisson structure on m), one recover to the cotangent lie algebroid of m by projecting onto the first factor [14]. given a lie algebroid (a, [ , ],ρ), the exterior algebra γ( ∧• a) of multisections of a together with the generalized schouten bracket, forms a gerstenhaber algebra [10]. moreover γ( ∧• a∗) together with the lie algebroid on generalized lie bialgebroids and jacobi groupoids 205 differential d forms a differential graded algebra. when a = tm is the usual tangent bundle lie algebroid, denote the differential of the lie algebroid (that is, de-rham differential of the manifold m) by δ. it is known that, lie algebroid structures on a vector bundle a are in oneto-one correspondence with linear poisson structures on the dual bundle a∗. this correspondence has been extended to the jacobi set up by introducing the notion of linear jacobi structure on a vector bundle [5]. we now describe these in details. given a vector bundle, a jacobi structure on the total space is called linear jacobi structure, if the jacobi bracket of two fibrewise linear functions is again linear, and the bracket of a linear function and the constant function 1 is a pull back function. let q : a → m be a vector bundle over m with dual bundle q∗ : a∗ → m. then any section x ∈ γa defines a fibrewise linear function lx on the dual bundle a ∗, namely lx(αx) = αx(xx), where αx ∈ a∗x, x ∈ m. conversely any fibrewise linear function on a∗ is of this form. then the space of linear functions and pull back functions (that is, of the form f ◦ q∗, for f ∈ c∞(m)) generates c∞(a∗). then we have the following theorem from [5]. theorem 2.7. let (a, [ , ],ρ) be a lie algebroid over m and ϕ ∈ γa∗ be a 1-cocycle. consider the bracket { , } on c∞(a∗) defined by {lx, ly } = l[x,y ], {lx,f ◦ q∗} = (ρ(x)f + ϕ(x)f) ◦ q∗, {f ◦ q∗,g ◦ q∗} = 0, for x,y ∈ γa and f,g ∈ c∞(m). then { , } defines a linear jacobi structure on a∗. moreover, this jacobi structure is given by λ(a∗,ϕ) = λa∗ + ∆ ∧ ϕ v, e(a∗,ϕ) = −ϕ v, where λa∗ is the linear poisson structure on a ∗ induced from the lie algebroid structure on a, ∆ is the liouville vector field on a∗ and ϕv ∈ x(a∗) is the vertical lift of ϕ ∈ γa∗. 206 a. das the converse part of the above theorem also holds true (see [5] for more details). a pair (a,ϕ) of a lie algebroid a and a 1-cocycle ϕ ∈ γa∗ of it, is referred as a jacobi algebroid. let (a, [ , ],ρ) be a lie algebroid over m and ϕ ∈ γa∗ be a 1-cocycle. then the vector bundle ã = a×r → m ×r carries a lie algebroid structure ([ , ]˜ϕ, ρ̃ϕ). for any x̃, ỹ ∈ γã considered as time dependent sections of a, the lie bracket and anchor are given by [x̃, ỹ ]˜ϕ = [x̃, ỹ ]˜ + ϕ(x̃) ∂ỹ ∂t − ϕ(ỹ ) ∂x̃ ∂t , ρ̃ϕ(x̃) = ρ̃(x̃) + ϕ(x̃) ∂ ∂t , where [x̃, ỹ ]˜(x,t) = [x̃t, ỹt](x) ρ̃(x̃)(x,t) = ρ(x̃t)(x), and ∂x̃ ∂t denotes the derivatie of x̃ with respect to time. now let ψ : ã → ã be the isomorphism of vector bundles over the identity on m × r defined by ψ(v,t) = (etv,t), for (v,t) ∈ ã = a × r. then using ψ and the lie algebroid structure ([ , ]˜ϕ, ρ̃ϕ) on ã, one can define a new lie algebroid structure ([ , ]ˆϕ, ρ̂ϕ) on ã such that the lie algebroids (ã, [ , ]˜ϕ, ρ̃ϕ) and (ã, [ , ]ˆϕ, ρ̂ϕ) are isomorphic. namely, we have [x̃, ỹ ]ˆϕ = e−t ( [x̃, ỹ ]˜ + ϕ(x̃) (∂ỹ ∂t − ỹ ) − ϕ(ỹ ) (∂x̃ ∂t − x̃ )) , ρ̂ϕ(x̃) = e−t ( ρ̃(x̃) + ϕ(x̃) ∂ ∂t ) , for all x̃, ỹ ∈ γã. thus the total space of the dual bundle ã∗ = a∗ × r → m × r carries a linear poisson structure λa∗×r and this poisson structure is the poissonization of the linear jacobi structure (λ(a∗,ϕ),e(a∗,ϕ)) of a ∗ [5]. that is, λa∗×r = e −t ( λ(a∗,ϕ) + ∂ ∂t ∧ e(a∗,ϕ) ) . the notion of morphism between lie algebroids was introduced in [2]. here we recall an alternative definition from [11]. definition 2.8. let a1 → m1 and a2 → m2 be two lie algebroids. then a vector bundle morphism f : a1 → a2 over f : m1 → m2 is called a on generalized lie bialgebroids and jacobi groupoids 207 lie algebroid morphism if the graph of (f,f), that is, c := { (ϕ,ψ) ∈ a∗1 × a ∗ 2 | ⟨ϕ,x⟩ = ⟨ψ,f(x)⟩, for all x ∈ a1 compatible with ϕ } is a coisotropic submanifold of a∗1×a∗2, where a ∗ 1 and a ∗ 2 are equipped with the linear poisson structures dual to the lie algebroids a1 and a2, respectively. if m1 = m2 and f = identity, then f is a lie algebroid morphism if and only if f preserves the lie brackets and commute with anchors. • lie bialgebroids definition 2.9. ([11]) a lie bialgebroid over m is a pair (a,a∗) of lie algebroids in duality over m, such that the differential d∗ on γ( ∧• a) defined by the lie algebroid structure of a∗ and the gerstenhaber bracket on γ( ∧• a) defined by the lie algebroid structure of a satisfies d∗[x,y ] = [d∗x,y ] + [x,d∗y ], for all x,y ∈ γa. then there is a following characterization of a lie bialgebroid [11]. theorem 2.10. let a be a lie algebroid over m such that its dual bundle a∗ also carries a lie algebroid structure. consider the composition π = λ ♯ a ◦ ra, t∗a∗ −→ t∗a −→ ta, where λa being the linear poisson structure on a coming from the lie algebroid a∗. then (a,a∗) is a lie bialgebroid if and only if t∗a∗ π // �� ta �� a∗ ρ∗ // tm is a lie algebroid morphism, where t∗a∗ → a∗ is the cotangent lie algebroid of the linear poisson structure on a∗ coming from the lie algebroid a, and ta → tm is the tangent lie algebroid of a. 208 a. das • generalized lie bialgebroids given a lie algebroid (a, [ , ],ρ) over m with 1-cocycle ϕ ∈ γa∗, there is an ϕ-deformed lie algebroid representation ρϕ : γa × c∞(m) → c∞(m) (x,f) 7→ (ρϕ(x))f = ρ(x)f + ϕ(x)f. thus one can define ϕ-deformed lie algebroid differential dϕ which is given by dϕ : γ( ∧• a∗) → γ( ∧•+1 a∗) α 7→ dα + ϕ ∧ α. then the ϕ-deformed lie derivative is defined using cartan formula lϕx : γ( ∧• a∗) → γ( ∧• a∗), α 7→ dϕιxα + ιxdϕα for x ∈ γa. one can also define ϕ-deformed schouten bracket on the multisections of a by the formula [p,q]ϕ = [p,q] + (p − 1)p ∧ (ιϕq) − (−1)p−1(q − 1)(ιϕp) ∧ q, for p ∈ γ( ∧p a),q ∈ γ( ∧q a) (see [1, 6]). let (a, [ , ],ρ) be a lie algebroid over m and ϕ0 ∈ γa∗ be a 1-cocyle. assume that the dual bundle a∗ also carries a lie algebroid structure ([ , ]∗,ρ∗) and x0 ∈ γa be its 1-cocyle. let the lie derivative of the lie algebroid a (resp. a∗) is denoted by l (resp. l∗). definition 2.11. ([6]) the pair ( (a,ϕ0),(a ∗,x0) ) is said to be a generalized lie bialgebroid over m if the following conditions are hold dx0∗ [x,y ] = [d x0 ∗ x,y ] ϕ0 + [x,dx0∗ y ] ϕ0, (1) lx0∗ϕ0p + l ϕ0 x0 p = 0 (2) for all x,y ∈ γa and p ∈ γ( ∧p a). remark 2.12. (i) the condition (2) of the above definition is equivalent to ϕ0(x0) = 0, ρ(x0) = −ρ∗(ϕ0), l∗ϕ0x + [x0,x] = 0, (3) for all x ∈ γa. these follows from condition (2) by applying p = f ∈ c∞(m) and p = x ∈ γa. (ii) when ϕ0 = 0 and x0 = 0, one recover the definition of a lie bialgebroid. on generalized lie bialgebroids and jacobi groupoids 209 example 2.13. given any smooth manifold m, the bundle tm×r → m has a lie algebroid structure whose bracket and anchor are given by [(x,f),(y,g)] = ( [x,y ],x(g) − y (f) ) , pr(x,f) = x, for (x,f),(y,g) ∈ x(m) ⊕ c∞(m). moreover (0,1) ∈ ω1(m) ⊕ c∞(m) = γ(t∗m × r) is a 1-cocycle of this lie algebroid. if (m,λ,e) is a jacobi manifold, then the 1-jet bundle t∗m × r → m also carries a lie algebroid structure (cf. example 2.6) and one can check that, (−e,0) ∈ x(m) ⊕ c∞(m) = γ(tm × r) is a 1-cocycle of it. for a jacobi manifold (m,λ,e), the pair ( (tm×r,(0,1)),(t∗m×r,(−e,0)) ) is a generalized lie bialgebroid over m [6]. another interesting class of examples of generalized lie bialgebroids are provided by strict jacobi-nijenhuis manifolds [7]. the relation between lie bialgebroid and generalized lie bialgebroid is given by the following result. proposition 2.14. ([6]) let (a, [ , ],ρ) be a lie algebroid over m with ϕ0 ∈ γa∗ be a 1-cocycle. suppose the dual bundle a∗ → m also carries a lie algebroid structure ([ , ]∗,ρ∗) and x0 ∈ γa be a 1-cocycle of it. consider the lie algebroids ã = (a × r, [ , ]˜ϕ0, ρ̃ϕ0) and ã∗ = (a∗ × r, [ , ]ˆx0∗ , ρ̂∗ x0) in duality over m × r. then, (i) if ( (a,ϕ0),(a ∗,x0) ) is a generalized lie bialgebroid over m, the pair (ã,ã∗) is a lie bialgebroid over m × r; (ii) if (ã,ã∗) is a lie bialgebroid over m × r, the pair ( (a,ϕ0),(a ∗,x0) ) is a generalized lie bialgebroid over m. thus using the duality of a lie bialgebroid, one can conclude the duality of a generalized lie bialgebroid. proposition 2.15. ([6]) if ( (a,ϕ0),(a ∗,x0) ) is a generalized lie bialgebroid over m, then so is the pair ( (a∗,x0),(a,ϕ0) ) . remark 2.16. one can also directly prove the duality of a generalized lie bialgebroid following the proof of kosmann-schwarzbach [13] for lie bialgebroids in the presence of cocycles, although these two proofs of the duality of a generalized lie bialgebroid can be shown to be equivalent in the presence of the proposition 2.14. 210 a. das given a generalized lie bialgebroid over m, it is proved in [6] that the base m carries a jacobi structure. let ( (a,ϕ0),(a ∗,x0) ) be a generalized lie bialgebroid over m. define a bracket { , } : c∞(m) × c∞(m) → c∞(m) by {f,g} = ⟨dϕ0f,dx0∗ g⟩. (4) proposition 2.17. the bracket defined above satisfies the following properties dϕ0{f,g} = [dϕ0f,dϕ0g]∗, (5) dx0∗ {f,g} = −[d x0 ∗ f,d x0 ∗ g]. (6) proof. since {f,g} = ⟨dϕ0f,dx0∗ g⟩ = ρx0∗ (dϕ0f)g = [dϕ0f,g]x0∗ , we have dϕ0{f,g} = [dϕ0f,dϕ0g]x0∗ = [d ϕ0f,dϕ0g]∗. similarly, {f,g} = ρϕ0(dx0∗ g)f = [dx0∗ g,f]ϕ0, therefore dx0∗ {f,g} = [d x0 ∗ g,d x0 ∗ f] ϕ0 = −[dx0∗ f,d x0 ∗ g]. theorem 2.18. ([6]) let ( (a,ϕ0),(a ∗,x0) ) be a generalized lie bialgebroid over m. then the bracket above defines a jacobi structure on m. remark 2.19. (i) from (4), we have {f,g} = ⟨df,d∗g⟩+fρ∗(ϕ0)g+gρ(x0)f, therefore the induced jacobi bivector field λm and the vector field em is given by λm(δf,δg) = ⟨df,d∗g⟩ = ( ρ∗ ◦ ρ∗(δf) ) (g), em = ρ∗(ϕ0) = −ρ(x0). (ii) let (m,λ,e) be a jacobi manifold. if we consider the generalized lie bialgebroid ( (tm × r,(0,1)),(t∗m × r,(−e,0)) ) given in example 2.6, the induced jacobi structure on m coincide with the original jacobi structure. (iii) the dual generalized lie bialgebroid ( (a∗,x0),(a,ϕ0) ) over m induces the opposite jacobi structure of the above. on generalized lie bialgebroids and jacobi groupoids 211 3. morphism between generalized lie bialgebroids in this section, we introduce a notion of morphism between generalized lie bialgebroids over a same base and prove that the induced jacobi structure on the base of a generalized lie bialgebroid is unique up to a morphism. • jacobi algebroid maps definition 3.1. let (a,ϕ) and (b,ψ) be two jacobi algebroids over m. then a bundle map φ : a → b over m is called a jacobi algebroid map if φ is a lie algebroid map and φ∗(ψ) = ϕ. proposition 3.2. let (a,ϕ) and (b,ψ) be two jacobi algebroids over m. then a bundle map φ : a → b over m is a jacobi algebroid map if and only if φ∗ : b∗ → a∗ is a jacobi map, under the dual linear jacobi structures. proof. let φ : a → b is a jacobi algebroid map. then for all x,y ∈ γa, we have {lx, ly } ◦ φ∗ = l[x,y ] ◦ φ ∗ = lφ[x,y ] = l[φ(x),φ(y )] = {lφ(x), lφ(y )} = {lx ◦ φ∗, ly ◦ φ∗}. for any f ∈ c∞(m), we also have {lx,f ◦ q∗a} ◦ φ ∗ = ( ρa(x)f + ϕ(x)f ) ◦ q∗a ◦ φ ∗ = ( ρb(φ(x))f + (ψ,φ(x))f ) ◦ q∗b = {lφ(x),f ◦ q ∗ b} = {lx ◦ φ∗,f ◦ q∗a ◦ φ ∗}. moreover for any f,g ∈ c∞(m), {f ◦ q∗a,g ◦ q ∗ a} ◦ φ ∗ = {f ◦ q∗b,g ◦ q ∗ b} = {f ◦ q ∗ a ◦ φ ∗,g ◦ q∗a ◦ φ ∗}, since both sides are equals to zero. therefore, φ∗ : b∗ → a∗ is a jacobi map. the converse part is similar. proposition 3.3. let (a,ϕ) and (b,ψ) be two jacobi algebroids over m. then a bundle map φ : a → b over m is a jacobi algebroid map if and only if φ̃ = φ × id : ( a × r, [ , ]˜ϕ, ρ̃a ϕ ) → ( b × r, [ , ]˜ψ, ρ̃b ψ ) is a lie algebroid morphism over m × r. 212 a. das proof. suppose φ is a jacobi algebroid map. then for any x̃, ỹ ∈ γ(ã) = γ(a × r), we have [ φ̃(x̃),φ̃(ỹ ) ]˜ψ = [ φ̃(x̃),φ̃(ỹ ) ]˜ + ψ ( φ̃(x̃) )∂(φ̃(ỹ )) ∂t − ψ ( φ̃(ỹ ) )∂(φ̃(x̃)) ∂t = φ̃ [ x̃, ỹ ]˜ + ( φ̃∗(ψ),x̃ ) φ̃ (∂ỹ ∂t ) − ( φ̃∗(ψ), ỹ ) φ̃ (∂x̃ ∂t ) = φ̃ ( [x̃, ỹ ]˜ϕ ) . also for any x̃ ∈ γ(ã), (ρ̃b ψ ◦ φ̃)(x̃) = ρ̃b ψ ( φ̃(x̃) ) = ρ̃b ( φ̃(x̃) ) + ψ ( φ̃(x̃) ) ∂ ∂t = ρ̃b ◦ φ(x̃) + ⟨ψ,φ̃(x̃)⟩ ∂ ∂t = ρ̃a(x̃) + ⟨φ∗ψ,x̃⟩ ∂ ∂t = ρ̃a(x̃) + ϕ(x̃) ∂ ∂t = ρ̃a ϕ (x̃). hence φ̃ defines a lie algebroid morphism. the converse part is similar. similarly φ is a jacobi algebroid map if and only if φ̃ = φ × id is a lie algebroid map from (a × r, [ , ]ˆϕ, ρ̂a ϕ ) to (b × r, [ , ]ˆψ, ρ̂b ψ ) over m × r. it is known that given a lie algebroid a, a section α ∈ γa∗ is a one cocycle of the lie algebroid if and only if the image of α in a∗ is a coisotropic submanifold of a∗ with respect to linear poisson structure [12]. next we give an analogues of this result to jacobi set up. proposition 3.4. let a be a lie algebroid over m and ϕ ∈ γa∗ be a 1-cocycle. then α ∈ γa∗ is a ϕ-deformed one cocycle of the lie algebroid (that is, dϕα = 0) if and only if the image of α in a∗ is a coisotropic submanifold of a∗ with respect to linear jacobi structure. proof. let α ∈ γa∗. then for any x ∈ γa, the function lx − ⟨α,x⟩ ◦ q∗ on a∗ vanishes on the image of α. in fact, the space of functions on a∗ which vanishes on the image of α is generated by such kind of functions. now for on generalized lie bialgebroids and jacobi groupoids 213 any x,y ∈ γa, we have{ lx − ⟨α,x⟩ ◦ q∗, ly − ⟨α,y ⟩ ◦ q∗ } = {lx, ly } − { ⟨α,x⟩ ◦ q∗, ly } − { lx,⟨α,y ⟩ ◦ q∗ } + { ⟨α,x⟩ ◦ q∗,⟨α,y ⟩ ◦ q∗ } = l[x,y ] + ( ρ(y )⟨α,x⟩ + ϕ(y )⟨α,x⟩ ) ◦ q∗ − ( ρ(x)⟨α,y ⟩ + ϕ(x)⟨α,y ⟩ ) ◦ q∗. therefore, { lx − ⟨α,x⟩ ◦ q∗, ly − ⟨α,y ⟩ ◦ q∗ } (α(m)) = ⟨α, [x,y ]⟩(m) + ( ρ(y )⟨α,x⟩ ) (m) + (ϕ(y ))(m)⟨α,x⟩(m) − ( ρ(x)⟨α,y ⟩ ) (m) − (ϕ(x))(m)⟨α,y ⟩(m) = − (dα)(x,y )(m) − (ϕ ∧ α)(x,y )(m) = − (dϕα)(x,y )(m). therefore, from proposition 2.4, it follows that the image of α is a coisotropic submanifold of a∗ with respect to linear jacobi structure if and only if dϕα = 0. • generalized lie bialgebroid morphisms definition 3.5. a morphism between two generalized lie bialgebroids( (a,ϕ0),(a ∗,x0) ) and ( (b,ψ0),(b ∗,y0) ) over m is a map φ : a → b of lie algebroids such that the dual map φ∗ : b∗ → a∗ is also a lie algebroid map and they preserves the cocycles. that is, φ(x0) = y0, φ ∗(ψ0) = ϕ0. proposition 3.6. let ( (a,ϕ0),(a ∗,x0) ) and ( (b,ψ0),(b ∗,y0) ) be two generalized lie bialgebroids over m. then a bundle map φ : a → b is a generalized lie bialgebroid morphism if and only if the following conditions hold: (i) φ : a → b is a jacobi algebroid map from (a,ϕ0) to (b,ψ0); (ii) φ : a → b is a jacobi map under the linear jacobi structures on a and b coming from the jacobi algebroids (a∗,x0) and (b ∗,y0) respectively. proposition 3.7. let ( (a,ϕ0),(a ∗,x0) ) and ( (b,ψ0),(b ∗,y0) ) be two generalized lie bialgebroids over m. then a bundle map φ : a → b is a generalized lie bialgebroid morphism if and only if φ̃ = φ × id is a lie bialgebroid morphism from (ã,ã∗) to (b̃,b̃∗) over m × r. 214 a. das proof. suppose φ is a generalized lie bialgebroid morphism. therefore φ is a jacobi algebroid map from (a,ϕ0) to (b,ψ0). hence from proposition 3.3, we have φ̃ = φ × id is a lie algebroid map from (a × r, [ , ]˜ϕ0, ρ̃a ϕ0) to (b × r, [ , ]˜ψ0, ρ̃b ψ0). moreover φ∗ is a jacobi algebroid map (a∗,x0) to (b∗,y0). therefore φ ∗ × id = φ̃∗ is a lie algebroid map from (a∗ × r, [ , ]ˆx0, ρ̂a x0) to (b∗ × r, [ , ]ˆy0, ρ̂b y0). hence φ̃ is a lie bialgebroid morphism from (ã,ã∗) to (b̃,b̃∗) over m × r. the converse part is similar. theorem 3.8. let ( (a,ϕ0),(a ∗,x0) ) be a generalized lie bialgebroid over m and (λm,em) denotes the induced jacobi structure on m. then the map φa : a → tm × r defined by φa(x) = (ρ(x),ϕ0(x)), for x ∈ γa, is a morphism between the generalized lie bialgebroids ( (a,ϕ0),(a ∗,x0) ) and ( (tm ×r,(0,1)),(t∗m × r,(−em,0)) ) , where ρ is the anchor of the lie algebroid a. moreover, if ( (a,ϕ0),(a ∗,x0) ) and ( (b,ψ0),(b ∗,y0) ) are two generalized lie bialgebroids over m and ψ : a → b is a generalized lie bialgebroid morphism, then the corresponding induced jacobi structures on the base manifold m are same. proof. the map φa is clearly a lie algebroid map and φa(x0) = (ρ(x0),0) = (−em,0). note that the dual map φ∗a : t ∗m × r → a∗ is such that φ∗a(α,f)(x) = ⟨ (α,f),φa(x) ⟩ = ⟨ (α,f),(ρ(x),ϕ0(x)) ⟩ = α(ρ(x)) + fϕ0(x), for any x ∈ γa. therefore, φ∗a(α,f) = ρ ∗(α) + fϕ0. hence, φ ∗ a(0,1) = ϕ0. it is also a direct calculation to show that φ∗a : ( t∗m × r, [ , ](λm,em ),ρ(λm,em ) ) → (a∗, [ , ]∗,ρ∗) preserves the lie bracket. it also commutes with the anchors, as ρ∗ ◦ φ∗a(α,f) = ρ∗(ρ ∗(α) + fϕ0) = ρ∗(ρ ∗(α)) + fρ∗(ϕ0) = λ ♯ m(α) + fem = ρ(λm,em )(α,f), where ρ∗ is the anchor of the lie algebroid a ∗. on generalized lie bialgebroids and jacobi groupoids 215 to prove the last part of the theorem, let the lie algebroid differential of a and a∗ (resp. b and b∗) be denoted by da and da∗ (resp. db and db∗). similarly the anchors are denoted by ρa and ρa∗ (resp. ρb and ρb∗). then, {f,g}(a,a∗) = ⟨d ϕ0 a f,d x0 a∗g⟩ = ρ x0 a∗ (d ϕ0 a f)g = ρ x0 a∗ ( φ∗a(δf,f) ) g = ρx0a∗ ( ψ∗φ∗b(δf,f) ) g = ρx0a∗ ψ ∗(φ∗b(δf,f))g = ρy0b∗(d ψ0 b f)g = {f,g}(b,b∗). hence the proof. therefore the induced jacobi structure on the base of a generalized lie bialgebroid is unique up to a morphism. 4. generalized lie bialgebroids and jacobi groupoids in this section, we give a characterization of generalized lie bialgebroids and using it, we show that generalized lie bialgebroids are infinitesimal form of jacobi groupoids. we begin with an example of twisted version of tangent lie algebroid. given a lie algebroid q : a → m, the bundle tq : ta → tm carries a lie algebroid structure, called tangent lie algebroid of a. if ϕ ∈ γa∗ is a 1-cocycle of the lie algebroid a, then one can define a twist on the tangent lie algebroid. example 4.1. let a → m be a lie algebroid and ϕ ∈ γa∗ be a 1-cocycle. thus ϕ defines a linear function on a. its complete lift defines a function on ta, which is linear with respect to the vector bundle structure ta → tm. hence it defines a section ϕ of the dual bundle (ta)∗ → tm. moreover ϕ ∈ γ(ta∗) becomes a 1-cocycle of the tangent lie algebroid ta → tm [4]. thus there is a lie algebroid structure on t̃a = ta× r over tm × r, whose sections are considered as dependent sections of the tangent lie algebroid ta → tm. the lie bracket and anchor of the lie algebroid t̃a are given by [x̃, ỹ ]˜ϕ = [x̃, ỹ ]˜ + ϕ(x̃) ∂ỹ ∂t − ϕ(ỹ ) ∂x̃ ∂t , ρ̃ϕ(x̃) = ρ̃ta(x̃) + ϕ(x̃) ∂ ∂t , where [x̃, ỹ ]˜(x,t) = [x̃t, ỹt](x), ρ̃ta(x̃)(x,t) = ρta(x̃t)(x), for x̃, ỹ ∈ γ(t̃a). 216 a. das the lie algebroid t̃a = ta×r → tm ×r defined above is called the twisted tangent lie algebroid of (a,ϕ). • an alternative characterization of generalized lie bialgebroids let (a, [ , ],ρ) be a lie algebroid over m and ϕ0 ∈ γa∗ be a 1-cocycle of it. suppose the dual bundle a∗ also carries a lie algebroid structure ([ , ]∗,ρ∗) and x0 ∈ γa be its 1-cocycle. let (λa,ea) be the linear jacobi structure on a coming from the lie algebroid a∗ and its 1-cocycle x0. then we have the following characterization of a generalized lie bialgebroid. theorem 4.2. the pair ( (a,ϕ0),(a ∗,x0) ) is a generalized lie bialgebroid over m if and only if the composition (λa,ea) ♯◦(ra,−id) : t∗a∗×r → t∗a × r → ta × r t∗(a∗) × r (λa,ea) ♯◦(ra,−id) // �� ta × r �� a∗ ( ρ∗( ),x0( ) ) // tm × r (7) is a lie algebroid morphism, where the domain t∗(a∗)×r −→ a∗ is the 1-jet lie algebroid of the linear jacobi manifold a∗ coming from the pair (a,ϕ0), and the range ta × r → tm × r is the twisted tangent lie algebroid of (a,ϕ0). proof. consider the pair of lie algebroids ( (a × r, [ , ]˜ϕ0, ρ̃ϕ0),(a∗ × r, [ , ]ˆx0∗ , ρ̂∗ x0) ) in duality over m × r and consider t∗(a∗ × r) (πa×r) ♯◦ra×r // �� t(a × r) �� a∗ × r // t(m × r) (8) where the left hand side is the cotangent lie algebroid of the linear poisson structure of a∗ × r coming from the lie algebroid (a × r, [ , ]˜ϕ0, ρ̃ϕ0), the right hand side is the tangent lie algebroid of a × r, and πa×r is the linear poisson structure on a × r coming from the lie algebroid structure (a∗ × on generalized lie bialgebroids and jacobi groupoids 217 r, [ , ]ˆx0∗ , ρ̂∗ x0). note that πa×r is the poissonization of the linear jacobi structure (λa,ea) of a. thus (πa×r) ♯(wa + γdt|t) = e−t ( λ ♯ a(wa) + γea(a),−wa(ea) ∂ ∂t ∣∣ t ) for wa + γdt|t ∈ t∗(a,t)(a × r), a ∈ a. then,{( (x,λ),(y,µ) )∣∣∣x ∈ tϕa∗,y ∈ tψa∗;λ,µ ∈ r} ⊆ (ta∗ × r) × (ta∗ × r) is in the graph of (7) if and only if for all (φ,ζ) ∈ t∗ϕ(a ∗) × r, we have⟨ (φ,ζ),(x,λ) ⟩ = ⟨ (λa,ea) ♯ ◦ (ra,−id)(φ,ζ),(y,µ) ⟩ = ⟨ (λa,ea) ♯(ra(φ),−ζ),(y,µ) ⟩ = ⟨( λ ♯ a(raφ) − ζea,−⟨raφ,ea⟩ ) ,(y,µ) ⟩ = ⟨ y,λ ♯ a(raφ) − ζea ⟩ − µ⟨raφ,ea⟩. on the other hand{(( x,λ d dt ∣∣ t ) , ( z,ξ d dt ∣∣ t ))∣∣∣∣x ∈ tϕa∗,y ∈ tψa∗;λ,µ ∈ r } ⊆ t(a∗ × r) × t(a∗ × r) is in the graph of (8) if and only if for all (φ,ζdt|t) ∈ t∗(ϕ,t)(a ∗ × r), we have⟨( φ,ζdt|t ) , ( x,λ d dt ∣∣ t )⟩ = ⟨( πa×r )♯ ◦ ra×r(φ,ζdt|t),(z,ξ d dt ∣∣ t )⟩ = ⟨( πa×r )♯( raφ,−ζdt|t ) , ( z,ξ d dt ∣∣ t )⟩ = ⟨ e−t ( λ ♯ a(raφ) − ζea,−⟨raφ,ea⟩ d dt ∣∣ t ) , ( z,ξ d dt ∣∣ t )⟩ =e−t ⟨ z,λ ♯ a(raφ) − ζea ⟩ − e−tξ⟨raφ,ea⟩. therefore, {( (x,λ),(y,µ) )∣∣x ∈ tϕa∗,y ∈ tψa∗;λ,µ ∈ r} is in the graph of (7) if and only if {( (x,λ d dt ∣∣ t ),(ety,etµ d dt ∣∣ t ) )∣∣x ∈ tϕa∗,y ∈ tψa∗;λ,µ ∈ r} is in the graph of (8). moreover using the corresponding dual linear poisson structures, one can show that{( (x,λ),(y,µ) )∣∣x ∈ tϕa∗,y ∈ tψa∗;λ,µ ∈ r} ⊆ (ta∗ × r) × (ta∗ × r) 218 a. das is a coisotropic submanifold if and only if{(( x,λ d dt ∣∣ t ) , ( ety,etµ d dt ∣∣ t ))∣∣∣x ∈ tϕa∗,y ∈ tψa∗;λ,µ ∈ r} ⊆ t(a∗ × r) × t(a∗ × r) is a coisotropic submanifold. thus we have that (7) is a lie algebroid morphism if and only if (8) is a lie algebroid morphism. hence the result follows from proposition 2.14 and theorem 2.10. • jacobi groupoids and generalized lie bialgebroids given a lie groupoid g ⇒ m, its tangent lie groupoid is given by tg ⇒ tm, whose structure maps are tα,tβ,tϵ and the composition is denoted by ⊕tg. definition 4.3. let g ⇒ m be a lie groupoid, η ∈ ω1(g) be a contact 1-form on g and σ : g → r be a multiplicative function on g. then the triple (g ⇒ m,η,σ) is called a contact groupoid if ηgh(xg ⊕tg yh) = ηg(xg) + eσ(g)ηh(yh) for all (xg,yh) ∈ (tg)(2). example 4.4. ([9]) given a lie groupoid g ⇒ m with lie algebroid ag → m, it is known that the cotangent bundle t∗g has a lie groupoid structure over a∗g. denote the structure maps of this groupoid by α̃, β̃, ϵ̃ and the composition by ⊕t∗g. now if σ : g → r be a multiplicative function on g, then one can twist the cotangent lie groupoid and get a lie groupoid structure on t∗g × r over a∗g whose structure maps are given by α̃σ(ωg,γ) = e −σ(g)α̃(ωg) β̃σ(νh,ζ) = β̃(νh) − ζ(δσ)aβ(h)g (ωg,γ) ⊕t∗g×r (νh,ζ) = ( (ωg + e σ(g)ζ(δσ)g) ⊕t∗g (eσ(g)νh),γ + eσ(g)ζ ) . then if we consider the canonical contact 1-form ηg on t ∗g × r and the multiplicative function σ◦pr, where pr : t∗g×r → g denotes the projection onto g, the triple (t∗g × r ⇒ a∗g,ηg,σ ◦ pr) is a contact groupoid. on generalized lie bialgebroids and jacobi groupoids 219 the lie algebroid of a contact groupoid is given by the following [9]. theorem 4.5. let (g ⇒ m,η,σ) be a contact groupoid. then the base m admits a unique jacobi structure (λ0,e0) such that (α,e σ) is a conformal jacobi map and β is an anti-jacobi map. moreover the lie algebroid of g ⇒ m is isomorphic to the 1-jet lie algebroid of the jacobi manifold (m,λ0,e0). let g ⇒ m be a lie groupoid with lie algebroid ag → m, and σ : g → r be a multiplicative function on g. then σ induces a 1-cocycle ϕ0 ∈ γ(a∗g) of the lie algebroid ag, defined by ϕ0|x(xx) = xx(σ), for xx ∈ axg, x ∈ m. if we consider the contact groupoid (t∗g × r ⇒ a∗g,ηg,σ ◦ pr) over a∗g, then the induced jacobi structure on a∗g is same as the linear jacobi structure on a∗g coming from the lie algebroid ag and its 1-cocycle ϕ0 [9]. thus the lie algebroid of the twisted cotangent groupoid t∗g×r → a∗g is isomorphic to the 1-jet lie algebroid t∗(a∗g) × r → a∗g, where a∗g is the linear jacobi manifold. moreover this isomorphism s is given by (j′g) −1 followed by (rag,−id), that is, s = (j′g) −1 ◦ (rag,−id), where j′g is the isomorphism between the lie algebroids a(t∗g × r) and t∗(ag) × r over a∗g. given a multiplicative function on a lie groupoid, one can also twist the tangent lie groupoid and get a lie groupoid structure on tg×r over tm×r. example 4.6. ([9]) let σ : g → r be a multiplicative function on g, then there is a lie groupoid structure on tg×r over tm ×r whose structure maps are given by (tα)σ(xg,λ) = ( (tα)(xg),xg(σ) + λ ) (tβ)σ(yh,µ) = ( (tβ)(yh),µ ) (xg,λ) ⊕tg×r (yh,µ) = (xg ⊕tg yh,λ). the lie algebroid of this lie groupoid tg × r ⇒ tm × r is given by tag × r → tm × r, the twisted tangent lie algebroid of (ag,ϕ0). let jg denote the isomorphism between tag × r and a(tg × r) over tm × r, generalizing the isomorphism between tag and atg over tm. definition 4.7. ([9]) (jacobi groupoid) let g ⇒ m be a lie groupoid with a jacobi structure (λ,e) on g and σ : g → r be a multiplicative 220 a. das function. then (g ⇒ m,λ,e,σ) is called a jacobi groupoid if the bundle map (λ,e) ♯ : t∗g × r → tg × r is a lie groupoid morphism t∗g × r (λ,e)♯ // �� tg × r �� a∗g // tm × r from the twisted cotangent lie groupoid given in example 4.4 to the twisted tangent lie groupoid given in example 4.6. example 4.8. contact groupoids are examples of jacobi groupoids [9]. jacobi lie groups [8] are just jacobi groupoids over a point [9]. suppose (g ⇒ m,λ,e,σ) be a jacobi groupoid. then the dual bundle a∗g ∼= (tm)0 also carries a lie algebroid structure whose bracket and anchor are given by [α,β]∗(x) = ( π1[(α̃,0),(β̃,0)](λ,e) ) (x), ρ∗(ωx) = λ ♯(ωx), for x ∈ m, where α,β ∈ γa∗g ∼= γ(tm)0; α̃, β̃ be their arbitrary extension to 1-forms on g, and t∗g×r → t∗g being the projection onto the first factor [9]. moreover the vector field e induces a 1-cocycle x0 ∈ γag of the lie algebroid a∗g and is defined by x0|x(ωx) = −⟨ωx,e(x)⟩, for ωx ∈ a∗xg, x ∈ m. let (λag,eag) be the linear jacobi structure on ag coming from the lie algebroid a∗g and its 1-cocycle x0. theorem 4.9. let (g ⇒ m,λ,e,σ) be a jacobi groupoid with lie algebroid ag. then ( (ag,ϕ0),(a ∗g,x0) ) is a generalized lie bialgebroid over m. proof. since (λ,e) ♯ : t∗g×r −→ tg×r is a morphism of lie groupoids, thus applying the lie functor, we get a morphism a ( (λ,e) ♯ ) : a(t∗g×r) −→ a(tg × r) a(t∗g × r) a ( (λ,e)♯ ) // �� a(tg × r) �� a∗g // tm × r on generalized lie bialgebroids and jacobi groupoids 221 between corresponding lie algebroids. then by an argument similar to [11] shows that the diagram a(t∗g × r) a ( (λ,e)♯ ) // j′ g �� a(tg × r) t∗(ag) × r (λag,eag) ♯ // t(ag) × r jg oo commutes, where jg : tag×r → a(tg×r) is the isomorphism of lie algebroids over tm × r, and j′g : a(t ∗g× r) → t∗(ag) × r is the isomorphism of lie algebroids over a∗g. thus it follows that, a(t∗g × r) a ( (λ,e)♯ ) // j′ g �� a(tg × r) t∗(ag) × r (λag,eag) ♯ // t(ag) × r jg oo t∗(a∗g) × r (rag,−id) 44jjjjjjjjjjjjjjjj s 55 (λag,eag) ♯◦(rag,−id) 55 also commutes, as a ( (λ,e) ♯ ) ◦ s = a ( (λ,e) ♯ ) ◦ (j′g) −1 ◦ (rag,−id) = jg ◦ (λag,eag)♯ ◦ (rag,−id). note that, s = (j′g) −1 ◦ (rag,−id) is an isomorphism of lie algebroids over a∗g, and also jg is an isomorphism of lie algebroids over tm ×r. from the diagram below, a(t∗g × r) xxqq qq qq qq qq a ( (λ,e)♯ ) // a(tg × r) &&nn nnn nnn nnn a∗g ( ρ∗( ),x0( ) ) // tm × r t∗(a∗g) × r ffmmmmmmmmmm s oo (λag,eag) ♯◦(rag,−id) // t(ag) × r 88ppppppppppp jg oo 222 a. das since the top row is a morphism between lie algebroids over ( ρ∗( ),x0( ) ) : a∗g → tm × r, the bottom row is also a morphism between lie algebroids over the map ( ρ∗( ),x0( ) ) : a∗g → tm × r. thus the result follows from theorem 4.2. 5. coisotropic subgroupoids of jacobi groupoid in this section we introduce the notion of coisotropic subgroupoids of a jacobi groupoid which is a straight forward generalization of coisotropic subgroupoids of a poisson groupoid [16]. we also study their infinitesimal counterpart. • coisotropic subgroupoids definition 5.1. let (g ⇒ m,λ,e,σ) be a jacobi groupoid. then a subgroupoid h ⇒ n of g ⇒ m is called a coisotropic subgroupoid if h is a coisotropic submanifold of g. example 5.2. (i) any poisson groupoid (g ⇒ m,λ) can be considered as a jacobi groupoid with e = 0 and σ = 0. then coisotropic subgroupoids of (g ⇒ m,λ) are the coisotropic subgroupoids of (g ⇒ m,λ,0,0). (ii) let (g ⇒ m,λ,e,σ) be a jacobi groupoid. then m is a coisotropic submanifold of g and carries an induced jacobi structure [9]. if n ↪→ m be a coisotropic submanifold, the subgroupoid g|n := α−1(n) ∩ β−1(n) is a coisotropic subgroupoid of (g ⇒ m,λ,e,σ). note that, the infinitesimal object corresponding to a jacobi groupoid (g ⇒ m,λ,e,σ) is the generalized lie bialgebroid ( (ag,ϕ0),(a ∗g,x0) ) . therefore it is natural to ask how the lie algebroid of a coisotropic subgroupoid h ⇒ n is related to the generalized lie bialgebroid ( (ag,ϕ0), (a∗g,x0) ) . to answer this question, we introduce coisotropic subalgebroids of a generalized lie bialgebroid and show that infinitesimal form of coisotropic subgroupoids of a jacobi groupoid appear as coisotropic subalgebroids of the corresponding generalized lie bialgebroid ( (ag,ϕ0),(a ∗g,x0) ) . definition 5.3. let ( (a,ϕ0),(a ∗,x0) ) be a generalized lie bialgebroid over m. then a lie subalgebroid b → n of a → m is called a coisotropic subalgebroid of ( (a,ϕ0),(a ∗,x0) ) if b ↪→ a is a coisotropic submanifold, where a is equipped with the linear jacobi structure coming from (a∗,x0). on generalized lie bialgebroids and jacobi groupoids 223 proposition 5.4. let a → m be a lie algebroid and ϕ0 ∈ γa∗ be a 1-cocycle. then a subbundle b → n of a → m is a lie subalgebroid (and hence ϕ0|n ∈ γb∗ is a 1-cocycle of it) if and only if b0 is a coisotropic submanifold of a∗, where a∗ is equipped with the linear jacobi structure and b0x = {γ ∈ a∗x ∣∣ γ(v) = 0, ∀v ∈ bx}, x ∈ n. proof. first suppose that, b is a lie subalgebroid of a → m. note that for any x ∈ γa, the function lx is a linear function on a∗. among the functions lx for x ∈ γa, those which vanishes on b0 are precisely those for which x ∣∣ n ∈ γb. let x,y ∈ γa be such that x ∣∣ n ,y ∣∣ n ∈ γb. then lx, ly are linear functions on a∗ vanishes on b0. we have the jacobi bracket {lx, ly } = l[x,y ]. since b → n is a lie subalgeboid of a → m, we have [x,y ] ∣∣ n ∈ γb. therefore the function {lx, ly } also vanishes on b0. among the pull back functions on a∗, those which vanishes on b0 are of the form f ◦ q∗, for some f ∈ c∞(m) with f ∣∣ n ≡ 0. therefore for any x ∈ γa and f ∈ c∞(m) with x ∣∣ n ∈ γb, f ∣∣ n ≡ 0, we have {lx,f ◦ q∗} = ( ρ(x)f + ϕ(x)f ) ◦ q∗. since ρ(x) ∣∣ n ∈ tn and f ∣∣ n ≡ 0 ( that is, (δf)|n ∈ (tn)0 ) , the function ρ(x)f +ϕ(x)f = ⟨ρ(x),δf⟩+ϕ(x)f is vanishes on n, and hence, {lx,f ◦q∗} is vanishes on b0. thus by proposition 2.4, we have b0 is a coisotropic submanifold of a∗. thus from the above proposition and since (b0)0 = b, we have the following. proposition 5.5. if b → n be a coisotropic subalgebroid of ( (a,ϕ0), (a∗,x0) ) , then b0 → n is a coisotropic subalgebroid of ( (a∗,x0),(a,ϕ0) ) . it is known that (cf. proposition 2.18), the base of a generalized lie bialgebroid carries an induced jacobi structure. the next result shows that the base of a coisotropic subalgebroid is a coisotropic submanifold with respect to this induced jacobi structure. proposition 5.6. let ( (a,ϕ0),(a ∗,x0) ) be a generalized lie bialgebroid over m and b → n be a coisotropic subalgebroid of ( (a,ϕ0),(a ∗,x0) ) . then n is a coisotropic submanifold of m. 224 a. das proof. if (λm,em) denote the induced jacobi structure on m, then from the remark 2.19, we have λ ♯ m = ρ∗ ◦ ρ ∗, where ρ and ρ∗ denote the anchors of the lie algebroids a and a ∗ respectively. we first prove that ρ∗(tn)0 ⊆ b0. this is true because, ⟨ρ∗ξx,v⟩ = ⟨ξx,ρ(v)⟩ = 0, for ξx ∈ (tn)0x and v ∈ bx. thus we have λ ♯ m(tn) 0 = ρ∗ ◦ ρ∗(tn)0 ⊆ ρ∗(b0) ⊆ tn, in the last inclusion we have used that b0 is a lie subalgebroid of a∗. therefore, n is a coisotropic submanifold of m. • infinitesimal form of coisotropic subgroupoids proposition 5.7. let (g ⇒ m,λ,e,σ) be a jacobi groupoid with generalized lie bialgebroid ( (ag,ϕ0),(a ∗g,x0) ) . let h ⇒ n be a coisotropic subgroupoid with lie algebroid ah → n. then ah → n is a coisotropic subalgebroid of ( (ag,ϕ0),(a ∗g,x0) ) . proof. since h ⇒ n is a lie subgroupoid of g ⇒ m, therefore ah → n is a lie subalgebroid of ag → m. next we claim that the anchor ρ∗ = λ♯ ∣∣ (tm)0 of the lie algebroid a∗g maps (ah)0 to tn. observe that, for any x ∈ n, (ah)0x = (tm) 0 x ∩ (th)0x and txn = txm ∩ txh. therefore, ρ∗(ah) 0 = λ♯(tm)0 ∩ λ♯(th)0 ⊆ tm ∩ th = tn, here we have used the fact that both m and h are coisotropic submanifolds of g. let θ,ϑ ∈ γa∗g ∼= (tm)0 be such that θ ∣∣ n ,ϑ ∣∣ n ∈ (ah)0. let θ̃, ϑ̃ be their respective extensions to 1-forms on g which are conormal to h. then the 1-form π1[(θ̃,0),(ϑ̃,0)](λ,e) on g is conormal to both m and h, as m and h are both coisotropic submanifolds of g. therefore, ( π1[(θ̃,0),(ϑ̃,0)](λ,e) )∣∣ n ∈ (tm)0 ∩ (th)0 = (ah)0. hence (ah)0 → n defines a lie subalgebroid of a∗g → m. therefore, by proposition 5.4, it follows that ah ↪→ ag is a coisotropic submanifold. thus, ah → n is a coisotropic subalgebroid of ( (ag,ϕ0),(a ∗g,x0) ) . on generalized lie bialgebroids and jacobi groupoids 225 corollary 5.8. let (g ⇒ m,λ,e,σ) be a jacobi groupoid and h ⇒ n be a coisotropic subgroupoid of it. then n is a coisotropic submanifold of m. acknowledgements the author would like to thank the referee for his comments on the earlier version of this manuscript. references [1] y. hagiwara, nambu-jacobi structures and jacobi algebroids, j. phys. a: math. gen. 37 (26) (2004), 6713 – 6725. 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[9] d. iglesias, j. c. marrero, jacobi groupoids and generalized lie bialgebroids, j. geom. phys. 48 (2-3) (2003), 385 – 425. [10] k. c. h. mackenzie, “lie groupoids and lie algebroids in differential geometry”, london mathematical society lecture note series 124, cambridge university press, cambridge, 2005. [11] k. c. h. mackenzie, p. xu, lie bialgebroids and poisson groupoids, duke math. j. 73 (2) (1994), 415 – 452. [12] k. c. h. mackenzie, p. xu, classical lifting process and multiplicative vector fields, quart. j. math. 49 (193) (1998), 59 – 85. [13] y. kosmann-schwarzbach, exact gerstenhaber algebras and lie bialgebroids, acta appl. math. 41 (1-3) (1995), 153 – 165. [14] i. vaisman, “lectures on the geometry of poisson manifolds”, progress in mathematics 118, birkhäuser verlag, basel, 1994. [15] a. weinstein, coisotropic calculus and poisson groupoids, j. math. soc. japan 40 (4) (1988), 705-727. [16] p. xu, on poisson groupoids, inter. j. math. 6 (1) (1995), 101 – 124. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 36, num. 1 (2021), 51 – 62 doi:10.17398/2605-5686.36.1.51 available online june 9, 2021 hurwitz components of groups with socle psl(3, q) h.m. mohammed salih department of mathematics, faculty of science, soran university kawa st. soran, erbil, iraq havalmahmood07@gmail.com received january 20, 2021 presented by a. turull accepted may 10, 2021 abstract: for a finite group g, the hurwitz space hinr,g(g) is the space of genus g covers of the riemann sphere p1 with r branch points and the monodromy group g. in this paper, we give a complete list of some almost simple groups of lie rank two. that is, we assume that g is a primitive almost simple groups of lie rank two. under this assumption we determine the braid orbits on the suitable nielsen classes, which is equivalent to finding connected components in hinr,g(g). key words: genus zero systems, braid orbits, connected components. msc (2020): 20b15 1. introduction let ω be a finite set of order n and g be a transitive subgroup of sn such that g = 〈x1,x2, . . . ,xr〉 , (1.1) r∏ i=1 xi = 1 , xi ∈ g# = g\{1} , i = 1, . . . ,r , (1.2) r∑ i=1 ind xi = 2(n + g − 1) , (1.3) where ind xi is the minimal number of 2-cycles needed to express xi as a product. we call g a group of genus g and the triple (g, ω,〈x1,x2, . . . ,xr〉) a genus g system. these conditions correspond to the existence of an n sheeted branched covering of riemann surface x of genus g with r-branch points and monodromy group g [9]. in [9], guralnick and thompson have conjectured that the set e∗(g) of possible isomorphism classes of composition factors of simple groups which issn: 0213-8743 (print), 2605-5686 (online) c© the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.36.1.51 mailto:havalmahmood07@gmail.com https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 52 h.m. mohammed salih are neither cyclic nor alternating, is finite for all g ≥ 0. furthermore, they have observed that the conjecture reduces to the consideration of the system (g, ω,〈x1,x2, . . . ,xr〉) where g is primitive on ω. a useful reference for more details is [9]. the primitive permutation representations of finite groups are determined by their maximal subgroups whose structure has been described by aschbacher and o’nan-scott theorem [3]. proposition 1.1. ([3]) suppose that g is a finite group and m is a maximal subgroup of g such that ⋂ g∈g mg = 1. let s be a minimal normal subgroup of g, let l be a minimal normal subgroup of s, and let ∆ = {l = l1,l2, . . . ,lm} be the set of the g-conjugates of l. then l is simple, s = 〈l1, . . . ,lr〉, g = ms and furthermore either (a) l is of prime order p; or l is non abelian simple group and one of the following hold: (b) f∗(g) = s ×r, where s ∼= r and m ∩s = 1; (c1) f∗(g) = s and m ∩s = 1; (c2) f∗(g) = s and m ∩s 6= 1 = m ∩l; (c3) f∗(g) = s and m ∩ s = m1 × m2 × ···× mt, where mi = m ∩ li, 1 ≤ i ≤ t. as far as we know (see [14, 10, 11, 12]), there are four types of classification of genus g system as follows: 1. up to signature. 2. up to ramification type. 3. up to the braid action and diagonal conjugation by aut(g). 4. up to the braid action and diagonal conjugation by inn(g). the weakest classification is up to signature (that is 1.) and the strongest one is up to the braid action and diagonal conjugation by inn(g) (that is 4.), because it includes all 1, 2 and 3. in [14, 15, 9, 1, 5, 4, 2], they have classified these cases (a), (b), (c1), (c2), (c3) up to signatures for genus zero. in [11, 12], they have produced a hurwitz components of groups with socle psl(3, q) 53 complete list of affine primitive genus 0, 1 and 2 groups up to the braid action and diagonal conjugation by inn(g). a group g is said to be almost simple if it contains a non-abelian simple group s and s ≤ g ≤ aut(s). in [10], kong works on almost simple groups of type projective special linear group psl(3,q). let g be a group such that psl(3,q) ≤ g ≤pγl(3,q) where pγl(3,q) is the projective semilinear group. g acts on points in the natural module, that is the set of projective points of 2-dimensional projective geometry pg(2,q). she gave a complete list for some almost simple groups of lie rank 2 up to ramification type in her phd thesis for a genus 0, 1 and 2 system. in this paper, we consider almost simple groups of lie rank 2 for genus zero and classify them up to the braid action and diagonal conjugation by inn(g). the equivalence classes of g-covers x of p1 with r branched points are called a hurwitz space and denoted by hinr,g(g) where in denotes an inner automorphism of g. note that x is a riemann surface of genus g. let ci be the conjugacy class of xi. then the multi set of non trivial conjugacy classes c = {c1, . . . ,cr} in g is called the ramification type of the g-covers x. for any r-tuple (x1, . . . ,xr) gives a ramification type c̄ with xi ∈ ci for i = 1, . . . ,r. let c̄ be a fixed ramification type, then the subset hinr (g,c̄) of hinr (g) consists of all [p,φ] with admissible surjective map φ: π(p1\p,p) → g sends the conjugacy class ∑ pi to the conjugacy class ci for i = 1, . . . ,r. it is a union of connected components in hinr (g). in this paper, we study the hurwitz space hinr (g). in particular we focus on the subset hinr (g,c̄) of hinr (g). we try to find the connected components hr(g,c̄) of g-curves x of genus 0 such that g(x/g) = g(p1) = 0. to do this, one needs to find corresponding braid orbits. the our main result is theorem 1.2, which gives the complete classification of primitive genus 0 systems of almost simple group of lie rank two. theorem 1.2. up to isomorphism, there exist exactly seven primitive genus zero groups with socle psl(3,q) for some q, 3 ≤ q ≤ 13. the corresponding primitive genus zero groups are enumerated in table 2 and table 3. in our situation, the computation shows that there is exactly 514 braid orbits of primitive genus 0 systems for some almost simple groups of lie rank two. the degree and the number of the branch points are given in table 1. 54 h.m. mohammed salih table 1: primitive genus zero systems: number of components degree number of group iso. types number of ramification types number of components r = 3 number of comp. r = 4 number of comp. r = 5 number of comp. total 13 1 45 93 14 5 112 21 4 76 204 26 4 234 31 1 15 92 2 94 57 1 8 72 2 74 totals 7 144 461 44 9 514 this paper consists of four sections as follows. section 2 sets out some notation and results that will be needed throughout the paper. we then discuss the relationship between connected components of hurwitz spaces and braid orbits on nielsen classes. in section 3, we describe our methodology which will be used to obtain the ramification types and braid orbits. furthermore, we give a particular example to explain this methodology. finally, several results are given about hurwitz spaces. 2. braid action on nielsen classes we begin this section with a formal definition of the artin braid group. definition 2.1. for r ≥ 2, the artin braid group br is generated by r−1 elements σ1,σ2, . . . ,σr−1 that satisfy the following relations: σiσj = σjσi for all i,j = 1, 2, . . . ,r − 1 with |i − j| ≥ 2, and σiσi+1σi = σi+1σiσi+1 for i = 1, 2, . . . ,r − 2. these relations are known as the braid relations. the braid σi acts on generating tuples x = (x1, . . . ,xr) of a finite group g with ∏r i=1 xi = 1 as follows: (x1, . . . ,xi,xi+1, . . . ,xr)σi = (x1, . . . ,xi+1,x −1 i+1xixi+1, . . . ,xr) (2.1) for i = 1, . . . ,r − 1. the braid orbit of x is the smallest set of tuples which contains x and is closed under the operations (2.1). applying φ: π(p1 \p,p) → g to the canonical generators of π1(p1 \p,p) gives the generators of a product one generating tuple in g that is, φ(λi) = xi. we define �r(g) = { (x1, . . . ,xr) : g = 〈x1, . . . ,xr〉 , r∏ i=1 xi = 1 , xi ∈ g# , i = 1, . . . ,r } . hurwitz components of groups with socle psl(3, q) 55 let a ≤ aut(g). then the subgroup a acts on �r(g) via sending (x1, . . . ,xr) to (a(x1), . . . ,a(xr)), for a ∈ a, which is known as the diagonal conjugation. this action commutes with the operations (2.1). thus a permutes the braid orbits. if a = inn(g), then it leaves each braid orbit invariant [16]. let �inr (g) = �r(g)/ inn(g). for a ramification type c̄, we define the subset n(c̄) = { (x1, . . . ,xr) : g = 〈x1, . . . ,xr〉, ∏r i=1 xi = 1, ∃σ ∈ sn such that xi ∈ ciσ for all i } which is called the nielsen class of c̄. the topology on har (g) is well defined. let or be the set of all r-tuples of distinct elements in p1, equipped with the product topology [16]. for the remaining of this section, we collect few results which will be used to explain the relationship between the braid orbits and their corresponding covers. lemma 2.2. ([16]) the map ψa : har (g) → or, ψa([p,φ])) = p , is covering. the fundamental group π1(or,p0) = br acts on ψ −1 a (p0) where p0 = {1, . . . ,r} is the base point in or via path lifting where the fiber is ψ−1a (p0) = { [p0,φ]a : φ : π1(p1 \p0,∞) → g is admissible }. this φ gives as product one generating tuple (x1, . . . ,xr) of g. lemma 2.3. ([16]) we obtain a bijection ψ−1a (p0) → � a r (g) by sending [p0,φ]a to the generators (x1, . . . ,xr) where xi = φ([γi]) for i = 1, . . . ,r. the image na(c̄) of n(c̄) in �ar (g) is the union of braid orbits. if ψa in lemma 2.2 restricts to a connected component h of har (g), then lemma 2.3 implies that the fiber in h over p0 corresponds to the set na(c̄). proposition 2.4. ([16]) let c̄ be a fixed ramification type in g, and the subset har (g,c̄) of har (g) consists of all [b,φ]a with b = {b1, . . . ,br}, φ: π1(p1 \p,∞) → g and φ(θbi )) ∈ ci for i = 1, . . . ,r. then h a r (g,c̄) is a union of connected components in har (g). under the bijection from lemma 2.3, the fiber in har (g,c̄) over b0 corresponds the set na(c̄). this yields a one to one correspondence between components of har (c) and the braid orbits on na(c̄). in particular, hinr (g,c) is connected if and only if there is only one braid orbit. 56 h.m. mohammed salih the following riemann existence theorem tells us there is a one to one correspondence between the equivalence classes of product one generating tuples (x1, . . . ,xr) of g and the equivalence classes of g-covers of type c̄ such that xi ∈ ci for i = 1, . . . ,r. proposition 2.5. ([8]) let g be a finite group and c̄ = {c1, . . . ,cr} be a ramification type. then there exists a g-cover of type c̄ if and only if there exists a generating tuple (x1, . . . ,xr) of g with ∏r i=1 xi = 1 and xi ∈ ci, for i = 1, . . . ,r. definition 2.6. ([8]) two generating tuples are braid equivalent if they lie in the same orbit under the group generated by the braid action and diagonal conjugation by inn(g). this means that if two generating tuples lie in the same braid orbit under either the diagonal conjugation or the braid action, then the corresponding covers are equivalent by riemann’s existence theorem. definition 2.7. two coverings µ1 : x1 → p1 and µ2 : x2 → p1 are equivalent if there exists a homeomorphism α: x1 → x2 with µ2α = µ1. as a consequence we have the following result. proposition 2.8. ([16]) two generating tuples are braid equivalent if and only if their corresponding covers are equivalent. to answer whether or not hr(g,c̄) is connected which is still an open problem, both computationally and theoretically. the mapclass package of james, magaard, shpectorov and volklein, is designed to perform braid orbit computations for a given finite group and given type. few results were known about it such as in [11] and [13]. 3. methodology and example: listing primitive genus zero systems the theory introduced in the previous section provides reformation of the geometric problem into the language of permutation groups. this leads us to work with permutation groups rather than with g-covers (see proposition 2.5). the following method shows that the existence primitive genus 0 system for a given group g and type c̄, and then computing braid orbits on the set hurwitz components of groups with socle psl(3, q) 57 of nielsen class na(c̄). proposition 2.4 yields a one to one correspondence between the braid orbits on na(c̄) and connected components of har (g,c̄). now we can decide whether or not har (g,c̄) connected, when g is a primitive almost simple groups of lie rank two and given type c̄. we are presenting our computations in tables 2 and 3. to obtain these tables we needed to do the following steps: • we extract all primitive permutation group g by using the gap function allprimitivegroups(degreeoperation,n). • for every almost simple group g, compute the conjugacy class representatives and permutation indices on n points. • for given n,g and g we use the gap function restrictedpartions to compute all possible ramification types satisfying the riemann-hurwitz formula. • compute the character table of g if possible and remove those types which have zero structure constant. • for each of the remaining types of length greater than or equal to 4, we use mapclass package to compute braid orbits, especially by using the function generatingmcorbits(g,0,tuple). for tuples of length 3 determine braid orbits via double cosets [8]. • we use the same rules for labeling and ordering conjugacy classes of g as in [13]. this will be done by both the proof in algebraic topology and calculations of gap (groups, algorithms, programming) software. also genus 0 generating tuples for almost simple groups of type psl(3,q) on their other primitive actions and genus 0 are given. the next example show that how to compute the ramification types and braid orbits for the group psl(3, 3). example 3.1. suppose that g = psl(3, 3) and |ω| = n = 3 3−1 3−1 = 13. gap> a:=allprimitivegroups(degreeoperation,13); [ c(13), d(2*13), 13:3, 13:4, 13:6, agl(1, 13), l(3, 3), a(13), s(13) ] gap> list(a,x->onanscotttype(x)); [ "1", "1", "1", "1", "1", "1", "2", "2", "2" ] gap> loadpackage("mapclass");; gap> read("qu1.g"); 58 h.m. mohammed salih gap> checkingthegroup(k); gap> k:=a[7]; l(3, 3) gap> checkingthegroup(k); gap> gt:=generatingtype(k,13,0); checking the ramification type 66 with 0 remaining [ [ 7, 8, 8 ], [ 7, 7, 8 ], [ 7, 7, 7 ], [ 6, 8, 5 ], [ 6, 8, 4 ], [ 6, 3, 5 ], [ 6, 3, 4 ], [ 3, 8, 8 ], [ 3, 7, 8 ], [ 3, 7, 7 ], [ 3, 3, 8 ], [ 3, 3, 7 ], [ 3, 3, 3 ], [ 2, 8, 12 ], [ 2, 8, 11 ], [ 2, 8, 10], [ 2, 8, 9 ], [ 2, 7, 12 ], [ 2, 7, 11 ], [ 2, 7, 10 ], [ 2, 7, 9 ], [ 2, 6, 6, 8 ], [ 2, 6, 6, 3 ], [ 2, 4, 5 ], [ 2, 3, 12 ], [ 2, 3, 11 ], [ 2, 3, 10 ], [ 2, 3, 9 ], [ 2, 2, 8, 8 ], [ 2, 2, 7, 8 ], [ 2, 2, 7, 7 ], [ 2, 2, 6, 5 ], [ 2, 2, 6, 4 ], [ 2, 2, 3, 8 ], [ 2, 2, 3, 7 ], [ 2, 2, 3, 3 ], [ 2, 2, 2, 12 ], [ 2, 2, 2, 11 ], [ 2, 2, 2, 10 ], [ 2, 2, 2, 9 ], [ 2, 2, 2, 6, 6 ], [ 2, 2, 2, 2, 8 ], [ 2, 2, 2, 2, 7 ], [ 2, 2, 2, 2, 3 ], [ 2, 2, 2, 2, 2, 2 ] ] gap> length(gt); 45 we can pick one of the generating tuple t and compute braid orbits as follows: gap> t:=list(gt[45],x->cc[x]); [ (2,13)(3,10)(4,9)(5,6), (2,13)(3,10)(4,9)(5,6), (2,13)(3,10)(4,9)(5,6), (2,13)(3,10)(4,9)(5,6), (2,13)(3,10)(4,9)(5,6), (2,13)(3,10)(4,9)(5,6) ] gap> orb:=generatingmcorbits(k,0,t);; total number of tuples: 183980160 collecting 20 generating tuples .. done cleaning done; 20 random tuples remaining orbit1: length=32760 generating tuple =[ ( 2,11)( 4,12)( 7,10)( 9,13), ( 1, 9)( 4, 7)( 6,10)( 8,12), ( 1, 6)( 2, 3)( 7,13)(10,12), ( 1, 9)( 2, 8)( 5, 7)(10,11), ( 2,10)( 5, 7)( 6,12)( 8,11), ( 3,11)( 4,10)( 7, 9)(12,13) ] centralizer size=1 0 tuples remaining cleaning a list of 20 tuples random tuples remaining: 0 cleaning done; 0 random tuples remaining computation complete : 1 orbits found. hurwitz components of groups with socle psl(3, q) 59 4. results in this section, we present some results which related to the connectedness of the hurwitz space for some almost simple groups of lie rank two for genus zero. proposition 4.1. if r ≥ 4 and g = psl(3,q) where q = 3, 5, then hinr (g,c̄) is connected. proof. since we have just one braid orbit for all types c̄ and the nielsen classes n(c̄) are the disjoint union of braid orbits. from proposition 2.4, we obtain that the hurwitz spaces hinr (g,c) are disconnected. proposition 4.2. if g = psl(3, 4).2 or g = psl(3,q) where q = 4, 9, then hinr (g,c̄) is disconnected. proof. since we have at least two braid orbits for some type c̄ and the nielsen classes n(c̄) are the disjoint union of braid orbits. from proposition 2.4, we obtain that the hurwitz spaces hinr (g,c) are disconnected. the proof of the following is analogous to the proof of proposition 4.1. proposition 4.3. if g = pgl(3,q) where q = 4, 7, then hinr (g,c) is connected. proposition 4.4. if g = pγl(3, 4) where r ≥ 4, then hinr (g,c) is connected. acknowledgements i would like thank to the referee for careful reading of the article and detailed report including suggestions and comments; and i appreciate his/her effort on reviewing the article. references [1] m. aschbacher, on conjectures of guralnick and thompson, j. algebra 135 (2) (1990), 277 – 343. [2] m. aschbacher, r. guralnick, k. magaard, rank 3 permutation characters and primitive groups of low genus, preprint. 60 h.m. mohammed salih [3] m. aschbacher, l. scott, maximal subgroups of finite groups, j. algebra 92 (1) (1985), 44 – 80. [4] d. frohardt, r. guralnick, k. magaard, genus 2 point actions of classical groups, preprint. [5] d. frohardt, r. guralnick, k. magaard, genus 0 actions of groups of lie rank 1, in “arithmetic fundamental groups and noncommutative algebra”, proceedings of symposia in pure mathematics, 70, ams, providence, rhode island, 2002, 449 – 483. [6] d. frohardt, k. magaard, composition factors of monodromy groups, ann. of math. 154 (2) (2001), 327 – 345. [7] the gap group, gap – groups, algorithms, and programming, version 4.6.2, 2013. http://www.gap-system.org [8] w. gehao, “genus zero systems for primitive groups of affine type”, phd thesis, university of birmingham, 2011. [9] r. m. guralnick, j. g. thompson, finite groups of genus zero, j. algebra 131 (1) (1990), 303 – 341. [10] x. kong, genus 0, 1, 2 actions of some almost simple groups of lie rank 2, phd thesis, wayne state university, 2011. [11] k. magaard, s. shpectorov, g. wang, generating sets of affine groups of low genus, in “computational algebraic and analytic geometry”, contemp. math., 572, ams, providence, rhode island, 2012, 173 – 192. [12] h. mohammed salih, “finite groups of small genus”, phd thesis, university of birmingham, 2014. [13] h. mohammed salih, connected components of affine primitive permutation groups, j. algebra 561 (2020), 355 – 373. [14] m. g. neubauer, “on solvable monodromy groups of fixed genus”, phd thesis, university of southern california, 1989. [15] t. shih, a note on groups of genus zero, comm. algebra 19 (10) (1991), 2813 – 2826. [16] h. völklein, “groups as galois groups”, cambridge studies in advanced mathematics, 53, cambridge university press, 1996. 5. appendix note that n.o means number of orbits, l.o means largest length of the orbit and gzs means genus 0 system. the following tables represent almost simple groups of lie rank two. http://www.gap-system.org hurwitz components of groups with socle psl(3, q) 61 table 2: part1: gzss for almost simple groups of lie rank two group ramification type n.o l.o ramification type n.o l.o (3e,3b,6b) 18 1 (3e,3d,6b) 8 1 (3e,3c,6a) 8 1 (3b,6b,5a) 1 1 (3b,6b,5b) 1 1 (3b,3c,5a) 1 1 (3b,3c,5b) 1 1 (3a,6a,5a) 1 1 (3a,6a,5b) 1 1 (3a,3d,5a) 1 1 pgl(3, 4) (2a,6b,15a) 1 1 (3a,3d,5b) 1 1 (2a,6b,15c) 1 1 (2a,6a,15d) 1 1 (2a,6a,15b) 1 1 (2a,3d,15d) 1 1 (2a,3c,15a) 1 1 (2a,3d,15b) 1 1 (2a,3c,15c) 1 1 (2a,2a,3b,6b) 1 24 (2a,2a,3b,3c) 1 18 (2a,2a,3a,6a) 1 24 (2a,2a,3a,3d) 1 18 table 3: part1: gzss for almost simple groups of lie rank two group ramification type n.o l.o ramification type n.o l.o (6a,3b,3b) 8 1 (6a,6a,3b) 12 1 (6a,6a,6a) 8 1 (2a,8a,8b) 1 1 (3a,3b,8a) 1 1 (3a,3b,8b) 1 1 (3a,4a,8a) 1 1 (3a,4b,8b) 6 1 (4a,3b,3b) 8 1 (4a,6a,3b) 8 1 (4a,6a,6a) 8 1 (4a,4a,3b) 12 1 (4a,4a,6a) 6 1 (4a,4a,4a) 1 1 (2a,3b,13a) 1 1 (2a,3b,13b) 1 1 (2a,3b,13c) 1 1 (2a,3b,13d) 1 1 (2a,6a,13a) 1 1 (2a,6a,13b) 1 1 (2a,6a,13c) 1 1 (2a,6a,13d) 1 1 (2a,4a,13a) 1 1 (2a,4a,13b) 1 1 psl(3, 3) (2a,4a,13c) 1 1 (2a,4a,13d) 1 1 (2a,3a,3a,4a) 1 12 (2a,3a,3a,3b) 1 12 (2a,2a,4a,4a) 1 124 (2a,2a,3b,4a) 1 120 (2a,2a,3b,3b) 1 108 (2a,2a,4a,6a) 1 144 (2a,2a,3b,6a) 1 144 (2a,2a,6a,6a) 1 132 (2a,2a,3a,8a) 1 8 (2a,2a,2a,8b) 1 8 (2a,2a,2a,13a) 1 13 (2a,2a,2a,13b) 1 13 (2a,2a,2a,13c) 1 13 (2a,2a,2a,13d) 1 13 (2a,2a,2a,3a,3a) 1 120 (2a,2a,2a,2a,4a) 1 2016 (2a,2a,2a,2a,3b) 1 1944 (2a,2a,2a,2a,6a) 1 2160 (2a,2a,2a,2a,2a) 1 32760 62 h.m. mohammed salih table 3 (continued): part1: gzss for almost simple groups of lie rank two group ramification type n.o l.o ramification type n.o l.o (2a,4c,7b) 2 1 (2a,4c,7a) 2 1 (2a,4b,7b) 2 1 (2a,4b,7a) 2 1 (2a,4a,7b) 2 1 (2a,4a,7a) 2 1 psl(3, 4) (2a,5a,5b) 6 1 (3a,4b,4c) 8 1 (3a,4a,4c) 8 1 (3a,4a,4b) 8 1 (3a,3a,5b) 12 1 (3a,3a,5a) 12 1 (2a,2a,2a,2a,2a) 2 756 (2a,2a,2a,5b) 2 30 (2a,8a,5b) 1 1 (2a,8a,10a) 1 1 (2a,8b,5b) 1 1 (2a,8b,10a) 1 1 (2a,6a,8a) 1 1 (2a,6a,8b) 1 1 psl(3, 5) (2a,3a,24b) 1 1 (2a,3a,24a) 1 1 (2a,3a,24c) 1 1 (2a,3a,24d) 1 1 (4c,4c,4c) 28 1 (3a,4c,4c) 26 1 (3a,3a,4c) 28 1 (2a,2a,2a,8a) 1 32 (2a,2a,2a,8b) 1 32 (2a,2a,2a,4a) 2 180 (3a,3a,4a) 48 1 psl(3, 7) (2a,4a,8b) 6 1 (2a,4a,8a) 6 1 (2a,4a,7b) 2 1 (2a,4a,7a) 2 1 (2a,4a,7c) 2 1 (2a,4a,14a) 6 1 (4b,4b,4c) 8 1 (3a,4b,6a) 10 1 (2b,4b,14a) 1 1 (2b,4b,14b) 1 1 (2a,4c,14a) 1 1 (2a,4c,14b) 1 1 pσl(3, 4) (2a,6a,7a) 2 1 (2a,6a,7b) 2 1 (2b,6a,8a) 8 1 (2a,5a,8a) 2 1 (2a,2b,3a,6a) 1 42 (2a,2b,2b,8a) 2 16 (2a,2a,3a,4c) 1 64 (2a,2a,2b,7a) 1 7 (2a,2a,2b,7b) 1 7 (2b,4b,21a) 1 1 (2b,4b,21b) 1 1 (2b,6a,14a) 3 1 (2b,6a,14b) 3 1 (2b,3b,21a) 1 1 (2b,3b,21b) 1 1 (2b,6b,15a) 2 1 (2b,6b,15b) 2 1 (4b,4b,6a) 12 1 (4b,4b,3b) 14 1 pγl(3, 4) (3a,6b,6b) 12 1 (2b,2b,3c,3b) 1 58 (2b,2b,3c,6a) 1 156 (2b,2b,3a,5a) 1 20 (2b,2b,4b,4b) 1 192 (2b,2b,2b,14a) 1 28 (2b,2b,2b,14b) 1 28 (2b,2b,2a,15b) 1 264 (2b,2b,2a,15a) 1 10 (2b,2a,3a,6b) 1 54 (2b,2b,2b,2b,3a) 1 1824 (2b,2b,2a,2a,3a) 1 192 introduction braid action on nielsen classes methodology and example: listing primitive genus zero systems results appendix � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 2 (2019), 255 – 268 doi:10.17398/2605-5686.34.2.255 available online april 29, 2019 ricci solitons on para-kähler manifolds sunil kumar yadav department of mathematics, poornima college of engineering isi-6, riico institutional area, sitapura, jaipur-302022, rajasthan, india prof sky16@yahoo.com received december 18, 2018 presented by anna maria fino accepted march 30, 2019 abstract: the main purpose of the paper is to study the nature of ricci soliton on para-kähler manifolds satisfying some certain curvature conditions. in particular, if we consider certain pseudosymmetric and parallel symmetric tensor on para-kähler manifolds we prove that v is solenoidal if and only if it is shrinking or steady or expanding depending upon the sign of scalar curvature for dimension m > 4, where (g,v,λ) be a ricci soliton in a paraholomorphic projectively, pseudosymmetric para-kähler manifolds. moreover, we obtain some results related to the quasi-conformal curvature tensor on such manifolds. key words: para-kähler manifold, ricci soliton, pseudosymmetry, paraholomorphic projective curvature, quasi-conformal curvature. ams subject class. (2010): primary 53c15; secondary 53c50 and 53c56. 1. introduction ricci flow is an excellent tool in simplifying the structure of the manifolds. it is defined for riemannian manifolds of any dimension. it is a process that deforms the metric of a riemannian manifold analogous to the diffusion of heat there by smoothing out the irregularity in the metric which is given by ∂g(t) ∂t = −2 ric(g(t)), where g is the riemannian metric dependent on time t and ric(g(t)) is the ricci tensor. we consider φt : m → m, t ∈ r, be a family of diffeomorphisms and (φt : t ∈ r) is a one parameter family of abelian group called flow. it generates a vector field xq given by xqf = df(φt(q)) dt , f ∈ c∞(m). if y is a vector field then lxy = limt→0 φ•ty−y t is known as lie-derivative of y with respect to x. ricci solitons move under the ricci flow under issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.2.255 mailto:prof_sky16@yahoo.com https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 256 sunil k. yadav φt : m → m of the initial metric, that is, they are stationary points of the ricci flow in the space of metric. if g0 is a metric on the co-domain, then g(t) = φ•tg0 is the pullback of g0, is a metric on the domain. thus if g0 is a soliton of the ricci flow on the co-domain, subjected to the condition lv g0 + 2 ric g0 + 2λg0 = 0 on the co-domain then g(t) is the soliton of ricci flow on the domain subjected to the condition lv g + 2 ric g + 2λg = 0 on the domain by [13] under suitable conditions. so g0 and g(t) are metrics which satisfy ricci flow. thus the following equation lv g + 2s + 2λg = 0, (1.1) is called ricci soliton. it is said to be shrinking, steady or expanding according as λ < 0, λ = 0 and λ > 0 respectively. therefore, ricci solitons are generalization of einstein manifolds and they are also known as quasi-einstein manifolds by theoretical physicists. para-kähler manifolds are examples of symplectic, locally product and semi-riemannian manifolds. some authors studied on paracomplex geometry [3]. besides, many authors considered the notion of “hyperbolic” instead of “para”. it was first used by prvanovic [8]. in this paper, the references [4, 5] have been our motivation in studying the para-kähler manifolds, where the structure tensor p is an almost complex and metric g is positive definite. this paper is organized as follows: in section 2, we give the basic concept of para-kähler manifolds and some certain curvature tensor. in section 3, we introduce certain pseudosymmetric conditions on such manifolds. in section 4, we discuss the parallel symmetric second order tensor field. in section 5, we consider quasi-conformally flat para-kähler manifolds. finally in section 6, we study the parallel quasi-conformal para-kähler manifolds. 2. para-kähler manifolds by a para-kählerian manifold we mean a triple (m,p,g), where m is a connected differentiable manifold of dimension n = 2m, p is a (1, 1)-tensor field and g is a pseudo-riemannian metric on m satisfying the conditions p 2 = i, g(px,py ) = −g(x,y ), ∇p = 0, (2.1) for any x,y ∈ℵ(m), where ℵ(m) is the lie algebra of vector fields on m, ∇ is the levi-civita connection of g and i is the identity tensor field. let (m,p,g) be a para-kählerian manifold. the riemann-christoffel curvature tensor r, the ricci curvature s and the scalar curvature r are defined ricci solitons on para-kähler manifolds 257 by r(x,y,z,w) = g(r(x,y )z,w), s(x,y ) = tr{z → r(z,x)y}, r = trg s. let q be the ricci operator given by s(x,y ) = g(qx,y ), for these tensor fields, the following identities are satisfied r(px,py ) = −r(x,y ), r(px,y ) = −r(x,py ), s(px,y ) = −s(py,x), s(px,py ) = −s(x,y ). (2.2) tr{z → r(x,y )pz} = −2s(x,py ), tr{z → r(pz,x)y} = s(x,py ), qy = − ∑ i εir(ei,y )ei. for any (0, 2)-type tensor field φ on m and x,y ∈ ℵ(m), we define the endomorphism x ∧φ y of ℵ(m) by (x ∧φ y )z = φ(y,z)x − φ(x,z)y, z ∈ℵ(m). the paraholomorphic projective curvature tensor p̃ of (m,p,g) is defined as follows ([7, 8, 9]): p̃(x,y ) = r(x,y )− 1 n + 2 {x∧sy−(px)∧s(py )+2g(qpx,y )p}. (2.3) we recall that p̃(x,y ) = −p̃(y,x), tr{z → p̃(z,x)y} = 0,∑ i εi p̃ (x,ei,ei,w) = 1 n + 2 {ns(x,w) −rg(x,w)}, (2.4) where {e1,e2, . . . ,en} is an orthonormal frame and εi is the indicator of ei, εi = g(ei,ei) = ±1. in [8], prvanovic defined the following (0, 4)-type tensor field: given x,y,z,v ∈ℵ(m), r0(x,y,z,v ) = 1 4 { g(x,z)g(y,v ) −g(x,v )g(y,z) −g(x,pz)g(y,pv ) + g(x,pv )g(y,pz) (2.5) − 2g(x,py )g(z,pv ) } . 258 sunil k. yadav for any q ∈ m, a subspace u ⊂ tqm is called non-degenerate if g restricted to u is non-degenerate. if {u,v} is a basis of a plane σ ⊂ tqm, then σ is non-degenerate if and only if g(u,u)g(v,v)−[g(u,v)]2 6= 0. thus the sectional curvature of σ = span{u,v} is k(σ) = r(u,v,u,v) g(u,u)g(v,v) − [g(u,v)]2 . from (2.1) it follows that x and px are orthogonal for any x ∈ℵ(tm). by a p-plane we mean a plane which is invariant by p . for any q ∈ m, a vector u ∈ tqm is isotropic if g(u,u) = 0. if u ∈ tqm is not isotropic, then the sectional curvature k(u) of the p-plane span{u,pu} is called the p-sectional curvature defined by u. when k(u) is constant, then (m,p,g) is called of constant p-sectional curvature, or a para-kähler space form. the notion of a quasi-conformal curvature tensor c is given by yano and sawaki [14] and is defined by c(x,y )z = αr(x,y )z + β [ s(y,z)x −s(x,z)y + g(y,z)qx −g(x,z)qy ] (2.6) − r n ( α n− 1 + 2β ) {g(y,z)x −g(x,z)y} , where α and β are constants. if α = 1 and β = − 1 n−2 , then (2.6) reduces to conformal curvature tensor [6]. a manifold (mn,g) (n > 3) is said to be quasiconformally flat manifold if c = 0. in [1], it is known that a quasi-conformally flat manifold is either conformally flat under α 6= 0 or einstein manifold under the conditions α = 0 and β 6= 0. the authors give no restrictions for α = 0 and β = 0. however, we consider the condition α 6= 0, or β 6= 0 in this study. in view of the equations (2.2), (2.4) and (2.6), we have∑ i εi g(c (pei,py )ei,w) = − α 2 g(py,w) + β[2s(py,pw)−τg(py,w)] − r n { α n− 1 + 2β } g(py,pw), (2.7) which implies∑ i εi c (pei,py )ei = − α 2 py + β[2qpy − τ py ] + r n { α n− 1 +2β } y, (2.8) ricci solitons on para-kähler manifolds 259 where τ is the special scalar curvature, which is defined as the trace of pq. it is remarked that ∑ i εig(pei,ei) = 0. for any (0,k)-type tensor (k ≥ 1) field t on a pseudo-riemannian manifold (m,g), we define a (0,k + 2)-tensor field r ·t by the following condition (r ·t)(u,v,x1, . . . ,xk) = − k∑ s=1 t(x1, . . . ,r(u,v )xs, . . . ,xk). (2.9) a pseudo-riemannian manifold (m,g) is called semisymmetric if r · r = 0; ricci-symmetric if r ·s = 0 ([2, 4, 11, 12]). we also define a (0,k + 2)-tensor (k ≥ 1) field q(g,t) as follows q(g,t)(u,v,x1, . . . ,xk) = − k∑ s=1 t(x1, . . . , (u ∧v )xs, . . . ,xk). (2.10) a pseudo-riemannian manifold (m,g) is ricci-pseudosemisymmetric [4] if there exists a function ls : m → r such that r ·s = lsq(g,s). (2.11) it is clear that every ricci-semisymmetric manifold is ricci-pseudosymmetric. in general, the converse is not true [4]. the riemannian curvature (1, 3)-tensor field associated to the levi-civita connection ∇ of g is given by r = [∇,∇]−∇. then r(x,y,z,v ) = −r(y,x,z,v ) = −r(x,y,v,z) = r(px,py,z,v ),∑ σ r(x,y,z,v ) = 0, (2.12) where σ represents the sum over all cyclic permutations. 3. ricci-pseudosymmetric and pseudosymmetric in this section, we study the ricci-pseudosymmetric and pseudodymmetric condition on para-kähler manifolds and deduce some results. theorem 3.1. every ricci-pseudosymmetric para-kählerian manifold is ricci-semisymmetric. 260 sunil k. yadav proof. let the manifold (m,p,g) be para-kähler satisfying the condition (r ·s)(x,y,u,v ) = lsq(g,s)(x,y,u,v ). (3.1) in view of (2.2) and (2.9), we have (r ·s)(px,py,u,v ) = −(r,s)(x,y,u,v ). (3.2) using (3.2) in (3.1), it follows that lsq(g,s)(x,y,u,v ) = −lsq(g,s)(px,py,u,v ). (3.3) since ls is non-zero at a certain point q ∈ m, from (3.3), we get q(g,s)(x,y,u,v ) = −q(g,s)(px,py,u,v ). in view of (2.10), we have s(x,v )g(y,u) −s(y,v )g(x,u) + s(x,u)g(y,v ) −s(y,u)g(x,v ) = −s(v,px)g(u,py ) + s(v,py )g(u,px) (3.4) −s(u,px)g(v,py ) + s(u,py )g(v,px). then taking contraction of (3.4) with respect to y and u, and using (2.2), we obtain s(x,v ) = r n g(x,v ). (3.5) which implies (m,p,g) is an einstein manifold with r · s = 0. thus it completes the proof. corollary 3.2. let (g,v,λ) be a ricci soliton in a ricci-pseudosymmetric para-kählerian manifold (m,p,g). then v is solenoidal iff it is shrinking or steady or expanding depending on the sign of the scalar curvature. proof. in view of (1.1) and (3.5), we get (lv g)(x,v ) + 2 r n g(x,v ) + 2λg(x,v ) = 0. (3.6) taking x = v = ei where {ei} is an orthonormal basis of the tangent space at each point of the manifold and taking summation over i (1 ≤ i ≤ n), we have (lv g)(ei,ei) + 2 r n g(ei,ei) + 2λg(ei,ei) = 0, (3.7) ricci solitons on para-kähler manifolds 261 which implies div v + r + λn = 0. (3.8) if v is solenoidal, then div v = 0. thus (3.8) reduces to λ = −( r n ). therefore, we obtain the desired result. a pseudo-riemannian manifold (m,g) is said to be pseudosymmetric [4] if there exists a function lr : m → r such that r ·r = lrq(g,r). (3.9) it is well-known that every semisymmetric manifold is also pseudosymmetrc but converse is not true, in general [4]. theorem 3.3. let (m,p,g) be a pseudosymmetric para-kähler manifold. then (i) (m,p,g) is ricci flat, for dim m = 4. (ii) (m,p,g) is semisymmetric, for dim m > 4. proof. let (m,p,g) be para-kähler manifold satisfying the condition (3.9). then using the analogy with theorem 3.1, we get (n− 4)r(x,u,v,w) − 2 s(v,pw)g(u,px) + s(w,pu)g(v,px) (3.10) −s(v,pu)g(w,px) + s(u,w)g(u,v ) −s(u,v )g(x,w) = 0. putting pu instead of u in (3.10) and contacting with respect to u and x we have s = 0, for n = 4. on the other hand, for n > 4, taking contraction (3.10) with respect to u and v we have s(x,w) = r n g(x,w), (3.11) which implies r ·s = 0. in view of (3.10), we get r ·r = 0. so theorem 3.3 is proved. corollary 3.4. let (g,v,λ) be a ricci soliton in a pseudosymmetric para-kähler manifold (m,p,g) (n = 4). then v is solenoidal iff it is always steady. corollary 3.5. let (g,v,λ) be a ricci soliton in a pseudosymmetric para-kählerian manifold (m,p,g) (n > 4). then v is solenoidal iff it is shrinking or steady or expanding depending on the sign of the scalar curvature. 262 sunil k. yadav a para-kähler manifold with paraholomorphic projective curvature tensor satisfies the condition r · p̃ = lpq(g,p̃), where lp is a function on m. such type of manifold is called paraholomorphic projective-pseudosymmetric. if r·p̃ = 0, then it is said to be paraholomorphic projective-semisymmetric manifold (see [7]). theorem 3.6. let (m,p,g) be a paraholomorphic projectivepseudosymmetric para-kähler manifold. then we have (i) (m,p,g) is ricci flat, for dim m = 4. (ii) (m,p,g) is semisymmetric, for dim m > 4. proof. it is well know that, if r·p̃ = 0 at a certain point q of the manifold m, then r ·r = 0 at this point (see [7]). now, we suppose that r ·p̃ 6= 0 and let the contraction of tensor p̃ is w̃ , defined by w̃(x,w) = ∑ i εip̃(x,ei,ei,w). (3.12) from (2.4), we have w̃(x,w) = 1 n + 2 {ns(x,w ) −rg(x,w)}. (3.13) thus (m,a,g) is paraholomorphic projective-pseudosymmetric. so we have (r · p̃)(x,y,z,u,v,w) = lpq(g,p̃)(x,y,z,u,v,w). (3.14) taking contraction of (3.14) with respect to u and v we get (r ·w̃)(x,y,z,w) = lpq(g,p̃)(x,y,z,w). also, from (2.9) and (3.13), we obtain (r ·s)(x,y,z,w) = lpq(g,s)(x,y,z,w). according to theorem 3.1, we get r · s = 0. since lp does not vanish at point q, we have q(g,s) = 0 by the help of above equation. from (2.4) and (3.14), we conclude that r·r = lpq(g,r), i.e., (m,p,g) is pseudosymmetric. therefore, the proof is completed. ricci solitons on para-kähler manifolds 263 corollary 3.7. let (g,v,λ) be a ricci soliton in a paraholomorphic projectively-pseudo symmetric para-kähler manifold (m,p,g) (n = 4). then v is solenoidal iff it is always steady. corollary 3.8. let (g,v,λ) be a ricci soliton in a paraholomorphic projectively pseudo symmetric para-kähler manifold (m,p,g) (n > 4). then v is solenoidal iff it is shrinking or steady or expanding depending on the sign of the scalar curvature. 4. parallel symmetric second order covariant tensor in this section, we study the second order parallel tensor on a para-kähler manifold. thus we give the following results. theorem 4.1. a second order parallel tensor in a para-kähler space form is a linear combination of para-kähler metric and para-kähler 2-form. proof. let ~ be a (0, 2)-tensor which is parallel in view of ∇, that is, ∇~ = 0. then from ricci identity [10], we have ~(r(x,y )z,v ) + ~(z,r(x,y )v ) = 0. (4.1) using (2.5) in (4.1) and replacing x = v = ei, 1 ≤ i ≤ n, on simplification, we get{ ~(y,z) −g(y,z)(trh) + ~(py,pz) −g(y,pz)(tr.hp) +2~(py,pz) + (n− 1)~(y,z) + 3~(z,p 2y ) } = 0, (4.2) where h is a (1, 1) tensor and tr.h = ∑n i=1 ~(ei,ei). using the notion of symmetrization and anti-symmetrization. then from (4.2), we obtain (n + 3)~(y,z) + 3~(py,pz) = g(y,z)(tr.h) (4.3) and (n + 3)~(y,z) + 3~(py,pz) = g(y,pz)(tr.hp). (4.4) replacing y and z by py and pz in (4.3) and (4.4) respectively, using (2.1), we have ~s(y,z) = − 1 (n + 6) (tr.h)g(y,z) (4.5) and ~a(y,z) = − 1 n (tr.hp)g(py,z). (4.6) 264 sunil k. yadav in view of (4.5) and (4.6), we get ~(y,z) = {ϑ (tr.h) g(y,z) + ω (tr.hp) ψ}, (4.7) where ϑ = − 1 (n+6) , ω = −1 n and ψ = g(py,z). thus it completes the proof. corollary 4.2. a locally ricci symmetric para-kähler space form is an einstein manifold. proof. according to our hypothesis, if we put h = s, in (4.7) then we have tr.h = r and tr.hp = 0. then follows from (4.7), we obtain s(y,z) = ϑrg(y,z). (4.8) which completes the proof. corollary 4.3. let (g,v,λ) be a ricci soliton in a para-kähler space form. then v is solenoidal iff it is shrinking or steady or expanding depending on the sign of the scalar curvature. proof. in view of (1.1) and (4.8), we get (lv g)(y,z) + 2ϑrg(y,z) + 2λg(y,z) = 0. (4.9) taking y = z = ei in (4.9), where {ei} is an orthonormal basis of the tangent space at each point of the manifold and taking summation over i (1 ≤ i ≤ n), we have (lv g)(ei,ei) + 2ϑrg(ei,ei) + 2λg(ei,ei) = 0, (4.10) which implies div v + ϑrn + λn = 0. (4.11) here, if v is solenoidal, then we get div v = 0. thus (4.11) reduces to λ = −ϑr. so this completes the proof. theorem 4.4. a ricci semi-symmetric para-kähler space form is an einstein manifold. proof. let the para-kähler space form holds the condition r·s = 0. then we have (r(x,y ) ·s)(v,u) = 0, (4.12) ricci solitons on para-kähler manifolds 265 which reduces to s(r(x,y )v,u) + s(v,r(x,y )u) = 0. (4.13) using (2.5) in (4.13) and putting y = v = ei, where {ei} is an orthonormal basis of the tangent space at each point of the manifold and again taking summation over i, 1 ≤ i ≤ n, we get s(x,u) = r (n + 2) g(x,u). (4.14) thus this proves the theorem. corollary 4.5. let (g,v,λ) be a ricci soliton in a ricci semi-symmetric para-kähler space form. then v is solenoidal iff it is shrinking or steady or expanding depending on the sign of the scalar curvature. proof. from (1.1) and (4.14), we get (lv g)(x,u) + 2 r (n + 2) g(x,u) + λg(x,u) = 0. (4.15) then putting x = u = ei in (4.15) and taking summation over i (1 ≤ i ≤ n), we have (lv g)(ei,ei) + 2 r (n + 2) g(ei,ei) + 2λg(ei,ei) = 0, (4.16) which implies div v + r (n + 2) n + λn = 0. (4.17) in view of (4.17), if v is solenoidal then we have div v = 0. thus (4.17) reduces to λ = − r (n+2) . so the proof is clear. corollary 4.6. if the (0, 2)-type tensor field lv + 2s is parallel, where v is a vector field on a para-kähler space form. then (g,v ) admits a ricci soliton if pv is solenoidal. moreover, it is shrinking or steady or expanding depending on the sign of the scalar curvature. 5. quasi-conformally flat para-kähler manifolds in this section, we consider the notion of quasi-conformally flat para-kähler space form, i.e., c(x,y )z = 0. thus we can state the following result. 266 sunil k. yadav theorem 5.1. let (g,v,λ) be a ricci soliton in a quasi-conformally flat para-kähler space form. then v is solenoidal iff it is always steady. proof. in view of (2.1), (2.2) and (2.7), we have α 2 g(py,w) + β [ 2s( y,w) + τg(py,w) ] − r n { α n− 1 + 2β } g(y,w) = 0. (5.1) contracting (5.1) over the pair of argument y and w , we obtain r { α (n− 1) } = 0, (5.2) which implies r = 0, α 6= 0. (5.3) in view of (5.1) and (5.3), we get s(y,w) = − { τ 2 + α 4β } g(py,w). (5.4) then using (5.3) and (5.4) in (2.6), we obtain r(x,y )z = β α [( τ 2 + α β )( g(py,z)x −g(px,p)y + g(y,z)px −g(x,z)py )] . (5.5) again from (1.1) and (5.4), it yields (lv g)(y,w) − 2 { τ 2 + α 4β } g(py,w) + 2λg(y,w) = 0. (5.6) taking contraction with respect to y and w for i (1 ≤ i ≤ n), we have (lv g)(ei,ei) − 2 { τ 2 + α 4β } g(pei,ei) + 2λg(ei,ei) = 0, (5.7) which reduces to div v + λn = 0. (5.8) suppose that v is solenoidal. then (5.8), takes the form λ = 0. thus the proof is obvious. corollary 5.2. in a quasi-conformally flat para-kähler space, the ricci and the curvature tensors have the shapes (5.4) and (5.5), respectively. ricci solitons on para-kähler manifolds 267 6. parallel quasiconformal curvature tensor this section deals with the parallelity condition (∇c = 0) of quasi-conformal curvature tensor on para-kähler space form. it is essential to state the following result. theorem 6.1. a para-kähler space form is quasi-conformally symmetric iff it is locally symmetric. proof. in view of the condition ∇c = 0, (2.7) reduces to − α 2 g(py,w) + β [ 2s(y,w) − τg(py,w) ] + r n { α n− 1 + 2β } g(y,w) = 0. (6.1) taking covariant derivation of (6.1) along the vector field z, we get β [ 2(∇zs)(y,w) −dτ(z)g(py,w) ] + dr(z) n { α n− 1 + 2β } g(y,w) = 0. (6.2) taking contraction in (6.2) with respect to y and w , multiplying by εi, we obtain dr(z) { α n− 1 + 4β } = 0. (6.3) thus (6.3) implies that dr(z) = 0. using this equation in (6.2), we have (∇zs)(y,w) = 1 2 dτ(z)g(py,w). (6.4) replacing y by ay in (6.4) and using (2.1), we get (∇zs)(py,w) = 1 2 dτ(z)g(y,w). (6.5) again taking contraction in (6.2) with respect to y and w , multiplying by εi, we obtain (∇zs)(py,w) = 0. (6.6) taking covariant derivation of (2.6) and using (6.6), we get (∇zc)(x,y )w = α(∇zr)(x,y )w, α 6= 0. (6.7) thus it completes the proof. 268 sunil k. yadav on the other-hand, if we consider the condition r ·c = 0, then we have r ·q = 0 by using (2.7). this implies that r ·s = 0. taking into account of r·c = 0 and r·s = 0, we get r·r = 0. thus we can state the following result. corollary 6.2. a para-kähler space form is quasi-conformally semisymmetric iff it is semisymmetric. acknowledgements the author would like to thank the referee for reading the manuscript in great detail and for his/her valuable suggestions and useful comments. references [1] k. amur, y.b. maralabhavi, on quasi-conformally flat spaces, tensor (n.s.) 31 (1977), 194 – 198. [2] e. boeckx, o. kowalski, l. vanhecke, “ riemannian manifolds of conullity two ”, world scientific publication co., inc., river edge, nj, 1996. [3] v. cruceanu, p. fortuny, p.m. gadea, a survey on paracomplex geometry, rocky mountain j. math. 26 (1996), 83 – 115. 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[11] z.i. szabo, structure theorems on riemannian spaces satisfying r(x,y )·r = 0, i, the local version, j. differential geom. 17 (1982), 531 – 582. [12] z.i. szabo, structure theorems on riemannian spaces satisfying r(x,y )·r = 0, ii, global version, geom. dedicata 19 (1985), 65 – 108. [13] p. topping, “ lectures on the ricci flow ”, london mathematical society lecturer note series, 325, cambridge university press, cambridge, 2006. [14] k. yano, s. sawaki, riemannian manifolds admitting a conformal transformation group, j. differential geometry 2 (1968), 161 – 184. introduction para-kähler manifolds ricci-pseudosymmetric and pseudosymmetric parallel symmetric second order covariant tensor quasi-conformally flat para-kähler manifolds parallel quasiconformal curvature tensor e extracta mathematicae vol. 31, núm. 1, 89 – 107 (2016) on the moduli space of donaldson-thomas instantons yuuji tanaka graduate school of mathematics, nagoya university furo-cho, chikusa-ku, nagoya, 464-8602, japan yu2tanaka@gmail.com presented by oscar garćıa prada received may 19, 2015 abstract: in alignment with a programme by donaldson and thomas, thomas [48] constructed a deformation invariant for smooth projective calabi-yau threefolds, which is now called the donaldson-thomas invariant, from the moduli space of (semi-)stable sheaves by using algebraic geometry techniques. in the same paper [48], thomas noted that certain perturbed hermitian-einstein equations might possibly produce an analytic theory of the invariant. this article sets up the equations on symplectic 6-manifolds, and gives the local model and structures of the moduli space coming from the equations. we then describe a hitchin-kobayashi style correspondence for the equations on compact kähler threefolds, which turns out to be a special case of results by álvarez-cónsul and garćıa-prada [1]. key words: gauge theory; the donaldson-thomas theory. ams subject class. (2010): 53c07. 1. introduction in [21], donaldson and thomas suggested higher-dimensional analogues of gauge theories, and proposed the following two directions: gauge theories on spin(7) and g2-manifolds; and gauge theories in complex 3 and 4 dimensions. the first ones could be related to “topological m-theory” proposed by nekrasov and others [39], [15]. the second ones are a “complexification” of the lower-dimensional gauge theories. in this direction, thomas [48] constructed a deformation invariant of smooth projective calabi-yau threefolds from the moduli space of (semi-)stable sheaves, which he called the holomorphic casson invariant because it can be viewed as a complex analogue of the taubes-casson invariant [47]. it is now called the donaldson-thomas invariant (d-t invariant for short), and further developed by joyce-song [28] and kontsevich-soibelman [32], [33], [34]. later, donaldson and segal [20] further promoted the programme, taking into account the progress made after the proposal. recently, more breakthroughs concerning the “categorification” of the d-t invariant by using perverse sheaves were made by a group led by joyce [7], [27], [8], [9], [4], also by kiem-li [29]. 89 90 y. tanaka let us mention here a conjecture (called the mnop conjecture) posed by maulik-nekrasov-okounkov-pandharipande [37], [38], which insists that the rank one d-t invariants (“counting” of ideal sheaves on a calabi-yau threefold) can be determined by only the betti numbers and the gromovwitten invariants. assuming the conjecture is true, one can observe that the rank one d-t invariants are symplectic invariants, as the gromov-witten invariants are symplectic invariants. one might further speculate that the full d-t invariants defined by joyce and song could be also symplectic invariants. one of our goals is to work toward proving this by using a gauge-theoretic equation (we call it the donaldson-thomas equation) on a compact symplectic 6-manifold, which ought to be an analytic counterpart of the notion of stable holomorphic vector bundles, as the problem is analytic in nature. perhaps, one might think of that a gauge-theoretic equation which would describe the d-t invariant could be the hermitian-einstein equations, as the hitchin-kobayashi correspondence [17], [18], [50], [51] (see also [31], [36]) insists that there is a one-to-one correspondence between the existence of the hermitian-einstein connection and the mumford-takemoto stability of an irreducible vector bundle over a compact kähler manifold. however, the hermitian-einstein equations do not form an elliptic system even with a gauge fixing equation in complex dimension three and more (see section 2.1), so this might cause a little problem. in order to work out this issue, donaldson and thomas [48] suggested a perturbation of the hermitian-einstein equations described below. this perturbation was also brought in by baulieu-kanno-singer [3] and iqbalnekrasov-okounkov-vafa [26] in string theory context. let z be a compact symplectic 6-manifold with symplectic form ω, p a principal u(r)-bundle on z, and e the associated unitary vector bundle on z. the equations we consider are ones for a connection a of p and an ad(p)-valued (0,3)-form u on z of the following form: f 0,2 a + ∂̄ ∗ au = 0 , f 1,1 a ∧ ω 2 + [u,ū] + 2πiµ(e)ideω 3 = 0 , where f 0,2 a and f 1,1 a are the (0,2) and (1,1) components of the curvature fa of a, and µ(e) := 1 r ∫ z c1(e) ∧ ω2. here we picked up an almost complex structure compatible with ω to get the splitting of the space of the complexified two forms. we call the equations the donaldson-thomas equations (d-t equations for short) and a solution moduli space of donaldson-thomas instantons 91 to the equations a donaldson-thomas instanton (d-t instanton for short). these equations with a gauge fixing equation form an elliptic system. we aim at developing an analytic theory concerning the d-t invariant by using the moduli space coming from these equations. in [44], [45], we studied some analytic properties of solutions to the equations on compact kähler threefolds. in [44], we proved that a sequence of solutions to the d-t equation has a subsequence which smoothly converges to a solution to the d-t equation outside a closed subset of the hausdorff dimension two. in [45], we proved some of singularities which appeared in the above weak limit can be removed. in this article, we describe the infinitesimal deformation and the kuranishi model of the moduli space of d-t instantons by using familiar techniques in gauge theory, for example, the corresponding results for the anti-self-dual instantons in real four dimensions were studied by atiyah-hitchin-singer [2] (see also [22], [19]), and for the hermitian-einstein connections by kim [30] (see also [31], [36]). we then describe a hitchin-kobayashi style correspondence for the d-t instanton on compact kähler threefolds, which turns out to be a special case of results by álvarez-cónsul and garćıa-prada [1]. the organisation of this article is as follows. in section 2, we briefly recall the hermitian-einstein connections, subsequently, we introduce the dt equations on symplectic 6-manifolds. we also mention a relation between the d-t equations and the complex anti-self-dual equations by dimensional reduction argument. in section 3, we give the kuranishi model of the space of the d-t instantons. in section 4, we describe a hitchin-kobayashi style correspondence for the d-t instanton on compact kähler threefolds. 2. the donaldson-thomas instantons 2.1. the hermitian-einstein connections on compact kähler manifolds. we first recall the notion of the hermitian-einstein connections on compact kähler manifolds. general references for the hermitian-einstein connections are [31] and [36]. let x be a compact kähler manifold of complex dimension n with kähler form ω, e a hermitian vector bundle over x with hermitian metric h. a metric preserving connection a of e is said to be a hermitian-einstein connection if a satisfies the following equations: f 0,2 a = 0 , iλf 1,1 a = 2nπµ(e)ide , (2.1) where f 0,2 a and f 1,1 a are the (0,2) and (1,1) components of the curvature fa 92 y. tanaka of a, λ := (ω)∗, and µ(e) := 1 r ∫ x c1(e) ∧ ωn−1. the existence of a solution to the equations (2.1) is related to the notion of stability for holomorphic vector bundles. in fact, donaldson [17], [18] and uhlenbeck-yau [50], [51] proved that there is a one-to-one correspondence between the existence of the hermitian-einstein connection and the mumfordtakemoto stability of an irreducible vector bundle over a compact kähler manifold (see also [31], [36]). the infinitesimal deformation of a hermitian-einstein connection a was studied by kim [30] (see also [31], [41]), and it is described by the following: 0 −→ ω0(x,u(e)) da−−→ ω1(x,u(e)) d+ a−−→ ω+(x,u(e)) d̄′ a−−→ a0,3(x,u(e)) d̄a−−→ a0,4(x,u(e)) d̄a−−→ · · · d̄a−−→ a0,n(x,u(e)) −→ 0, (2.2) where a0,q(x,u(e)) := c∞(u(e) ⊗ a0,q), u(e) = end(e,h) is the bundle of skew-hermitian endomorphisms of e, a0,p is the space of real (0,p)-forms (see [42, pp. 32-33]) over x, defined by a0,p ⊗r c = λ0,p ⊕ λp,0, ω+(x,u(e)) := a0,2(x,u(e)) ⊕ ω0(x,u(e))ω = { ϕ + ϕ̄ + fω : ϕ ∈ ω0,2(x,u(e)), f ∈ ω0(x,u(e)) } , d̄a : a 0,p(x,u(e)) → a0,p+1(x,u(e)) is defined by d̄aα = ∂̄aα 0,p + ∂aα0,p for α = α0,p + α0,p, where α0,p ∈ ω0,p(x,u(e)), and d+a := π + ◦ da , d̄′a := d̄a ◦ π 0,2, where π+ and π0,2 are respectively the orthogonal projections from ω2 to ω+ and a0,2. kim proved that (2.2) is an elliptic complex if a is a hermitian-einstein connection. however, it is obviously not the atiyah-hitchin-singer type complex [2] if n ≥ 3, since there are additional terms such as a0,3(x,u(e)) and so on. hence, the hermitian-einstein connections would not work for an analytic construction of the donaldson-thomas invariant just as it is. but, in moduli space of donaldson-thomas instantons 93 [48], thomas noted a perturbed hermitian-einstein equation, which basically corresponds to a “holding” of the extra term a0,3(x,u(e)) in (2.2) (we shall see it in section 3.1), could possibly work for an analytic definition of the donaldson-thomas invariant. we introduce that perturbed equation in the next subsection. 2.2. the donaldson-thomas instantons on compact symplectic 6-manifolds. let z be a compact symplectic 6-manifold with symplectic form ω, and e a unitary vector bundle of rank r over z. we take an almost complex structure on z compatible with the symplectic form ω. then the almost complex structure induces the splitting of the complexified two forms as λ2 ⊗ c = λ2,0 ⊕ λ0,2 ⊕ λ1,1. we consider the following equations for a connection a of e, which preserves the hermitian structure of e, and a u(e)-valued (0,3)-form u on z. f 0,2 a + ∂̄ ∗ au = 0 , (2.3) f 1,1 a ∧ ω 2 + [u,ū] + 2πiµ(e)ide ω 3 = 0 , (2.4) where f 0,2 a and f 1,1 a are the (0,2) and (1,1) components of the curvature fa of a, and µ(e) := 1 r ∫ z c1(e) ∧ ω2. we call these equations (2.3), (2.4) the donaldson-thomas equations, and a solution (a,u) to these equations a donaldson-thomas instanton (d-t instanton for short). one may think of these equations as the hermitian-einstein equations with a perturbation u. however, we think of u as a higgs field, namely, a new variable. one of advantages of bringing in the new field u is that the donaldsonthomas equations form an elliptic system after fixing a gauge transformation, despite the fact that the hermitian-einstein equations on compact kähler threefolds do not form it in the same way. these equations (2.3), (2.4) were also studied in physics such as in [3]. in that context, these equations are interpreted as a bosonic part of dimensional reduction equations of the n = 1 super yang-mills equation in 10 dimensions to 6 dimensions (see also [26], [40]). the equations in the kähler case. if the almost complex structure is integrable, then we have ∂̄af 0,2 a = 0 by the bianchi identity. hence 94 y. tanaka ∂̄a∂̄ ∗ au = 0 by (2.3), thus we have ∂̄ ∗ au = 0 on compact kähler threefolds. therefore, the donaldson-thomas equations (2.3), (2.4) becomes ∂̄∗au = 0, f 0,2 a = 0 , f 1,1 a ∧ ω 2 + [u,ū] + 2πiµ(e)ide ω 3 = 0 . the above equations could be thought of as a generalization of the hitchin equation on riemann surfaces [24] to kähler threefolds in the same way as the vafa-witten equations on kähler surfaces as mentioned in [46]. in section 4 to this article, we describe the corresponding hitchin-kobayashi correspondence in this setting, which turns out to be a special case of results by álvarez-cónsul and garćıa-prada [1]. 2.3. the complex asd and the donaldson-thomas instantons. in this section, we see that the donaldson-thomas equations on calabi-yau threefolds can be thought of as the dimensional reduction of the complex asd equations on calabi-yau fourfolds, this was pointed out by tian [49], and it is analogous to the hitchin pair [24]. complex asd equations on calabi-yau fourfolds. let x be a compact calabi-yau fourfold with kähler form ω and holomorphic (4,0)-form θ. we assume the normalization condition θ ∧ θ̄ = 16 4! ω4 on ω and θ. let e be a hermitian vector bundle over x. by using the holomorphic (4,0)-form θ, we define the complex hodge operator ∗θ : λ0,2 → λ0,2 by tr(ϕ ∧ ∗θψ) = ⟨ϕ,ψ⟩θ̄ , ϕ,ψ ∈ λ0,2. then ∗2θ = 1, and the space of (0,2)-forms further decomposes into λ 0,2 = λ 0,2 + ⊕ λ 0,2 − , where λ 0,2 + = {ϕ ∈ λ 0,2 : ∗θϕ = ϕ} , λ 0,2 − = {ϕ ∈ λ 0,2 : ∗θϕ = −ϕ}. note that the operator ∗θ is an anti-holomorphic map, hence λ 0,2 + and λ 0,2 − are real subspaces of λ0,2. we consider the following equations for connections of e: (1 + ∗θ)f 0,2 a = 0 , iλf 1,1 a = 8πµ(e)ide, (2.5) where µ(e) := 1 r ∫ x c1(e) ∧ ω3. moduli space of donaldson-thomas instantons 95 we call these equations complex asd equations, and a solution to these equations a complex asd instanton. these were brought in by donaldson and thomas in [21]. these equations with a gauge fixing equation form an elliptic system. analytic properties of the complex asd instantons were studied by tian [49]. note that the complex asd instantons are special cases of spin(7)-instantons on spin(7)-manifolds (see [43, §3.1]). more recently, donaldson-thomas style invariants for calabi-yau fourfolds, which concerns the moduli space of the solutions to the above complex asd equations, were defined by borisov-joyce [5], cao [10] and cao-leung [11] (see also [12], [13], [14]). dimensional reduction. we describe a relation between the donaldson-thomas equations (2.3), (2.4) and the complex asd equations (2.5) by dimensional reduction argument. this was pointed out by tian [49]. let z be a compact calabi-yau threefold with kähler form ω0 and holomorphic (3,0)-form θ0, and t 2 a torus of complex dimension one. we consider the direct product of z and t2, and denote it by x, namely, x := z × t2. we define a kähler form ω and a holomorphic (4,0)-form on x by ω := ω0 + dz ∧ dz̄ , θ := θ0 ∧ dz , where dz is the standard flat (1,0) form on t2. let e be a hermitian vector bundle with structure group su(r) over z, and p : x = z × t2 → z. we then consider t2-invariant solutions to the complex asd equations (2.5) on p∗e → x. then these solutions satisfy the donaldson-thomas equations on z. in fact, if we write a connection a on x = z × t2 as ax = a + ϕdz + ϕ̄dz̄, where a is the z-component of the connection ax and ϕ ∈ γ(z,su(e)), then the curvature becomes fax = fa + daϕ ∧ dz + daϕ̄ ∧ dz̄ + [ϕ,ϕ̄]dz ∧ dz̄ . hence, if we put u := ϕθ̄0 ∈ ω0,3(z,su(e)), then a and u satisfy the donaldson-thomas equations, provided that this ax is a t 2-invariant solution to the complex asd equations. 3. local model for the moduli space of donaldson-thomas instantons let z be a compact symplectic 6-manifold with symplectic form ω, (e,h) a hermitian vector bundle over z with hermitian metric h. 96 y. tanaka we denote by a(e) = a(e,h) the set of all connections of e which preserve the hermitian structure of e, and put c(e) := a(e) × ω0,3(z,u(e)). we denote by g(e) = g(e,h) the gauge group, the group of unitary automorphism of (e,h), where the action of the gauge group on c(e) is defined by g(a,u) = (a − (dag)g−1,g−1ug). these spaces c(e), g(e) can be seen as fréchet spaces with c∞-norms, but we shall use sobolev completions of them in section 3.2. we denote by γ(a,u) the stabilizer at (a,u) ∈ c(e) of the gauge group g(e), namely, γ(a,u) := { g ∈ g(e) : g(a,u) = (a,u) } . we call (a,u) ∈ c(e) irreducible if γ(a,u) coincides with the centre of the structure group of e, and reducible otherwise. we denote by c∗(e) the set of all irreducible pair (a,u) ∈ c(e). note that the action of g(e) is not free on c∗(e), but the action of ĝ(e) = g(e)/u(1) is free on c∗(e). we denote by d(e) the set of all d-t instantons of e, and by d∗(e) the set of all irreducible d-t instantons of e. we call m(e) = d(e)/g(e) the moduli space of the donaldson-thomas instantons. 3.1. linearization. the infinitesimal deformation of a d-t instanton (a,u) is described by the following sequence: 0 −→ ω0(z,u(e)) d(a,u) −−−−−−→ ω1(z,u(e)) ⊕ a0,3(z,u(e)) d+ (a,u) −−−−−−→ ω+(z,u(e)) −→ 0 , (3.1) where d(a,u)(s) = (das, [ũ,s]) , ũ = u + ū , d+ (a,u) (α,υ) = d+aα + λ 2([u,ῡ] + [υ,ū]) + d̄∗aυ for s ∈ ω0(z,u(e)) and (α,υ) ∈ ω1(z,u(e)) ⊕ a0,3(z,u(e)). if (a,u) is a d-t instanton, then (3.1) is a complex. in fact, d+ (a,u) d(a,u) = 0 follows directly from the equations (2.3), (2.4). the complex (3.1) can be seen as “holding” of the a0,3(z,u(e))-term in (2.2), namely, it is equivalent to consider the following complex instead of (3.1): 0 −→ ω0(x,u(e)) da−−→ ω1(x,u(e)) d+ a−−→ ω+(x,u(e)) d̄′ a−−→ a0,3(x,u(e)) −→ 0 . (3.2) moduli space of donaldson-thomas instantons 97 this is the same as that of the hermitian-einstein connections in section 2.2, but it still makes sense in the almost complex setting. hence the following just reduces to the case in (3.2), and it was proved by reyes carrión [41]. proposition 3.1. if (a,u) ∈ d(e), then the complex (3.1) is elliptic. we denote by hi (a,u) = hi (a,u) (z,u(e)) the i-th cohomology of the complex (3.1) for i = 0,1,2. the complex (3.2) has the associated dolbeault complex as kim [30] described it in the kähler case (see also [31, chapter vii, §2]): 0 −−−−→ ω0 da−−−−→ ω1 d+ a−−−−→ ω+ d̄′ a−−−−→ a0,3 d̄a−−−−→ 0yj0 yj1 yj2 yj3 0 −−−−→ ω0,0 ∂̄a−−−−→ ω0,1 ∂̄a−−−−→ ω0,2 ∂̄a−−−−→ ω0,3 ∂̄a−−−−→ 0 , (3.3) where j0 is injective, j1 is bijective, j2 is surjective with the kernel {βω : β ∈ ω0}, and j3 is bijective. hence the index of the complex (3.2), thus that of the complex (3.1), can be expressed by that of the dolbeault complex above, which is given by ∫ z â(z)∧ch ( k 1/2 z ) ∧ch(u(e)) (see [23, §3.5]). in the kähler case, the index can be computed as ∫ z c1(z) ∧ ( r − 1 2 c1(e) 2 − rc2(e) ) + r2 3∑ i=0 (−1)i dimh0,i(z) . note that the index is zero if z is a calabi-yau threefold. 3.2. kuranishi model and the local description of the moduli space. we denote by ck(e), c∗k(e), dk(e), d ∗ k(e) the l 2 k-completions of c(e), c∗(e), d(e), d∗(e) respectively, and by gk+1(e) the l2k+1-completion of g(e). we take k sufficiently large so that gk+1 becomes a hilbert lie group acting smoothly on ck(e), the quotient topology ck(e)/gk+1(e) becomes hausdorff (see e.g. [22, §3]), and to use implicit function theorems for the sobolev spaces. a general reference for the sobolev spaces and the implicit function theorems on them for our purpose is, for example, [52]. 98 y. tanaka slice. we define slice s(a,u),ε at (a,u) in ck(e) by s(a,u),ε := { (α,υ) ∈ l2k ( u(e) ⊗ ( λ1 ⊕ a0,3 )) : d∗(a,u)(α,υ) = 0 , ||(α,υ)||l2k ≤ ε } . this set s(a,u),ε is transverse to the gk+1-orbit through (a,u) as kerd∗(a,u) is orthogonal, with respect to the l2-norm in l2k ( u(e) ⊗ (λ1 ⊕ a0,3) ) , to imd(a,u). there is a natural map p(a,u),ε : s(a,u),ε −→ ck(e)/gk+1(e) , (α,υ) 7−→ [(a + α,u + υ ′)] , where υ′ = j3(υ), and j3 : a 0,3 → ω0,3 is the map in (3.3). in the following, we take (a,u) ∈ c∗k(e) for simplicity. proposition 3.2. let (a,u) ∈ c∗k(e). then there exists ε > 0 such that s(a,u),ε is diffeomorphic to p(a,u),ε ( s(a,u),ε ) in c∗k(e)/ĝk+1(e). proof. this is a familiar claim in gauge theory, the proof is a modification of known results for the asd and the hermitian-einstein connections (cf. [16, theorem 6], [22, theorem 3.2 and theorem 4.4], [31, chapter vii, §4, theorem 4.16], and [36, proposition 4.2.1]). we divide the proof into two steps: step 1. we consider a map f(a,u) : s(a,u),ε × ĝk+1(e) → c∗k(e) defined by f(a,u)((α,υ),g) = g(a+α,u+υ ′). then the differential of f(a,u) at ((0,0), id) is given by df(a,u)|((0,0),id)((β,φ),s) = (β,φ) + d(a,u)(s) . as imd(a,u) and kerd ∗ (a,u) are l2-orthogonal in l2k ( u(e) ⊗ (λ1 ⊕ a0,3) ) , df(a,u)|((0,0),id) is injective if (a,u) is irreducible. on the other hand, associated to the operator d∗(a,u)d(a,u) : l 2 k+1(u(e) ⊗ λ 0)/u(1) −→ l2k−1(u(e) ⊗ λ 0)/u(1) , where l2k+1(u(e) ⊗ λ 0)/u(1) = { s ∈ l2k+1(u(e) ⊗ λ 0) : ∫ z tr (s) volg = 0 } , there exist the green operator g0 : l2k(u(e)⊗λ 0)/u(1) → l2k(u(e)⊗λ 0)/u(1) and the harmonic projection h0 : l2k(u(e) ⊗ λ 0)/u(1) → l2k(u(e) ⊗ λ 0)/u(1) with the identity: id = h0 + d∗(a,u)d(a,u) ◦ g 0 moduli space of donaldson-thomas instantons 99 (see e.g. [52, chapter iv, §5]). from the identity, we obtain d∗(a,u)((γ,χ) − d(a,u)g 0d∗(a,u)(γ,χ)) = 0 for any (γ,χ) ∈ l2k(u(e) ⊗ (λ 1 ⊕ a0,3)). thus, for a given (γ,χ) ∈ l2k(u(e) ⊗ (λ1⊕a0,3)), we take (β,φ) = (γ,χ)−d(a,u)g0d∗(a,u)(γ,χ), s = g 0d∗ (a,u) (γ,χ) to get (γ,χ) = (β,φ) + d(a,u)(s). therefore df(a,u)|((0,0),id) is surjective. we then use an inverse mapping theorem for the hilbert spaces (see e.g. [35, chapter 6]) to deduce that around (a,u), c∗k(e) is locally diffeomorphic to a neighbourhood of ((a,u), id) in s(a,u),ε × ĝk+1(e). step 2. we then prove that if for (α1,υ1),(α2,υ2) ∈ s(a,u),ε there exists g ∈ gk+1(e) such that (a + α1, ũ + υ1) = g(a + α2, ũ + υ2) , (3.4) then cg is close to ide in l 2 k+1 for some c ∈ u(1). since we assume that (a,u) is irreducible, we can take c ∈ u(1) so that g′ = cg − ide ∈ ker ( d(a,u) )⊥ . from (3.4), we get dag ′ = α1g ′ − g′α2 + (α1 − α2) , [ũ,g′] = g′υ1 − υ2g′ + υ1 − υ2 . hence, d(a,u)g ′ = ( α1g ′ − g′α2 + α12,g′υ1 − υ2g′ + υ12 ) , (3.5) where α12 = α1 − α2,υ12 = υ1 − υ2. since g′ lies in ( kerd(a,u) )⊥ , there exists a constant c > 0 independent of (a,u) and g′ such that ||g′||l2 k+1 ≤ c||d(a,u)g′||l2 k . thus, using (3.5), we obtain ||g′||l2 k+1 ≤ c ( ||g′||l2 k ( ||α1||l2 k + ||α2||l2 k + ||υ2||l2 k ) + ||α12||l2 k + ||υ12||l2 k ) . hence, ||g′||l2 k+1 ≤ c 1 − 3εc ( ||α12||l2 k + ||υ12||l2 k ) for ε < 1/3c. thus, we get ||cg − ide||l2 k+1 < c′ε for ε small, where c′ is a positive constant. from this, the assertion of the lemma is reduced to step 1. remark 3.3. by modifying the proof of lemma 3.2, one can prove that for (a,u) ∈ ck(e), there exists ε > 0 such that s(a,u),ε/γ̂(a,u) is diffeomorphic to p(a,u) ( s(a,u),ε/γ̂(a,u) ) in ck(e)/ĝk+1(e), where γ̂(a,u) = γ(a,u)/u(1), following, for example, [22, theorem 4.4]. 100 y. tanaka kuranishi model. this is also a familiar picture in gauge theory. we describe it for the donaldson-thomas instanton case, modifying known results in the asd and hermitian-einstein connections (cf. [16, proposition 8], [31, chapter vii, §4, theorem 4.20], and [36, proposition 4.5.3]). we take (a,u) ∈ dk(e), and consider a deformation (a + α,u + υ′) ∈ dk(e), where (α,υ) ∈ l2k ( u(e) ⊗ (λ1 ⊕ a0,3) ) . then, (α,υ) satisfies the following: d+aα + π +(α ∧ α) + bu(υ) + λ2[υ,ῡ] + d̄∗aυ + ∗̄α∗̄υ = 0 , (3.6) where bu(υ) := λ 2([u,ῡ] + [υ,ū]). associated to the operator d+ (a,u) (d+ (a,u) )∗ : l2k(u(e) ⊗ λ +) −→ l2k(u(e) ⊗ λ +) , there exist the green operator g2 : l2k(u(e) ⊗ λ +) → l2k(u(e) ⊗ λ +) and the harmonic projection h : l2k(u(e) ⊗ λ +) → l2k(u(e) ⊗ λ +) with the identity: id = h + d+ (a,u) (d+ (a,u) )∗ ◦ g2 (see e.g. [52, chapter iv, §5]). using these, we define a map k(a,u) : l 2 k(u(e) ⊗ (λ 1 ⊕ a0,3)) −→ l2k(u(e) ⊗ (λ 1 ⊕ a0,3)) by k(a,u)(α,υ) := ( α + ( d+a )∗ ◦ g2 ◦ (π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ) , υ + ( d̄′a + (b ∗ u) ′) ◦ g2 ◦ (π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ)) , where (b∗u) ′ = b∗u ◦ πω, b∗u : ω0ω → a0,3 is the adjoint of bu, and πω is the orthogonal projection from ω2 to ω0ω. lemma 3.4. a pair (α,υ) ∈ l2k(u(e) ⊗ (λ 1 ⊕ a0,3)) satisfies (3.6) if and only if it satisfies d+ (a,u) k(a,u)(α,υ) = 0 and h ( π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ ) = 0 . moduli space of donaldson-thomas instantons 101 proof. using the identity id = h + d+ (a,u) ( d+ (a,u) )∗ ◦ g2, we rewrite the left-hand side of (3.6) as d+a ( α + (d+a) ∗ ◦ g2 ◦ ( π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ )) + bu(υ) + d̄∗a ( υ + ( d̄′a + (b ∗ u) ′) ◦ g2 ◦ (π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ)) + h ◦ ( π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ ) = d+ (a,u) k(a,u) + h ◦ ( π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ ) . (3.7) hence, if d+ (a,u) k(a,u) = 0 and h ◦ ( π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ ) = 0, then (3.6) holds. conversely, if (3.6) holds, then from (3.7) we get d+ (a,u) k(a,u) + h ◦ ( π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ ) = 0 . thus, ( d+ (a,u) )∗ d+ (a,u) k(a,u) = 0. this implies ∥∥d+ (a,u) k(a,u) ∥∥ l2 k−1(u(e)⊗λ +) = 0, hence, d+ (a,u) k(a,u) = 0 and h ◦ ( π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ ) = 0. we put sd (a,u),ε := { (α,υ) ∈ s(a,u),ε : (α,υ) satisfies (3.6) } , and denote by hi (a,u) (z,u(e)) (i = 0,1,2) the harmonic spaces of the complex (3.1). lemma 3.5. k(a,u) ( sd (a,u),ε ) ⊂ h1 (a,u) (z,u(e)) . proof. from the definition of the map k(a,u), we have d∗(a,,u)k(a,,u)(α,υ) = d ∗ (a,u)(α,υ) + d∗(a,u)(d + (a,u) )∗ ( g2 ◦ ( π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ )) for (α,υ) ∈ sd (a,u),ε . this is equal to 0, because d∗ (a,u) (α,υ) = 0 for (α,υ) ∈ sd (a,u),ε , and d∗ (a,u) ( d+ (a,u) )∗ = 0 as d+ (a,u) d(a,u) = 0. from lemma 3.4, we also have d+ (a,u) k(a,u) = 0. thus lemma 3.5 holds. from lemmas 3.4 and 3.5, we deduce the following. lemma 3.6. a pair (α,υ) ∈ l2k ( u(e) ⊗ (λ1 ⊕ a0,3) ) lies in sd (a,u),ε if and only if k(a,u)(α,υ) ∈ h1(a,u)(z,u(e)) and h ( π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ ) = 0 . 102 y. tanaka we now prove the following. theorem 3.7. let (a,u) ∈ d∗(e). then there exists a neighbourhood u of 0 in h1 (a,u) (z,u(e)) such that around [(a,u)] the moduli space m∗(e) = d∗(e)/ĝ(e) is locally modeled on the zero set of a real analytic map κ(a,u) : u → h2(a,u)(z,u(e)) with κ(a,u)(0) = 0, and the first derivative of κ(a,u) at 0 also vanishes. proof. from the definition of the map k(a,u), we have k(a,u)(0) = 0. since the differential of k(a,u) at 0 is identity, we can deduce, from the inverse mapping theorem on the hilbert spaces (see e.g. [35, chapter 6]), that there exist a neighbourhood u of 0 in h1 (a,u) (z,u(e)) and a map k−1 (a,u) : u −→ l2k ( u(e) ⊗ ( λ1 ⊕ a0,3 )) such that k−1 (a,u) is a diffeomorphism between u and k−1 (a,u) (u). we then define a map κ(a,u) : u → h2(a,u) by κ(a,u) = ψ ◦ k −1 (a,u) , where ψ : h1 (a,u) → h2 (a,u) is defined by ψ(α,υ) = h ( π+(α ∧ α) + λ2[υ,ῡ] + ∗̄α∗̄υ ) . we now take ε sufficiently small so that all the following hold. firstly, from lemma 3.6, the zero set of κ(a,u) is mapped by k −1 (a,u) diffeomorphically to an open subset in sd (a,u),ε . next, from proposition 3.2, sd (a,u),ε is diffeomorphic to p(a,u),ε(s d (a,u),ε ) in d∗k(e)/ĝk+1(e). hence, the zero set of κ(a,u) is diffeomorphic to a neighbourhood of [(a,u)] in d∗k(e)/ĝk+1(e). moreover, from the elliptic regularity, the harmonic elements are actually smooth, therefore the neighbourhood of [(a,u)] in d∗k(e)/ĝk+1(e) is isomorphic to a neighbourhood of [(a,u)] in m∗(e). the assertions that κ(a,u) = 0 and the derivative of κ(a,u) at 0 is zero just follow from the definition κ(a,u) = ψ ◦k −1 (a,u) and the fact that the differential of k(a,u) at 0 is the identity. from theorem 3.7, one can deduce that m∗(e) is smooth around [(a,u)] if h2 (a,u) (z,u(e)) = 0. but, as in the case of the hermitian-einstein connections (cf. [30], [31, chapter vii, §4], [25, chapter 2, §2.1], [36, chapter 4, §4.5]), it can be improved in the following way. firstly, we note that, corresponding to the decomposition of u(r) = ir ⊕ su(r), the bundle u(e) naturally decomposes into r and u(e)0 over z, where u(e)0 is the bundle of trace-free skewhermitian endmorphisms of e, and there is a subcomplex of the complex (3.1), moduli space of donaldson-thomas instantons 103 which is defined by using the bundle u(e)0 instead of u(e). the decomposition is preserved by the operators of the complex, hence it induces a corresponding splitting of hi (a,u) (z,u(e)) (i = 0,1,2). for (αc,υc) ∈ λ1(z) ⊕ a0,3(z), it is always h ( π+(αc ∧αc) + λ2[υc, ῡc] + ∗̄αc∗̄υc ) = 0, hence the map κ(a,u) values in h2(z,u(e)0). in particular, we obtain the following. corollary 3.8. around [(a,u)] ∈ m∗(e) with h2 (a,u) (z,u(e)0) = 0, the moduli space m∗(e) is smooth. remark 3.9. around (a,u) ∈ d(e), which is not irreducible, one can prove that h1 (a,u) (z,u(e)) and h2 (a,u) (z,u(e)) are γ(a,u)-invariant, and the map κ(a,u) is γ(a,u)-equivariant. hence, combining the claim in remark 3.3, one can deduce that around [(a,u)] the moduli space m(e) is locally modeled on κ−1 (a,u) (0)/γ(a,u). 4. the hitchin-kobayashi correspondence for the donaldson-thomas instantons on compact kähler threefolds perhaps one might ask what kind of a hitchin-kobayashi style correspondence would hold for the donaldson-thomas instanton on compact kähler threefolds. in this section, we describe this, which actually follows from a result by álvarez-cónsul and garćıa-prada [1]. let z be a compact kähler threefold, and e = (e,h) a hermitian vector bundle over z with hermitian metric h. if (a,u) is a d-t instanton on e, then the connection a defines a holomorphic structure ∂̄a on e as f 0,2 a = 0, thus, we can think of e as a locally free sheaf o(e,∂̄a). in addition, the end(e)-valued (0,3)-form u is naturally identified with a section of the bundle end(e) ⊗ k−1z , so ∗̄u is a section of the bundle end(e) ⊗ kz. the equation ∂̄∗au = 0 implies ∂̄a∗̄u = 0, hence, φ := ∗̄u is a holomorphic section of end(e) ⊗ kz. we then consider a pair (e,φ) consisting of a torsion-free sheaf e and a holomorphic section φ of end(e) ⊗ kz. a subsheaf f of e is said to be a φ-invariant if φ(f) ⊂ f ⊗kz. we define a slope µ(f) of a coherent subsheaf f of e by µ(f) := 1 rank(f) ∫ z c1(det f) ∧ ω2 . definition 4.1. a pair (e,φ) consisting of a torsion-free sheaf e and a holomorphic section φ of end(e)⊗kz is called semi-stable if µ(f) ≤ µ(e) for 104 y. tanaka any φ-invariant coherent subsheaf f with rank(f) < rank(e). a pair (e,φ) is called stable if µ(f) < µ(e) for any φ-invariant coherent subsheaf f with rank(f) < rank(e). definition 4.2. a pair (e,φ) consisting of a torsion-free sheaf e and a holomorphic section φ of end(e)⊗kz is said to be poly-stable if it is a direct sum of stable sheaves with the same slopes in the sense of definition 4.1. then the correspondence can be stated as a one-to-one correspondence between a pair (e,φ), where e is a locally-free sheaf on a kähler threefold z and a holomorphic section φ of end(e) ⊗ kz, which is stable in the sense of definition 4.1; and the existence of a solution to the donaldson-thomas equations on e. this fits into a setting studied by álvarez-cónsul and garćıaprada [1] (see also [6]), and it is stated as a special case of their results as the case of a twisted quiver bundle with one vertex and one arrow, whose head and tail conincide, and with twisting sheaf the anti-canonical bundle. we state it in our setting as follows. theorem 4.3. ([1]) let z be a compact kähler threefold with kähler form ω. let (e,φ) be a pair consisting of a locally-free sheaf e on z and a holomorphic section φ of end (e)⊗kz. then, (e,φ) is poly-stable if and only if e admits a unique hermitian metric h satisfying λfh + λ 3[φ,φ̄h] + 6πiµ(e)ide = 0 , where fh is the curvature form of h, and λ := (∧ω)∗. note that the equation ∂̄∗au = 0 in the donaldson-thomas equations on a compact kähler threefold is implicitly addressed in theorem 4.3 by saying that φ = ∗̄u is a holomorphic section of end(e) ⊗ kz. one more remark is that a proof of the hitchin-kobayashi correspondence using the mehtaramanathan argument for the vafa-witten equations in [46] could also apply to the donaldson-thomas instanton on smooth projective threefold as mentined in [46]. acknowledgements i would like to thank mikio furuta, ryushi goto, ryoichi kobayashi, hiroshi ohta for valuable comments, and referees for many useful advice. i am also grateful to katrin wehrheim for wonderful encouragement. a part of this article was written when i had visited beijing international center for mathematical research, peking university in 2008 – 2009, moduli space of donaldson-thomas instantons 105 i am very grateful to gang tian and the institute for their support and hospitality. a part of revision was made during my visit to institut des hautes études scientifiques in february to march of 2012. i would like to thank the institute for the support and giving me an excellent research environment. last but not least, i would like to thank dominic joyce for enlightening me on these subjects over the years. this work was partially supported by jsps grant-in-aid for scientific research no. 15h02054. references [1] l. álvarez-cónsul, o. garćıa-prada, hitchin-kobayashi correspondence, quivers and vortices, comm. math. phys. 238 (2003), 1 – 33. 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[52] r.o. wells, jr., “ differential analysis on complex manifolds ”, third edition, with a new appendix by oscar garcia-prada, graduate texts in mathematics, 65, springer, new york, 2008. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 36, num. 2 (2021), 147 – 155 doi:10.17398/2605-5686.36.2.147 available online june 17, 2021 rosenthal `∞-theorem revisited l. drewnowski ∗ faculty of mathematics and computer science adam mickiewicz university uniw. poznańskiego 4, 61–614 poznań, poland drewlech@amu.edu.pl received april 27, 2021 presented by m. gonzález accepted may 20, 2021 abstract: a remarkable rosenthal `∞-theorem is extended to operators t : `∞(γ,e) → f , where γ is an infinite set, e a locally bounded (for instance, normed or p-normed) space, and f any topological vector space. key words: vector-valued `∞-function space, locally bounded space, p-normed space, quasi-normed space, isomorphism, rosenthal `∞-theorem. msc (2020): 46a16. 1. introduction and the standard rosenthal `∞-theorem i begin by quoting my antique result from 1975 which i dare to call standard rosenthal `∞-theorem. it was first placed in [4] and, after a short time, with a much refined proof, inspired very essentially by ideas from a work of j. kupka [11](1974), in [3] (the order of appearance happened to be just the reverse). it was quite a surprising common extension of the original `∞(γ) and c0(γ) results of haskell p. rosenthal [18, proposition 1.2; theorem 3.4 and remark 1 after it] (announced in [17]) proved for operators acting from these spaces to any banach space f , and of similar in spirit results of n.j. kalton [8, theorem 3.2;3.3, 4.3; theorem 2.3] for γ = n and any tvs f. consult [4, a comment after (r:n) on p. 209] for more precise information. also see n.j. kalton [9]. in the `∞-case rosenthal imposed a much stronger condition on t than (r-0), namely, that “t|c0(γ) is an isomorphism”, while in the c0case his condition was precisely (r-0). i now return to that old work of mine with further extensions. the previous variants of rosenthal’s theorem have found significant applications to banach spaces [18, 1, 14] and to more general spaces ([6, theorem 1.2], [7, p. 314 and ff.], [5]), and to vector measures ([12], ∗to my wife krystyna and our daughters monika and karolina issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.36.2.147 mailto:drewlech@amu.edu.pl https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 148 l. drewnowski [13], [2, lemma 4.1.37, lemma 4.1.39, lemma 4.1.41], [15, proposition 3]). it is therefore natural to expect that the new, more general variants will have an even wider range of applicability. a feature that all the rosenthaltype results encountered in this paper share is that they concern operators t : x → f which send ‘primitive’ vectors in x that are away from zero to vectors that are away from zero in f. in an unbelievable way ‘primitive’ can be removed (though, in general, there is no good functional analytic road in x from “primitive” vectors to those arbitrary ones), leading to the conclusion that the restriction of t to a large subspace of x must be an isomorphism. this means that these results are of a fundamentally basic nature. theorem 1.1. (standard rosenthal `∞-theorem) let f be a tvs and x be a subspace of `∞(γ) such that eγ ∈ x for all γ ∈ γ. also, let t : x → f be an operator such that (r-0) for some 0-neighborhood u in f and all γ ∈ γ: t(eγ) /∈ u. then γ has a subset a of the same cardinality m as γ for which the restriction t|x(a) is an isomorphism. 1.1. notation, conventions, terminology, some basic facts. to ensure proper understanding of was said above, and of the new material, i clarify here a few points. the acronym tvs stands for (nonzero and hausdorff) topological vector space over the field of scalars k ∈ {r,c}, and operator always means continuous linear operator. for a tvs e and an infinite set γ, `∞(γ,e) is the space of bounded functions x : γ → e with the topology of uniform convergence on γ. thus for any 0-neighborhood b in e the corresponding 0-neighborhood in `∞(γ,e) is b• := {x ∈ `∞(γ,e) : x(γ) ⊂ b}. clearly, if e = (e,‖·‖) is an fnormed space, then this topology is defined by the f-norm ‖·‖∞ given by ‖x‖∞ = supγ∈γ ‖x(γ)‖. the notation bc and b•c for the complement of b in e, and of b• in `∞(γ,e), resp., will also be used in the sequel. c0(γ,e) is the closed subspace of `∞(γ,e) consisting of those x that have limit 0 along the filter of cofinite subsets of γ. if x,y ∈ `∞(γ,e) and supp(x) ∩ supp(y) = ∅, then x,y are said to be disjoint, and this is indicated by writing x ⊥ y. notice that whenever x,y ∈ b• and x ⊥ y, also x + y ∈ b•. this property will be essential in what follows. if x is a set of functions x : γ → e and a ⊂ γ, then x(a) := {x ∈ x : supp(x) ⊂ a}; 1a stands for the characteristic function of a. if e = (k, | · |), then the notation for the spaces is simplified to `∞(γ) and c0(γ). for γ ∈ γ, eγ = 1{γ} are the standard unit vectors in `∞(γ). in rosenthal `∞-theorem revisited 149 the general case, their role will be taken over by the primitive vectors xeγ (x ∈ e, γ ∈ γ). no completeness or convexity assumptions will be imposed; they are simply not needed in this paper. finishing, we recall that by a well known aoki-rolewicz theorem (see [16, theorem 3.2.1]), the topology of any locally bounded space e may be defined by some p-norm ‖·‖ (0 < p 6 1). (‖ax‖ = |a|p‖x‖ for all x ∈ e,a ∈ k). see also [10, chapter i.2] for a thorough discussion of this and other types of normlike functionals (e.g., quasi-norms) that are often used to define the topology of a locally bounded space, and for the relevant terminology concerning spaces used in such contexts. the author’s preference are p-norms. in principle, however, it is a matter of taste and/or convenience (or, sometimes, necessity), whether one works with locally bounded spaces or with, for instance, p-normed spaces. a more general type of vector-valued `∞-spaces will appear in theorem 2.4. additional notation of temporary character will be introduced when needed. 2. new extensions of rosenthal `∞-theorem as the main result (also in view of the versatility of its proof) of the paper i consider the following. theorem 2.1. let γ be an infinite set, e a locally bounded tvs, f any tvs, and x a subspace of `∞(γ,e) such that for all (γ,x) ∈ γ×e: xeγ ∈ x. also, let t : x → f be an operator and assume that (r-1) for some bounded 0-neighborhood b in e there is a 0-neighborhood u in f such that for all (γ,x) ∈ γ ×bc : t(xeγ) /∈ u. then there exists a subset a of γ of cardinality m = |γ| such that t|x(a) is an isomorphism. proof. the reasoning is an adaptation, with appropriate modifications and a very cautious treatment of the most ‘delicate’ points, of the proof of theorem 1.1 given in [3]. step 1: denote z := x ∩b• and notice the following: (i) t(z) is a bounded subset of f. (ii) ∀(γ,x) ∈ γ ×bc : xeγ ∈ x(γ) ∩b•c, hence t(xeγ) /∈ u. (iii) if x,y ∈ z and x ⊥ y, then x + y ∈ z. (iv) if x ∈ z,γ ∈ γ and xγ := x(γ), then x−xγeγ ∈ z. 150 l. drewnowski step 2: next, fix a base u of balanced neighborhoods of zero in f , choose v ∈ u with v + v ⊂ u, then r ∈ n such that t(z) ⊂ rv , and finally w ∈ u for which w (r) := w+ r) · · · +w ⊂ rv . the assertion of the theorem will be reached once we show that: (a1) there exists a set a ⊂ γ such that |a| = |γ| = m and tx 6∈ w for all x ∈ x(a) ∩ b•c. beware: the validity of the statement concerning the role of (a1) crucially depends on the local boundedness of x(a) implied by the assumption that e has this property. before proceeding, observe that (a1) is equivalent to the following: (a2) there exists a set ∆ ⊂ γ with |∆| = m such that for any ∆′ ⊂ ∆ with |∆′| = m one can find (α,s) ∈ ∆′×bc with the property: ∀x ∈ x(∆r∆′)∩b•: t(x + seα) /∈ w . (a1)⇒(a2): simply take ∆ = a. (a2)⇒(a1): since m · m = m, one can partition ∆ into m (disjoint) sets ci (i ∈ i, |i| = m), each of cardinality m. according to (a2), for each i ∈ i one can find (γi,s) ∈ ci×bc such that t(x+seγi ) /∈ w for all x ∈ z(∆rci). then a := {γi : i ∈ i} is as required in (a1). now we are ready to show that (a1) or, equivalently, (a2) does indeed hold. suppose (a2) is false so that: non(a2) for every set ∆ ⊂ γ with |∆| = m there is ∆′ ⊂ ∆ with |∆′| = m and such that ∀(α,s) ∈ ∆′ ×bc: one can find x ∈ z(∆ r ∆′) with the property: t(x + seα) ∈ w . then, applying non(a2) r times, one arrives at a decreasing sequence of sets γ = ∆0 ⊃ ∆1 ⊃ . . . ⊃ ∆r such that, for every k ∈ [r] = {1, . . . ,r}, |∆k| = m, and for each α ∈ ∆k and s ∈ bc there exists x ∈ z(∆k−1 r ∆k) for which t(x+seα) = tx+t(seα) ∈ w . now, fix α ∈ ∆r and s ∈ bc, and use them in each of the r steps above. this will give for each k ∈ [r]: xk ∈ z(∆k−1 r ∆k) so that t(xk + seα) = txk + t(seα) ∈ w . summation over k ∈ [r] leads to the following: rt(seα) + t ( r∑ k=1 xk ) ∈ w (r) ⊂ rv. since the xk’s are pairwise disjoint, x := ∑r k=1 xk ∈ z, hence tx ∈ rv . it follows that rt(seα) ∈ rv + rv ⊂ ru. hence t(seα) ∈ u, contradicting condition (r-1). rosenthal `∞-theorem revisited 151 remarks 2.2. (i) in condition (r-1) any other bounded neighborhood b′ of zero in e can be used as well. in fact, if u is ‘good’ for b, then since b ⊂ kb′ for some k ∈ n, k−1u is good for b′. (ii) the most important examples of subspaces x are `∞(γ) and c0(γ) and, of somewhat lesser importance, their further subspaces consisting of elements x with either |supp(x)| 6 n or |supp(x)| < n, where n is any infinite cardinal number 6 m. note that for these x, if a ⊂ γ and |a| = |γ|, then x(a) is isomorphic to x. (iii) the theorem admits a seemingly stronger form, with condition (r-1) requiring that for a bounded 0-neighborhood b in e there is a 0-neighborhood u in e such that the set γ′ := {γ ∈ γ : ∀x ∈ bc : t(xeγ) /∈ u} is infinite, and the assertion saying that then there exists a subset a of γ′ of the same cardinality as γ′ such that t|x(a) is an isomorphism. in reality, as easily understood, both versions are of the same strength. a similar remark applies to theorem 2.3 below. it is obvious that theorem 2.1 can be equivalently restated for e being a p-normed space (e,‖·‖) (0 < p 6 1), using the ball b = {x ∈ e : ‖x‖ 6 ε} for some ε > 0 in condition (r-1). we won’t do that here. yet two other variants of theorem 2.1, formulated in the framework of p-normed spaces are worth explicit stating. in the first (and in its proof), close in spirit to the standard rosenthal theorem we will use the following notation. b := {x ∈ e : ‖x‖ 6 1}, s := {x ∈ e : ‖x‖ = 1}, ‖·‖∞ for the associated p-norm in `∞(γ,e), b∞ for the closed unit ball, and s∞ for the unit sphere in `∞(γ,e), resp. additionally, we let ŝ∞ := {x ∈ s∞ : x(γ) ∈ s for some γ ∈ γ}. theorem 2.3. let e = (e,‖·‖) be a p-normed space (0 < p 6 1), f a tvs, and γ an infinite set. further, let x be a subspace of the p-normed space `∞(γ,e) such that ∀(γ,x) ∈ γ × e: xeγ ∈ x. also, let t : x → f be an operator such that (r-2) for some 0-neighborhood u in f, t(seγ) /∈ u for all (γ,s) ∈ γ ×s. then γ has a subset a of the same cardinality m as γ for which t|x(a) is an isomorphism. proof. the result follows from theorem 2.1. this will be achieved by showing that the present condition (r-2) implies condition (r-1) in that theorem. fix ε > 0, and take any x ∈ e with λ = ‖x‖ > 1 + ε. next, let y = x/λ1/p. then for any γ ∈ γ: t(yeγ) /∈ u. hence t(xeγ) /∈ λ1/pu ⊃ (1 + ε)1/pu =: u ′. 152 l. drewnowski thus (r-1) in theorem 2.1 holds for the bounded 0-neighborhood (1 + ε)b∞ in e and the 0-neighborhood u ′ in f. to complete the story, let us yet see that also theorem 2.3 implies theorem 2.1: assume that in (r-1) of the latter theorem we want to use 1 2 b in place of b (cf. remark 2.2(1)). so take any γ ∈ γ and x ∈ e with λ = ‖x‖ > 1 2 . set y = x/λ1/p. then ‖y‖ = 1, so that t(yeγ) /∈ u (u as in (r-2)). hence t(xeγ) /∈ λ1/pu ⊃ 12p u =: u ′. thus t(xeγ) /∈ u ′ and (r-2) is satisfied with u ′ in place of u. it may be of some interest to have an independent and direct proof of the last result. here it goes. another proof. proceed precisely as in the proof of theorem 2.1, replacing b• and b•c by b∞ and ŝ∞, keeping steps 1 and 2 unchanged, and then replacing conditions (a1), (a2) by the following two, resp. (b1) there exists a set a ⊂ γ such that |a| = |γ| = m and tx 6∈ w for all x ∈ z(a) ∩ ŝ∞. (b2) there exists a set ∆ ⊂ γ with |∆| = m such that for any ∆′ ⊂ ∆ with ∆′| = m one can find (α,s) ∈ ∆′×s with the property: ∀x ∈ z(∆ r ∆′): t(x + seα) /∈ w , verifying that they are equivalent, and after that showing that under the assumptions of the theorem (b1) does indeed hold. having done that select w ′ ∈ u with w ′ + w ′ ⊂ w , take the smallest m ∈ n with m > 1/p, and next select 0 < ε < 1 so that for η := mpεp there holds t((ηb∞)∩x) ⊂ w ′ (by the continuity of t), and then proceed to verify that (b̂1) for the set a from (b1) one has: tx 6∈ w ′ for all x ∈ z(a) ∩s∞. so, take any x ∈ z(a)∩s∞, and then choose α ∈ a so that writing xα = x(α) and λ = ‖xα‖, λ > 1−ε. then let yα = xα/λ1/p, and define y := x−xαeα + yαeα. clearly, y ∈ z(a) ∩ ŝ∞ and hence ty /∈ w . since z := x − y = (xα −yα)eα ∈ x(a), ‖z‖∞ = ‖xα −yα‖ = (1 −λ1/p)p 6 (1 −λm)p = [(1 −λ)(1 + λ + · · · + λm−1)]p < mpεp = η. it follows that tz ∈ w ′. since y = x − z and ty /∈ w while tz ∈ w ′, we conclude that tx /∈ w ′. to finish, take any x ∈ x(a) with λ := ‖x‖∞ > 1. then y := λ−1/px ∈ s∞. by (b̂1), ty = λ−1/ptx /∈ w ′. hence tx /∈ λ1/pw ′, and tx /∈ w ′, too. from this the assertion follows. rosenthal `∞-theorem revisited 153 the second promised variant, suggested by the referee, deals with “multispace” type `∞-spaces. theorem 2.4. let γ be an infinite set and, for a fixed p ∈ (0, 1], let eγ = (eγ : γ ∈ γ) be a family of p-normed spaces, each eγ with its own p-norm ‖·‖γ. consider the vector subspace `∞(eγ) of the product ∏ γ∈γ eγ consisting of elements x = (xγ : γ ∈ γ) (with xγ written often as x(γ)) such that ‖x‖∞ := supγ ‖xγ‖γ < ∞, and equip it with the p-norm ‖·‖∞ defined by this equality. next, let x be a subspace of `∞(eγ) such that xeγ ∈ x for all γ ∈ γ and all x ∈ eγ. further, let f be any tvs, t : x → f be an operator, and assume that (r-3) for some ε > 0 there is a 0-neighborhood u in f such that t(xeγ) /∈ u for all γ ∈ γ and all x ∈ eγ with ‖x‖γ > ε. then there exists a subset a of γ of cardinality m = |γ| such that t|x(a) is an isomorphism. proof. again, we argue as in the proof of theorem 2.1, with necessary changes. to avoid any ambiguity, all details are given. we first fix the additional notation to be used in the present proof. thus, we denote b(eγ) =: {x ∈ eγ : ‖x‖γ 6 ε}, bc(eγ) =: {x ∈ eγ : ‖x‖γ‖ > ε}, b• =: {x ∈ `∞(eγ) : ‖x‖∞ 6 ε}, b•c =: {x ∈ `∞(eγ) : ‖x‖∞ > ε} . next, keep steps 1 and 2 unchanged, and replace conditions (a1), (a2) by the following two, resp. (c1) there exists a set a ⊂ γ such that |a| = |γ| = m and tx 6∈ w for all x ∈ x(a) ∩b•c. (c2) there exists a set ∆ ⊂ γ with |∆| = m such that for any ∆′ ⊂ ∆ with |∆′| = m one can find (α,s) ∈ ∆′ × bc(eα) with the property: ∀x ∈ x(∆ r ∆′) ∩b•: t(x + seα) /∈ w . then we show that (c1) ⇐⇒ (c2). (c1)⇒(c2): simply take ∆ = a. (c2)⇒(c1): since m·m = m, one can partition ∆ into m (disjoint) sets ci (i ∈ i, |i| = m), each of cardinality m. according to (c2), for each i ∈ i one can find (γi,s) ∈ ci×bc(eγi ) such that t(x+seγi ) /∈ w for all x ∈ z(∆rci). then a := {γi : i ∈ i} is as required in (c1). then we go on to show that (c1) or, equivalently, (c2) does indeed hold. suppose (c2) is false so that: 154 l. drewnowski non(c2) for every set ∆ ⊂ γ with |∆| = m there is ∆′ ⊂ ∆ with |∆′| = m and such that ∀(α,s) ∈ ∆′ × bc(eα): one can find x ∈ z(∆ r ∆′) with the property: t(x + seα) ∈ w . then, applying non(c2) r times, one arrives at a decreasing sequence of sets γ = ∆0 ⊃ ∆1 ⊃ . . . ⊃ ∆r such that, for every k ∈ [r] = {1, . . . ,r}, |∆k| = m, and for each α ∈ ∆k and s ∈ bc(eα) there exists x ∈ z(∆k−1 r ∆k) for which t(x + seα) = tx + t(seα) ∈ w . now, fix α ∈ ∆r and s ∈ bc(eα), and use them in each of the r steps above. this will give for each k ∈ [r]: xk ∈ z(∆k−1 r ∆k) so that t(xk + seα) = txk + t(seα) ∈ w . summation over k ∈ [r] leads to the following: rt(seα) + t ( r∑ k=1 xk ) ∈ w (r) ⊂ rv. since the xk’s are pairwise disjoint, x := ∑r k=1 xk ∈ z, hence tx ∈ rv . it follows that rt(seα) ∈ rv + rv ⊂ ru. hence t(seα) ∈ u, contradicting condition (r-3). acknowledgements i am grateful to iwo labuda, zbigniew lipecki, and marek nawrocki for their interest in this work and numerous helpful suggestions. i also thank the editors of extracta mathematicae, especially jesús m.f. castillo, and the referee, for their criticism and engagement in improving the look of this publication. references [1] s.a. argyros, j.m.f. castillo, a.s. granero, m. jimenezsevilla, j.p. moreno, complementation and embeddings of c0(i) in banach spaces, proc. london math. soc. 85 (2002), 742 – 768. 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[16] s. rolewicz, “ metric linear spaces ”, second edition, pwnùpolish scientific publishers, warsaw; d. reidel publishing co., dordrecht, 1984. [17] h.p. rosenthal, on complemented and quasi-complemented subspaces of quotients of c(s) for stonian s, proc. nat. acad. sci. u.s.a. 60 (1968), 1165 – 1169. [18] h.p. rosenthal, on relatively disjoint families of measures, with some applications to banach space theory, studia math. 37 (1970), 13 – 36. (correction, ibid., 311 – 313.) introduction and the standard rosenthal -theorem notation, conventions, terminology, some basic facts. new extensions of rosenthal -theorem � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 1 (2019), 85 – 97 doi:10.17398/2605-5686.34.1.85 available online april 3, 2019 bounded normal generation is not equivalent to topological bounded normal generation philip a. dowerk 1,∗, françois le mâıtre 2,†,@ 1 institut für geometrie, tu dresden, 01062 dresden, germany 2 université paris diderot, sorbonne université, cnrs, institut de mathématiques de jussieu-paris rive gauche, imj-prg, f-75013, paris, france philip.dowerk@tu-dresden.de , f.lemaitre@math.univ-paris-diderot.fr received december 7, 2018 presented by valentin ferenczi accepted february 20, 2019 abstract: we show that some derived l1 full groups provide examples of non simple polish groups with the topological bounded normal generation property. in particular, it follows that there are polish groups with the topological bounded normal generation property but not the bounded normal generation property. key words: topological group, topological bounded normal generation, topological simplicity, full group, conjugacy classes. ams subject class. (2010): 20e45, 22a05, 28d15. 1. introduction the study of how quickly the conjugacy class of a non-central element generates the whole group has often turned out to be useful in understanding both the algebraic and topological structure of the given group. for example, such results on involutions by broise [2] were crucial in de la harpe’s proof of simplicity of projective unitary groups of some von neumann factors [3], while analogous results by ryzhikov [12] are used to prove automatic continuity of homomorphisms for full groups of measure-preserving equivalence relations by kittrell and tsankov [9]. a group g has bounded normal generation (bng) if the union of the conjugacy classes of any nontrivial element and of its inverse generate g in finitely many steps. observe that any group with bounded normal generation is simple. bounded normal generation already appears in fathi’s result that ∗ research supported by erc cog no. 614195 and by erc cog no. 681207. † research supported by project anr-14-ce25-0004 gamme and projet anr-17ce40-0026 agrume. @ corresponding author issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.1.85 mailto:philip.dowerk@tu-dresden.de mailto:f.lemaitre@math.univ-paris-diderot.fr mailto:f.lemaitre@math.univ-paris-diderot.fr https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 86 p.a. dowerk, f. le mâıtre the group of measure-preserving transformations of a standard probability space is simple [7], but the name was only given recently in [4, 5]. obtaining bounded normal generation with a sharp number of steps is a crucial point of the seminal article [11] by liebeck and shalev concerning bounds on the diameter of the cayley graph of a finite simple group. other applications of bounded normal generation can be found in the first named author’s work together with thom [4, 5] concerning invariant automatic continuity and the uniqueness of polish group topologies for projective unitary groups of certain operator algebras, in particular for separable type ii1 von neumann factors. the notion of bounded normal generation can be topologized: a topological group g has topological bounded normal generation if the conjugacy classes of every nontrivial element and of its inverse generate a dense subset of g in finitely many steps. this definition was introduced in [4] to give a clean structure to the proof of bounded normal generation for projective unitary groups of ii1 factors: prove topological bounded normal generation via some finite-dimensional approximation methods in the strong operator topology, then use some commutator tricks and results on symmetries to obtain bounded normal generation. however, it is a priori not clear whether bounded normal generation and topological bounded normal generation are the same properties. basic examples of non simple topologically simple groups such as the symmetric group over an infinite set or the projective unitary group of an infinite dimensional hilbert space actually fail topological bounded normal generation (see corollary 2.6). in fact, this problem was left open in [4] and is the main point of this article: we provide examples that separate bounded normal generation from its topological counterpart. actually, we show that there are polish groups which have topological bounded normal generation but which are not even simple. the groups that serve our purposes are called derived l1 full groups, introduced by the second named author in [10] as a measurable analogue the derived groups of small topological full groups. let us now state our main result, see section 2 for definitions. theorem a. (see theorem 4.4) the derived l1 full group of any hyperfinite ergodic graphing has topological bounded normal generation. in the above theorem we can moreover estimate how many products of conjugacy classes of a non-trivial element and its inverse in the derived l1 full bng is not equivalent to tbng 87 group we need to generate a dense subset, see theorem 4.4. we do not know wether one can remove the hyperfiniteness asumption in theorem a. as a concrete example of this result, one can consider the derived l1 full group of a measure-preserving ergodic transformation, which was already known to be non-simple but topologically simple [10]. using reconstruction-type results, we can show that this provides many examples of non simple groups with topological bounded normal generation. theorem b. (see corollary 5.4) there are uncountably many pairwise non-isomorphic polish groups which have topological bounded normal generation but fail to be simple (in particular they fail bounded normal generation). the article is structured as follows: in section 2 we provide preliminaries around bounded normal generation and define derived l1 full groups of graphings, in section 3 we show that for hyperfinite graphings these groups contain a dense subgroup in which every element is a product of two involutions, in section 4 we use this to prove theorem a (see theorem 4.4) and finally in section 5 we prove a reconstruction theorem for derived l1 full groups and use it to prove theorem b (see corollary 5.4). 2. preliminaries 2.1. topological bounded normal generation. if g is a group and g ∈ g, we write g±g = { hgh−1 : h ∈ g } ∪ { hg−1h−1 : h ∈ g } . given a subset a of g, we moreover write a·n for the set of products of n elements of a, that is a·n = {a1 · · ·an : a1, . . . ,an ∈ a}. the following definitions are taken from [4, 5]. definition 2.1. a group g has bounded normal generation (bng) if for every non-trivial g ∈ g, there is n(g) ∈ n such that( g±g )·n(g) = g. a function n : g\{1} → n satisfying the above assumption is then called a normal generation function. 88 p.a. dowerk, f. le mâıtre observe that bng implies simplicity. an example of a simple group without bng is given by finitely supported permutations of signature 1 on an infinite set. let us now define the topological version of bng. definition 2.2. a topological group g has topological bounded normal generation (tbng) if for every non-trivial g ∈ g, there is n(g) ∈ n such that (g±g)·n(g) = g. we still call a function n : g \ {1} → n satisfying the above assumption a normal generation function. the purpose of this paper is to exhibit examples of non-simple topological groups with tbng. let us first explain why the first examples of non simple topologically simple groups which come to mind actually fail tbng. definition 2.3. let g be a topological group. a lower semi-continuous invariant length function on g is a map l : g → [0, +∞] such that (i) l(1g) = 0; (ii) l(g−1) = l(g) for all g ∈ g; (iii) l(gh) ≤ l(g) + l(h) for all g,h ∈ g; (iv) l(ghg−1) = l(h) for all g,h ∈ g; (v) for all r ≥ 0, the set l−1([0,r]) is closed. such a function is unbounded if for all k ∈ n there is g ∈ g such that l(g) ∈]k, +∞[. example 2.4. on the symmetric group over a discrete set x endowed with the pointwise convergence topology, one can let l(g) be the cardinality of the support of g. if x is infinite, then such a length function is unbounded. on the unitary group of a hilbert space endowed with the strong topology, one can let l(g) be the minimum over λ ∈ s1 of the rank of (g −λidh) or of the trace class norm of (g−λidh). if the hilbert space is infinite dimensional, then such a length function is unbounded and quotients down to an unbounded lower semi-continuous length function on the projective unitary group. observe that if we have an invariant length function on g then the group of elements of finite length is a fσ normal subgroup of g. moreover, we have the following straightforward proposition. bng is not equivalent to tbng 89 proposition 2.5. if a topological group g admits an unbounded lower semi-continuous invariant length function, then it fails tbng. proof. let l be an unbounded lower semi-continuous invariant length function on g. by unboundness we find g ∈ g with l(g) ∈]0, +∞[ (in particular g is non-trivial). fix n ∈ n and let n = n · l(g). observe that (g±g)·n ⊆ l−1([0,n]), and since the latter is closed we also have (g±g)·n ⊆ l−1([0,n]). since l is unbounded, l−1([0,n]) is not equal to g and we conclude that for every n ∈ n we have (g±g)·n 6= g. so g fails tbng. applying this proposition to our two examples, we get the following. corollary 2.6. the symmetric group over an infinite discrete set fails tbng for the topology of pointwise convergence. the projective unitary group of an infinite-dimensional hilbert space fails tbng for the strong operator topology. remark. the fact that the projective unitary group of an infinite-dimensional hilbert space fails tbng can also be seen via generalized projective s-numbers as in [4, pp. 80-81]. let us conclude this section by introducing another natural topological version of bng. define strong topological bounded normal generation (stbng) for a topological group g by asking that for every non-trivial g ∈ g, there is n(g) ∈ n such that ( g±g )·n(g) = g. since we always have ( g±g )·n ⊆ (g±g)·n, stbng implies tbng. in a compact group finite pointwise products of closed subsets are compact hence closed, so we actually always have ( g±g )·n = (g±g)·n. hence stbng is equivalent to tbng for compact groups. however it is not true in general that products of closed sets are closed (for instance in g = r consider a = b = n∪− √ 2n), so the following question is natural. question 2.7. is it true in general that tbng is equivalent to stbng? 2.2. derived l1 full groups. let (x,µ) be a standard probability space. we will ignore null sets, in particular we identify two borel functions on (x,µ) if they are the same up 90 p.a. dowerk, f. le mâıtre to a null set. given a,b borel subsets of x, a partial isomorphism of (x,µ) of domain a and range b is a borel bijection f : a → b which is measure preserving for the measures induced by µ on a and b respectively. we denote by dom f = a its domain, and by rng f = b its range. note that in particular, µ(dom f) = µ(rng f). when dom f = rng f = x we say that f is a measurepreserving transformation and we denote by aut(x,µ) the group of such transformations. a measure-preserving transformation is periodic when all its orbits are finite. given a measure-preserving transformation t and a borel t-invariant subset a of x, we denote by ta the measure-preserving transformation defined by ta(x) = { t(x) if x ∈ a, x otherwise . a graphing is a countable set φ of partial isomorphisms of (x,µ). it generates a measure preserving equivalence relation rφ, defined to be the smallest equivalence relation such that ϕ(x) rφ x for every ϕ ∈ φ. every graphing φ induces a natural graph structure on x where we connect x to y whenever there is ϕ ∈ φ such that y = ϕ(x). the connected components of this graph are precisely the rφ classes, and we thus have a graph metric dφ : rφ → n given by dφ(x,y) = min { n ∈ n : ∃ϕ1, . . . ,ϕn ∈ φ, �1, . . . ,�n ∈{−1, +1}, y = ϕ�nn ◦ · · · ◦ϕ �1 1 (x) } . the full group of the measure-preserving equivalence relation rφ is the group of all measure-preserving transformations t such that for all x ∈ x we have (x,t(x)) ∈rφ. we denote it by [rφ]. the l1 full group of the graphing φ is the group of all t ∈ [rφ] such that∫ x dφ(x,t(x))dµ(x) < +∞ . it is denoted by [φ]1. it has a natural complete separable right-invariant metric d̃φ given by d̃φ(t,u) = ∫ x dφ(t(x),u(x))dµ(x) . we refer the reader to [10] for proofs and more background. bng is not equivalent to tbng 91 for the next definition, recall that the derived group of a group g is the subgroup generated by elements of the form ghg−1h−1 where g,h ∈ g. it is the smallest normal subgroup n of g such that g/n is abelian. definition 2.8. the derived l1 full group of a graphing φ is the closure of the derived group of the l1 full group of φ. it is denoted by [φ]′1. the derived l1 full group of a graphing φ is by construction the smallest closed normal subgroup n of [φ]1 such that [φ]1/n is abelian. a graphing is aperiodic if all its connected components are infinite. an important fact is that as soon as φ is an aperiodic graphing, every periodic element of the l1 full group of φ actually belongs to the derived l1 full group of φ (see [10, lemma 3.11]). moreover, we have the following result which will be strengthened in the next section for hyperfinite graphings. theorem 2.9. ([10, theorem 3.13]) let φ be an aperiodic graphing. then its derived l1 full group [φ]′1 is topologically generated by involutions. 3. products of two involutions in the derived l1 full group of hyperfinite graphings definition 3.1. a measure-preserving equivalence relation is called finite if all its equivalence classes are finite. it is hyperfinite if it can be written as an increasing union of measurepreserving equivalence relations rn whose equivalence classes are finite. a graphing φ is hyperfinite if the equivalence relation it generates is hyperfinite. let φ be a hyperfinite graphing, our aim is to show that its derived l1 full group contains a dense subgroup all of whose elements are products of two involutions. lemma 3.2. let φ be an aperiodic graphing and suppose that rφ is written as an increasing union of equivalence relations rn. then ⋃ n∈n[rn]∩ [φ] ′ 1 is dense in [φ]′1. proof. by theorem 2.9 we only need to be able to approximate involutions from [φ]′1 by elements from ⋃ n∈n[rn] ∩ [φ] ′ 1. let u ∈ [φ] ′ 1 be an involution, then the u-invariant sets an = {x ∈ x : xrnu(x)} satisfy ⋃ n∈n an = x since ⋃ nrn = rφ. by the dominated convergence theorem, we then have 92 p.a. dowerk, f. le mâıtre uan → u. moreover uan is an involution so uan ∈ [φ]′1 and by construction uan ∈ [rn] so uan ∈ [rn] ∩ [φ]′1 and the lemma is proved. definition 3.3. given a graphing φ, we say that the rφ-classes have a uniformly bounded φ-diameter if there is n ∈ n such that for any (x,y) ∈rφ we have dφ(x,y) ≤ n. lemma 3.4. let φ be an aperiodic graphing, let r be a finite subequivalence relation of rφ whose equivalence classes have a uniformly bounded φ-diameter. then [r] is a subgroup of [φ]′1. proof. the fact that there is a uniform bound m on the diameter of the r-classes ensures that [r] is a subgroup of [φ]1. moreover all the elements of [r] are periodic and hence belong to [φ]′1. lemma 3.5. let r be an equivalence relation. suppose that there is n ∈ n such that all the r-classes have cardinality at most n. then [r] is locally finite. proof. let u1, . . . ,uk ∈ [r] for some arbitrary k ∈ n. by using a borel linear order on x, we can identify in an r-invariant manner the r orbit of each x ∈ x with a set of the form {1, . . . ,mx} with mx ≤ n. for each x ∈ x there are only finitely many ways that the marked group generated by 〈u1, . . . ,uk〉 can act on the finite set {1, . . . ,mx}. so we get a finite partition of x into r-invariant sets such that on each atom of this partition, the marked group 〈u1, . . . ,uk〉 acts on each orbit the same way, and we thus embed 〈u1, . . . ,uk〉 in a finite product of finite permutation groups. theorem 3.6. let φ be a hyperfinite graphing. then [φ]′1 contains a dense locally finite subgroup, all whose elements are products of two involutions. proof. by definition we may write rφ as an increasing union of finite equivalence relations rn. it is well known that we can moreover assume that each rn has equivalence classes of size at most n. let us show that we can also assume each rn has equivalence classes of uniformly bounded φ-diameter and cardinality. we first find an increasing sequence of integers (ϕ(n))n∈n such that for all n ∈ n, µ ({ x ∈ x : ∣∣[x]rn∣∣ > ϕ(n) or diamφ ([x]rn) > ϕ(n)}) < 12n . bng is not equivalent to tbng 93 then by the borel-cantelli lemma, for almost all x ∈ x there are only finitely many n ∈ n such that |[x]rn| > ϕ(n) or diamφ([x]rn ) > ϕ(n). so if we define new equivalence relations sn by (x,y) ∈sn if (x,y) ∈rn and diamφ([x]rn ) ≤ ϕ(n) and |[x]rn| ≤ ϕ(n), we still have ⋃ n∈n sn = r and the sn-classes have a uniformly bounded diameter and cardinality as wanted. by lemma 3.2, ⋃ n∈n[sn] ∩ [φ] ′ 1 is dense in [φ] ′ 1. but lemma 3.4 yields that [sn]∩ [φ]′1 = [sn]. since sn is finite and all its classes have cardinality at most ϕ(n), [sn] is locally finite by lemma 3.5. finally each element of [sn] is a product of two involutions by [8, sublemma 4.3]. so the subgroup ⋃ n∈n[sn] is as wanted. 4. topological bounded normal generation for derived l1 full groups in this section we will prove that derived l1 full groups of ergodic amenable graphings have tbng. a key tool will be theorem 3.6, but we also need a better understanding of products of conjugates of involutions. lemma 4.1. ([10, lemma 3.22]) let φ be an ergodic graphing, and let u ∈ [φ]1 be an involution whose support has measure α ≤ 1/2. then the closure of the [φ]′1 conjugacy class of u contains all involutions in [φ]1 whose support has measure α and is disjoint from the support of u. lemma 4.2. let φ be an ergodic graphing, let u ∈ [φ]1 be an involution whose support has measure α < 1/3. then the closure of the [φ]′1 conjugacy class of u contains all involutions in [φ]1 whose support has measure α. proof. let v ∈ [φ]1 have a support of measure α. observe that µ(supp u∪ supp v ) < 2/3. by ergodicity, we find an involution w ∈ [rφ] whose support has measure 1/3 and is disjoint from supp u ∪supp v . by shrinking down w , we may actually assume that w ∈ [φ]1 and that µ(supp w) = α. by the previous lemma w belongs to the closure of the conjugacy class of u and v belongs to the closure of the conjugacy class of w so v belongs to the closure of the conjugacy class of u. lemma 4.3. let φ be an ergodic graphing and let u ∈ [φ]1 be an involution whose support has measure α < 1/3. then every involution whose support has measure ≤ 2α is the product of 2 elements from the closure of the [φ]′1 conjugacy class of u. 94 p.a. dowerk, f. le mâıtre proof. let v ∈ [φ]1 be an involution whose support has measure β ≤ 2α. cut the support of v into two disjoint v -invariant sets a1 and a2 of measure β/2, and let w ∈ [φ]1 be an involution with support of measure α − β/2 disjoint from the support of v (in particular, w commutes with v ). then v = va1va2 = (va1w)(va2w) and by the previous lemma both va1w and va2w belong to the closure of the conjugacy class of u. theorem 4.4. let φ be a hyperfinite ergodic graphing. then [φ]′1 has tbng. a normal generation function is given by n(t) = 2 + 2d 7 2µ(supp t) e. proof. let t ∈ [φ]′1. let a ⊆ supp t be a maximal subset such that a and t(a) are disjoint. then by maximality supp t ⊆ t−1(a)∪a∪t(a), and thus µ(a) ≥ µ(supp t)/3. we may now find an involution u ∈ [φ]1 whose support is contained in a and has measure µ(supp t)/7. now [t,u] is an involution whose support has measure µ(supp[t,u]) = 2µ(supp t)/7 < 1/3. let v be an arbitrary involution, we cut down its support into v -invariant pieces of measure µ(supp[t,u]) plus a remaining piece of measure less than µ(supp[t,u]). observe that v is equal to the product of the transformations it induces on these pieces. there are at most d 1 µ(supp[t,u]) e = d 7 2µ(supp t) e such pieces, and by lemma 4.2 and lemma 4.3 applied to the involutions induced by v on the pieces, we can then write v as a product of at most 1+d 7 2µ(supp t) e elements of the closure of the conjugacy class of [t,u]. since there is a dense subgroup of [φ]′1 whose elements are products of 2 involutions, we have a dense subgroup of [φ]′1 all whose elements are in (t±[φ] ′ 1 )k where k = 2 ( 1 + ⌈ 7 2µ(supp t) ⌉) . so [φ]′1 has tbng. remark. to get stbng via the same approach, we would need to know that each element of [φ]′1 is a product of at most k involutions for some fixed k. we do not know wether this is true, and we also cannot exclude that every element of the derived l1 full group is a product of two involutions. corollary 4.5. let t be an ergodic measure-preserving transformation. then [t ]′1 has tbng but is not simple. bng is not equivalent to tbng 95 proof. by [6, theorem 1] the graphing {t} is hyperfinite, so by theorem 4.4 the derived l1 full group [tα] ′ 1 has tbng. but by [10, theorem 4.26] we also have that [t ]′1 is not simple. 5. reconstruction for the derived l1 full group recall that two measure-preserving transformations t and t ′ are flip conjugate if there is a measure-preserving transformation s such that t = st ′s−1 or t−1 = st ′s−1. in [10] it was shown that when the l1 full groups of two ergodic measure-preserving transformations t and t ′ are isomorphic, then t and t ′ are flip conjugate. the purpose of this section is to prove the same result for the derived l1 full group. let us first note that by [10, propoposition 3.17] and a reconstruction theorem of fremlin as stated in [10, theorem 3.18], we have the following result: lemma 5.1. let φ and ψ be a aperiodic graphings, and let ρ : [φ]′1 → [ψ] ′ 1 be a group isomorphism. then there is a non-singular transformation s such that for all u ∈ [φ]′1 we have ρ(u) = sus −1. in the ergodic case, we can upgrade this as in [10, corollary 3.19]. corollary 5.2. let φ and ψ be a ergodic graphings, and let ρ : [φ]′1 → [ψ]′1 be a group isomorphism. then there is a measure-preserving transformation s such that for all u ∈ [φ]′1 we have ρ(u) = sus −1. proof. same as the proof of [10, corollary 3.18]. we then have the following reconstruction theorem, whose proof is inspired from a similar result of bezuglyi and medynets for topological full groups [1, lemma 5.12]. theorem 5.3. let ta and tb be two ergodic measure-preserving transformations. then the following are equivalent: (i) ta and tb are flip conjugate; (ii) [ta] ′ 1 and [tb] ′ 1 are topologically isomorphic; (iii) [ta] ′ 1 and [tb] ′ 1 are abstractly isomorphic. 96 p.a. dowerk, f. le mâıtre proof. the only non-trivial implication is (iii)⇒(i). let ρ : [ta]′1 → [tb] ′ 1 be a group isomorphism. by the previous corollary we find a measure-preserving transformation s such that for all u ∈ [ta]′1 one has ρ(u) = sus−1. we then find a,b ⊆ x such that x = atta(a) tb tta(b) tt 2a (b) (see e.g. [10, proposition 2.7]) and let a1 = a, a2 = ta(a) , a3 = b , a4 = ta(b) , a5 = t 2 a (b) . consider then the involutions u1, · · · ,u5 defined by ui(x) =   ta(x) if x ∈ ai , t−1a (x) if x ∈ ta(ai) , x else . these involutions all belong to the l1 full group of ta, so they actually belong to [ta] ′ 1 by [10, lemma 3.10]. now for every x ∈ ai we have ta(x) = ui(x) so for every x ∈ s(ai) we have stas−1(x) = suis−1(x). since for each i ∈ {1, . . . , 5} we have suis−1 ∈ [tb]′1 ≤ [tb]1, we conclude that stas −1 can be obtained by cutting and pasting finitely many elements of [tb]1, hence stas −1 ∈ [tb]1. by the same argument we also have tb ∈ [stas−1]1 so tb and stas−1 share the same orbits. belinskaya’s theorem (as stated in [10, theorem 4.1]) now implies that ta and tb are flip conjugate. corollary 5.4. there is a continuum of pairwise non-isomorphic nonsimple polish groups having tbng. in particular, tbng does not imply bng. proof. consider, for an irrational α ∈]0, 1/2[, the rotation tα(x) = x mod 1. by corollary 4.5 the derived l1 full groups [tα] ′ 1 have tbng and are not simple. moreover all those groups are non-isomorphic by the previous theorem and the fact that two irrational rotations tα and tβ are conjugate iff α = ±β mod 1. bng is not equivalent to tbng 97 references [1] s. bezuglyi, k. medynets, full groups, flip conjugacy, and orbit equivalence of cantor minimal systems, colloq. math. 110 (2) (2008), 409 – 429. 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[9] j. kittrell, t. tsankov, topological properties of full groups, ergodic theory dynam. systems 30 (2) (2010), 525 – 545. [10] f. le mâıtre, on a measurable analogue of small topological full groups, adv. math. 332 (2018), 235 – 286. [11] m.w. liebeck, a. shalev, diameters of finite simple groups: sharp bounds and applications, ann. of math. (2) 154 (2) (2001), 383 – 406. [12] v.v. ryzhikov, representation of transformations preserving the lebesgue measure, in the form of a product of periodic transformations, mat. zametki 38 (6) (1985), 860 – 865. introduction preliminaries topological bounded normal generation. derived l1 full groups. products of two involutions in the derived l1 full group of hyperfinite graphings topological bounded normal generation for derived l1 full groups reconstruction for the derived l1 full group � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 1 (2019), 61 – 76 doi:10.17398/2605-5686.34.1.61 available online february 3, 2019 additivity of elementary maps on gamma rings b.l. macedo ferreira comission of mathematicians, federal technological university of paraná professora laura pacheco bastos avenue, 800, 85053–510, guarapuava, brazil brunoferreira@utfpr.edu.br received january 11, 2018 presented by alexandre turull accepted october 15, 2018 abstract: let m and m′ be gamma rings, respectively. we study the additivity of surjective elementary maps of m × m′. we prove that if m contains a non-trivial γ-idempotent satisfying some conditions, then they are additive. key words: elementary maps, gamma rings, additivity. ams subject class. (2010): 16y99. 1. gamma rings and elementary maps let m and γ be two abelian groups. we call m a γ-ring if the following conditions are satisfied: (i) xαy ∈ m, (ii) (x + y)αz = xαz + yαz, xα(y + z) = xαy + xαz, (iii) x(α + β)y = xαy + xβy, (iv) (xαy)βz = xα(yβz), for all x,y,z ∈ m and α,β ∈ γ. n. nobusawa introduced the notion of a γ-ring, more general than a ring in his paper entitled “on a generalization of the ring theory”. for those readers who are not familiar with this language of γ-rings we recommend “on a generalization of the ring theory” and “on the γ-rings of nobusawa” [2] and [1] respectively. our purpose in this paper is the study of the additivity of a specific application on γ-rings, for this we will address some preliminary definitions. a nonzero element 1 ∈ m is called a multiplicative γ-identity of m or γ-unity element (for some γ ∈ γ) if 1γx = xγ1 = x for all x ∈ m. a nonzero element e1 ∈ m is called a γ1-idempotent (for some γ1 ∈ γ) if e1γ1e1 = e1 and issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.1.61 mailto:brunoferreira@utfpr.edu.br https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 62 b.l.m. ferreira a nontrivial γ1-idempotent if it is a γ1-idempotent different from multiplicative γ1-identity element of m. let γ, γ′, m and m′ be additive groups such that m is a γ-ring and m′ is a γ′-ring. let m : m → m′ and m∗ : m′ → m be two maps and φ : γ → γ′, φ∗ : γ′ → γ two bijective maps. we call the ordered pair (m,m∗) an elementary map of m×m′ if m(aαm∗(x)βb) = m(a)φ(α)xφ(β)m(b), m∗(xµm(a)νy) = m∗(x)φ∗(µ)aφ∗(ν)m∗(y) for all α,β ∈ γ, a,b ∈ m, µ,ν ∈ γ′ and x,y ∈ m′. we say that the elementary map (m,m∗) of m × m′ is additive (resp., injective, surjective, bijective) if both maps m and m∗ are additive (resp., injective, surjective, bijective). let m and γ be two abelian groups such that m is a γ-ring and e1 ∈ m a nontrivial γ1-idempotent. let us consider e2 : γ × m → m, e′2 : m × γ → m two m-additive maps such that e2(γ1,a) = a− e1γ1a, e′2(a,γ1) = a−aγ1e1. let us denote e2αa = e2(α,a), aαe2 = e ′ 2(a,α), 11αa = e1αa + e2αa, aα11 = aαe1 + aαe2 and suppose (aαe2)βb = aα(e2βb) for all α,β ∈ γ and a,b ∈ m. then 11γ1a = aγ111 = a and (aα11)βb = aα(11βb), for all α,β ∈ γ and a,b ∈ m, and m has a peirce decomposition m = m11 ⊕m12 ⊕m21 ⊕m22, where mij = eiγ1mγ1ej (i,j = 1, 2), satisfying the multiplicative relations: (i) mijγmjl ⊆ mil(i,j, l = 1, 2); (ii) mijγ1mkl = 0 if j 6= k (i,j,k, l = 1, 2). if a and b are subsets of a γ-ring m and θ ⊆ γ, we denote aθb the subset of m consisting of all finite sums of the form ∑ i aiγibi where ai ∈ a, γi ∈ θ and bi ∈ b. a right ideal (resp., left ideal) of a γ-ring m is an additive subgroup i of m such that iγm ⊆ i (resp., mγi ⊆ i). if i is both a right and a left ideal of m, then we say that i is an ideal or two-side ideal of m. an ideal p of a γ-ring m is called prime if for any ideals a,b ⊆ m, aγb ⊆ p implies that a ⊆ p or b ⊆ p. a γ-ring m is said to be prime if the zero ideal is prime. theorem 1.1. ([9, theorem 4]) if m is a γ-ring, the following conditions are equivalent: (i) m is a prime γ-ring; (ii) if a,b ∈ m and aγmγb = 0, then a = 0 or b = 0. additivity of elementary maps on gamma rings 63 the first result about the additivity of maps on rings was given by martindale iii in an excellent paper [10]. he established a condition on a ring m such that every multiplicative bijective map on m is additive. li and lu [8] also considered this question in the context of prime associative rings containing a nontrivial idempotent. they proved the following theorem. theorem 1.2. let m and m′ be two associative rings. suppose that m is a 2-torsion free ring containing a family {eα : α ∈ λ} of idempotents which satisfies: (i) if x ∈ m is such that xm = 0, then x = 0; (ii) if x ∈ m is such that eαmx = 0 for all α ∈ λ, then x = 0 (and hence mx = 0 implies x = 0); (iii) for each α ∈ λ and x ∈ m, if eαxeαm(1 −eα) = 0 then eαxeα = 0. then every surjective elementary map (m,m∗) of m×m′ is additive. during the last decade, many mathematicians devoted to study the additivity of maps on associative rings. however, is very difficult to say anything when these applications are defined on arbitrary rings which are not necessarily associative. for the reader interested in applications defined in non-associative rings we recommend some papers [3, 4, 5, 6, 7]. thus this motivated us in the present paper takes up the special case of an γ-ring. we investigate the problem of when a elementary map must be an additive map on the class of γ-rings. 2. the main result we will prove that every surjective elementary map (m,m∗) of m × m′ is additive for this we will assume that m contains a family {eα : α ∈ λ} of γα-idempotents satisfying some conditions. our main result reads as follows. theorem 2.1. let γ, γ′, m and m′ be additive groups such that m is a γ-ring and m′ is a γ′-ring. suppose that m contains a family {eα : α ∈ λ} of γα-idempotents which satisfies: (i) if x ∈ m is such that xγm = 0, then x = 0; (ii) if x ∈ m is such that eαγαmγx = 0 for all α ∈ λ, then x = 0 (and hence mγx = 0 implies x = 0); 64 b.l.m. ferreira (iii) for each α ∈ λ and x ∈ m, if (eαγαxγαeα)γmγ(1α − eα) = 0 then eαγαxγαeα = 0. then every surjective elementary map (m,m∗) of m×m′ is additive. the following lemmas has the same hypotheses of theorem 2.1 and we need these lemmas for the proof of this theorem. thus, let us consider e1 ∈ {eα : α ∈ λ} a nontrivial γ1-idempotent of m and 11 = e1 + e2. we begin with the following trivial lemma lemma 2.1. m(0) = 0 and m∗(0) = 0. proof. m(0) = m(0αm∗(0)β0) = m(0)φ(α)0φ(β)m(0) = 0. similarly, we have m∗(0) = 0. lemma 2.2. m and m∗ are bijective. proof. it suffices to prove that m and m∗ are injective. first show that m is injective. let x1 and x2 be in m and suppose that m(x1) = m(x2). since m ∗(uµm(xi)νv) = m ∗(u)φ∗(µ)xiφ ∗(ν)m∗(v) (i = 1, 2) for all µ,ν ∈ γ′ and u,v ∈ m′, it follows that m∗(u)φ∗(µ)x1φ∗(ν)m∗(v) = m∗(u)φ∗(µ)x2φ ∗(ν)m∗(v). hence from the surjectivity of φ∗ and m∗ and conditions (i) and (ii) we conclude that x1 = x2. now we turn to proving the injectivity of m∗. let u1 and u2 be in m ′ and suppose m∗(u1) = m ∗(u2). since m∗m(xαm∗(ui)βy) = m ∗(m(x)φ(α)uiφ(β)m(y)) = m∗ ( m(x)φ(α)mm−1(ui)φ(β)m(y) ) = m∗m(x)φ∗φ(α)m−1(ui)φ ∗φ(β)m∗m(y) for all α,β ∈ γ and x,y ∈ m, it follows that m∗m(x)φ∗φ(α)m−1(u1)φ ∗φ(β)m∗m(y) = m∗m(x)φ∗φ(α)m−1(u2)φ ∗φ(β)m∗m(y). noting that φ∗φ and m∗m are also surjective, we see that m−1(u1) = m−1(u2), by conditions (i) and (ii). consequently u1 = u2. additivity of elementary maps on gamma rings 65 lemma 2.3. the pair (m∗−1,m−1) is an elementary map of m × m′, that is, the maps m∗−1 : m → m′ and m−1 : m′ → m satisfy m∗ −1( aαm−1(x)βb ) = m∗ −1 (a)φ∗ −1 (α)xφ∗ −1 (β)m∗ −1 (b), m−1 ( xµm∗ −1 (a)νy ) = m−1(x)φ−1(µ)aφ−1(ν)m−1(y) for all α,β ∈ γ, µ,ν ∈ γ′, a,b ∈ m and x,y ∈ m′. proof. the first equality can follow from m∗ ( m∗ −1 (a)φ∗ −1 (α)xφ∗ −1 (β)m∗ −1 (b) ) = m∗ ( m∗ −1 (a)φ∗ −1 (α)mm−1(x)φ∗ −1 (β)m∗ −1 (b) ) = aφ∗ ( φ∗ −1 (α) ) m−1(x)φ∗ ( φ∗ −1 (β) ) b = aαm−1(x)βb and the second equality follows in a similar way. lemma 2.4. let s,a,b ∈ m such that m(s) = m(a) + m(b). then (i) m(sαxβy) = m(aαxβy) + m(bαxβy) for α,β ∈ γ and x,y ∈ m; (ii) m(xαyβs) = m(xαyβa) + m(xαyβb) for α,β ∈ γ and x,y ∈ m; (iii) m∗−1(xαsβy) = m∗−1(xαaβy) + m∗−1(xαbβy) for α,β ∈ γ and x,y ∈ m for x,y ∈ m. proof. (i) let α,β ∈ γ and x,y ∈ m. then m(sαxβy) = m ( sαm∗m∗ −1 (x)βy ) = m(s)φ(α)m∗ −1 (x)φ(β)m(y) = ( m(a) + m(b) ) φ(α)m∗ −1 (x)φ(β)m(y) = m(a)φ(α)m∗ −1 (x)φ(β)m(y) + m(b)φ(α)m∗ −1 (x)φ(β)m(y) = m(aαxβy) + m(bαxβy). (ii) the proof is similar to (i). 66 b.l.m. ferreira (iii) let x,y ∈ m. by lemma 2.3 m∗ −1 (xαsβy) = m∗ −1 (xαm−1m(s)βy) = m∗ −1 (x)φ∗ −1 (α)m(s)φ∗ −1 (β)m∗ −1 (y) = m∗ −1 (x)φ∗ −1 (α) ( m(a) + m(b) ) φ∗ −1 (β)m∗ −1 (y) = m∗ −1 (x)φ∗ −1 (α)m(a)φ∗ −1 (β)m∗ −1 (y) + m∗ −1 (x)φ∗ −1 (α)m(b)φ∗ −1 (β)m∗ −1 (y) = m∗ −1 (xαaβy) + m∗ −1 (xαbβy). the proof is complete. lemma 2.5. the following are true: (i) m(a11 + a12 + a21 + a22) = m(a11) + m(a12) + m(a21) + m(a22); (ii) m∗−1(a11 + a12 + a21 + a22) = m ∗−1(a11) + m ∗−1(a12) + m ∗−1(a21) + m∗−1(a22). proof. by the surjectivity of m, there exists s ∈ m such that m(s) = m(a11) + m(a12) + m(a21) + m(a22). now, for arbitrary α,β ∈ γ, xi1 ∈ mi1 and y1j ∈ m1j, we have m∗ −1 (xi1αe1γ1sγ1e1βy1j) = m∗ −1 (xi1αe1γ1a11γ1e1βy1j) + m ∗−1(xi1αe1γ1a12γ1e1βy1j) + m∗ −1 (xi1αe1γ1a21γ1e1βy1j) + m ∗−1(xi1αe1γ1a22γ1e1βy1j) = m∗ −1 (xi1αe1γ1a11γ1e1βy1j), which implies xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e1βy1j = 0. (2.1) in a similar way, for y2j ∈ m2j we get that xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e1βy2j = 0. (2.2) from (2.1) and (2.2) we conclude xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e1βy = 0. additivity of elementary maps on gamma rings 67 in a similar way, for y1j ∈ m1j and y2j ∈ m2j we get that xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e2βy1j = 0, xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e2βy2j = 0, respectively, which implies xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ1e2βy = 0. thus, xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γ111βy = 0, for all β ∈ γ, y ∈ m, that is, xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) γm = 0. by condition (i) of the theorem we have xi1αe1γ1 ( s− (a11 + a12 + a21 + a22) ) = 0. repeating the above arguments, for xi2 ∈ mi2 we get that xi2αe1γ1 ( s− (a11 + a12 + a21 + a22) ) = 0 which implies xαe1γ1 ( s− (a11 + a12 + a21 + a22) ) = 0. similarly, we obtain xαe2γ1 ( s− (a11 + a12 + a21 + a22) ) = 0, which yields xα ( s− (a11 + a12 + a21 + a22) ) = 0. thus, mγ ( s− (a11 + a12 + a21 + a22) ) = 0 which results in s = a11 + a12 + a21 + a22. the proof of (ii) is similar, since the pair (m∗−1; m−1) is also an elementary map of m×m′. lemma 2.6. the following are true: (i) m(a12 + b12αa22) = m(a12) + m(b12αa22); 68 b.l.m. ferreira (ii) m(a11 + a12αa21) = m(a11) + m(a12αa21); (iii) m(a21 + a22αb21) = m(a21) + m(a22αb21). proof. (i) from lemma 2.5-(i) and (ii) we have m(a12 + b12αa22) = m ( e1γ1(e1 + b12)γ1(a12 + e2αa22) ) = m(e1)φ(γ1)m ∗−1(e1 + b12)φ(γ1)m(a12 + e2αa22) = m(e1)φ(γ1) ( m∗ −1 (e1) + m ∗−1(b12) ) φ(γ1) ( m(a12) + m(e2αa22) ) = m(e1)φ(γ1)m ∗−1(e1)φ(γ1)m(a12) + m(e1)φ(γ1)m ∗−1(e1)φ(γ1)m(e2αa22) + m(e1)φ(γ1)m ∗−1(b12)φ(γ1)m(a12) + m(e1)φ(γ1)m ∗−1(b12)φ(γ1)m(e2αa22) = m(e1γ1e1γ1a12) + m(e1γ1e1γ1a22) + m(e1γ1b12γ1a12) + m(e1γ1b12γ1e2αa22) = m(a12) + m(b12αa22). note that we use properties (i) and (ii) in m(a12 + e2αa22) = m(a12) + m(e2αa22) and m∗ −1 (e1 + b12) = m ∗−1(e1) + m ∗−1(b12), respectively. so (i) follows. observing that a11 + a12αa21 = (a11 + a12αe2)γ1(e1 + a21)γ1e1, a21 + a22αa21 = (a21 + a22αe2)γ1(e1 + b21)γ1e1, then (ii) and (iii) can be proved similarly. lemma 2.7. m(a21γ1a12 + a22γ1b22) = m(a21γ1a12) + m(a22γ1b22). proof. we first claim that m(a21γ1a12γ1c22 + a22γ1b22γ1c22) = m(a21γ1a12γ1c22) + m(a22γ1b22γ1c22) (2.3) additivity of elementary maps on gamma rings 69 holds for all c22 ∈ m22. indeed, from lemma 2.5-(i) and (ii), we see that m(a21γ1a12αc22 + a22γ1b22αc22) = m ( (a21 + a22)γ1(a12 + b22)αc22 ) = m(a21 + a22)φ(γ1)m ∗−1(a12 + b22)φ(α)m(c22) = ( m(a21) + m(a22) ) φ(γ1) ( m∗ −1 (a12) + m ∗−1(b22) ) φ(α)m(c22) = m(a21)φ(γ1)m ∗−1(a12)φ(α)m(c22) + m(a21)φ(γ1)m ∗−1(b22)φ(α)m(c22) + m(a22)φ(γ1)m ∗−1(a12)φ(α)m(c22) + m(a22)φ(γ1)m ∗−1(b22)φ(α)m(c22) = m(a21γ1a12αc22) + m(a21γ1b22αc22) + m(a22γ1a12αc22) + m(a22γ1b22αc22) = m(a21γ1a12αc22) + m(a22γ1b22αc22), as desired. now we find s ∈ m such that m(s) = m(a21γ1a12) +m(a22γ1b22). for arbitrary element x21 ∈ m21, by lemma 2.4-(i) and lemma 2.6-(iii), m(sαx21) = m(sαx21γ1e1) = m ( (a21γ1a12)αx21γ1e1 ) + m ( (a22γ1b22)αx21γ1e1 ) = m(a21γ1a12αx21 + a22γ1b22αx21). it follows that ( s− (a21γ1a12 + a22γ1b22) ) αx21 = 0, (2.4) for all α ∈ γ. our next step will be to prove that( s− (a21γ1a12 + a22γ1b22) ) αx22 = 0 (2.5) holds for every x22 ∈ m22. first, for y21, by lemmas 2.4-(i) and lemma 2.6-(iii) m(sαx22βy21) = m ( (a21γ1a12)αx22βy21 ) + m ( (a22γ1b22)αx22βy21 ) = m ( (a21γ1a12)αx22βy21 + (a22γ1b22)αx22βy21 ) , which implies that sαx22βy21 = (a21γ1a12)αx22βy21 + (a22γ1b22)αx22βy21. it follows that ( s− (a21γ1a12 + a22γ1b22) ) αx22 ·βy21 = 0. 70 b.l.m. ferreira for y22 ∈ m22, by lemma 2.4-(i) and (2.3) we have m(sαx22βy22) = m ( (a21γ1a12)αx22βy22 ) + m ( (a22γ1b22)αx22βy22 ) = m ( (a21γ1a12)αx22βy22 ) + m ( (a22γ1b22)αx22βy22 ) = m ( a21γ1a12αx22βy22 ) + m ( (a22γ1b22)αx22βy22 ) = m ( a21γ1a12αx22βy22 + (a22γ1b22)αx22βy22 ) = m ( (a21γ1a12)αx22βy22 + (a22γ1b22)αx22βy22 ) yielding that sαx22βy22 = (a21γ1a12 + a22γ1b22)αx22βy22. it follows that( s− (a21γ1a12 + a22γ1b22) ) αx22βy22 = 0. hence, we obtain that( s− (a21γ1a12 + a22γ1b22) ) αx22 · γm = 0. so eq. (2.5) follows by theorem 2.1 condition (i). from eqs. (2.4) and (2.5), we can get that ( s−(a21γ1a12+a22γ1b22) ) γm = 0. therefore, s = a21γ1a12 + a22γ1b22 by theorem 2.1 condition (i) again. taking lemma 2.3 into account, still hold true when m is replaced by m∗−1, that is lemma 2.8. the following are true: (i) m∗−1(a12 + b12αa22) = m ∗−1(a12) + m ∗−1(b12αa22). (ii) m∗−1(a11 + a12αa21) = m ∗−1(a11) + m ∗−1(a12αa21). (iii) m∗−1(a21 + a22αb21) = m ∗−1(a21) + m ∗−1(a22αb21). (iv) m∗−1(a21γ1a12 + a22γ1b22) = m ∗−1(a21γ1a12) + m ∗−1(a22γ1b22). lemma 2.9. m(a12 + b12) = m(a12) + m(b12). proof. let s ∈ m such that m(s) = m(a12) + m(b12). for x1j ∈ m1j, applying lemma 2.4-(i), m(sγ1e1αx1j) = m(a12γ1e1αx1j) + m(b12γ1e1αx1j) = 0. these equations show that sγ1e1αx1j = 0 = (a12 + b12)γ1e1αx1j. hence,( s− (a12 + b12) ) γ1e1αx1j = 0. for all x2j ∈ m2j m∗ −1 (sγ1e1αx2j) = m ∗−1(a12γ1e1αx2j) + m ∗−1(b12γ1e1αx2j) = 0, additivity of elementary maps on gamma rings 71 which implies that ( s− (a12 + b12) ) γ1e1αx2j = 0. thus ( s− (a12 + b12) ) γ1e1αx = 0, for all α ∈ γ and x ∈ m, which implies( s− (a12 + b12) ) γ1e1γm = 0. for y11 ∈ m11, applying lemma 2.4-(i),(ii), m(y11βe1γ1sγ1e2αx22) = m(y11βe1γ1a12γ1e2αx22) + m(y11βe1γ1b12γ1e2αx22) = m(y11βe1γ1a12γ1e2αx22) + m(y11βe1γ1b12γ1e2αx22) = m(y11βe1γ1(a12 + b12)γ1e2αx22) these equations show that y11βe1γ1 ( s− (a12 + b12) ) γ1e2αx22 = 0. for all y21 ∈ m21 m∗ −1 (y21βe1γ1sγ1e2αy21) = m ∗−1(y21βe1γ1a12γ1e2αx22) + m∗ −1 (y21βe1γ1b12γ1e2αx22) = m∗ −1 (y21βe1γ1a12γ1e2αx22) + m∗ −1 (y21βe1γ1y21βe1γ1b12γ1e2αx22) = m∗ −1 (y21βe1γ1(a12 + b12)γ1e2αx22) = m∗ −1( y21βe1γ1(a12 + b12)γ1e2αx22 ) which implies that y21βe1γ1 ( s− (a12 + b12) ) γ1e2αx22 = 0. for yi2 ∈ mi2, applying lemma 2.4-(i),(ii), m(yi2βe1γ1sγ1e2αx22) = m(yi2βe1γ1a12γ1e2αx22) + m(yi2βe1γ1b12γ1e2αx22) = 0, 72 b.l.m. ferreira which implies yi2βe1γ1sγ1e2αx22 = 0 = yi2βe1γ1(a12 + b12)γ1e2αx22, yβe1γ1 ( s−(a12 + b12) ) γ1e2αx22 = 0. for yij ∈ mij, applying lemma 2.4-(i),(ii), m(yijβe2γ1sγ1e2αx22) = m(yijβe2γ1a12γ1e2αx22) + m(yijβe2γ1b12γ1e2αx22) = 0, yijβe2γ1sγ1e2αx22 = 0 = yijβe2γ1(a12 + b12)γ1e2αx22. these equations show that yijβe2γ1 ( s− (a12 + b12) ) γ1e2αx22 = 0, yβe2γ1 ( s− (a12 + b12) ) γ1e2αx22 = 0, mγ ( s− (a12 + b12) ) γ1e2αx22 = 0,( s− (a12 + b12) ) γ1e2αx22 = 0,( s− (a12 + b12) ) γ1e2αx = 0,( s− (a12 + b12) ) γm = 0, s = a12 + b12. lemma 2.10. m(a11 + b11) = m(a11) + m(b11). proof. choose s = s11 + s12 + s21 + s22 ∈ m such that m(s) = m(a11) + m(b11). m(s) = m(e1γ1a11γ1e1) + m(e1γ1b11γ1e1) = m(e1)φ(γ1)m ∗−1(a11)φ(γ1)m(e1) + m(e1)φ(γ1)m ∗−1(b11)φ(γ1)m(e1) = m(e1)φ(γ1) ( m∗ −1 (a11) + m ∗−1(b11) ) φ(γ1)m(e1) = m(e1)φ(γ1) ( m∗ −1 (e1γ1a11γ1e1) + m ∗−1(e1γ1b11γ1e1) ) φ(γ1)m(e1) = m(e1)φ(γ1)m ∗−1(e1γ1sγ1e1)φ(γ1)m(e1) = m(e1γ1e1γ1sγ1e1γ1e1) = m(e1γ1sγ1e1) = m(s11). additivity of elementary maps on gamma rings 73 it follows that s = s11. hence s− (a11 + b11) ∈ m11. first we let x11 ∈ m12 be arbitrary. applying lemma 2.4-(i) we get that m(sαe1γ1x11γ1e1βe2) = m(a11αe1γ1x11γ1e1βe2) + m(b11αe1γ1x11γ1e1βe2) = m(a11αe1γ1x11γ1e1βe2 + b11αe1γ1x11γ1e1βe2) = m ( (a11 + b11)αe1γ1x11γ1e1βe2 ) yielding that sαe1γ1x11γ1e1βe2 = (a11 + b11)αe1γ1x11γ1e1βe2. therefore( s− (a11 + b11) ) αe1γ1x11γ1e1βe2 = 0. this implies ( s− (a11 + b11) ) αe1γ1xγ1e1βe2 = 0. (2.6) second we let x12 ∈ m12 be arbitrary. applying lemma 2.4-(i) we get that m(sαe1γ1x12γ1e2βe2) = m(a11αe1γ1x12γ1e2βe2) + m(b11αe1γ1x12γ1e2βe2) = m(a11αe1γ1x12γ1e2βe2 + b11αe1γ1x12γ1e2βe2) = m ( (a11 + b11)αe1γ1x12γ1e2βe2 ) yielding that sαe1γ1x12γ1e2βe2 = (a11 + b11)αe1γ1x12γ1e2βe2. therefore( s− (a11 + b11) ) αe1γ1x12γ1e2βe2 = 0. this implies ( s− (a11 + b11) ) αe1γ1xγ1e2βe2 = 0. (2.7) third we let x21 ∈ m21 be arbitrary. applying lemma 2.4-(i) we get that m(sαe2γ1x21γ1e1βe2) = m(a11αe2γ1x21γ1e1βe2) + m(b11αe2γ1x21γ1e1βe2) = m(a11αe2γ1x21γ1e1βe2 + b11αe2γ1x21γ1e1βe2) = m ( (a11 + b11)αe2γ1x21γ1e1βe2 ) yielding that sαe2γ1x21γ1e1βe2 = (a11 + b11)αe2γ1x21γ1e1βe2. therefore( s− (a11 + b11) ) αe2γ1x21γ1e1βe2 = 0. 74 b.l.m. ferreira this implies ( s− (a11 + b11) ) αe2γ1xγ1e1βe2 = 0. (2.8) lastly we let x22 ∈ m22 be arbitrary. applying lemma 2.4-(i) we get that m(sαe2γ1x22γ1e2βe2) = m(a11αe2γ1x22γ1e2βe2) + m(b11αe2γ1x22γ1e2βe2) = m(a11αe2γ1x22γ1e2βe2 + b11αe2γ1x22γ1e2βe2) = m ( (a11 + b11)αe2γ1x22γ1e2βe2 ) yielding that sαe2γ1x22γ1e2βe2 = (a11 + b11)αe2γ1x22γ1e2βe2. therefore( s− (a11 + b11) ) αe2γ1x22γ1e2βe2 = 0. this implies ( s− (a11 + b11) ) αe2γ1xγ1e2βe2 = 0. (2.9) from (2.6)-(2.9) we have( s− (a11 + b11) ) α11γ1xγ111βe2 = 0, which implies ( s− (a11 + b11) ) αxβe2 = 0, for all α,β ∈ γ and x ∈ m, which yields( s− (a11 + b11) ) γmγe2 = 0. so ( s− (a11 + b11) ) αmβ(11 −e1) = 0 which implies( e1γ1(s− (a11 + b11)γ1e1 ) γmγ(11 −e1) = 0. it follows, from theorem 2.1 condition (iii), that s = a11 + b11. lemma 2.11. m is additive on e1γ1m = m11 + m12. proof. the proof is the same as that of martindale iii (1969, lemma 5) and is included for the sake of completeness. in fact, let a11,b11 ∈ m11 and a12,b12 ∈ m12. making use of lemmas 2.5, 2.9 and 2.10 we can see that m ( (a11 + a12) + (b11 + b12) ) = m ( (a11 + b11) + (a12 + b12) ) = m(a11 + b11) + m(a12 + b12) = m(a11) + m(b11) + m(a12) + m(b12) = m(a11 + a12) + m(b11 + b12). holds true, as desired. additivity of elementary maps on gamma rings 75 proof of theorem 2.1. suppose that a,b ∈ m and choose s ∈ m such that m(s) = m(a) + m(b). for all α ∈ λ, m is additive on eαγαm because of lemma 2.11. thus, for every r ∈ m, we have m(eαγαrµs) = m(eα)φ(γα)m ∗−1(r)φ(µ)m(s) = m(eα)φ(γα)m ∗−1(r)φ(µ) ( m(a) + m(b) ) = m(eα)φ(γα)m ∗−1(r)φ(µ)m(a) + m(eα)φ(γα)m ∗−1(r)φ(µ)m(b) = m(eαγαrµa) + m(eαγαrµb) = m(eαγαrµa + eαγαrµb) = m ( eαγαrµ(a + b) ) . so eαγαrµs = eαγαrµ(a + b). therefore eαγαmγ ( s− (a + b) ) = 0 holds for every α ∈ λ. we then conclude that s = a + b from theorem 2.1 condition (ii). this shows that m is additive on m. to prove the additivity of m∗, let x,y ∈ m′. for a,b ∈ m, by using the additivity of m, we have m ( aλ ( m∗(x) + m∗(y) ) µb ) = m ( aλm∗(x)µb ) + m ( aλm∗(y)µb ) = m(a)φ(λ)xφ(µ)m(b) + m(a)φ(λ)yφ(µ)m(b) = m(a)φ(λ)(x + y)φ(µ)m(b) = m ( aλm∗(x + y)µb ) . it follows that aλ ( m∗(x) + m∗(y)−m∗(x + y) ) µb = 0 holds for all a,b ∈ m, that is, aλ ( m∗(x) + m∗(y) −m∗(x + y) ) γm = 0 holds for all a ∈ m, which implies aλ ( m∗(x) + m∗(y) −m∗(x + y) ) = 0 holds for all a ∈ m, which implies mγ ( m∗(x) + m∗(y) −m∗(x + y) ) = 0 which forces m∗(x + y) = m∗(x) + m∗(y) because of theorem 2.1 conditions (i) and (ii). this completes the proof. 76 b.l.m. ferreira corollary 2.1. let γ, γ′, m and m′ be additive groups such that m is a γ-ring and m′ is a γ′-ring such that m is a prime γ-ring containing a non-trivial γ-idempotent (m need not have an γ-identity element), where γ ∈ γ. suppose e2 : γ×m → m, e′2 : m×γ → m two m-additive maps such that e2(γ1,a) = a − e1γ1a, e′2(a,γ1) = a − aγ1e1. denote e2αa = e2(α,a), aαe2 = e ′ 2(a,α), 11αa = e1αa + e2αa, aα11 = aαe1 + aαe2 and suppose (aαe2)βb = aα(e2βb) for all α,β ∈ γ and a,b ∈ m. then every surjective elementary map (m,m∗) of m×m′ is additive. proof. the result follows directly from theorem 2.1 and theorem 1.1. corollary 2.2. let γ, γ′, m and m′ be additive groups such that m is a γ-ring and m′ is a γ′-ring such that m is a prime γ-ring containing a non-trivial γ-idempotent and a γ-unity element, where γ ∈ γ. then every surjective elementary map (m,m∗) of m×m′ is additive. references [1] w.e. barnes, on the γ-rings of nobusawa, pacific j. math. 18 (1966), 411 – 422. [2] n. nobusawa, on a generalization of the ring theory, osaka j. math. 1 (1964), 81 – 89. [3] j.c.m. ferreira, b.l.m. ferreira, additivity of n-multiplicative maps on alternative rings, comm. algebra 44 (2016), 1557 – 1568. [4] b.l.m. ferreira, j.c.m. ferreira, h. guzzo, jr., jordan triple maps of alternative algebras, jp j. algebra number theory appl. 33 (2014), 25 – 33. [5] b.l.m. ferreira, h. guzzo jr., j.c.m. ferreira, additivity of jordan elementary maps on standard rings, algebra discrete math. 18 (2014), 203 – 233. [6] b.l.m. ferreira, r. nascimento, derivable maps on alternative rings, recen 16 (2014), 9 – 15. [7] b.l.m. ferreira, j.c.m. ferreira, h. guzzo jr., jordan triple elementary maps on alternative rings, extracta math. 29 (2014), 1 – 18. [8] p. li, f. lu, additivity of elementary maps on rings, comm. algebra 32 (2004), 3725 – 3737. [9] s. kyuno, on prime gamma rings, pacific j. math. 75 (1978), 185 – 190. [10] w.s. martindale iii, when are multiplicative mappings additive?, proc. amer. math. soc. 21 (1969), 695 – 698. gamma rings and elementary maps the main result � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 36, num. 1 (2021), 63 – 80 doi:10.17398/2605-5686.36.1.63 available online may 5, 2021 stability of some essential b-spectra of pencil operators and application a. ben ali, m. boudhief, n. moalla department of mathematics, science faculty of sfax university of sfax, tunisia boudhiafmon198@gmail.com , nedra.moalla@ipeis.rnu.tn received january 1, 2021 presented by pietro aiena accepted april 6, 2021 abstract: in this paper, we give some results on the essential b-spectra of a linear operator pencil, which are used to determine the essential b-spectra of an integro-differential operator with abstract boundary conditions in the banach space lp([−a,a] × [−1, 1]), p ≥ 1 and a > 0. key words: operator pencil, finite-rank and power finite-rank perturbations, essential b-spectra, transport operator. msc (2020): 47a53, 47a55. 1. introduction let x be a banach space. we will denote by c(x) (resp. l(x)) the set of all closed linear (resp. the algebra of all bounded) linear operators from x into x. for t ∈ c(x), we write d(t) ⊂ x for the domain, n(t) ⊂ x for the null space and r(t) ⊂ x for the range of t. we denote by α(t) the dimension of n(t) and β(t) the codimension of r(t) in x. for t ∈ c(x) and m ∈ l(x), we define the resolvent of the linear operator pencil λm −t, where λ ∈ c, or the m-resolvent of t by ρm (t) := {λ ∈ c : λm −t has a bounded inverse} , and its spectrum by σ(m,t) = c\ρm (t) . for t ∈c(x), we define the set 4(t) = { n ∈ n : ∀m ∈ n, m ≥ n ⇒ r(tn) ∩n(t) ⊂ r(tm) ∩n(t) } . the degree of stable iteration of t is defined as dis(t) = inf 4(t), where dis(t) = ∞ if 4(t) = ∅. issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.36.1.63 mailto:boudhiafmon198@gmail.com mailto:nedra.moalla@ipeis.rnu.tn https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 64 a. ben ali, m. boudhief, n. moalla we define the set of upper semi-fredholm operators by φ+(x) = { t ∈c(x) : α(t) < ∞ and r(t) is closed in x } , and the set of lower semi-fredholm operators by φ−(x) = { t ∈c(x) : β(t) < ∞ and r(t) is closed in x } . φ(x) := φ+(x) ∩ φ−(x) will denote the set of fredholm operators from x into x. the index of a fredholm operator t is defined by i(t) = α(t)−β(t). according to [17], an operator t ∈c(x) is called quasi-fredholm of degree d ∈ n if the following three conditions are satisfied: (i) dis(t) = d ; (ii) r(td) ∩n(t) is a closed and complemented subspace of x; (iii) r(t) + n(td) is a closed and complemented subspace of x. this set of operators will be denoted by qf(d). following [10, definition 2.4], an operator t ∈ c(x) is called upper semi b-fredholm (resp. lower semi b-fredholm) if there exists an integer d ∈ n such that t ∈ qf(d) and such that n(t)∩r(td) is of finite dimension (resp. r(t) + n(td) is of finite codimension). these sets are denoted respectively by φ+b(x) and φ − b(x). we denote by φb(x) := φ + b(x) ∩ φ − b(x), the set of b-fredholm operators from x into x. in this case, the index of t is defined as the integer: ind(t) = dim(n(t) ∩ r(td)) − codim(r(t) + n(td)). an operator t ∈ c(x) is called b-weyl if it is a b-fredholm operator of index zero. we will denote this set by bw(x). for t ∈c(x), we define the ascent a(t) of t by a(t) = inf { n ∈ n : n ( tn ) = n ( tn+1 )} , and the descent d(t) of t by d(t) = inf { n ∈ n : r ( tn ) = r ( tn+1 )} . we define respectively the set of drazin invertible operators, left drazin invertible operators and right drazin invertible operators as follows: dr(x) := {t ∈c(x) : a(t) and d(t) are both finite} , ld(x) := { t ∈c(x) : a(t) is finite and r ( ta(t)+1 ) is closed } , rd(x) := { t ∈c(x) : d(t) is finite and r ( td(t) ) is closed } . an operator t ∈ c(x) is of kato type if there exist an integer d ∈ n and a pair of two closed subspaces (n1,n2) of x such that: stability of some essential b-spectra 65 (i) x = n1 ⊕n2; (ii) t(n1) ⊂ n1 and t/n1 is semi-regular; (iii) t(n2) ⊂ n2 and (t/n2 ) d = 0, i.e., t/n2 is nilpotent. note that, in the case of hilbert spaces, labrousse in [17, theorem 3.2.2] has shown that, the set of quasi-fredholm operators coincides with the kato type operators for the class of closed operators. according to [17, p. 206, remark], this equivalence is also true in the case of banach spaces. for t ∈c(x) and m ∈ l(x), we define the b-fredholm spectrum, drazin spectrum, the upper semi b-fredholm spectrum, the lower semi b-fredholm spectrum, the left drazin spectrum, the right drazin spectrum, the b-weyl spectrum, the closed-range spectrum and the kato spectrum of the linear operator pencil λm −t, where λ ∈ c, or the pair (m,t) as follows: σbf (m,t) = { λ ∈ c : λm −t /∈ φb(x) } , σd(m,t) = { λ ∈ c : λm −t /∈ dr(x) } , σbf+ (m,t) = { λ ∈ c : λm −t /∈ φ+b(x) } , σbf−(m,t) = { λ ∈ c : λm −t /∈ φ−b(x) } , σld(m,t) = { λ ∈ c : λm −t /∈ ld(x) } , σrd(m,t) = { λ ∈ c : λm −t /∈ rd(x) } , σbw (m,t) = { λ ∈ c : λm −t /∈ bw(x) } , σec(m,t) = { λ ∈ c : r(λm −t) is not closed } , σek(m,t) = { λ ∈ c : λm −t is not kato } . the point spectrum, the residual spectrum and the continuous spectrum of the pair (m,t), when t ∈c(x) and m ∈ l(x), are defined respectively by: σp(m,t) = { λ ∈ c : λm −t is not injective } , σr(m,t) = { λ ∈ c : n(λm −t) = {0} and r(λm −t) x } , σc(m,t) = { λ ∈ c : n(λm −t) = {0}, r(λm −t) = x and r(λm −t) 6= x } , where, r(λm −t) is the closure of r(λm − t). the collection {σp(m,t), σr(m,t),σc(m,t)} forms a partition of the spectrum σ(m,t), which means that they are pairwise disjoint and σ(m,t) = σp(m,t)∪σr(m,t)∪σc(m,t). 66 a. ben ali, m. boudhief, n. moalla for t ∈ c(x) and m ∈ l(x), the upper semi b-fredholm, the lower semi b-fredholm and the b-fredholm resolvent of the linear operator pencil λm −t, where λ ∈ c, are defined respectively by ρbf+ (m,t) = c\σbf+ (m,t) , ρbf−(m,t) = c\σbf−(m,t) , ρbf (m,t) = ρbf+ (m,t) ∩ρbf−(m,t) . the present paper is a generalization of the results obtained by a. jeribi et al. in [15] and some of stability results obtained by m. berkani et al. in [6] for the usual essential b-spectra. it generalizes also the works obtained by a. jeribi in [13, 14] about the invariance of the s-essential spectra under weakly compact or strictly singular perturbations, which are not applied in the b-fredholm theory. so that, we can use, by adding some hypothesis, the perturbations of the b-fredholm spectra under finite-rank and power finite-rank commuting operators. more precisely, let t1,t2 ∈ c(x) be two commuting closed linear operators such that the bounded linear operator m commutes in the resolvent sense with t1 and t2 (see definition 2.2) and satisfying (λm −t1)−1 −(λm −t2)−1 ∈f(x) (resp. (λm −t1)−1 −(λm −t2)−1 ∈ fp(x) or nilpotent) for some λ ∈ ρm (t1) ∩ ρm (t2). then, we prove that σ∗(m,t1) = σ∗(m,t2), where σ∗(m,.) ∈ { σbf (m,.),σbf+ (m,.),σbf−(m,.), σbw (m,.),σld(m,.),σrd(m,.),σd(m,.) } . these perturbation results are needed to extend the results obtained in [15], on the s-essential spectra of closed densely defined linear operators to essential b-spectra of operator pencil λm −t, when t ∈c(x), m ∈ l(x) and λ ∈ c. moreover, under the additional hypothesis m(c(t)) = c(t) (see definition 2.1), we give the relationship between the closed-range spectrum and the kato spectrum of the linear operator pencil λm −t, when t ∈ c(x), m ∈ l(x) and λ ∈ c (see proposition 2.4), which generalizes a result obtained in [9, proposition 3.2] in the case of bounded operators and [8, corollary 3] for closed densely defined linear operators for the usual spectrum. we establish also, the equality between the closed-range spectrum and some essential b-spectra of operator pencil acting on a banach space (theorem 2.5). the obtained results, are finally used to describe the essential b-spectra of the operator pencil of the following integro-differential operator with abstract boundary conditions in the banach space xp := lp ( [−a,a] × [−1, 1],dxdy ) , a > 0, 1 ≤ p < ∞, ah = th + k , stability of some essential b-spectra 67 where th, k and m are defined by  th : d(th) ⊆ xp −→ xp ψ 7−→ −ξ ∂ψ ∂x (x,ξ) −σ(ξ)ψ(x,ξ) , d(th) = { ψ ∈wp : ψi = hψo } , where wp := { ψ ∈ xp : ξ∂ψ∂x ∈ xp } ,  k : xp −→ xp u 7−→ ∫ 1 −1 k(x,ξ,ν)u(x,ν)dν , and { m : xp −→ xp ϕ 7−→ m(ϕ)(x,ξ) = η(ξ)ϕ(x,ξ) , where σ(.) and η(.) are in l∞(−1, 1), k(., ., .) is a measurable function, and h is the boundary operator connecting the outgoing and the incoming fluxes. the outline of this work is organized in the following way: in section 2, we give some stability results of some essential b-spectra of linear operator pencil. the main results of this section are theorem 2.3 and theorem 2.5. in section 3, we apply the results developed in section 2 to characterize the b-essential spectra of a transport operator with abstract boundary conditions on lp-spaces, 1 ≤ p < ∞. 2. stability of some essential b-spectra of pencil operators we are interested, in this section, in some of stability results of the essential b-spectra of an operator pencil λm − t , where m ∈ l(x), t ∈ c(x) and λ ∈ c. since, the kato decomposition theorem, remains true in the case of banach spaces as shown in [17, p. 206], we can directly use the following proposition inspired from [10], when necessary, in the case of banach spaces without proof. proposition 2.1. let t ∈ c(x). if t is a semi b-fredholm operator, then there exist two closed subspaces x0 and x1 of x such that (i) x = x0 ⊕x1, (ii) t(x0) ⊂ x0 and t0 = t/x0 is a semi-fredholm operator, (iii) t(x1) ⊂ x1 and t1 = t/x1 is a nilpotent operator. 68 a. ben ali, m. boudhief, n. moalla first we recall the following subspace, introduced by p. saphar in [20], and it was defined by p. aiena in [1] in purely algebraic terms. definition 2.1. the algebraic core c(t) of a linear operator t is defined to be the greatest subspace n ⊂ d(t) for which t(n) = n. for more details for the algebraic core c(t), we can refer to [1]. theorem 2.1. let t ∈c(x) and m ∈ l(x) such that m(c(t)) = c(t) and ρm (t) 6= ∅. if t is a semi b-fredholm operator, then there exists ε > 0 such that t − µm is a semi-fredholm operator, for each µ ∈ d(0,ε)\{0}. moreover, we have α(t −µm) and β(t −µm) are constants on d(0,ε)\{0}. proof. if m = i, then we obtain the result established in [7]. if m 6= i, then the fact that t is a semi b-fredholm operator, this implies from proposition 2.1, the existence of two t-invariant closed subspaces x0 and x1 such that • x = x0 ⊕x1, • t(x0) ⊂ x0 and t0 = t/x0 is a semi-fredholm operator, • t(x1) ⊂ x1 and t1 = t/x1 is nilpotent. since m(c(t)) = c(t) and t(c(t)) = c(t), then we can conclude, by using the definition of c(t), that x0 and x1 are invariants under the operator m. so, we can consider m0 = m/x0 and m1 = m/x1 such that m = m0⊕m1. case 1: if x0 = {0}, then we get m0 = 0, t0 = 0, m = m1 and t = t1 is nilpotent. since, the operator t1 is nilpotent then we have t1 − µi1 is invertible, for µ 6= 0, where i1 = i/x1 . we have t1 −µm1 = (t1 −µi1)[i1 −µ(t1 −µi1)−1(m1 − i1)] is invertible for µ such that 0 < |µ| < 1 ‖(t1 −µi1)−1(m1 − i1)‖ = γ . hence, t − µm = t1 − µm1 is a semi-fredholm operator, for µ such that 0 < |µ| < γ. moreover, we have α(t − µm) = α(t1 − µm1) = 0 and β(t −µm) = β(t1 −µm1) = 0 on d(0,γ)\{0}. case 2: suppose x0 6= {0}. • if m0 = 0, then by using case 1, we obtain that t−µm = t0⊕t1−µm1 is a semi-fredholm operator for µ such that 0 < |µ| < γ. • if m0 6= 0, then t − µm = t0 − µm0 ⊕ t1 − µm1. it follows from [19, theorem 7.9], that t0 − µm0 is a semi-fredholm for µ such that stability of some essential b-spectra 69 µ ∈ d ( 0, ε ′ ‖m0‖ ) \{0}, for some ε′ > 0. set ε = min ( ε′ ‖m0‖ ,γ ) . therefore, the operator t − µm is semi-fredholm for µ such that µ ∈ d(0,ε)\{0}. again, from [19, theorem 7.9], we get α(t − µm) = α(t0 − µm0) and β(t −µm) = β(t0 −µm0) are constants on d(0,ε)\{0}. using theorem 2.1, we can deduce the following result which is a generalization of [15, proposition 2.1] corollary 2.1. let t ∈ c(x) and m ∈ l(x) such that m(c(t)) = c(t) and ρm (t) 6= ∅. then, (i) ρbf+ (m,t), ρbf−(m,t) and ρbf (m,t) are open subsets of c; (ii) ind(λm−t) is constant on any component of ρbf+ (m,t), ρbf−(m,t) and ρbf (m,t). proof. (i) let λ0 ∈ ρbf+ (m,t), then from theorem 2.1 there exists an ε > 0 such that t − µm is an upper semi-fredholm operator, for each µ ∈ d(λ0,ε)\{0}. this implies that ρbf+ (m,t) is an open subset of c. the same proof is used to show that, ρbf−(m,t) and ρbf (m,t) are open subsets of c. (ii) let ω be a component of ρbf+ (m,t) (resp. ρbf−(m,t)), λ0 ∈ ω be a fixed point and λ1 ∈ ω be an arbitrary point that are connected by a polygonal line γ contained in ω. it follows from the assertion (i) of this corollary that, for each µ ∈ γ, there exists an open disc d(µ,ε), such that ind(µm − t) = ind(λm − t), for each λ ∈ d(µ,ε). by the heine-borel theorem, there exist a finite number of open discs that cover γ. this allows us to deduce that ind(λ0m −t) = ind(λ1m −t). now, we recall the following definition considered in [12] for bounded linear operators and it remains also true in the general case of closed linear operators: definition 2.2. let m ∈ l(x) and t ∈c(x) such that ρm (t) 6= ∅. we say that m and t commute in the sense of resolvent if for all λ ∈ ρm (t), m(t −λm)−1 = (t −λm)−1m . remarks 2.1. (a) if m and t commute in the sense of the resolvent, the assumption m(c(t)) ⊂ c(t) is verified. indeed, let x ∈ c(t). then, from [1, theorem 1.8], there exists a sequence (un) ⊂ d(t) such that x = u0 and tun+1 = un, for every n ∈ z+. set yn = mun. the commutativity 70 a. ben ali, m. boudhief, n. moalla of the resolvent of the operators m and t, permits us to deduce that m = (λm − t)−1m(λm − t) on d(t) and tm = mt on d(t), which entails that mun ∈ d(t). thus, we get y0 = mu0 = mx and tyn+1 = tmun+1 = mtun+1 = mun = yn, for every n ∈ z+. this, implies that mx ∈ m(c(t)) and finally we obtain that m(c(t)) ⊂ c(t). (b) if m and t commute in the sense of the resolvent and m is invertible, then we get m−1(c(t)) ⊂ c(t) and finally we can conclude that c(t) ⊂ m(c(t)). proposition 2.2. let m ∈ l(x) be an invertible operator and t ∈c(x) such that 0 ∈ ρm (t). if mt−1 = t−1m, then (i) for λ 6= 0 and n ≥ 1: (t −λm)n(t−1)n = (−λ)nmn ( t−1 −λ−1m−1 )n ; (ii) for all n ≥ 1: r ( (t −λm)n ) = r ( (t−1 −λ−1m−1)n ) . proof. (i) for n = 1, the equality is obvious. let n ≥ 1 and assume that (t −λm)n(t−1)n = (−λ)nmn(t−1 −λ−1m−1)n, then (t −λm)n+1(t−1)n+1 = (t −λm) [ (t −λm)n ( t−1 )n] t−1 = (t −λm) [ (−λ)nmn ( t−1 −λ−1m−1 )n] t−1 = (t −λm)t−1 [ (−λ)nmn ( t−1 −λ−1m−1 )n] = (−λ)n+1mn+1 ( t−1 −λ−1m−1 )n+1 . (ii) it follows from (i), that r [ mn(t−1 −λ−1m−1)n ] = r [ (t−1 −λ−1m−1)n ] ⊆ r [ (t −λm)n ] . conversely, if y ∈ r((t − λm)n), then there exists x ∈ d(tn) such that y = (t − λm)nx. the fact that (t−1)n(x) = d(tn), this enable us the existence of t ∈ x such that x = (t−1)n(t). hence, y = (t −λm)n ( t−1 )n (t) and finally y = (−λ)nmn ( t−1 −λ−1m−1 )n (t) ∈ r [ (t−1 −λ−1m−1)n ] . proposition 2.3. let m ∈ l(x) be an invertible operator and t ∈c(x) such that 0 ∈ ρm (t). if mt−1 = t−1m, then (i) for λ 6= 0 and n ≥ 1: (t−1)n(t −λm)n = (−λ)nmn ( t−1 −λ−1m−1 )n and tn(t−1 −λ−1m−1)n = (−λ−1)n(m−1)n(t −λm)n on d(tn); (ii) for all n ≥ 1: n((t −λm)n) = n ( (t−1 −λ−1m−1)n ) . stability of some essential b-spectra 71 proof. (i) since t is invertible and d(t −λm) = d(t), then r ( (t−1)n ) = d(tn) and r(tn) = x. therefore, the operators (t−1)n(t − λm)n and tn ( t−1 − λ−1m−1 )n are well defined. then, we verify directly that (t−1)n(t −λm)n = (−λ)nmn ( t−1 −λ−1m−1 )n and tn ( t−1 −λ−1m−1 )n = (−λ−1)n(m−1)n(t −λm)n on d(tn). (ii) it is a direct consequence of (i). remark 2.2. if m = i, we recover the results obtained in [10]. now, we state the main result of this section. theorem 2.2. let m ∈ l(x) be an invertible operator and t ∈ c(x) such that 0 ∈ ρm (t). if mt−1 = t−1m, then σ∗(m,t) = { λ−1 : λ ∈ σ∗ ( m−1,t−1 ) \{0} } , where σ∗(m,t) ∈ { σbf+ (m,t),σbf−(m,t),σbf (m,t), σd(m,t),σbw (m,t),σld(m,t),σrd(m,t) } . proof. let λ 6= 0. by using proposition 2.2 and proposition 2.3, we get that λm−1 − t−1 is a b-fredholm operator if and only if λ−1m − t is a b-fredholm one. the same arguments are used to prove the other b-spectra. in order to give, the stability result of some essential b-spectra of operator pencil by means of some class of commuting perturbations which generalizes some results established in [6], we shall define the following class of operators: we say that a linear operator is of finite-rank if its range is of finite dimension. if there exists an integer p ∈ n∗ such that dim r(tp) < ∞, then it is called a power finite-rank operator. we will denote by f(x) (resp. fp(x)) the set of all finite-rank linear bounded (resp. power finite-rank) operators. in many applications (see section 3) the perturbed operator is not of finite rank but we have some information about the difference of the resolvent, so the usual result. theorem 2.3. let m ∈ l(x) be an invertible operator and t1,t2 ∈ c(x) such that m commutes with t1 and t2 in the sense of resolvent. if for some λ ∈ ρm (t1)∩ρm (t2), the operator (t1−λm)−1−(t2−λm)−1 ∈f(x), 72 a. ben ali, m. boudhief, n. moalla then σ∗(m,t1) = σ∗(m,t2) , where, σ∗(m,.) ∈{σbf (m,.),σbf+ (m,.),σbf−(m,.),σbw (m,.)}. proof. without loss of generality, we can assume that λ = 0, then t−11 − t −1 2 ∈ f(x). let µ 6= 0. the use of theorem 2.2 shows that, µm − t1 is a b-fredholm operator if and only if µ−1m−1 − t−11 is a bfredholm one. since t−11 −t −1 2 ∈f(x), then by using [3, corollary 3.10], we get µ−1m−1 −t−12 is a b-fredholm operator. again by theorem 2.2, this is equivalent to µm−t2 is also a b-fredholm operator, which finish the proof of the b-fredholm spectrum equality. for the upper semi b-fredholm spectrum, the lower semi b-fredholm spectrum and the b-weyl spectrum, we use the same technique as above and [11, proposition 2.7]. definition 2.3. ([18]) let x be a banach space, a : d(a) ⊂ x → x and t : d(t) ⊂ x → x two linear operators. we say that a commutes with t , and we denote at = ta, if (i) d(a) ⊂ d(t); (ii) tx ∈ d(a) whenever x ∈ d(a); (iii) at = ta on {x ∈ d(a) : ax ∈ d(t)}. under the additional hypothesis of commutativity of operators, we get a stronger version of theorem 2.3: theorem 2.4. let m ∈ l(x) be an invertible operator and t1,t2 ∈ c(x) such that m commutes with t1 and t2 in the resolvent sense and t1t2 = t2t1. if for some λ ∈ ρm (t1) ∩ ρm (t2), the operator (t1 − λm)−1 − (t2 − λm)−1 ∈fp(x), then σ∗(m,t1) = σ∗(m,t2) , where σ∗(m,t) ∈ { σbf (m,.),σbf+ (m,.),σbf−(m,.), σbw (m,.),σld(m,.),σrd(m,.),σd(m,.) } . proof. without loss of generality, we can assume that λ = 0, then t−11 − t−12 ∈ fp(x). let µ 6= 0. it follows from theorem 2.2, that µm − t1 is stability of some essential b-spectra 73 a b-fredholm operator if and only if µ−1m−1 − t−11 is a b-fredholm one. since, t−11 −t −1 2 ∈fp(x), then from [16], we obtain that µ −1m−1 −t−12 is also a b-fredholm operator, which is equivalent to µm −t1 is a b-fredholm operator by theorem 2.2. this, shows that σbf (m,t1) = σbf (m,t2). for the other equalities, we use the same technique as above. corollary 2.2. let m ∈ l(x) be an invertible operator and t1,t2 ∈ c(x) such that m commutes with t1 and t2 in the resolvent sense and t1t2 = t2t1. if for some λ ∈ ρm (t1) ∩ ρm (t2), the operator (t1 − λm)−1 − (t2 − λm)−1 is nilpotent, then σ∗(m,t1) = σ∗(m,t2) , where σ∗() ∈ { σbf (m,t),σbf+ (m,t),σbf−(m,t), σbw (m,t),σld(m,t),σrd(m,t),σd(m,t) } . corollary 2.3. let t ∈c(x), m ∈ l(x) be an invertible operator and q ∈ l(x) a nilpotent operator such that ρm (t) 6= ∅. if tq = qt on d(t), mq = qm and m,t commute in the resolvent sense, then σ∗(m,t) = σ∗(m,t + q) , where σ∗() ∈ { σbf (m,t),σbf+ (m,t),σbf−(m,t), σbw (m,t),σld(m,t),σrd(m,t),σd(m,t) } . proof. since, tq = qt and q is a nilpotent operator, then (λm−t)−1q is nilpotent, for all λ ∈ ρm (t). thus, its spectral radius is equal to zero, which implies that λ ∈ ρm (t + q) and that (λm −t −q)−1 = (λm −t)−1(m − (λi −t)−1q)−1 = (λm −t)−1 n∑ k=0 ((λm −t)−1q)k = (λm −t)−1 + (λm −t)−1q n−1∑ k=1 ((λm −t)−1)kqk−1 74 a. ben ali, m. boudhief, n. moalla with n is the nilpotent-index of (λm−t)−1q. hence, (λm−t−q)−1−(λm− t)−1 is nilpotent. so, we deduce from corollary 2.2, that σ∗(m,t + q) = σ∗(m,t). the following proposition is proved in [12] for bounded linear operators and it holds also true in the general case of closed densely-defined linear operators. proposition 2.4. let t ∈ c(x) be densely-defined linear operator and m ∈ l(x) such that m(c(t)) = c(t). if λ ∈ σec(m,t) is non-isolated, then λ ∈ σek(m,t). remark 2.3. proposition 2.4 is also true if we replace σek(m,t) by σqf (m,t), where σqf (m,t) = {λ ∈ c : λm − t s not a quazi-fredholm operator}. the following theorem, shows the equality between the closed-range spectrum and some essential b-spectra of operator pencil acting on the banach space. theorem 2.5. let t ∈c(x) be densely-defined linear operator and m ∈ l(x) such that m(c(t)) = c(t). if σec(m,t) = σ(m,t) and every λ ∈ σec(m,t) is non-isolated. then σ(m,t) = σbf (m,t) = σbw (m,t) = σbf+ (m,t) = σbf−(m,t) = σd(m,t) . proof. since σbf (m,t) ⊂ σ(m,t), it suffices to show that σ(m,t) ⊂ σbf (m,t). let λ ∈ σ(m,t), then from proposition 2.4, we have λ ∈ σek(m,t). since, a b-fredholm operator is a quasi-fredholm one, this shows that σek(m,t) ⊂ σbf (m,t). the same arguments are used for the upper semi b-fredholm, the lower semi b-fredholm and the b-weyl spectrum. the fact that, a drazin invertible operator is a b-fredholm one, then by using the same arguments as above we can prove that, σ(m,t) = σd(m,t). finally, we conclude that σ(m,t) = σec(m,t) = σbf (m,t) = σbw (m,t) = σbf+ (m,t) = σbf−(m,t) = σd(m,t). 3. application in this section, we will use the previous results to treat the essential bspectra of a transport operator with abstract boundary conditions. let xp := lp ( (−a,a) × (−1, 1),dxdξ ) , a > 0 , 1 ≤ p < ∞ . stability of some essential b-spectra 75 we consider the following integro-differential operator with abstract boundary conditions: ah = th + k , where th is defined by  th : d(th) ⊆ xp −→ xp ψ 7−→ thψ(x,ξ) = −ξ ∂ψ ∂x (x,ξ) −σ(ξ)ψ(x,ξ) , d(th) = { ψ ∈wp : ψi = hψo } , where wp := { ϕ ∈ xp : ξ∂ϕ∂x ∈ xp } and σ(.) ∈ l∞(−1, 1); ψo,ψi are, respectively, the outgoing and the incoming fluxes related by the boundary operator h (“o” for the outgoing and “i” for the incoming), and given by  ψi(ξ) = ψ(−a,ξ) , ξ ∈ (0, 1) , ψi(ξ) = ψ(a,ξ) , ξ ∈ (−1, 0) , ψo(ξ) = ψ(−a,ξ) , ξ ∈ (−1, 0) , ψo(ξ) = ψ(a,ξ) , ξ ∈ (0, 1) . the bounded collision operator k is defined by  k : xp −→ xp u 7−→ k(u)(x,ξ) = ∫ 1 −1 k(x,ξ,ν)u(x,ν) dν , where the kernel k : (−a,a) × (−1, 1) × (−1, 1) −→ r is assumed to be measurable. the following boundary spaces denoted by xop and x i p are defined as follows: xop := lp [ {−a}× (−1, 0); |ξ|dξ ] ×lp [ {a}× (0, 1); |ξ|dξ ] := xo1,p ×x o 2,p equipped with the norm ∥∥u◦,x◦p∥∥ := (∥∥u◦1,x◦1,p∥∥p + ∥∥u◦2,x◦2,p∥∥p)1p = [∫ 0 −1 |u(−a,v)|p |v|dv + ∫ 1 0 |u(a,v)|p |v|dv ]1 p , 76 a. ben ali, m. boudhief, n. moalla and xip := lp [ {−a}× (0, 1); |ξ|dξ ] ×lp [ {a}× (−1, 0); |ξ|dξ ] := xi1,p ×x i 2,p equipped with the norm ∥∥ui,xip∥∥ := (∥∥ui1,xi1,p∥∥p + ∥∥ui2,xi2,p∥∥p)1p = [∫ 1 0 |u(−a,v)|p |v|dv + ∫ 0 −1 |u(a,v)|p |v|dv ]1 p . in this part, we will determine the essential b-spectra of the pair (m,ah), where m is the operator defined by{ m : xp −→ xp ϕ 7−→ m(ϕ)(x,ξ) = η(ξ)ϕ(x,ξ) , where η(.) ∈ l∞(−1, 1). to clarify our subsequent analysis, we define the following bounded operators introduced in [15]:  mλ : x i p −→ xop , mλu := ( m+λ u,m − λ u ) , with (m+λ u)(−a,ξ) := u(−a,ξ) e −2a |ξ| (λη(ξ)+σ(ξ)) , 0 < ξ < 1 , (m−λ u)(a,ξ) := u(a,ξ) e −2a |ξ| (λη(ξ)+σ(ξ)) , −1 < ξ < 0 ,   bλ : x i p −→ xp , bλu := χ(−1,0)(ξ)b − λ u + χ(0,1)(ξ)b + λ u, with (b−λ u)(x,ξ) := u(a,ξ) e −(λη(ξ)+σ(ξ))|ξ| |a−x| , −1 < ξ < 0 , (b+λ u)(x,ξ) := u(−a,ξ) e −(λη(ξ)+σ(ξ))|ξ| |a+x| , 0 < ξ < 1 ,   gλ : xp −→ xop , gλϕ := ( g+λ ϕ,g − λ ϕ ) , with g−λ ϕ(a,ξ) := 1 |ξ| ∫ a −a e −(λη(ξ)+σ(ξ)) |ξ| |a+x|ϕ(x,ξ) dx, −1 < ξ < 0 , g+λ ϕ(−a,ξ) := 1 |ξ| ∫ a −a e −(λη(ξ)+σ(ξ)) |ξ| |a−x|ϕ(x,ξ) dx, 0 < ξ < 1 , stability of some essential b-spectra 77 and finally  cλ : xp −→ xp , cλϕ = χ(−1,0)c − λ ϕ + χ(0,1)c + λ ϕ, with c−λ ϕ(x,ξ) := 1 |ξ| ∫ a x e −(λη(ξ)+σ(ξ)) |ξ| |x−x ′| ϕ(x′,ξ) dx′ , −1 < ξ < 0 , c+λ ϕ(x,ξ) := 1 |ξ| ∫ x −a e −(λη(ξ)+σ(ξ)) |ξ| |x−x ′| ϕ(x′,ξ) dx′ , 0 < ξ < 1 , where χ(0,1)(.) and χ(−1,0)(.) denote, respectively, the characteristic functions of the intervals (0, 1) and (−1, 0). note that, from [15, proposition 3.1], the operators mλ, bλ, gλ and cλ are bounded respectively by exp(−2aµ∗ re λ), (pµ∗ re λ)−1/p, (µ∗ re λ)−1/q and (µ∗ re λ)−1 where q is the conjugate of p. in what follows, we will show that the m-spectrum of t0 (i.e., th with h = 0) is the continuous spectrum σc(m,t0) of the pair (m,t0). lemma 3.1. (i) the point spectrum of the pair (m,t0) is empty. (ii) the residual spectrum of the pair (m,t0) is empty. proof. (i) we consider for λ ∈ c such that re(λ) ≤ 0, the eigenvalue problem (λm − t0)ψ = 0, where the unknown ψ must be in d(t0). his solution is formally given by ψ(x,ξ) = k(ξ)e −1 |ξ| [λη(ξ)+σ(ξ)]x. moreover, since ψ ∈ d(t0), then we get ψi = 0. so we obtain k(ξ) = 0 on (−1, 1). consequently, ψ = 0. (ii) to prove that the residual spectrum σr(m,t0) is also empty, we shall determine the point spectrum of the adjoint operator pencil densely defined λm −t0, where λ ∈ c. the adjoint operators t∗0 and m ∗ are, respectively, given by:  t∗0 : d(t ∗ 0 ) ⊆ xq −→ xq ψ 7−→ t∗0 ψ(x,ξ) = ξ ∂ψ ∂x (x,ξ) −σ(ξ)ψ(x,ξ) , d(t∗0 ) = { ψ ∈wq : ψo = 0 } , where q is the conjugate of p ( 1 p + 1 q = 1 ) ,{ m∗ : xq −→ xq ϕ −→ m∗(ϕ)(x,v) = η(v)ϕ(x,v) . 78 a. ben ali, m. boudhief, n. moalla let us consider the eigenvalue problem (λm∗ −t∗0 )ψ = 0. (3.1) in view of the boundary conditions, a straightforward estimation shows that, the problem (3.1) admits only the trivial solution. then, we obtain σp(m ∗,t∗0 ) = ∅. since, σr(m,t0) ⊆ σp(m ∗,t∗0 ), then we can easily obtain the desired result. now, by using the previous results, we can deduce the following theorem: theorem 3.1. let m ∈ l(x). then, σ(m,t0) = σc(m,t0) = σec(m,t0) = { λ ∈ c : re(λ) ≤ 0 } . proof. since, σ(m,t0) = σp(m,t0)∪σr(m,t0)∪σc(m,t0), then by using lemma 3.1, we deduce σ(m,t0) = σc(m,t0). on the other hand, it follows from [15, theorem 3.1], that σ(m,t0) = {λ ∈ c : re(λ) ≤ 0}. combining these two results, we can obtain the assertion of the present theorem. now, we are able to express the b-spectra of the pair (m,th): theorem 3.2. if the boundary operator h is of finite rank and commutes with m, and every point λ in σec(m,t0) is non-isolated, then σ∗(m,th) = σ∗(m,t0) = { λ ∈ c : re(λ) ≤ 0 } , where σ∗(m,.) ∈ { σbf (m,.),σbf+ (m,.),σbf−(m,.),σbw (m,.) } . proof. according to [15], we have (λm −th)−1 = h ∑ n≥0 bλ(mλh) ngλ + cλ, (3.2) where cλ = (λm −t0)−1. since (λm −th)−1 − (λm −t0)−1 ∈f(x), this implies by theorem 2.3 that σ∗(m,th) = σ∗(m,t0). the fact that m and th commute in the sense of the resolvent and m is invertible this allows us, by the use of remarks 2.1. theorem 2.5 and theorem 3.1 to conclude that σ∗(m,t0) = { λ ∈ c : re(λ) ≤ 0 } , where σ∗(m,.) ∈ { σbf (m,.),σbf+ (m,.),σbf−(m,.),σbw (m,.) } stability of some essential b-spectra 79 theorem 3.3. suppose that the boundary operator h and the collision operator k are of finite rank. if mh = hm and every point λ in σec(m,t0) is non-isolated, then we get σ∗(m,ah) = { λ ∈ c : re(λ) ≤ 0 } , where σ∗(m,.) ∈ { σbf (m,.),σbf+ (m,.),σbf−(m,.),σbw (m,.) } . proof. since, the collision operator k is finite rank, then it follows from theorem 2.3 and theorem 3.2, that σ∗(m,ah) = σ∗(m,th+k) = σ∗(m,th) = σ∗(m,t0), and finally we obtain that σ∗(m,ah) = { λ ∈ c : re(λ) ≤ 0 } , where σ∗(m,.) ∈ { σbf (m,.),σbf+ (m,.),σbf−(m,.),σbw (m,.)}. references [1] p. aiena, “ fredholm and local spectral theory ii, with applications to weyltype theorems ”, lecture notes in mathematics, 2235, springer, cham, 2018. 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[20] p. saphar, contribution à l’étude des applications linéaires dans un espace de banach, bull. soc. math. france 92 (1964), 363 – 384. introduction stability of some essential b-spectra of pencil operators application � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 1 (2019), 77 – 83 doi:10.17398/2605-5686.34.1.77 available online february 20, 2019 non-additive lie centralizer of strictly upper triangular matrices driss aiat hadj ahmed centre régional des metiers d’education et de formation (crmef) tangier, morocco ait hadj@yahoo.com received december 31, 2018 presented by consuelo mart́ınez accepted february 4, 2019 abstract: let f be a field of zero characteristic, let nn(f) denote the algebra of n×n strictly upper triangular matrices with entries in f, and let f : nn(f) → nn(f) be a non-additive lie centralizer of nn(f), that is, a map satisfying that f([x,y ]) = [f(x),y ] for all x,y ∈ nn(f). we prove that f(x) = λx + η (x) where λ ∈ f and η is a map from nn(f) into its center z (nn(f)) satisfying that η([x,y ]) = 0 for every x,y in nn(f). key words: lie centralizer, strictly upper triangular matrices, commuting map. ams subject class. (2010): 16s50, 15a27,16u80, 15b99, 47b47, 16r60. 1. introduction consider a ring r. an additive mapping t : r → r is called a left (respectively right) centralizer if t(ab) = t(a)b (respectively t(ab) = at(b)) for all a,b ∈ r. the map t is called a centralizer if it is a left and a right centralizer. the characterization of centralizers on algebras or rings has been a widely discussed subject in various areas of mathematics. in [13] zalar proved the following interesting result: if r is a 2-torsion free semiprime ring and t is an additive mapping such that t(a2) = t(a)a (or t(a2) = at(a)), then t is a centralizer. vukman [12] considered additive maps satisfying similar conditions, namely 2t(a2) = t(a)a + at(a) for any a ∈ r, and showed that if r is a 2-torsion free semiprime ring then t is also a centralizer. since then, the centralizers have been intensively investigated by many mathematicians (see, e.g., [3, 4, 5, 6, 8]). let r be a ring. an additive map f : r → r, is called a lie centralizer of r if f([x,y]) = [f(x),y] for all x,y ∈ r, (1.1) where [x,y] is the lie product of x and y. issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.1.77 mailto:ait_hadj@yahoo.com https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 78 driss aiat hadj ahmed recently, ghomanjani and bahmani [9] dealt with the structure of lie centralizers of trivial extension algebras, whereas fošner and jing [7] studied lie centralizers of triangular rings. the inspiration of this paper comes from the articles [1, 5, 7] in which the authors deal with the lie centralizer maps of triangular algebras and rings. in this note we will consider non-additive lie centralizers on strictly upper triangular matrices over a field of zero characteristic. throughout this article, f is a field of zero characteristic. let mn(f) and nn(f) denote the algebra of all n × n matrices and the algebra of all n × n strictly upper triangular matrices over f, respectively. we use diag(a1,a2, . . . ,an) to represent a diagonal matrix with diagonal (a1,a2, . . . ,an) where ai ∈ f. the set of all n × n diagonal matrices over f is denoted by dn(f). let in be the identity in mn(f), j = n−1∑ i=1 ei,i+1 and {eij : 1 ≤ i,j ≤ n} the canonical basis of mn(f), where eij is the matrix with 1 in the (i,j) position and zeros elsewhere. by cnn(f)(x) we will denote the centralizer of the element x in the ring nn(f). the notation f : nn(f) → nn(f) means a non-additive map satisfying f([x,y ]) = [f(x),y ] for all x,y ∈ nn(f. notice that it is easy to check that z (nn(f)) = fe1n. the main result in this paper is the following: theorem 1.1. let f be a field of zero characteristic. if f : nn(f) → nn(f) is a non-additive lie centralizer then there exists λ ∈ f and a map η : nn(f) → z (nn(f)) satisfying η([x,y ]) = 0 for every x,y in nn(f) such that f(x) = λx + η (x) for all x in nn(f). notice that the converse is trivially true: every map f(x) = λx + η (x) with η satisfying the condition in theorem 1.1 is a (non-additive) lie centralizer. 2. proofs let’s start with some basic properties of lie centralizers. lemma 2.1. let f be a non-additive lie centralizer of nn(f). then: (1) f(0) = 0; (2) for every x,y ∈ nn(f), we have f([x,y ]) = [x,f(y )]; non-additive lie centralizer 79 (3) f is a commuting map, i.e., f(x)x = xf(x) for all x ∈ nn(f). proof. to prove (1) it suffices to notice that f(0) = f([0, 0]) = [f(0), 0] = 0. (2) observe that if f([x,y ]) = [f(x),y ], then we have f(xy −y x) = f(x)y −y f(x). interchanging x and y in the above identity, we have f(y x −xy ) = f(y )x −xf(y ). replacing x with −x in the above relation, we arrive at f(xy − y x) = xf(y ) −f(y )x which can be written as f([x,y ]) = [x,f(y )]. from (1) one also gets (3): [f(x),x] = f([x,x]) = f(0) = 0. remark 2.1. let f be a non-additive lie centralizer of nn(f) and x ∈ cnn(f)(y ). then f(x) ∈ cnn(f)(y ). indeed, if x ∈ cnn(f)(y ), then [x,y ] = 0 and 0 = f(0) = f([x,y ]) = [f(x),y ]. lemma 2.2. let f be a non-additive lie centralizer of nn(f). then: (1) f ( n−1∑ i=1 aiei,i+1 ) = n−1∑ i=1 biei,i+1; (2) there exists λ ∈f such that f(j) = λj. proof. let d0 = n∑ i=1 (n− i) ei,i. (1) consider a ∈ mn(f). it is well known that [d0,a] = a if and only if a = n−1∑ i=1 aiei,i+1. hence, if a = n−1∑ i=1 aiei,i+1, we have [d0,a] = a. thus f ([d0,a]) = [d0,f (a)] = f (a). therefore f(a) = n−1∑ i=1 biei,i+1. 80 driss aiat hadj ahmed (2) as in (1), consider a = n−1∑ i=1 aiei,i+1 for some ai ∈f. then [j,a] = 0 if and only if a = aj for some a ∈f. indeed, f(j) = n−1∑ i=1 aiei,i+1 by (1). thus, 0 = f(0) = f([j,j]) = [j,f (j)]. hence, there exists λ ∈f such that f(j) = λj. we will need the following lemma. lemma 2.3. (lemma 2.1, [14]) suppose that f is an arbitrary field. if g,h ∈ utn(f) are such that gi,i+1 = hi,i+1 6= 0 for all 1 ≤ i ≤ n−1, then g and h are conjugated in utn(f). here utn(f) is the multiplicative group of n×n upper triangular matrices with only 1’s in the main diagonal. from the lemma above we obtain the following corollary. corollary 2.1. let f be a field. for every a = ∑ 1≤i i + 1, (b2)ij = { bi if j = i + 1, 0 otherwise, where bi is an element in f different from ai,i+1. it is easy to see that b1, b2 are in s, and a = b1 + b2, so we wanted. 82 driss aiat hadj ahmed lemma 2.6. let f be a field. for arbitrary elements a,b of nn(f), there exists λa,b ∈ f such that f(a + b) = f(a) + f(b) + λa,be1n. proof. for any a,b,x of nn(f), we have [f(a + b),x] = f ([a + b,x]) = [a + b,f(x)] = [a,f(x)] + [b,f(x)] = [f(a),x] + [f(b),x] = [f(a) + f(b),x], which implies that f(a + b) −f(a) −f(b) ∈z (nn(f)). thus, there exists λa,b ∈f such that f(a + b) = f(a) + f(b) + λa,be1n. now we can prove the main theorem. proof of theorem 1.1. let a, b ∈ s be two non-commuting elements. by lemma 2.4, f(a) = λaa, f(b) = λbb , λa,λb ∈f. since f is a non-additive lie centralizer, we get, f ([a,b]) = [f (a) ,b] = λa[a,b] = [a,f(b)] = λb[a,b]. then, [a,b] 6= 0 implies that λa = λb. if a, b ∈s commute, then we take c ∈s that does not commute neither with a nor with b. as we have just seen, λa = λc and λb = λc. so λa = λb = λ for arbitrary elements a,b ∈ s. given x ∈ nn(f) we know, by lemma 2.5, that there exists a,b ∈ s such that x = a + b (we can assume that x /∈ s). then f(x)−f(a)−f(b) ∈z (nn(f)) by lemma 2.6. that is f(x) − λaa − λbb = f(x) − λx ∈ z (nn(f)) for λ ∈ f such that f(a) = λa for each a ∈ s. we can define η : nn(f) → z (nn(f)) such that η (x) = f(x) − λx, that is, f(x) = λx + η (x). notice that η(a) = 0 for each a ∈ s. furthermore, if x,y ∈ nn(f), then f ([x,y ]) = λ [x,y ] + η ([x,y ]) = [f (x) ,y ] = [λx + η (x) ,y ] = λ [x,y ] , since η (x) ∈z (nn(f)). consequently, η ([x,y ]) = 0 and theorem 1.1 is proved. non-additive lie centralizer 83 acknowledgements the author would like to thank the referee for providing useful suggestions which served to improve this paper. references [1] j. bounds, commuting maps over the ring of strictly upper triangular matrices, linear algebra appl. 507 (2016), 132 – 136. [2] m. brešar, centralizing mappings on von neumann algebra, proc. amer. math. soc. 111 (1991), 501 – 510. [3] m. brešar, centralizing mappings and derivations in prime rings, j. algebra 156 (1993), 385 – 394. [4] m. brešar, commuting traces of biadditive mappings, commutativitypreserving mappings and lie mappings, trans. amer. math. soc. 335 (1993), 525 – 546. [5] w.-s. cheung, commuting maps of triangular algebras, j. london math. soc. (2) 63 (2001), 117 – 127. [6] d. eremita, commuting traces of upper triangular matrix rings, aequationes math. 91 (2017), 563 – 578. [7] a. fošner, w. jing, lie centralizers on triangular rings and nest algebras, adv. oper. theory 4 (2) (2019), 342 – 350. [8] w. franca, commuting maps on some subsets of matrices that are not closed under addition, linear algebra appl. 437 (2012), 388 – 391. [9] f. ghomanjani, m.a. bahmani, a note on lie centralizer maps, palest. j. math. 7 (2) (2018), 468 – 471. [10] t.k. lee, derivations and centralizing mappings in prime rings, taiwanese j. math. 1 (3) (1997), 333 – 342. [11] t.k. lee, t.c. lee, commuting additive mappings in semiprime rings, bull. inst. math. acad. sinica 24 (1996), 259 – 268. [12] j. vukman, an identity related to centralizers in semiprime rings, comment. math. univ. carolin. 40 (3) (1999), 447 – 456. [13] b. zalar, on centralizers of semiprime rings, comment. math. univ. carolin. 32 (4) (1991), 609 – 614. [14] r. slowik, expressing infinite matrices as products of involutions, linear algebra appl. 438 (2013), 399 – 404. [15] l. chen, j.h. zhang, nonlinear lie derivations on upper triangular matrices, linear multilinear algebra 56 (2008), 725 – 730. [16] d. aiat hadj ahmed, r. slowik, m-commuting maps of the rings of infinite triangular and strictly triangular matrices (in preparation). introduction proofs e extracta mathematicae vol. 33, núm. 2, 145 – 147 (2018) a note on a paper of s.g. kim diana m. serrano-rodŕıguez departamento de matemáticas, universidad nacional de colombia 111321 bogotá, colombia dmserrano0@gmail.com , diserranor@unal.edu.co presented by ricardo garćıa received august 20, 2018 abstract: we answer a question posed by s.g. kim in [3] and show that some of the results of his paper are immediate consequences of known results. key words: bohnenblust-hille inequality. ams subject class. (2000): 46g25, 30b50. the recent paper [3] deals with extreme multilinear forms and polynomials and the constants of the bohnenblust-hille inequalities. in this note we answer a question posed in [3] and show that two theorems stated in [3] are immediate corollaries of well known results of this field. let k be r or c. the multilinear bohnenblust-hille inequality asserts that, given a positive integer m, there is an optimal constant c (m : k) ≥ 1 such that   ∞∑ i1,...,im=1 ∣∣u(ei1 , . . . , eim)∣∣ 2mm+1   m+1 2m ≤ c (m : k) ∥u∥ , for all bounded m-linear forms u : c0 × · · · × c0 → k. the case of complex scalars was first investigated in [1] and the case of real scalars seems to have been just explored more recently. it is well known that the exponent 2m/ (m + 1) is sharp, so one of the main goals of the research in this field is to investigate the constants involved. the following result was proved in [4]: theorem 1. ([4, corollary 5.4], 2018) let m ≥ 2 be a positive integer. if the optimal constant c (m : r) is attained in a certain t : c0×· · ·×c0 → r, then the quantity of non zero monomials of t is bigger than 4m−1 − 1. as an immediate corollary we conclude that if n1, . . . , nm ≥ 1 are positive integers such that m∏ j=1 nj ≤ 4m−1 − 1 , 145 146 d.m. serrano-rodŕıguez then sup  n1,...,nm∑ i1,...,im=1 ∣∣t (ei1 , . . . , eim)∣∣ 2mm+1   m+1 2m < c (m : r) , where the sup runs over all norm one m-linear forms t : ℓn1∞ ×· · ·×ℓnm∞ → r. in particular, sup   2∑ i,j,k=1 |t (ei, ej, ek)| 6 4   4 6 < c (3 : r) , where the sup runs over all norm one m-linear forms t : ℓ2∞ × ℓ2∞ × ℓ2∞ → r, and this is the content of [3, theorem 4.9]. the polynomial bohnenblust-hille inequality for real scalars asserts that, given a positive integer m, there is an optimal constant cp (m : r) ≥ 1 such that ( ∑ |α|=m |aα| 2m m+1 )m+1 2m ≤ cp (m : r) ∥q∥ , for all n ≥ 1 and for all m-homogeneous polynomials q : ℓn∞ (r) → r given by q(z) = ∑ |α|=m aαz α. to the best of our knowledge, the case of real scalars became unexplored until the publication of the paper [2] in 2015, where it is proved that the constants cp (m : r) cannot be chosen with a sub-exponential growth. more precisely, theorem 2. ([2, theorem 2.2], 2015) cp (m : r) > ( 2 4 √ 3 √ 5 )m > (1.177) m , for all positive integers m ≥ 2. the above result is, obviously, by far, rather precise than [3, theorem 4.5], which states that cp (m : r) ≥ 2 m+1 2m , a note on a paper of s.g. kim 147 for all positive integers m ≥ 2. the only case that deserves a little bit more of attention is the case m = 2, since( 2 4 √ 3 √ 5 )2 < 2 3 4 , but in the case m = 2 a quick look at the proof of [2, proof of theorem 2.2] shows that cp (2 : r) ≥ 3 3 4 5 4 ≈ 1. 823 6 > 2 3 4 , and this answers in the negative the question (2) posed by the author in [3, question (2)]. references [1] h.f. bohnenblust, e. hille, on the absolute convergence of dirichlet series, ann. of math. (2) 32 (1931), 600 – 622. [2] j.r. campos, p. jiménez-rodŕıguez, g.a. muñoz-fernández, d. pellegrino, j.b. seoane-sepúlveda, on the real polynomial bohnenblust-hille inequality, linear algebra appl. 465 (2015), 391 – 400. [3] s.g. kim, the geometry of l ( 3ℓ2∞ ) and optimal constants in the bohnenblusthille inequality for multilinear forms and polynomials, extracta math. 33 (1) (2018), 51 – 66. [4] d. pellegrino, e. teixeira, towards sharp bohnenblust-hille constants, commun. contemp. math. 20 (2018), 1750029, 33 pp. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 36, num. 2 (2021), 279 – 288 doi:10.17398/2605-5686.36.2.279 available online november 3, 2021 characterization of symplectic forms induced by some tangent g-structures of higher order p.m. kouotchop wamba 1, g.f. wankap nono 2 1 department of mathematics, higher teacher training college university of yaoundé 1, po.box 47 yaoundé, cameroon 2 department of mathematics and computer science, faculty of science university of ngaoundéré, po.box 454 ngaoundéré, cameroon wambapm@yahoo.fr , georgywan@yahoo.fr received september 6, 2021 presented by j.c. marrero accepted september 30, 2021 abstract: let (m,ω) be a symplectic manifold induced by an integrable g-structure p on m. in this paper, we characterize the symplectic manifolds induced by the tangent lifts of higher order r ≥ 1 of g-structure p , from m to trm. key words: complete lifts of differential forms, prolongations of functions, vector fields and gstructures. msc (2020): 53c15; secondary 53c75, 53d05, 53d17. 1. introduction let m be a smooth manifold of dimension n ≥ 1. the tangent bundle of order r of m is the n(r + 1)-dimensional manifold trm of r-jets at 0 ∈ r of differential mapping % : r → m. we denote by πr,m : trm → m the canonical projection defined by πr,m (j r 0%) = %(0). let (u,x i) be a local coordinate system of m, we denote by (xi,xiα) the adapted local coordinate of t rm over the open set tru, we have:{ xi(jr0%) = x i(%(0)), xiβ(j r 0%) = 1 β! dβ dtβ (xi ◦%)(t) |t=0 . the differential geometry of the tangent bundles of higher order has been extensively studied by many authors, for instance i. kolar ([1]), morimoto ([3] and [4]). it plays an essential role in the description of a lagrangian formalism of higher order. on the other hand, we denote by fm the frame bundle of m, it is gl(n)-principal bundle, where gl(n) is a linear lie group of rn. let g be issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.36.2.279 mailto:wambapm@yahoo.fr mailto:georgywan@yahoo.fr https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 280 p.m.k. wamba, g.f. wankap nono a lie sub-group of linear group, we recall that, a g-structure on m is a sub-gprincipal bundle (p,m,pm ) of fm. we know that some structures of classical differential geometry can be described by some g-structures. for instance, the almost complex structures, the riemannian structures, the symplectic structures and regular foliations. that why, the study of tangent lifts of higher order of these structures has their importance in the calculus of variations and some problems connected as the hamiltonization problems of higher order. by the canonical linear action ρn : gl(n)×rn → rn, we define an injective morphism of lie groups j 〈r〉 n : t r(gl(n)) → gl(n(r + 1)) such that, for any manifold m, we have (see [3]) a natural principal bundle morphism j 〈r〉 m : trfm → ftrm over idtrm . the mapping is an embedding. let (p,m,pm ) be a g-structure, we set gr = j〈r〉n (t rg), t rp = j〈r〉m (t rp). the principal bundle (t rp,trm,gr) is a gr-structure on trm. it is called tangent g-structure of order r. in [3], it has showed that the tangent gstructure of order r is integrable if and only if the initial structure is integrable. in particular, this prolongation generalize simultaneously the tangent lifts of higher order of almost complex structures, riemannian structures, symplectic structures and regular foliations. in the particular case where r = 1, a. morimoto has shown the following result. theorem 1. ([2]) let g be a lie group generated by all elements u ∈ gl(n) invariant with respect to a bilinear form f on r2m. we denote by (t p,tm,t pm ) the tangent lift of integrable g-structure (p,m,pm ). let ωg be a symplectic form induced by p and ωg1 the symplectic form induced by (t p,tm,t pm ). we have: ωg1 = (ωg) (c) (1) where (ωg) (c) is the complete lift of ωg from m to tm. in the case where r ≥ 2, we do not have similar result. in this paper, we propose a generalization of this result. thus, this paper is structured as follows. in section 2, we recall some prolongations of differential forms from a manifold m to trm. in section 3, we prove the main result of this paper. characterization of symplectic forms 281 2. complete lift of differential forms in this section we recall briefly the main result of a. morimoto [4], about lifts of functions, vector fields and differential forms to the tangent bundle of higher order. these results will be used in the main section. let m be a smooth manifold of dimension m ≥ 1. let f be a function of class c∞ and α ∈{0, . . . ,r}, we define the α-lift f(α) as the function on trm given by the formula f(α)(jr0%) = 1 α! dα dtα (f ◦%)(t) |t=0 . if α is negative, then we set f(α) = 0. the family of α-lifts of functions is very important because, if x and y are vector fields such that x(f(α)) = y (f(α)) for all functions f on m and α = 0, . . . ,r, then x = y . proposition 1. let x ∈ x(m) and 0 ≤ α ≤ r. there is one and only one vector field x(α) on trm such that: x(α) ( f(β) ) = (xf)(β−α) for any f ∈ c∞(m) and 0 ≤ β ≤ r. proof. see [4]. the vector field x(α) is called α-prolongation of x. for some properties of x(α), see for instance [1] or [4]. let (u,xi) be a local coordinate system of m and (xi,xiβ) be a local coordinate system on t rm over tru induced by (u,xi) such that x = ai ∂ ∂xi , with ai ∈ c∞(u) we have: x(α) = (ai)(β−α) ∂ ∂xiβ . proposition 2. let ω be a differential form of degree p. it exists on trm one and only one differential form of degree p denoted by ω(c) verifying: ω(c) ( x (β1) 1 , . . . ,x (βp) p ) = ( ω(x1, . . . ,xp) )(r−(β1+···+βp)) (2) for all x1, . . . ,xp ∈ x(m) and β1, . . . ,βp ∈{0, . . . ,r}. proof. see [4]. 282 p.m.k. wamba, g.f. wankap nono definition 1. the differential form ω(c) is called complete lift of ω from m to trm. let { x1, . . . ,xn } be a local coordinates system of m such that, the local expression of ω is given by: ω = ωi1···ipdx i1 ∧·· ·∧dxip. the local expression of ω(c) is given by: ω(c) = ∑ β1+···+βp+β=r (ωi1···ip) (β)dxi1β1 ∧·· ·∧dx ip βp . (3) in the particular case where p = 2, i.e., ω = ωijdx i∧dxj, then the differential form ω(c) has the matrix form  (ωij) (r) · · · ωij ... .. . ... ωij · · · 0   . it is called matrix representation of the complete lift ω(c). in particular, if ω is non degenerate, then ω(c) is also non degenerate. corollary 1. let ω be a differential form of degree p on m. we have: d ( ω(c) ) = (dω)(c). (4) proof. see [4]. remark 1. this corollary shows that, if ω is closed (resp. exact) then ω(c) is closed (resp. exact). thus, if ω is a symplectic form on m then ω(c) is a symplectic form on trm. on the other hand, the method employed for the description of the complete lifts of differential forms can be extended to a symmetric tensor. in particular for a symmetric tensor g of type (0, 2) on m, if locally g = gijdxi ⊗dxj, then the complete lift g(c) is given by: g(c) = (gij) (r−α−β)dxiα ⊗dx j β. characterization of symplectic forms 283 3. the main result let f be a bilinear symmetric (resp. skew symmetric n = 2m) non degenerate form on rn. in [3], the author shows that: if g is a lie sub-group in i(f), where i(f) is the lie sub-group of all elements u ∈ gl(n) such that, for any x,y ∈ rn, f ( u(x),u(y) ) = f(x,y), then gr is the lie sub-group in the lie group i(f〈r〉), where τr : t r(r) → r is defined by τr(j r 0%) = 1 r! dr(%) dtr (t) |t=0 and the bilinear form f〈r〉 = τr ◦trf is symmetric (resp. skew symmetric). remark 2. let (p,m,pm ) be a g-structure, where g ⊂ i(f) as above, f is skew symmetric and non degenerate (n = 2m). let φ : m → p be a section, for any x ∈ m the map txm ×txm −→ r (ux,vx) 7−→ f ( φ(x)−1(ux),φ(x) −1(vx) ) is bilinear, skew symmetric and does not depend of φ. we denote it by ωg(x), in particular we obtain an almost symplectic form ωg on the manifold m. if (p,m,pm ) is integrable, then ωg is a symplectic form on m. it is called symplectic form induced by (p,m,pm ). theorem 2. let f and f〈r〉 be as above. let g be a lie subgroup of the lie group i(f). let (p,m,pm ) be an integrable g-structure on m (dim m = n = 2m). we denote by ωg and ωgr the symplectic forms induced by p and t rp respectively. we have: ωgr = (ωg) (c). (5) let φ ∈ γ(p), where γ(p) denote the space of smooth sections of p , we have: φ(x)(ej) = φ i j(x) ( ∂ ∂xi ) x . (6) by the same process defined in [3], we prolong φ from p to t rp and we obtain the section φ on t rp . the matrix form of φ is given by:  φij · · · 0 ... ... ... (φij) (r) · · · φij   . 284 p.m.k. wamba, g.f. wankap nono thus, we deduce that, for any i,j ∈{1, . . . ,n} and α,β ∈{0, . . . ,r}, we have: φi+nαj+nβ = ( φij )(α−β) . (7) lemma. we denote by (e`+nα)`,α the canonical basis of rn(r+1). using the matrix form of φ, we deduce that: φ(ej+nα) = ( φij )(α−β) ∂ ∂xiβ (8) for any j ∈{1, . . . ,n} and α ∈{0, . . . ,r}. proof. it comes from previous equation. proof of theorem 2. we denote by f : rn×rn → r the bilinear skew symmetric, non degenerate form and (aij)1≤i,j≤n its matrix form (n = 2m). the matrix form of the bilinear skew symmetric form f〈r〉 : rn(r+1) ×rn(r+1) → r is given by:   0 · · · aij... ... ... aij · · · 0   . we have, for any i,j ∈{1, . . . ,n} and α,β ∈{0, . . . ,r}, f〈r〉(ei+nα,ej+nβ) = δ α r−βaij (9) where δαr−β is the kronecker symbol. we denote by ωgr the symplectic form induced by an integrable gr-structure t rp . for any x ∈ trm and α,β ∈{0, . . . ,r}, f〈r〉(ei+nα,ej+nβ) = = f〈r〉 ( φ(x)−1 (( φki )(γ−α) ∂ ∂xkγ ) , φ(x)−1 (( φsj )(µ−β) ∂ ∂xsµ )) = ωgr (( φki )(γ−α) ∂ ∂xkγ , ( φsj )(µ−β) ∂ ∂xsµ ) = ( φki )(γ−α) ·(φsj)(µ−β)ωgr( ∂∂xkγ , ∂∂xsµ ) . characterization of symplectic forms 285 we deduce that: ( φki )(γ−α) ·(φsj)(µ−β)ωgr( ∂∂xkγ , ∂∂xsµ ) = δαr−βaij. (10) we set ωgr ( ∂ ∂xkγ , ∂ ∂xsµ ) = $k+nγ,s+nµ. (11) in this case the equation 10 becomes: ( φki )(γ−α) ·(φsj)(µ−β)$k+nγ,s+nµ = δαr−βaij. (12) fact 1: for α = r. if β = r, then the equation 12 becomes φki ·φ s j$k+nr,s+nr = 0. (13) we deduce that $k+nr,s+nr = 0. (14) let β 6= 0, we suppose that, for any µ > β , $k+nr,s+nµ = 0. we have: φki · ( φsj )(µ−β) $k+nr,s+nµ = = φki ·φ s j$k+nr,s+nβ + r∑ µ=β+1 φki · ( φsj )(µ−β) $k+nr,s+nµ = φki ·φ s j$k+nr,s+nβ = 0. we deduce that: $k+nr,s+nβ = 0 for any β 6= 0. for β = 0, we have: r∑ µ=0 φki · ( φsj )(γ) $k+nr,s+nγ = φ k i ·φ s j$k+nr,s + r∑ µ=1 φki · ( φsj )(γ) $k+nr,s+nγ = φki ·φ s j$k+nr,s = aij. now φki ·φ s j$k,s = aij, we deduce that: $k+nr,s = $k,s. (15) 286 p.m.k. wamba, g.f. wankap nono fact 2: if α = r − 1. by a similar calculus we obtain: for any β ∈ {1, . . . ,r}, we have: { $k+n(r−1),s+nβ = 0 if β ≥ 2, $k+n(r−1),s+nβ = ωks if β = 1. for β = 0, in the equation 12, we have:( φki )(γ−α) ·(φsj)(µ)$k+nγ,s+nµ = = ( φki )(1) ·(φsj)(µ)$k+nr,s+nµ + φki ·(φsj)(µ)$k+n(r−1),s+nµ = ( φki )(1) ·φsj$k+nr,s + φki ·(φsj)(1)$k+n(r−1),s+n + φki ·φsj$k+n(r−1),s = ( φki )(1) ·φsj$k,s + φki ·(φsj)(1)$k,s + φki ·φsj$k+n(r−1),s = −φki ·φ s j($k,s) (1) + φki ·φ s j$k+n(r−1),s = 0. we deduce that: $k+n(r−1),s = ($k,s) (1). (16) fact 3: we conjecture that $k+nα,s+nβ = ($k,s) (r−α−β). by induction, we fix α and we suppose that for any γ > α and λ > β, we have: $k+nγ,s+nλ = ($k,s) (r−γ−λ). (17) thus,( φki )(γ−α) ·(φsj)(µ−β)$k+nγ,s+nµ = φki ·φ s j$k+nα,s+nβ + ∑ γ,λ>α,β ( φki )(γ−α) ·(φsj)(λ−β)$k+nγ,s+nλ = φki ·φ s j$k+nα,s+nβ + ∑ γ,λ>α,β ( φki )(γ−α) ·(φsj)(λ−β)($k,s)(r−γ−λ) = φki ·φ s j$k+nα,s+nβ + ( φki ·φ s j$k,s )(r−α−β) −φki ·φsj($k,s)(r−α−β) = φki ·φ s j$k+nα,s+nβ + δ α r−βaij −φ k i ·φ s j($k,s) (r−α−β). using the equation 10, we deduce that: φki ·φ s j$k+nα,s+nβ = φ k i ·φ s j($k,s) (r−α−β). (18) characterization of symplectic forms 287 therefore $k+nα,s+nβ = ($k,s) (r−α−β). thus, we deduce that ωgr = $i+nα,j+nβdx i α ∧dx j β = ($k,s) (r−α−β)dxiα ∧dx j β. thus ωgr = (ωg) (c). intrinsic second proof of theorem 2. let (p,m,pm ) be an integrable g-structure as above. because of the integrability of p and the naturality, we may assume m = rn and p = rn × g ⊂ rn × gl(n) = lrn. then trn = rn × rn and ωg = idrn ×f : trn ×rn trn = rn × (rn × rn) → r. more trrn = rn(r+1), t rp = trrn ×gr ⊂ trrn ×gl(trrn) = l(trrn) and t(trrn) = trrn × trrn and ωgr = idtrrn × f〈r〉 : trrn × (trrn × trrn) = t(trrn) ×trrn t(trrn) → r, where f〈r〉 + τr ◦ trf : trrn × trrn → r and τr : trr → r is the respective functional. further, (ωg)(c) + τr ◦ tr(ωg) modulo the exchange isomorphism t(trm) = tr(tm) and the product preserving identification tr(tm×m tm) = tr(tm)×trm tr(tm). then (ωg) (c) = τr ◦tr ( idrn ×f ) = idtrrn × ( τr ◦trf ) = idtrrn ×f〈r〉 = ωgr. the proof is complete. corollary. let g be a lie group generated by all elements of linear group invariant with respect to some bilinear symmetric non degenerate form f. let (p,m,pm ) be a g-structure on m. we denote by gg and ggr the pseudo riemannian metric on m and trm induced by p and t rp respectively. we have: ggr = (gg) (c). (19) proof. the proof is similar to the proof of theorem 2. acknowledgements the authors would like to thank the anonymous reviewers for their valuable suggestions and remarks which improved the quality of this paper. 288 p.m.k. wamba, g.f. wankap nono references [1] i. kolar, p. michor, j. slovak, “natural operations in differential geometry”, springer-verlag, berlin 1993. [2] a. morimoto, prolongations of g-structures to tangent bundles, nagoya math. j. 32 (1968), 67 – 108. [3] a. morimoto, prolongations of g-structures to tangent bundles of higher order, nagoya math. j. 38 (1970), 153 – 179. [4] a. morimoto, liftings of some types of tensors fields and connections to tangent bundles of pr-velocities, nagoya math. j. 40 (1970), 13 – 31. introduction complete lift of differential forms the main result � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 2 (2019), 237 – 254 doi:10.17398/2605-5686.34.2.237 available online october 2, 2019 the µ-topological hausdorff dimension hela lotfi department of mathematics, faculty of sciences of monastir 5000-monastir, tunisia helalotfi@hotmail.fr received may 13, 2019 presented by m. mbekhta accepted june 18, 2019 abstract: in 2015, r. balkaa, z. buczolich and m. elekes introduced the topological hausdorff dimension which is a combination of the definitions of the topological dimension and the hausdorff dimension. in our manuscript, we propose to generalize the topological hausdorff dimension by combining the definitions of the topological dimension and the µ-hausdorff dimension and we call it the µ-topological hausdorff dimension. we will present upper and lower bounds for the µ-topological hausdorff dimension of the sierpiński carpet valid in a general framework. as an application, we give a large class of measures µ, where the µ-topological hausdorff dimension of the sierpiński carpet coincides with the lower and upper bounds. key words: hausdorff dimension, topological hausdorff dimension. ams subject class. (2010): 28a78, 28a80. 1. introduction different notions of dimensions have been introduced since the appearance of the hausdorff dimension by f. hausdorff in 1918 (see e.g. [10], [16] and [15]), such as the topological dimension, (see e.g. [4] and [7]). in 1975 when mandelbrot coined the word fractal (see [13]). he did so to denote an object whose hausdorff dimension was strictly greater than its topological dimension, but he abandoned this definition later, (see e.g. [6], [13] and [14]). in the euclidean space rn, there has been no generally accepted definition of a fractal, even though fractal sets have been widely used as models for many physical phenomena (see e.g. [9],[11] and [12]). the idea behind these models is that of self-similarity (see e.g. [5] and [12]). then billingsley defined the hausdorff measure in a probability space (see e.g. [2] and [3]). in [1], r. balka, z. buczolich and m. elekes introduced a new dimension concept for metric spaces, called the topological hausdorff dimension. it was defined by a very natural combination of the definitions of the topological dimension and the hausdorff dimension. the value of the topological hausdorff dimension was issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.2.237 mailto:helalotfi@hotmail.fr https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 238 hela lotfi always between the topological dimension and the hausdorff dimension. in particular, this dimension was a non-trivial lower estimate for the hausdorff dimension. p. billingsley introduced a dimension defined by a measure µ, (see e.g. [2] and [3]) called the µ-hausdorff dimension, which was a generalization of the hausdorff dimension. in the same vein, we propose to generalize the topological hausdorff dimension by combining the definitions of the topological dimension and the µ-hausdorff dimension, so-called the µ-topological hausdorff dimension. the paper is organized as follows. in section 2, we recall the topological hausdorff dimension. afterwards, we state the basic properties of this dimension and we cite some examples. in section 3, we introduce the µ-topological hausdorff dimension. finally in section 4, we give an estimation of the µtopological hausdorff dimension of the sierpiński carpet and then we provide a class of measures µ for which we compute the later dimension. 2. topological hausdorff dimension let (x,d) be a metric space. we denote by a the closure of subset a and ∂a its boundary in the metric space x. if b ⊆ a then ∂ab designates the boundary of subset b in the metric space a with an induced topology. let b(x,r) = { y ∈ x : d(x,y) < r } be the open ball of radius r centered at point x. for a bounded subset u of x we denote the diameter by: |u| = sup { d(x,y) : x,y ∈ u } . for two metric spaces (x,d) and (y,d′), a function f : x → y is lipschitz if there exists constant λ ∈ r+ such that d′ ( f(x),f(y) ) 6 λ d(x,y), for all x,y ∈ x. we begin by recalling the definition of the hausdorff dimension. definition 2.1. let a be a subset of a separable metric space x, and α be a positive number. for any ε > 0, we define: hαε (a) = inf  ∑ j |uj|α : a ⊂ ⋃ j uj, |uj| < ε   . we also define: hα(a) = lim ε→0+ hαε (a). the µ-topological hausdorff dimension 239 then the hausdorff dimension of a is given as follows: dimh(a) = inf{α > 0 : hα(a) < +∞}. the topological hausdorff dimension of a non-empty separable metric space x, introduced in [1], is defined as: dimth(x) = inf { d : x has basis u such that dimh(∂u) 6 d− 1 for every u ∈u } , with convention dimh(∅) = −1. in [1], the authors proved that dimt(x) 6 dimth(x) 6 dimh(x) where dimt denotes the topological dimension of a non-empty separable metric space defined by: dimt(x) = inf { d : x has basis u such that dimt(∂u) 6 d− 1 for every u ∈u } where convention dimt(∅) = −1, (see [4] or [7]). moreover, the authors gave an alternative recursive definition of the topological dimension as follows: dimt(x) = min { d : there is a ⊆ x such that dimt(a) 6 d− 1 and dimt(x\a) 6 0 } . notice that balka et al. defined the topological hausdorff dimension of a subset a of x by considering a as a metric space and equipping it with a dimension induced from that of x. next, we recall some properties of the topological hausdorff dimension of x given in [1]. proposition 2.2. let x be a separable metric space. (i) if a ⊆ b ⊆ x, then dimth(a) 6 dimth(b). (ii) if x = ⋃ n∈n xn, where xn (n ∈ n) are closed subsets of x, then: dimth(x) = sup n∈n dimth(xn). 240 hela lotfi (iii) let y be a separable metric space and f : x → y be a lipschitz homeomorphism, so: dimth(y ) 6 dimth(x). particularly, if f is bi-lipschitz, then dimth(y ) = dimth(x). example 2.3. (a) let x = r, so we have: dimt(r) = dimth(r) = dimh(r) = 1. in addition, we have: dimth(q) = dimth(r\q) = 0. it is noticeable that: dimth(r) 6= sup (dimth(q), dimth(r\q)) . indeed, q is not a closed set of r. (b) let x = r2. let d be the von koch snowflake curve. then: dimt(d) = dimth(d) = 1 < dimh(d) = ln 4 ln 3 . let s be the sierpiński triangle. thereby: dimt(s) = dimth(s) = 1 < dimh(s) = ln 3 ln 2 . let t be the sierpiński carpet. thus: dimt(t) = 1 < dimth(t) = ln 6 ln 3 < dimh(t) = ln 8 ln 3 . remark. the topological hausdorff dimension is not a topological notion. indeed, the following property was established in [1]: dimth(x × [0, 1]) = 1 + dimh(x). (2.1) the µ-topological hausdorff dimension 241 based on property (2.1), we can build two homeomorphic spaces such that their topological hausdorff dimensions are different as follows: consider x,y ⊆ [0, 1] two cantor sets such that: dimh(x) 6= dimh(y ). since these two sets are totally discontinuous then x and y are homeomorphic to the middle-thirds cantor set (see [8]). therefore, there exists homeomorphism ϕ : x → y , so: x × [0, 1] −→ y × [0, 1] (x,t) 7−→ (ϕ(x), t) is a homeomorphism. thus, using property (2.1), we obtain: dimth(x × [0, 1]) 6= dimth(y × [0, 1]). 3. µ-topological hausdorff dimension in the following, we propose to generalize the topological hausdorff dimension. we begin by recalling the dimension defined by billingsley in [2] and [3]. let x be a metric space, f a countable set of subsets of x, and µ a non-negative function defined on f and satisfying the following property: for each x ∈ x and ε > 0, there is u ∈f such that x ∈ u and µ(u) < ε. (3.1) let a be a non-empty subset of x and α be a positive number. for any ε > 0, we define: hαµ,ε(a) = inf  ∑ j µ(uj) α : a ⊂ ⋃ j uj, uj ∈f, µ(uj) < ε   (3.2) with convention 0α = 0. as ε decreases, the class of permissible covers of a in (3.2) is reduced. then the infimum hαµ,ε(a) increases, and so approaches a limit as ε → 0. we write the following: hαµ(a) = lim ε→0 hαµ,ε(a). therefore, the µ-hausdorff dimension of a non-empty subset a of x relative to f, as defined by billingsley, is given by: dimµ(a) = inf { α > 0 : hαµ(a) < +∞ } . 242 hela lotfi it is noted that the µ-hausdorff dimension and the hausdorff dimension have the same properties. similar to the definition of the topological hausdorff dimension, we introduce the µ-topological hausdorff dimension of x relative to f. definition 3.1. let x be a metric space, f a countable set of subsets of x, and µ a non-negative function defined on f and satisfying (3.1). then the µ-topological hausdorff dimension of x relative to f is given by: dimtµ(x) = inf { α : x has basis u such that, for every u ∈u, dimµ(∂u) 6 α− 1 } with convention dimµ(∅) = −1. notice that the µ-topological hausdorff dimension of a subset a of x is defined by considering a as a metric space and equipping it with a dimension induced from that of x. moreover, the µ-topological hausdorff dimension is monotonous in the sense of inclusion. remark. we find the topological hausdorff dimension in the following particular case: let q+\{0} be the set of all positive rational numbers. if x is a separable metric space, f = {b(xn,r) : n ∈ n,r ∈ q+\{0}} where {xn : n ∈ n} is dense in x, and µ is the function such that µ ( b(xn,r) ) = 2r. then: dimtµ(x) = dimth(x). (3.3) indeed, let ε > 0 and x ∈ x. we choose r ∈ q+\{0} where r < ε2 . then there exists n ∈ n such that d(xn,x) < r. hence, x ∈ b(xn,r) and µ ( b(xn,r) ) = 2r < ε. as a consequence, µ satisfies (3.1). now, to find (3.3), it must be showed that for all subset a of x: dimµ(a) = dimh(a). firstly, it is clear to see that: dimµ(a) > dimh(a). for the second inequality, let α > 0 and ε > 0. consider {uj}j, a ε-cover of a. for all j ∈ n we pick λj ∈ q+\{0} such that: |uj|α < λαj < |uj| α + ε 2j . the µ-topological hausdorff dimension 243 let yj ∈ uj, then there exists nj ∈ n such that d(yj,xnj ) < λj. whence: uj ⊂b(xnj, 2λj). since a ⊂ ⋃ j uj, a ⊂ ⋃ j b(xnj, 2λj). thus: hα µ,4(εα+ε) 1 α (a) 6 ∑ j µ ( b(xnj, 2λj) )α = 4α ∑ j λαj < 4α ∑ j |uj|α + 2 · 4αε. as a result, hα µ,4(εα+ε) 1 α (a) 6 4αhαε (a) + 2 ·4αε. when ε approaches to zero, we obtain hαµ(a) 6 4αhα(a). finally, dimµ(a) 6 dimh(a). 4. calculating µ-topological hausdorff dimension of sierpiński carpet in this section, we give an estimation of the µ-topological hausdorff dimension of the sierpiński carpet t. let x = r2, f = ⋃ n>1 fn where fn is the triadic squares set of the n-th generation, and µ is a non-negative function defined on f and satisfying (3.1). let us recall that a triadic square of the n-th generation is defined by: c = i ×j ⊆ r2, where i and j are two triadic intervals of the n-th generation. 4.1. lower bound of µ-topological hausdorff dimension of sierpiński carpet t. now we establish a lower estimation of the µtopological hausdorff dimension of the sierpiński carpet t. for µ, we associate functions w1 and w2, defined on the set of triadic intervals i in r by: if i is a triadic interval of the n-th generation, contained in [0, 1[, then: w1(i) = inf j µ(i ×j) and w2(i) = inf j µ(j × i) (4.1) 244 hela lotfi where the lower bound is taken on all the triadic intervals j of the n-th generation contained in [0, 1[. else: w1(i) = w2(i) = 0. seeing that µ satisfies (3.1), then the w1(respectively w2)-hausdorff dimension is well-defined. indeed, it must be proved that w1 and w2 satisfy (3.1). it is clear that w1 satisfy (3.1) when x 6∈ [0, 1[. let x ∈ [0, 1[ and ε > 0. as µ satisfies (3.1), then for all y ∈ [0, 1[ there exists c ∈ f such that (x,y) ∈ c = i ×j and µ(c) < ε. therefore, x ∈ i and w1(i) < ε. consequently, w1 satisfies (3.1), and similarly we prove that w2 satisfies (3.1). thereby, we have the following result. theorem 4.1. we have dimtµ(t) > 1 + sup ( dimw1 (k), dimw2 (k) ) where k is the middle-thirds cantor set. proof. we will establish that dimtµ(t) > 1 + dimw1 (k). the other inequality dimtµ(t) > 1 + dimw2 (k) can be proved in a similar way. for this purpose, we need the following intermediate result. lemma 4.2. let s < dimw1 (k). then there exists xs ∈ k satisfying dimw1 ( ]xs −r,xs + r[∩k ) > s for each r > 0. (4.2) proof of lemma. assume, on the contrary, that for all x ∈ k, there exists rx > 0 such that dimw1 (]x−rx,x + rx[∩k) 6 s. it is clear to see that k ⊂ ⋃ x∈k ]x−rx,x + rx[. as k is compact and according to the compactness of subsets, we have k ⊂ p⋃ i=1 ]xi−rxi,xi + rxi[. then the middle-thirds cantor set can be written as: k = p⋃ i=1 (]xi −rxi,xi + rxi[∩k) . hence: dimw1 (k) = sup 16i6p dimw1 (]xi −rxi,xi + rxi[∩k) 6 s. this contradicts the fact that dimw1 (k) > s. the µ-topological hausdorff dimension 245 now, we return to the proof of theorem 4.1. for a fixed s < dimw1 (k), from lemma 4.2, there exists xs ∈ k such that for all r > 0 we have dimw1 ( ]xs −r,xs + r[∩k ) > s. let u be an open basis of t. since k × [0, 1] ⊂ t and as we remark that: dimtµ(x) = 1 + inf u sup u∈u dimµ(∂u) (4.3) where the lower bound is taken on all basis u of x, then it sufficient to demonstrate that there exists u ∈ u such that dimµ(∂tu ∩ k × [0, 1]) > s, where ∂t is the boundary in the sierpiński carpet t. 1. first case: xs ∈ k\{0}. the point (xs, 1) ∈ ] 0, 7 6 [ × ] 0, 7 6 [ ∩t = ⋃ i ui = u with ui ∈u. consider i such that ui contains point (xs, 1). note u instead of ui. we put ys = inf {y 6 1 : (xs,y) ∈ u}, then (xs,ys) ∈ ∂tu ∩k × [0, 1]. on the other hand, there exists rs > 0 such that: ]xs −rs,xs + rs[×]1 −rs, 1 + rs[∩t ⊂ u and xs −rs > 0. therefore: ]xs −rs,xs + rs[∩k ⊂ p (∂tu ∩k × [0, 1]) , with p : r2 → r, (x,y) 7→ x. fix ε > 0 and let ⋃ i ci be a covering of ∂tu ∩ k × [0, 1] by triadic squares, where ci = ii × ji while satisfying µ(ci) < ε. it follows that⋃ i ii is a covering of ]xs −rs,xs + rs[∩k satisfying w1(ii) < ε. thus:∑ i µ (ci) s > ∑ i w1(ii) s > hsw1,ε (]xs −rs,xs + rs[∩k) . as a consequence: hsµ,ε (∂tu ∩k × [0, 1]) > h s w1,ε (]xs −rs,xs + rs[∩k) . accordingly, when ε approaches to zero we obtain: hsµ (∂tu ∩k × [0, 1]) > h s w1 (]xs −rs,xs + rs[∩k) . based on dimw1 (]xs −rs,xs + rs[∩k) > s, we have: dimµ (∂tu ∩k × [0, 1]) > s. 246 hela lotfi 2. second case: xs = 0. the point (0, 0) ∈ ] −1 6 , 1 [ × ] −1 6 , 1 [ ∩ t = ⋃ i ui = u with ui ∈ u. consider i such that ui contains point (0, 0). note u instead of ui. next, we put ys = sup{y > 0 : (0,y) ∈ u}, then (0,ys) ∈ ∂tu ∩ k × [0, 1]. moreover, there exists rs > 0 such that: ] −rs,rs[× ]ys −rs,ys + rs[∩t ⊂ u. hence, ] −rs,rs[∩k ⊂ p (∂tu ∩ k × [0, 1]) with p : r2 → r, (x,y) 7→ x. indeed, if t = 0 ∈ ] − rs,rs[∩k, then according to the above reasoning, we have (0, 0) ∈ u, and there exists y0 ∈ [0, 1] such that: (0,y0) ∈ ∂tu ∩k × [0, 1] where y0 = sup{y > 0 : (0,y) ∈ u}. therefore: 0 = p (0,y0) ∈ p (∂tu ∩k × [0, 1]) . furthermore, if t ∈ ] − rs,rs[∩k and t 6= 0, i.e. t ∈ ]0,rs[∩k, then according to the above reasoning, we have (t, 0) ∈ u, and there exists y1 ∈ [0, 1] such that (t,y1) ∈ ∂tu ∩k × [0, 1], where y1 = sup{y > 0 : (t,y) ∈ u} , so t ∈ p (∂tu ∩k × [0, 1]). let ε > 0 and let ⋃ i ci be a covering of ∂tu ∩ k × [0, 1] by triadic squares where ci = ii × ji, while satisfying µ(ci) < ε. it follows that⋃ i ii is a covering of ] −rs,rs[∩k satisfying w1(ii) < ε. thus:∑ i µ (ci) s > ∑ i w1(ii) s > hsw1,ε (] −rs,rs[∩k) . hence: hsµ,ε (∂tu ∩k × [0, 1]) > h s w1,ε (] −rs,rs[∩k) . then when ε approaches to zero, and since dimw1 (] −rs,rs[∩k) > s, we have: dimµ (∂tu ∩k × [0, 1]) > s. the µ-topological hausdorff dimension 247 4.2. upper bound of µ-topological hausdorff dimension of sierpiński carpet t. now we establish an upper bound of the µ-topological hausdorff dimension of t. in the following, we assume that µ satisfies this condition: for each (x,y) ∈ r2, ε > 0, and n ∈ n\{0}, there exists p > n and c ∈fp verifiying (x,y) ∈ c with µ(c) < ε. (4.4) it is easy to see that µ satisfies (3.1). we can observe that a finite, non-atomic and borelian measure µ, defined on r2, satisfies (4.4). in what follows, we will essentially consider triadic squares contained in [0, 1[2. these squares are usually coded as follows: let an be the n-fold cartesian product of a = {0, 1, 2} and a∗ = ⋃ n>1 an. to concatenate two words a and b in a∗, we put b at the end of a. the resulting word is denoted by ab. for an element a in a∗ we denote by |a| the length of a where |a| = n, such that a ∈an. let i = i1i2 . . . in ∈an, so we associate a triadic interval as follows: ii = [ n∑ k=1 ik 3k , n∑ k=1 ik 3k + 1 3n [ ⊂ [ 0, 1 [ . for all i,j ∈an, we consider a triadic square of fn defined by: ci,j = ii × ij ⊂ [0, 1[2. for all i,j ∈ a∗, such that |i| = |j|, we consider the following functions ν1i,j and ν 2 i,j associated to µ and defined on the set of triadic intervals i in r as follows: if i 6⊂ [0, 1[, then ν1i,j(i) = ν 2 i,j(i) = 0, and for all i = ik ⊂ [0, 1[, where k ∈an: ν1i,j(ik) = µ(ci1,jk) ν2i,j(ik) = µ(cik,j1) (4.5) with 1 = 11 . . . 1 ∈an. we observe that ν1i,j and ν 2 i,j satisfy (3.1). indeed, let i,j ∈a ∗, which are written as i = i1 . . . in0 and j = j1 . . .jn0 , so case x 6∈ [0, 1[ is trivial. we fix x ∈ [0, 1[, and we consider the triadic development of x given by: x = +∞∑ i=1 xi 3i where x ∈ ix1...xn for all n ∈ n ∗. 248 hela lotfi put: x′ = n0∑ k=1 ik 3k + 1 2 · 3n0 ∈ [0, 1[ and y′ = n0∑ k=1 jk 3k + ∞∑ k=n0+1 xk−n0 3k ∈ [0, 1[. let ε > 0, seeing that µ satisfies (4.4), then there exists a positive integer p > n0 such that (x ′,y′) ∈ ci1,jx1...xp−n0 and µ(ci1,jx1...xp−n0 ) < ε. thus, ν1i,j(ix1...xp−n0 ) = µ(ci1,jx1...xp−n0 ) < ε. therefore, ν1i,j satisfies (3.1) since x ∈ ix1...xp−n0 . in the same way, we prove that ν 2 i,j satisfies (3.1). consequently, the ν1i,j (respectively ν 2 i,j)-hausdorff dimension is welldefined. hence, we have the following result. theorem 4.3. given a function µ satisfying (4.4) and vanishing on the triadic squares that are not contained in [0, 1[2, we have: dimtµ(t) 6 1 + lim inf n→+∞ sup |i| = |j| = n l = 1, 2 dimνli,j (k) where k is the middle-thirds cantor set. proof. for n ∈ n\{0} and u,v ∈ z, let (zun,zvn) be the center of a triadic square [ u 3n , u+1 3n [ × [ v 3n , v+1 3n [ from fn. denote by hn the set of intervals which are written as ] zun,z u+2 n [ where u ∈ z. put un = { i ×j : i,j ∈hn } , clearly u = ⋃ n>1 un is a countable open basis of r2. to establish theorem 4.3, considering the fact ∂t (u ∩t) ⊂ ∂u ∩t for all u ∈ r2, where ∂t is the boundary in the sierpiński carpet t and taking into account (4.3), it suffices to show that for all u ∈up we have: dimµ(t ∩∂u) 6 inf n>p sup |i| = |j| = n l = 1, 2 dimνli,j (k). (4.6) let, for u ∈up, ∂u be the union of four cloisters. we choose c as one of these cloisters. we first treat the case where c is a vertical cloister. let n > p and ci,j be a triadic square of the n-th generation such that ci,j ∩ t ∩ c 6= ∅. we also choose {ik}k, a ε-cover of k ∩ [0, 1[ by triadic intervals, i.e. k ∩ [0, 1[⊂ ⋃ k ik with ν 1 i,j(ik) < ε for each k. it follows that (see figure 1) ci,j ∩ t ∩c ⊂ ⋃ k ci1,jk, where 1 = 11 . . . 1 and |1| = |k|. the µ-topological hausdorff dimension 249 figure 1: sierpiński carpet square u ∈u2. triadic square ci,j of third generation such that ci,j ∩∂u ∩t 6= ∅. triadic squares ci1,j0 and ci1,j2. moreover, µ ( ci1,jk ) = ν1i,j(ik) < ε for all k with |k| = |1|. for α > 0, we have: hαµ,ε(ci,j ∩t ∩c) 6 ∑ k ( µ(ci1,jk) )α = ∑ k ( ν1i,j(ik) )α therefore: hαµ,ε(ci,j ∩t ∩c) 6 h α ν1i,j,ε (k ∩ [0, 1[), when ε goes to zero, we obtain: hαµ(ci,j ∩t ∩c) 6 h α ν1i,j (k ∩ [0, 1[). subsequently, dimµ(ci,j ∩t ∩c) 6 dimν1i,j (k ∩ [0, 1[) 6 dimν1i,j (k). clearly, taking account of convention dimµ(∅) = −1, the previous inequality is still valid if ci,j ∩t ∩c = ∅. 250 hela lotfi on the other hand: t ∩ c∩ [0, 1[2 ⊂ ⋃ |i|=|j|=n ci,j thus: t ∩ c∩ [0, 1[2 = ⋃ |i|=|j|=n ci,j ∩t ∩c then we obtain: dimµ(t ∩c∩ [0, 1[2) = sup |i|=|j|=n dimµ(ci,j ∩t ∩c) 6 sup |i|=|j|=n dimν1i,j (k). since µ vanishes on the triadic squares that are not contained in [0, 1[2, then: dimµ(t ∩c) = dimµ(t ∩c∩ [0, 1[2). thus: dimµ(t ∩c) 6 sup |i|=|j|=n dimν1i,j (k). as a consequence: dimµ(t ∩c) 6 inf n>p sup |i|=|j|=n dimν1i,j (k). if c is a horizontal cloister, we analogously obtain: dimµ(t ∩c) 6 inf n>p sup |i|=|j|=n dimν2i,j (k). finally, we have: dimµ(t ∩∂u) 6 inf n>p sup |i| = |j| = n l = 1, 2 dimνli,j (k). 4.3. equality case. let us recall that we have proved in theorem 4.1 and theorem 4.3 the following inequalities: 1 + sup ( dimw1 (k), dimw2 (k) ) 6 dimtµ(t) 6 1 + lim inf n→+∞ sup |i| = |j| = n l = 1, 2 dimνli,j (k). (4.7) the µ-topological hausdorff dimension 251 in this section, we give an example of measure µ, where the equality holds between the upper and lower bounds of the µ-topological hausdorff dimension of t . for this purpose, let ( pi,j ) i,j∈a be a square matrix of order 3 such that for each i,j ∈a, pi,j > 0 and ∑ 06i,j62 pi,j = 1. we consider the bernoulli measure supported on [0, 1[2 and defined by: µ(ci,j) = n∏ k=1 pik,jk where i = i1i2 . . . in and j = j1j2 . . .jn. choose δ and β as two positive real numbers such that: pδ1,0 + p δ 1,2 = 1 and p β 0,1 + p β 2,1 = 1. theorem 4.4. (equality case) if matrix (pi,j) satisfies p0,1 6 min(p0,0,p0,2), p2,1 6 min(p2,0,p2,2), p1,0 6 min(p0,0,p2,0), p1,2 6 min(p0,2,p2,2), (4.8) then dimtµ(t) = 1 + sup(β,δ). remark. a class of matrices satisfying (4.8) is: a =   1−4a 5 a 1−4a 5 a 1−4a 5 a 1−4a 5 a 1−4a 5   , where 0 < a 6 1 9 . note that this class of matrices contains lebesgue measure (case when a = 1 9 .) proof. the proof of theorem 4.4 is split into three steps: step 1. we begin by proving that for all i,j ∈a∗ such that |i| = |j|, we have: dimν1i,j (k) = δ and dimν2i,j (k) = β. let k̃ be the middle-thirds cantor set deprived of extremities of triadic intervals. therefore, we obtain: dimν1i,j (k) = dimν1i,j (k̃). 252 hela lotfi let i = i1i2 . . . iq, j = j1j2 . . .jq ∈ a∗. it is observed that if ik1k2...kn is a triadic interval crossing k̃, then for all l ∈ {1, 2, . . .n}, we have kl ∈{0, 2} and ( ν1i,j(ik1k2...kn) )δ = ξδ pδ1,k1 p δ 1,k2 . . .pδ1,kn, where ξ = q∏ k=1 pik,jk > 0. thus, function (ν 1 i,j) δ behaves as a measure. then for all disjoint covering {is}s of k̃ by triadic intervals we have:∑ s ( ν1i,j(is) )δ = ξδ whence, dimν1i,j (k) = δ. similarly, we prove that dimν2i,j (k) = β. step 2. now we verify that dimw1 (k) = α and dimw2 (k) = ρ. put for all k ∈{0, 2}: lk = min 06j62 pk,j and tk = min 06j62 pj,k. choose α and ρ as two positive real numbers such that: lα0 + l α 2 = 1 and t ρ 0 + t ρ 2 = 1. by step 1, we have: dimw1 (k) = dimw1 (k̃). it is remarkable that if ii1i2...in is a triadic interval that crosses k̃, then for all k ∈{1, 2, . . .n}, we have ik ∈{0, 2}. therefore: (w1(ii1i2...in)) α = lαi1 l α i2 . . . lαin. hence, wα1 behaves as a measure. consequently, if {is}s is a covering of k̃ by disjoint triadic intervals, we have:∑ s (w1(is)) α = 1. it results that hαw1 (k̃) = 1. thus, dimw1 (k̃) = α, and then: dimw1 (k) = α. analogously, we prove that dimw2 (k) = ρ. the µ-topological hausdorff dimension 253 step 3. finally, from conditions (4.8), we obtain: l0 = min 06j62 p0,j = p0,1 l2 = min 06j62 p2,j = p2,1 and t0 = min 06j62 pj,0 = p1,0 t2 = min 06j62 pj,2 = p1,2 by hypothesis, we have: lα0 + l α 2 = 1 p β 0,1 + p β 2,1 = 1 and t ρ 0 + t ρ 2 = 1 pδ1,0 + p δ 1,2 = 1 then: α = β and ρ = δ. thus, based on (4.7), we obtain: 1 + sup(α,ρ) 6 dimtµ(t) 6 1 + sup(δ,β). hence, the result yields. corollary 4.5. dimth(t) = ln 6 ln 3 . proof. matrix (pi,j)i,j∈a, where pi,j = 1 9 for all i,j ∈ a satisfies (4.8). then by theorem 4.4, and seeing that α = ρ = ln 2 2 ln 3 , we have: dimtµ(t) = 1 + ln 2 2 ln 3 . let us recall that triadic squares allow the calculation of the hausdorff dimension (see [16]). it is noticeable that if we choose µ(c) = 1 2 |c|2, where c is a triadic square in [0, 1[2, we have for all a ⊂ [0, 1[2: dimh(a) = 2 dimµ(a). as a result: dimth(t) = 2 dimtµ(t) − 1. this achieves the proof of this corollary. 254 hela lotfi references [1] r. balka, z. buczolich, m. elekes, a new fractal dimension: the topological hausdorff dimension, adv. math. 274 (2015), 881 – 927. 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[15] j. peyrière, calculs de dimensions de hausdorff, duke math. j. 44 (1977), 591 – 601. [16] j. peyrière, mesures singulières associées à des découpages aléatoires d’un hypercube, colloq. math. 51 (1987), 267 – 276. introduction topological hausdorff dimension -topological hausdorff dimension calculating -topological hausdorff dimension of sierpinski carpet lower bound of -topological hausdorff dimension of sierpinski carpet t. upper bound of -topological hausdorff dimension of sierpinski carpet t. equality case. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 1 (2019), 29 – 40 doi:10.17398/2605-5686.34.1.29 available online february 3, 2019 browder essential approximate pseudospectrum and defect pseudospectrum on a banach space aymen ammar, aref jeribi, kamel mahfoudhi department of mathematics, faculty of sciences of sfax, university of sfax route de soukra km 3.5, b.p. 1171, 3000 sfax, tunisia ammar aymen84@yahoo.fr , aref.jeribi@fss.rnu.tn , kamelmahfoudhi72@yahoo.com received january 18, 2018 presented by pietro aiena accepted july 5, 2018 abstract: in this paper, we introduce and study the browder essential approximate pseudospectrum and the browder essential defect pseudospectrum of bounded linear operators on a banach space. moreover, we characterize these spectra and will give some results concerning the stability of them under suitable perturbations. key words: pseudospectrum, browder essential approximate pseudospectrum, browder essential defect pseudospectrum. ams subject class. (2010): 39b82, 44b20, 46c05. 1. introduction let x be an infinite-dimensional banach space, let l(x) be the set of all bounded linear operators acting on x, and let k(x) be its ideal of compact operators on x. let t ∈ l(x). then d(t), n(t), α(t), r(t), β(t), t ′ and σ(t) are, respectively, used to denote the domain, the kernel, the nullity, the range, the defect, the adjoint and the spectrum of t. if the range r(t) is closed and α(t) < ∞ (resp. β(t) < ∞) then t is said to be an upper semifredholm operator (resp. a lower semi-fredholm operator). the set of upper semi-fredholm operators (resp. lower semi-fredholm operators) is denoted by φ+(x) (resp. φ−(x)). the set of all semi-fredholm operators is defined by φ±(x) := φ+(x) ∪ φ−(x), and the class φ(x) of all fredholm operators is defined by φ(x) := φ+(x) ∩ φ−(x). issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.1.29 mailto:ammar_aymen84@yahoo.fr mailto:aref.jeribi@fss.rnu.tn mailto:kamelmahfoudhi72@yahoo.com https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 30 a. ammar, a. jeribi, k. mahfoudhi the index of a semi-fredholm operator t is defined by i(t) = α(t) −β(t). an operator f ∈l(x) is called a fredholm perturbation if t + f ∈ φ(x) whenever t ∈ φ(x). the set of fredholm perturbations is denoted by f(x). an operator f ∈l(x) is called an upper semi-fredholm perturbation (resp. a lower semi-fredholm perturbation) if t + f ∈ φ+(x) (resp. t + f ∈ φ−(x)) whenever t ∈ φ+(x) (resp. t ∈ φ−(x)). the set of upper semi-fredholm perturbations (resp. lower semi-fredholm perturbations) is denoted by f+(x) (resp. f−(x)). now, we define the minimum modulus m(t) := inf { ‖tx‖ : x ∈d(x), ‖x‖ = 1 } , and the defect modulus q(t) := sup { r > 0 : rbx ⊂ tbx } , where bx is the closed unit ball of x. for more information see [16] and [20]. note that m(t) > 0 if and only if t is bounded below, i.e. t is injective and t has closed range and q(t) > 0 if and only if t is surjective. recall also that m(t∗) = q(t) and q(t∗) = m(t). the ascent (resp. descent) of t ∈l(x) is the smallest nonnegative integer a := asc(t) (resp. d := desc(t)) such that n(ta) = n(ta+1) (resp. r(td) = r(td+1)). if such an integer does not exist, then asc(t) = ∞ (resp. desc(t) = ∞). we also introduce some special parts of pseudospectrum having valuable spectral properties such as σap(t) := { λ ∈ c : m(λ−t) = 0 } , σδ(t) := { λ ∈ c : q(λ−t) = 0 } . the spectrum σap(t) (resp. σδ(t)) is called the approximate spectrum (resp. defect spectrum). the browder essential spectrum of t is defined as σeb(t) := ⋂ kt (x) σ(t + k), (1.1) the browder essential approximate point spectrum of t is defined as σeab(t) := ⋂ kt (x) σap(t + k), (1.2) browder essential approximate pseudospectrum 31 and the browder essential defect spectrum of t is defined as σeδb(t) := ⋂ kt (x) σδ(t + k), (1.3) where kt (x) := { k ∈ k(x) : tk = kt } . for more information on the browder essential approximate spectrum and his essential defect spectrum one may refer to [1, 9, 17, 18]. it is clear that σeb(t) = σeab(t) ∪σeδb(t). the pseudospectrum of bounded linear operators t on a banach space x can be split into subsets in many different ways, depending on the purpose one has in mind. we may refer to [2, 3, 5, 7, 13] as examples. definition 1.1. let t ∈l(x) and ε > 0. we define the following sets: (i) the pseudospectrum σε(t) = ⋃ dt (x) σ(t + d), (ii) the approximate pseudospectrum σap,ε(t) = ⋃ dt (x) σap(t + d), (iii) the defect pseudospectrum σδ,ε(t) = ⋃ dt (x) σδ(t + d), where dt (x) = { d ∈l(x) : ‖d‖ < ε, td = dt } . in this paper we study some parts of the pseudospectrum of bounded linear operators on a banach space from the viewpoint of fredholm theory. in particular, we study the browder essential approximate pseudospectrum and the browder essential defect pseudospectrum. we have already mentioned that (1.1), (1.2) and (1.3) inherit ε-versions, which are the browder essential 32 a. ammar, a. jeribi, k. mahfoudhi pseudospectrum σeb,ε(·), the browder essential approximate pseudospectrum σeab,ε(·) and the browder essential defect pseudospectrum σeδb,ε(·) defined by σeb,ε(t) = ⋂ kt (x) σε(t + k), σeab,ε(t) = ⋂ kt (x) σap,ε(t + k), σeδb,ε(t) = ⋂ kt (x) σδ,ε(t + k). this paper is divided into three sections. in the second one we recall some facts which are helpful to prove the main results. throughout the third section we characterize browder essential approximate pseudospectrum and the browder essential defect pseudospectrum. finally, we prove the invariance of the browder essential approximate pseudospectrum and his essential defect pseudospectrum and establish some results of perturbation on the context of linear operators on a banach space. 2. auxiliary results in order to prove our main results we begin by introducing some well known perturbation results lemma 2.1. ([16, theorem 9]) let t,k ∈l(x). we have (i) if t ∈ φ+(x) and k ∈k(x) then t +k ∈ φ+(x) and i(t +k) = i(t). (ii) if t ∈ φ−(x) and k ∈k(x) then t +k ∈ φ−(x) and i(t +k) = i(t). the following result was proved in [11]. lemma 2.2. let t ∈ l(x) and k ∈ k(x) such that k commutes with t. we have (i) if t ∈ φ+(x) then asc(t) < ∞ if, and only if, asc(t + k) < ∞. (ii) if t ∈ φ−(x) then desc(t) < ∞ if, and only if, desc(t + k) < ∞. a bounded operator r ∈l(x) on a banach space x is said to be a riesz operator if λ−t ∈ φ(x) for every λ ∈ c\{0}. the class of all riesz operators is denoted by r(x). browder essential approximate pseudospectrum 33 lemma 2.3. ([15, theorem 3.5]) let r ∈r(x) which commutes with t. we have (i) if t ∈ φ+(x) then asc(t) < ∞ if and only if asc(t + r) < ∞. (ii) if t ∈ φ−(x) then desc(t) < ∞ if and only if desc(t + r) < ∞. lemma 2.4. ([14, theorem 3.9]) let t ∈ φ+(x). the following statements are equivalent: (i) i(t) ≤ 0. (ii) t can be expressed in the form t = s + k where k ∈ k(x) and s ∈l(x) is an operator with closed range and α(s) = 0. 3. main results in this section we establish an useful result for the browder essential approximate pseudospectrum and the browder essential defect pseudospectrum. we start our characterization with the following theorem: theorem 3.1. let t ∈l(x) and ε > 0. then (i) λ /∈ σeab,ε(t) if and only if, for all d ∈ l(x) such that ‖d‖ < ε, we have λ−t −d ∈ φ+(x), i(λ−t −d) ≤ 0 and asc(λ−t −d) < ∞. (ii) λ /∈ σeδb,ε(t) if and only if, for all d ∈ l(x) such that ‖d‖ < ε, we have λ−t −d ∈ φ−(x), i(λ−t −d) ≥ 0 and desc(λ−t −d) < ∞. proof. (i) let λ /∈ σeap,ε(t). then there exists a compact operator k on x such that tk = kt and λ /∈ σap,ε(t +k). according to the definition 1.1, we obtain that λ /∈ σap(t + d + k) for all d ∈l(x) such that ‖d‖ < ε and d commutes with t + k. therefore, λ−t−d−k ∈ φ+(x), i(λ−t−d−k) ≤ 0 and asc(λ−t−d−k) = 0 for all d ∈l(x) such that ‖d‖ < ε. since k commutes with λ−t −d−k, from lemma 2.2 we obtain asc(λ−t−d) < ∞. using lemma 2.1, we deduce that λ−t −d ∈ φ+(x) and i(λ−t −d) ≤ 0. 34 a. ammar, a. jeribi, k. mahfoudhi to prove the converse, suppose that for all d ∈ l(x) such that ‖d‖ < ε we have λ−t −d ∈ φ+(x), i(λ−t −d) ≤ 0 and asc(λ−t −d) < ∞. there are two possible cases: 1stcase : if λ /∈ σap,ε(t) then λ /∈ σap,ε(t + k), so the proof is completed. 2ndcase : if λ ∈ σap,ε(t) then from [16, theorem 10] we infer that the space x is decomposed into a direct sum of two closed subspaces x0 and x1 such that dim x0 < ∞, (λ − t − d)(xi) ⊆ xi for i ∈ {1, 2}, (λ − t − d)\x0 is nilpotent operator and (λ − t − d)\x1 is injective operator. let k be the finite rank operator defined by{ k = i on x0, k = 0 on x1. it is clear that k is a compact operator commuting with t and d such that λ−t −d −k is an injective operator (i.e. α(λ−t −d −k) = 0). then, from lemma 2.4 there exists a constant c > 0 such that ‖(λ−t −d −k)x‖≥ c‖x‖, for all x ∈d(t). this proves that infx∈x, ‖x‖=1 ‖(λ − t − d − k)x‖ ≥ c > 0. thus λ /∈ σap(t +d+k). moreover, (t +d)k = k(t +d) and by using definition 1.1 we infer that λ /∈ σap,ε(t + k). hence λ /∈ σeab,ε(t). (ii) reasoning in the same way as (i), it suffices to replace φ+(·), σeab,ε(·), σap,ε(·), i(·) ≤ 0 and (λ − t − d)\x1 , which is injective, by φ−(·), σeδb,ε(·), σδ,ε(·), i(·) ≥ 0 and (λ−t −d)\x1 , which is surjective, respectively. remark 3.1. it follows immediately from theorem 3.1 (i) that σeab,ε(t) = ⋃ ‖d‖<ε σeab(t + d). moreover, it follows from theorem 3.1 (ii) that σeδb,ε(t) = ⋃ ‖d‖<ε σeδb(t + d). browder essential approximate pseudospectrum 35 next, the browder essential approximate pseudospectrum and the browder essential defect pseudospectrum will be characterized by means of semifredholm perturbation. we set f+t (x) = { f ∈f+(x) : tf = ft } , and f−t (x) = { f ∈f−(x) : tf = ft } . theorem 3.2. let t ∈l(x) and ε > 0. then (i) σeab,ε(t) = ⋂ f∈f+ t (x) σap,ε(t + f). (ii) σeδb,ε(t) = ⋂ f∈f− t (x) σδ,ε(t + f). proof. (i) for the first inclusion, it is clear that kt (x) ⊂f+t (x). then,⋂ f∈f+ t (x) σap,ε(t + f) ⊂ ⋂ f∈kt (x) σap,ε(t + f) := σeab,ε(t). for the second inclusion, let λ /∈ ⋂ f∈f+ t (x) σap,ε(t + f), then there exists f ∈f+(x) such that tf = ft and λ /∈ σap,ε(t + f). using definition 1.1, we have λ /∈ σap(t + f + d) for all d ∈ l(x) such that ‖d‖ < ε and d commutes with t and f . hence, λ−t−d−f ∈ φ+(x), i(λ−t−d−f) ≤ 0 and asc(λ−t−d−f) = 0. since f commutes with λ−t −d −f, from lemma 2.3, it follows that asc(λ−t −d) < ∞ and from lemma 2.1 we deduce that λ−t −d ∈ φ+(x) and i(λ−t −d) ≤ 0. hence λ /∈ σeab,ε(t). (ii) the proof is similar to that of the first part. 36 a. ammar, a. jeribi, k. mahfoudhi if we set rt (x) = { r ∈r(x) : tr = rt } , then theorem 3.2 remains true if f+t (x) and f − t (x) are replaced by rt (x). we then have σeab,ε(t) = ⋂ rt (x) σap,ε(t + r) and σeδb,ε(t) = ⋂ rt (x) σδ,ε(t + r). definition 3.1. an operator t ∈ l(x) is said to be quasi-compact operator (t ∈ qk(x)) if there exists a compact operator k and an integer m such that ‖tm −k‖ < 1. if t ∈l(x), we define the set qkt (x) = { k ∈qk(x) : tk = kt } we invite the reader to [6] for more information about the quasi-compactness operators. we have the following inclusions kt (x) ⊂rt (x) ⊂qkt (x). if t ∈l(x) we define the sets sεt (x) = { k ∈l(x) : k commutes with t + d and (λ−t −d −k)−1k ∈qkt (x) for all d ∈l(x) such that ‖d‖ < ε and λ ∈ ρ(t + d + k) } , and lεt (x) = { k ∈l(x) : k commutes with t + d and k(λ−t −d −k)−1 ∈qkt (x) for all d ∈l(x) such that ‖d‖ < ε and λ ∈ ρ(t + d + k) } . theorem 3.3. let t ∈l(x) with nonempty resolvent set. then, σeab,ε(t) = ⋂ k∈ sε t (x) σap,ε(t + k). browder essential approximate pseudospectrum 37 proof. let λ /∈ ⋂ k∈sε t (x) σap,ε(t + k), then there exists k ∈s ε t (x) such that for every ‖d‖ < ε and λ ∈ ρ(t + d + k), we have (λ−t −d −k)−1k ∈qkt (x) and λ /∈ σap,ε(t + k). using [6, theorem 1.6] we obtain that i + (λ−t −d −k)−1k ∈ φ(x) and i ( i + (λ−t −d −k)−1k ) = 0. since we can write λ−t −d = (λ−t −d −k) ( i + (λ−t −d −k)−1k ) . according to definition 1.1, we have for all d ∈l(x) such that ‖d‖ < ε, (t + k)d = d(t + k) and λ /∈ σap(t + d + k). we conclude for all d ∈ l(x) such that ‖d‖ < ε that λ−t −d ∈ φ+(x). also, we have i(λ−t −d) = i(λ−t −d −k) ≤ 0. it remains to show that asc(λ − t − d) < 0 for all d ∈ l(x) such that ‖d‖ < ε. let k commutes with t + d, then k commutes with λ−t −d for every λ ∈ c. then (λ−t −d)n = (λ−t −d −k)n ( i + (λ−t −d −k)−1k )n = ( i + (λ−t −d −k)−1k )n (λ−t −d −k)n for every n ∈ n. use the fact that (λ−t −d)n is injective ( i.e., 0 belongs to n((λ−t −d)n) ) , this implies that λ−t −d is injective ( n(λ−t −d) ⊂ n((λ − t − d)n) for every n ) . consequently, the ascent of λ − t − d is 0. then asc(λ − t − d) < ∞. this prove that λ /∈ σeab,ε(t). the opposite inclusion follows from kt (x) ⊆sεt (x). then⋂ k∈ sε t (x) σap,ε(t + k) ⊆ ⋂ k∈kt (x) σap,ε(t + k). corollary 3.1. let t ∈l(x) with nonempty resolvent set. then, σeab,ε(t) = ⋂ k∈ lε t (x) σap,ε(t + k). 38 a. ammar, a. jeribi, k. mahfoudhi proposition 3.1. let t ∈l(x) with nonempty resolvent set. then, σeδ,ε(t) = ⋂ k∈ sε t (x) σδ,ε(t + k) = ⋂ k∈ lε t (x) σδ,ε(t + k). remark 3.2. let t ∈l(x) and ε > 0. (i) let ut (x), (resp. vt (x)) be a subset of l(x). if kt (x) ⊂ ut (x) ⊂ sεt (x), (resp. kt (x) ⊂vt (x) ⊂l ε t (x) ) then σeab,ε(t) = ⋂ k∈ut (x) σap,ε(t + k) = ⋂ k∈vt (x) σap,ε(t + k). ( resp. σeδ,ε(t) = ⋂ k∈ut (x) σδ,ε(t + k) = ⋂ k∈vt (x) σδ,ε(t + k) ) . (ii) if for all j,j2 ∈ ut (x) (resp. vt (x)) we have j ± j2 ∈ ut (x) (resp. vt (x)) then for each j ∈ut (x) (resp. vt (x)) we have σeab,ε(t + j) = σeab,ε(t) and σeδ,ε(t + j) = σeδ,ε(t). in the next theorem we will give a fine characterization of σeab,ε(·) and σeδ,ε(·) by means of t + d-bounded perturbations. definition 3.2. an operator t ∈l(x) is called t-bounded if there exist c > 0 such that ‖bx‖≤ c(‖x‖ + ‖tx‖) for all x ∈d(t) ⊂d(b). we define for all t ∈l(x) the set hεt (x) = { k ∈sεt (x) : k is (t + d)-bounded } . theorem 3.4. let t ∈l(x) and ε > 0. then, σeab,ε(t) = ⋂ k∈hε t (x) σap,ε(t + k). proof. because kt (x) ⊆hεt (x), then⋂ k∈hε t (x) σap,ε(t + k) ⊆ ⋂ k∈kt (x) σap,ε(t + k) := σeab,ε(t). browder essential approximate pseudospectrum 39 conversely, let λ /∈ ⋂ k∈hε t (x) σap,ε(t + k), then there exists k ∈ h ε t (x) such that λ /∈ σap,ε(t + k), which means that for all d ∈l(x) such that ‖d‖ < ε we have λ−t −d−k is injective. using [6, theorem 1.6] we obtain that i + (λ−t −d −k)−1k ∈ φ(x) and i ( i + (λ−t −d −k)−1k ) = 0. we can write λ−t −d = (λ−t −d −k) ( i + (λ−t −d −k)−1k ) . the proof of our statement is then obtained by using the same argument of the proof of theorem 3.3. by using analogous arguments to those of the proof of theorem 3.4 we obtain: theorem 3.5. let t ∈l(x) and ε > 0. then, σeδ,ε(t) = ⋂ k∈hε t (x) σδ,ε(t + k). acknowledgements the authors wish to warmly thank the referee for all his suggestions. references [1] p. aiena, “fredholm and local spectral theory with applications to multipliers”, kluwer academic publishers, dordrecht, 2004. 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[22] m.p.h. wolff, discrete approximation of unbounded operators and approximation of their spectra, j. approx. theory 113 (2) (2001), 229 – 244. introduction auxiliary results main results � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 1 (2019), 41 – 60 doi:10.17398/2605-5686.34.1.41 available online march 4, 2019 upper and lower bounds for the difference between the weighted arithmetic and harmonic operator means s.s. dragomir 1,2 1 mathematics, college of engineering & science, victoria university po box 14428, melbourne city, mc 8001, australia sever.dragomir@vu.edu.au , http://rgmia.org/dragomir 2 school of computer science & applied mathematics, university of the witwatersrand private bag 3, johannesburg 2050, south africa received october 11, 2018 presented by mostafa mbekhta accepted february 4, 2019 abstract: in this paper we establish some new upper and lower bounds for the difference between the weighted arithmetic and harmonic operator means under various assumption for the positive invertible operators a, b. some applications when a, b are bounded above and below by positive constants are given as well. key words: young’s inequality, convex functions, arithmetic mean-harmonic mean inequality, operator means, operator inequalities. ams subject class. (2010): 47a63, 47a30, 15a60,.26d15; 26d10. 1. introduction throughout this paper a, b are positive invertible operators on a complex hilbert space (h,〈·, ·〉). we use the following notations for operators a∇νb := (1 −ν) a + νb, the weighted operator arithmetic mean, a]νb := a 1/2 ( a−1/2ba−1/2 )ν a1/2, the weighted operator geometric mean and a!νb := ( (1 −ν) a−1 + νb−1 )−1 the weighted operator harmonic mean, where ν ∈ [0, 1]. when ν = 1 2 , we write a∇b, a]b and a!b for brevity, respectively. issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.1.41 mailto:sever.dragomir@vu.edu.au http://rgmia.org/dragomir https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 42 s.s. dragomir the following fundamental inequality between the weighted arithmetic, geometric and harmonic operator means holds a!νb ≤ a]νb ≤ a∇νb (1.1) for any ν ∈ [0, 1]. for various recent inequalities between these means we recommend the recent papers [3]-[6], [8]-[12] and the references therein. in the recent work [7] we obtained between others the following result: theorem 1. let a, b be positive invertible operators and m > m > 0 such that ma ≥ b ≥ ma. (1.2) then for any ν ∈ [0, 1] we have rk (m,m) a ≤ a∇νb −a!νb ≤ rk (m,m) a, (1.3) where r = min{ν, 1 −ν}, r = max{ν, 1 −ν} and the bounds k (m,m) and k (m,m) are given by k(m,m) (1.4) :=   (m− 1)2 (m + 1)−1 if m < 1, max { (m− 1)2 (m + 1)−1 , (m − 1)2 (m + 1)−1 } if m ≤ 1 ≤ m, (m − 1)2 (m + 1)−1 if 1 < m, and k (m,m) :=   (m − 1)2 (m + 1)−1 if m < 1, 0 if m ≤ 1 ≤ m, (m− 1)2 (m + 1)−1 if 1 < m. (1.5) in particular, 1 2 k (m,m) a ≤ a∇b −a!b ≤ 1 2 k (m,m) a. (1.6) let a, b positive invertible operators and positive real numbers m, m′, m, m ′ such that the condition 0 < mi ≤ a ≤ m′i < m ′i ≤ b ≤ mi holds. put h := m m and h′ := m ′ m′ , then for any ν ∈ [0, 1] we have [7] r ( h′ − 1 )2 ( h′ + 1 )−1 a ≤ a∇νb −a!νb ≤ r (h− 1)2 (h + 1)−1 a, (1.7) upper and lower bounds 43 where r = min{ν, 1 −ν}, r = max{ν, 1 −ν} and, in particular, 1 2 ( h′ − 1 )2 ( h′ + 1 )−1 a ≤ a∇b −a!b ≤ 1 2 (h− 1)2 (h + 1)−1 a. (1.8) let a, b positive invertible operators and positive real numbers m, m′, m, m ′ such that the condition 0 < mi ≤ b ≤ m′i < m ′i ≤ a ≤ mi holds. then for any ν ∈ [0, 1] we also have [7] r ( h′ − 1 )2 ( h′ + 1 )−1 ( h′ )−1 a ≤ a∇νb −a!νb ≤ r (h− 1)2 (h + 1)−1 h−1a, (1.9) and, in particular, 1 2 ( h′ − 1 )2 ( h′ + 1 )−1 ( h′ )−1 a ≤ a∇b −a!b ≤ 1 2 (h− 1)2 (h + 1)−1 h−1a. (1.10) motivated by the above facts, in this paper we establish some new upper and lower bounds for the difference a∇νb−a!νb for ν ∈ [0, 1] under various assumption for the positive invertible operators a, b. some applications when a, b are bounded above and below by positive constants are given as well. a graphic comparison for upper bounds is provided as well. 2. min and max bounds the following lemma is of interest in itself. lemma 1. for any a, b > 0 and ν ∈ [0, 1] we have ν (1 −ν) (b−a)2 max{b,a} ≤ aν (a,b) −hν (a,b) ≤ ν (1 −ν) (b−a)2 min{b,a} , (2.1) where aν (a,b) and hν (a,b) are the scalar weighted arithmetic mean and harmonic mean, respectively, namely aν (a,b) := (1 −ν) a + νb and hν (a,b) := ab (1 −ν) b + νa . 44 s.s. dragomir in particular, 1 4 (b−a)2 max{b,a} ≤ a (a,b) −h (a,b) ≤ 1 4 (b−a)2 min{b,a} , (2.2) where a (a,b) := a + b 2 and h (a,b) := 2ab b + a . proof. consider the function ξν : (0,∞) → (0,∞) defined by ξν(x) = 1 −ν + νx− x (1 −ν) x + ν , where ν ∈ [0, 1]. then ξν (x) = (1 −ν + νx) [(1 −ν) x + ν] −x (1 −ν) x + ν = (1 −ν)2 x + ν (1 −ν) x2 + ν (1 −ν) + ν2x−x (1 −ν) x + ν = ν (1 −ν) x2 − 2ν (1 −ν) x + ν (1 −ν) (1 −ν) x + ν = ν (1 −ν) (x− 1)2 (1 −ν) x + ν , (2.3) for any x > 0 and ν ∈ [0, 1]. if we take in the definition of ξν, x = b a > 0, then we have ξν ( b a ) = 1 a [aν (a,b) −hν (a,b)] . from the equality (2.3) we also have ξν ( b a ) = ν (1 −ν) (b−a)2 aaν (b,a) . therefore, we have the equality aν (a,b) −hν (a,b) = ν (1 −ν) (b−a)2 aν (b,a) (2.4) for any a, b > 0 and ν ∈ [0, 1]. upper and lower bounds 45 since for any a, b > 0 and ν ∈ [0, 1] we have min{a,b}≤ aν (b,a) ≤ max{a,b} then ν (1 −ν) (b−a)2 max{a,b} ≤ ν (1 −ν) (b−a)2 aν (b,a) ≤ ν (1 −ν) (b−a)2 min{a,b} (2.5) and by (2.4) we get the desired result (2.1). remark 1. we show that there is no constant k1 > 1 and k2 < 1 such that ν (1 −ν) (b−a)2 max{b,a} ≤ aν (a,b) −hν (a,b) ≤ ν (1 −ν) (b−a)2 min{b,a} , (2.6) for some ν ∈ (0, 1) and any a, b > 0. assume that there exist k1, k2 > 0 such that k1ν (1 −ν) (b−a)2 max{b,a} ≤ aν (a,b) −hν (a,b) ≤ k2ν (1 −ν) (b−a)2 min{b,a} , (2.7) for some ν ∈ (0, 1) and any a, b > 0. let ε > 0 and write the inequality (2.7) for a > 0 and b = a + ε to get, via (2.4) that k1ν (1 −ν) ε2 a + ε ≤ ν (1 −ν) ε2 (1 −ν) ε + a ≤ k2ν (1 −ν) ε2 a . (2.8) if we divide by ν (1 −ν) ε2 > 0 in (2.8), then we get k1 1 a + ε ≤ 1 (1 −ν) ε + a ≤ k2 1 a , (2.9) for any a > 0 and ε > 0. by letting ε → 0+ in (2.9), we get k1 ≤ 1 ≤ k2 and the statement is proved. 46 s.s. dragomir we have the following operator double inequality: theorem 2. let a, b be positive invertible operators and m > m > 0 such that the condition (1.2). then for any ν ∈ [0, 1] we have ν (1 −ν) c (m,m) a ≤ ν (1 −ν) max{m, 1} (b −a) a−1 (b −a) ≤ a∇νb −a!νb ≤ ν (1 −ν) min{m, 1} (b −a) a−1 (b −a) ≤ ν (1 −ν) c (m,m) a, (2.10) where c (m,m) :=   (m − 1)2 if m < 1, 0 if m ≤ 1 ≤ m, (m−1)2 m if 1 < m, and c (m,m) :=   (m−1)2 m if m < 1, 1 m max { (m− 1)2 , (m − 1)2 } if m ≤ 1 ≤ m, (m − 1)2 if 1 < m. in particular, 1 4 c (m,m) a ≤ 1 4 max{m, 1} (b −a) a−1 (b −a) ≤ a∇b −a!b ≤ 1 4 min{m, 1} (b −a) a−1 (b −a) ≤ 1 4 c (m,m) a. (2.11) proof. if we write the inequality (2.1) for a = 1 and b = x, then we get ν (1 −ν) (x− 1)2 max{x, 1} ≤ 1 −ν + νx− ( (1 −ν) + νx−1 )−1 ≤ ν (1 −ν) (x− 1)2 min{x, 1} (2.12) for any x > 0 and for any ν ∈ [0, 1]. upper and lower bounds 47 if x ∈ [m,m] ⊂ (0,∞), then max{x, 1} ≤ max{m, 1} and min{m, 1} ≤ min{x, 1} and by (2.12) we get ν (1 −ν) minx∈[m,m] (x− 1) 2 max{m, 1} ≤ ν (1 −ν) (x− 1)2 max{m, 1} ≤ 1 −ν + νx− ( (1 −ν) + νx−1 )−1 ≤ ν (1 −ν) (x− 1)2 min{m, 1} ≤ ν (1 −ν) maxx∈[m,m] (x− 1) 2 min{m, 1} (2.13) for any x ∈ [m,m] and for any ν ∈ [0, 1]. observe that min x∈[m,m] (x− 1)2 =   (m − 1)2 if m < 1, 0 if m ≤ 1 ≤ m, (m− 1)2 if 1 < m, and max x∈[m,m] (x− 1)2 =   (m− 1)2 if m < 1, max { (m− 1)2 , (m − 1)2 } if m ≤ 1 ≤ m, (m − 1)2 if 1 < m. then minx∈[m,m] (x− 1) 2 max{m, 1} =   (m − 1)2 if m < 1, 0 if m ≤ 1 ≤ m, (m−1)2 m if 1 < m, = c (m,m) and maxx∈[m,m](x− 1)2 min{m, 1} =   (m−1)2 m if m < 1, 1 m max { (m− 1)2 , (m − 1)2 } if m ≤ 1 ≤ m, (m − 1)2 if 1 < m, = c (m,m) . 48 s.s. dragomir using the inequality (2.13) we have ν (1 −ν) c (m,m) ≤ ν (1 −ν) (x− 1)2 max{m, 1} ≤ 1 −ν + νx− ( (1 −ν) + νx−1 )−1 ≤ ν (1 −ν) (x− 1)2 min{m, 1} ≤ ν (1 −ν) c (m,m) (2.14) for any x ∈ [m,m] and for any ν ∈ [0, 1]. if we use the continuous functional calculus for the positive invertible operator x with mi ≤ x ≤ mi, then we have from (2.14) that ν (1 −ν) c (m,m) i ≤ ν (1 −ν) max{m, 1} (x − i)2 ≤ (1 −ν) i + νx − ( (1 −ν) i + νx−1 )−1 ≤ ν (1 −ν) min{m, 1} (x − i)2 ≤ ν (1 −ν) c (m,m) i (2.15) for any ν ∈ [0, 1]. if we multiply (1.2) both sides by a−1/2 we get mi ≥ a−1/2ba−1/2 ≥ mi. by writing the inequality (2.15) for x = a−1/2ba−1/2 we obtain ν(1−ν)c(m,m)i ≤ ν(1 −ν) max{m, 1} ( a−1/2ba−1/2 − i )2 (2.16) ≤ (1 −ν)i + νa−1/2ba−1/2 −a−1/2 ( (1 −ν)a−1 + νb−1 )−1 a−1/2 ≤ ν(1 −ν) min{m, 1} ( a−1/2ba−1/2 − i )2 ≤ ν(1 −ν)c(m,m)i for any ν ∈ [0, 1]. upper and lower bounds 49 if we multiply the inequality (2.16) both sides with a1/2, then we get ν (1 −ν) c (m,m) a ≤ ν (1 −ν) max{m, 1} a1/2 ( a−1/2ba−1/2 − i )2 a1/2 ≤ (1 −ν) a + νb − ( (1 −ν) a−1 + νb−1 )−1 ≤ ν (1 −ν) min{m, 1} a1/2 ( a−1/2ba−1/2 − i )2 a1/2 ≤ ν (1 −ν) c (m,m) a, (2.17) and since a1/2 ( a−1/2 ba−1/2 − i )2 a1/2 = a1/2 ( a−1/2 (b −a) a−1/2 )2 a1/2 = a1/2a−1/2 (b −a) a−1/2a−1/2 (b −a) a−1/2a1/2 = (b −a) a−1 (b −a) , then by (2.17) we get the desired result (2.10). when the operators a and b are bounded above and below by constants we have the following result as well: corollary 1. let a, b be two positive operators and m, m′, m, m ′ be positive real numbers. put h := m m and h′ := m ′ m′ . (i) if 0 < mi ≤ a ≤ m′i < m ′i ≤ b ≤ mi, then ν (1 −ν) (h′ − 1)2 h a ≤ ν (1 −ν) h (b −a) a−1 (b −a) ≤ a∇νb −a!νb ≤ ν (1 −ν) (b −a) a−1 (b −a) ≤ ν (1 −ν) (h− 1)2 a, (2.18) and, in particular, (h′ − 1)2 4h a ≤ 1 4h (b −a) a−1 (b −a) ≤ a∇b −a!b ≤ 1 4 (b −a) a−1 (b −a) ≤ 1 4 (h− 1)2 a. (2.19) 50 s.s. dragomir (ii) if 0 < mi ≤ b ≤ m′i < m ′i ≤ a ≤ mi, then ν (1 −ν) ( h′ − 1 h′ )2 a ≤ ν (1 −ν) (b −a) a−1 (b −a) ≤ a∇νb −a!νb ≤ ν (1 −ν) h (b −a) a−1 (b −a) ≤ ν (1 −ν) (h− 1)2 h a (2.20) and, in particular, 1 4 ( h′ − 1 h′ )2 a ≤ 1 4 (b −a) a−1 (b −a) ≤ a∇b −a!b ≤ 1 4 h (b −a) a−1 (b −a) ≤ (h− 1)2 4h a. (2.21) proof. we observe that h, h′ > 1 and if either of the condition (i) or (ii) holds, then h ≥ h′. if (i) is valid, then we have a < h′a = m ′ m′ a ≤ b ≤ m m a = ha, (2.22) while, if (ii) is valid, then we have 1 h a ≤ b ≤ 1 h′ a < a. (2.23) if we use the inequality (2.10) and the assumption (i), then we get (2.18). if we use the inequality (2.10) and the assumption (ii), then we get (2.20). 3. bounds in term of kantorovich’s constant we consider the kantorovich’s constant defined by k (h) := (h + 1) 2 4h , h > 0. (3.1) the function k is decreasing on (0, 1) and increasing on [1,∞), k(h) ≥ 1 for any h > 0 and k(h) = k( 1 h ) for any h > 0. upper and lower bounds 51 observe that for any h > 0 k(h) − 1 = (h− 1)2 4h = k ( 1 h ) − 1. observe that k ( b a ) − 1 = (b−a)2 4ab for a,b > 0. since, obviously ab = min{a,b}max{a,b} for a,b > 0, then we have the following version of lemma 1: lemma 2. for any a, b > 0 and ν ∈ [0, 1] we have 4ν (1 −ν) min{a,b} [ k ( b a ) − 1 ] ≤ aν (a,b) −hν (a,b) ≤ 4ν (1 −ν) max{a,b} [ k ( b a ) − 1 ] . (3.2) for positive invertible operators a, b we define a∇∞b := 1 2 (a + b) + 1 2 a1/2 ∣∣∣a−1/2 (b −a) a−1/2∣∣∣a1/2, a∇−∞b := 1 2 (a + b) − 1 2 a1/2 ∣∣∣a−1/2 (b −a) a−1/2∣∣∣a1/2. if we consider the continuous functions f∞, f−∞ : [0,∞) → [0,∞) defined by f∞ (x) = max{x, 1} = 1 2 (x + 1) + 1 2 |x− 1| , f−∞ (x) = max{x, 1} = 1 2 (x + 1) − 1 2 |x− 1| , then, obviously, we have a∇±∞b = a1/2f±∞ ( a−1/2ba−1 ) a1/2. (3.3) if a and b are commutative, then a∇±∞b = 1 2 (a + b) ± 1 2 |b −a| = b∇±∞a. 52 s.s. dragomir theorem 3. let a, b be positive invertible operators and m > m > 0 such that the condition (1.2) holds. then we have 4ν (1 −ν) g (m,m) a∇−∞b ≤ a∇νb −a!νb ≤ 4ν (1 −ν) g (m,m) a∇∞b, (3.4) where g (m,m) :=   k (m) − 1 if m < 1, 0 if m ≤ 1 ≤ m, k (m) − 1 if 1 < m, g (m,m) :=   k (m) − 1 if m < 1, max{k (m) ,k (m)}− 1 if m ≤ 1 ≤ m, k (m) − 1 if 1 < m. in particular, g (m,m) a∇−∞b ≤ a∇b −a!b ≤ g (m,m) a∇∞b. (3.5) proof. from (3.2) we have for a = 1 and b = x that 4ν (1 −ν) min{1,x} [k (x) − 1] ≤ 1 −ν + νx− ( (1 −ν) + νx−1 )−1 ≤ 4ν (1 −ν) max{1,x} [k (x) − 1] (3.6) for any x > 0. from (3.6) we then have 4ν (1 −ν) f−∞(x) min x∈[m,m] [k (x) − 1] ≤ 1 −ν + νx− ( (1 −ν) + νx−1 )−1 ≤ 4ν (1 −ν) f∞ (x) max x∈[m,m] [k (x) − 1] (3.7) for any x ∈ [m,m]. observe that max x∈[m,m] [k (x) − 1] =   k (m) − 1 if m < 1, max{k (m) ,k (m)}− 1 if m ≤ 1 ≤ m, k (m) − 1 if 1 < m, = g (m,m) upper and lower bounds 53 and min x∈[m,m] [k (x) − 1] =   k (m) − 1 if m < 1, 0 if m ≤ 1 ≤ m, k (m) − 1 if 1 < m. = g (m,m) . therefore by (3.7) we get 4ν (1 −ν) f−∞ (x) g (m,m) ≤ 1 −ν + νx− ( (1 −ν) + νx−1 )−1 ≤ 4ν (1 −ν) f∞ (x) g (m,m) (3.8) for any x ∈ [m,m] and ν ∈ [0, 1]. if we use the continuous functional calculus for the positive invertible operator x with mi ≤ x ≤ mi, then we have from (3.8) that 4ν (1 −ν) f−∞ (x) g (m,m) ≤ (1 −ν) i + νx − ( (1 −ν) + νx−1 )−1 ≤ 4ν (1 −ν) f∞ (x) g (m,m) (3.9) for any x ∈ [m,m] and ν ∈ [0, 1]. by writing the inequality (3.9) for x = a−1/2ba−1/2 we obtain 4ν (1 −ν) f−∞ ( a−1/2ba−1/2 ) g (m,m) (3.10) ≤ (1 −ν) i + νa−1/2ba−1/2 −a−1/2 ( (1 −ν) a−1 + νb−1 )−1 a−1/2 ≤ 4ν (1 −ν) f∞ ( a−1/2ba−1/2 ) g (m,m) for any ν ∈ [0, 1]. if we multiply (3.10) both sides by a1/2 we get 4ν (1 −ν)a1/2f−∞ ( a−1/2ba−1/2 ) a1/2g (m,m) ≤ (1 −ν) a + νb − ( (1 −ν) a−1 + νb−1 )−1 ≤ 4ν (1 −ν) a1/2f∞ ( a−1/2ba−1/2 ) a1/2g (m,m) for any ν ∈ [0, 1], which, by (3.3) produces the desired result (3.4). we have: 54 s.s. dragomir corollary 2. let a, b be two positive operators and m, m′, m, m ′ be positive real numbers. put h := m m and h′ := m ′ m′ . if either of the conditions (i) or (ii) from corollary 1 holds, then 4ν (1 −ν) [ k ( h′ ) − 1 ] a∇−∞b ≤ a∇νb −a!νb (3.11) ≤ 4ν (1 −ν) [k (h) − 1] a∇∞b. in particular,[ k ( h′ ) − 1 ] a∇−∞b ≤ a∇b −a!b ≤ [k (h) − 1] a∇∞b. (3.12) proof. if (i) is valid, then we have a < h′a = m ′ m′ a ≤ b ≤ m m a = ha. by using the inequality (3.4) we get (3.11). if (ii) is valid, then we have 1 h a ≤ b ≤ 1 h′ a < a. by using the inequality (3.4) we get 4ν (1 −ν) [ k ( 1 h′ ) − 1 ] a∇−∞b ≤ a∇νb −a!νb ≤ 4ν (1 −ν) [ k ( 1 h ) − 1 ] a∇∞b, and since k ( 1 h′ ) = k (h′) and k ( 1 h ) = k (h), the inequality (3.11) is also obtained. 4. further bounds the following result also holds: theorem 4. let a, b be positive invertible operators and m > m > 0 such that the condition (1.2) holds. then we have pν (m,m) a ≤ a∇νb −a!νb ≤ pν (m,m) a (4.1) upper and lower bounds 55 for any ν ∈ [0, 1], where pν(m,m) :=   ν(1−ν)(m−1)2 (1−ν)m+ν if m < 1, 0 if m ≤ 1 ≤ m, ν(1−ν)(m−1)2 (1−ν)m+ν if 1 < m, pν (m,m) :=   ν(1−ν)(m−1)2 (1−ν)m+ν if m < 1, max { ν(1−ν)(m−1)2 (1−ν)m+ν , ν(1−ν)(m−1)2 (1−ν)m+ν } if m ≤ 1 ≤ m, ν(1−ν)(m−1)2 (1−ν)m+ν if 1 < m. proof. consider the function ξν : (0,∞) → (0,∞) defined by ξν (x) = 1 −ν + νx− x (1 −ν) x + ν , where ν ∈ [0, 1]. taking the derivative, we have ξ′ν(x) = ν − (1 −ν)x + ν −x(1 −ν) [(1 −ν)x + ν]2 = ν [(1 −ν)x + ν]2 − 1 [(1 −ν)x + ν]2 = ν(1 −ν)(x− 1) [(1 −ν)x + ν + 1] [(1 −ν)x + ν]2 for any x ≥ 0 and ν ∈ [0, 1]. this shows that the function is decreasing on [0, 1] and increasing on (1,∞). we have ξν (0) = 1 −ν, ξν (1) = 0 and limx→∞ξν (x) = ∞. since, by (2.3) ξν (x) = ν (1 −ν) (x− 1)2 (1 −ν) x + ν , x ≥ 0, then for [m,m] ⊂ [0,∞) we have min x∈[m,m] ξν (x) =   ν(1−ν)(m−1)2 (1−ν)m+ν if m < 1, 0 if m ≤ 1 ≤ m, ν(1−ν)(m−1)2 (1−ν)m+ν if 1 < m, = pν (m,m) 56 s.s. dragomir and max x∈[m,m] ξν(x) =   ν(1−ν)(m−1)2 (1−ν)m+ν if m < 1, max { ν(1−ν)(m−1)2 (1−ν)m+ν , ν(1−ν)(m−1)2 (1−ν)m+ν } if m ≤ 1 ≤ m, ν(1−ν)(m−1)2 (1−ν)m+ν if 1 < m, = pν (m,m) . therefore pν (m,m) ≤ 1 −ν + νx− ( (1 −ν) + νx−1 )−1 ≤ pν (m,m) (4.2) for any x ∈ [m,m] and ν ∈ [0, 1]. if we use the continuous functional calculus for the positive invertible operator x with mi ≤ x ≤ mi, then we have from (4.2) that p (m,m) i ≤ (1 −ν) i + νx − ( (1 −ν) i + νx−1 )−1 ≤ pν (m,m) i (4.3) for any ν ∈ [0, 1]. if we multiply (1.2) both sides by a−1/2 we get mi ≥ a−1/2ba−1/2 ≥ mi. by writing the inequality (4.3) for x = a−1/2ba−1/2 we obtain p(m,m)i ≤ (1 −ν)i + νa−1/2ba−1/2 −a−1/2 ( (1 −ν)a−1 + νb−1 )−1 a−1/2 ≤ pν (m,m) i (4.4) for any ν ∈ [0, 1]. if we multiply (4.4) both sides by a1/2 we get p (m,m) a ≤ (1 −ν) a + νb − ( (1 −ν) a−1 + νb−1 )−1 ≤ pν (m,m) a for any ν ∈ [0, 1]. upper and lower bounds 57 remark 2. if we consider p (m,m) :=   (m−1)2 2(m+1) if m < 1, 0 if m ≤ 1 ≤ m, (m−1)2 2(m+1) if 1 < m, p (m,m) :=   (m−1)2 2(m+1) if m < 1, max { (m−1)2 2(m+1) , (m−1)2 2(m+1) } if m ≤ 1 ≤ m, (m−1)2 2(m+1) if 1 < m, then by (4.1) we have p (m,m) a ≤ a∇b −a!b ≤ p (m,m) a, (4.5) provided that a, b are positive invertible operators and m > m > 0 are such that the condition (1.2) holds. corollary 3. let a, b be two positive operators and m, m′, m, m ′ be positive real numbers. put h := m m and h′ := m ′ m′ . (i) if 0 < mi ≤ a ≤ m′i < m ′i ≤ b ≤ mi, then for any ν ∈ [0, 1] ν (1 −ν) (h′ − 1)2 (1 −ν) h′ + ν a ≤ a∇νb −a!νb ≤ ν (1 −ν) (h− 1)2 (1 −ν) h + ν a (4.6) and, in particular, (h′ − 1)2 2 (h′ + 1) a ≤ a∇b −a!b ≤ (h− 1)2 2 (h + 1) a. (4.7) (ii) if 0 < mi ≤ b ≤ m′i < m ′i ≤ a ≤ mi, then for any ν ∈ [0, 1] ν (1 −ν) (h′ − 1)2 h′ (1 −ν + νh′) a ≤ a∇νb −a!νb ≤ ν (1 −ν) (h− 1)2 h (1 −ν + νh) a (4.8) 58 s.s. dragomir and, in particular, (h′ − 1)2 2h′ (1 + h′) a ≤ a∇b −a!b ≤ (h− 1)2 2h (1 + h) a. (4.9) proof. we observe that h, h′ > 1 and if either of the condition (i) or (ii) holds, then h ≥ h′. if (i) is valid, then we have a < h′a = m ′ m′ a ≤ b ≤ m m a = ha, while, if (ii) is valid, then we have 1 h a ≤ b ≤ 1 h′ a < a. if we use the inequality (4.1) and the assumption (i), then we get (4.6). if we use the inequality (4.1) and the assumption (ii), then we get (4.8). 5. a comparison we observe that an upper bound for the difference a∇νb −a!νb as provided in (1.3) is b1 (ν,m,m) a := max{ν, 1 −ν}×   (m−1)2 m+1 a if m < 1, max { (m−1)2 m+1 , (m−1)2 m+1 } a if m ≤ 1 ≤ m, (m−1)2 m+1 a if 1 < m while the one from (2.10) is b2 (ν,m,m) a := ν (1 −ν)×   (m−1)2 m a if m < 1, 1 m max { (m− 1)2, (m − 1)2 } a if m ≤ 1 ≤ m, (m − 1)2 a if 1 < m, where a, b are positive invertible operators and m > m > 0 such that the condition (1.2) holds. upper and lower bounds 59 we consider for x = m ∈ (0, 1) and y = ν ∈ [0, 1] the difference d1 (x,y) = max{y, 1 −y} (x− 1)2 x + 1 −y (1 −y) (x− 1)2 x that has the 3d plot on the box [0.3, 0.6] × [0, 1] depicted in figure 1 showing that it takes both positive and negative values, meaning that neither of the bounds b1 (ν,m,m) a and b2 (ν,m,m) a is better in the case 0 < m < m < 1. figure 1: plot of difference d1(x,y) figure 2: plot of difference d2(x,y) 60 s.s. dragomir we consider for x = m ∈ (1,∞) and y = ν ∈ [0, 1] the difference d2 (x,y) = max{y, 1 −y} (x− 1)2 x + 1 −y (1 −y) (x− 1)2 that has the 3d plot on the box [1, 3]×[0, 1] depicted in figure 2 showing that it takes both positive and negative values, meaning that neither of the bounds b1 (ν,m,m) a and b2 (ν,m,m) a is better in the case 1 < m < m < ∞. similar conclusions may be derived for lower bounds, however the details are left to the interested reader. references [1] s.s. dragomir, bounds for the normalised jensen functional, bull. austral. math. soc. 74 (3) (2006), 417 – 478. [2] s.s. dragomir, a note on young’s inequality, preprint rgmia res. rep. coll. 18 (2015), art. 126, http://rgmia.org/papers/v18/v18a126.pdf. [3] s.s. dragomir, some new reverses of young’s operator inequality, preprint rgmia res. rep. coll. 18 (2015), art. 130, http://rgmia.org/papers/v18/v18a130.pdf. [4] s.s. dragomir, on new refinements and reverses of young’s operator inequality, preprint rgmia res. rep. coll. 18 (2015), art. 135, http://rgmia.org/papers/v18/v18a135.pdf. [5] s.s. dragomir, some inequalities for operator weighted geometric mean, preprint rgmia res. rep. coll. 18 (2015), art. 139, http://rgmia.org/papers/v18/v18a139.pdf. [6] s.s. dragomir, some reverses and a refinement of hölder operator inequality, preprint rgmia res. rep. coll. 18 (2015), art. 147, http://rgmia.org/papers/v18/v18a147.pdf. [7] s.s. dragomir, some inequalities for weighted harmonic and arithmetic operator means, preprint rgmia res. rep. coll. 19 (2016), art. 5, http://rgmia.org/papers/v19/v19a05.pdf. [8] s. furuichi, refined young inequalities with specht’s ratio, j. egyptian math. soc. 20 (2012), 46 – 49. [9] s. furuichi, on refined young inequalities and reverse inequalities, j. math. inequal. 5 (2011), 21 – 31. [10] w. liao, j. wu, j. zhao, new versions of reverse young and heinz mean inequalities with the kantorovich constant, taiwanese j. math. 19 (2) (2015), 467 – 479. [11] m. tominaga, specht’s ratio in the young inequality, sci. math. japon. 55 (2002), 583 – 588. [12] g. zuo, g. shi, m. fujii, refined young inequality with kantorovich constant, j. math. inequal. 5 (2011), 551 – 556. http://rgmia.org/papers/v18/v18a126.pdf http://rgmia.org/papers/v18/v18a130.pdf http://rgmia.org/papers/v18/v18a135.pdf http://rgmia.org/papers/v18/v18a139.pdf http://rgmia.org/papers/v18/v18a147.pdf http://rgmia.org/papers/v19/v19a05.pdf introduction min and max bounds bounds in term of kantorovich's constant further bounds a comparison � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 1 (2019), 19 – 28 doi:10.17398/2605-5686.34.1.19 available online february 3, 2019 on self-circumferences in minkowski planes mostafa ghandehari, horst martini department of mathematics, university of texas at arlington, tx 76019, u.s.a. faculty of mathematics, university of technology, 09107 chemnitz, germany ghandeha@uta.edu , horst.martini@mathematik.tu-chemnitz.de received july 5, 2018 presented by javier alonso accepted august 1, 2018 abstract: this paper contains results on self-circumferences of convex figures in the frameworks of norms and (more general) also of gauges. let δ(n) denote the self-circumference of a regular polygon with n sides in a normed plane. we will show that δ(n) is monotonically increasing from 6 to 2π if n is twice an odd number, and monotonically decreasing from 8 to 2π if n is twice an even number. calculations of self-circumferences for the case that n is odd as well as inequalities for the self-circumference of some irregular polygons are also given. in addition, properties of the mixed area of a plane convex body and its polar dual are used to discuss the self-circumference of convex curves. key words: gauge, minkowski geometry, normed plane, polygonal gauges, radon curve, selfcircumference, self-perimeter. ams subject class. (2010): 52a21, 46b20, 52a10. 1. introduction the concept of minkowski distance defined by means of a convex body centred at the origin was developed by h. minkowski [14]. minkowski spaces are finite dimensional real banach spaces, with the planar subcase of normed or minkowski planes, and the geometry of such spaces and planes is usually called minkowski geometry (cf. [1] and [19]). the article [13] and the whole monograph [19] contain useful background material referring to minkowski geometry and, in particular, to those parts of the theory of convex sets which are needed. in this article we will deal with the self-circumference (or self-perimeter) of unit circles of normed or minkowski planes and (more general) of gauges. related inequalities for polygons can be found in [5], [9], and [10], and further results in this direction are presented in [3], [17], and [11]. we will also use properties of mixed areas of planar convex bodies and their polars to discuss self-circumferences of some types of (also non-polygonal) convex curves, including radon curves (see [12]). issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.1.19 mailto:ghandeha@uta.edu mailto:horst.martini@mathematik.tu-chemnitz.de https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 20 m. ghandehari, h. martini by a planar convex body k we mean a compact, convex subset of the euclidean plane having non-empty interior. we shall take as unit circle for the considered minkowski plane a convex body k centered at the origin. the minkowskian distance d(x,y) from x to y is defined by d(x,y) = de(x,y) r , (1) where de(x,y) is the euclidean distance from x to y, and r is the euclidean radius of k in the direction of the vector y−x. we will refer to the standard plane with this new metric as normed or minkowski plane. the minkowskian length of a polygonal path is obtained by adding the minkowskian lengths of the corresponding line segments. the minkowskian length of a curve is defined by taking the supremum over all polygonal paths inscribed to that curve. the self-circumference of the unit circle k is the minkowskian length of it measured with respect to k itself. in other words, the length of the boundary of k using the metric induced by k itself is called the self-circumference of k and denoted δ(k). it was first proved in [6] that 6 ≤ δ(k) ≤ 8 . (2) equality is attained on the left if, and only if, k is the affine image of a regular hexagon, and on the right if, and only if, k is a parallelogram. another proof is given in [16], and chakerian and talley [3] established a number of properties of self-circumferences and raised interesting questions. martini and shcherba (cf. [9] and [10]) discussed self-perimeters of quadrangles and pentagons for the more general concept of gauges, where the unit circle is still a convex curve, but not necessarily centered at the origin, and (1)) holds analogously; see also [11]. 2. preliminaries let k be a plane convex body with the origin as interior point. for each angle θ, 0 ≤ θ < 2π, we let r(k,θ) be the radius of k in direction (cos θ, sin θ), such that the boundary of k has the equation r = r(k,θ) in polar coordinates. the distance from the origin to the supporting line of k with outward normal (cos θ, sin θ) is denoted by h(k,θ). this is the restriction of the support function of k to the euclidean unit circle. since k is convex, it has a welldefined unique tangent line at all but at most a countable number of points. we let ds(k,θ) represent the element of euclidean arclength of the boundary on self-circumferences in minkowski planes 21 of k at a point where the unit normal is given by (cos θ, sin θ). then we have for the length of the boundary of k l(k) = 2π∫ 0 h(k,θ)dθ , (3) while the euclidean area of k is given by a(k) = 1 2 2π∫ 0 h(k,θ)ds(k,θ) . (4) the polar dual of k, denoted by k∗, is another plane convex body having the origin as interior point, and it is defined by h(k∗,θ) = 1 r(k,θ) and r(k∗,θ) = 1 h(k,θ) . (5) the mixed area a(k1,k2) of two convex sets k1,k2 is defined by a(k1,k2) = 1 2 2π∫ 0 h(k1,θ)ds(k2,θ) . (6) it turns out that the mixed area is symmetric in its arguments. the following result (due to firey [4]) will be used: the mixed area of a plane convex body and its polar dual is at least π. the unit circle k of a minkowski plane is referred to as its indicatrix. we define the isoperimetrix of that plane to be the convex body t such that h(t,θ) = 1 r(k,θ + π 2 ) , (7) cf. [1] and [19]. the boundary of a centrally symmetric set is called a radon curve if it coincides with the corresponding isoperimetrix. for further properties of radon curves we refer to [12]. we now discuss the definition of self-circumference and give some properties. if k is a centrally symmetric convex body centered at the origin, then by (1) and the preceding discussion, the self-circumference δ(k) is given by δ(k) = ∫ ds(k,θ) r(k,θ + π 2 ) . (8) 22 m. ghandehari, h. martini if z is any point interior to k and k is not necessarily centered at the origin (thus really yielding a gauge), then the positive and the negative selfcircumference (depending on orientation) of k relative to z are defined by δ+(k,z) = ∫ ds(k,θ) r(k,θ + π 2 ) (9) and δ−(k,z) = ∫ ds(k,θ) r(k,θ − π 2 ) , (10) where the origin of the coordinate system is at z. both δ+(k,z) and δ−(k,z) reduce to δ(k) in case that k is symmetric with respect to z. for k defining a gauge, go lab [6] conjectured that δ(k,z) ≥ 6, for all interior points z, and max δ(k,z) ≤ 9. the latter conjecture was settled by grünbaum [7]. and in [17] the lower bound for the general case was confirmed. if k1 and k2 are plane convex bodies with the origin as an interior point, then the length of the positively oriented boundary of k1 with respect to k2 is given by δ+(k1,k2) = ∫ ds(k1,θ) r(k2,θ + π 2 ) , (11) and the length of the negatively oriented boundary by δ−(k1,k2) = ∫ ds(k1,θ) r(k2,θ − π2 ) . (12) schäffer [16] and independently thompson [18] proved that for k centered at the origin δ+(k) = δ−(k ∗) and δ−(k) = δ+(k ∗) hold. more generally, chakerian [2] used the concept of mixed areas to prove that δ+(k1,k2) = δ−(k ∗ 2,k ∗ 1 ) and δ−(k1,k2) = δ+(k ∗ 2,k ∗ 1 ) . (13) 3. polygons we study first the self-circumference of regular polygons. after that we give inequalities for positive and negative self-circumferences of quadrangles. in [9] and [10] results on self-circumferences of quadrangles and pentagons were obtained, and in [5] for polygons. theorem 1 below was also proved in [5]; for the sake of completeness we include it here. on self-circumferences in minkowski planes 23 theorem 1. let δ(n) denote the self-circumference of an affine image of a regular polygon with n sides. then δ(n) is monotonically increasing from 6 to 2π if n is twice an odd number, and monotonically decreasing from 8 to 2π if n is twice an even number. the formula for δ(n) in theorem 1 is δ(n) = 2n sin π n for n being a doubled odd number. since sinπ n assures rational values only for n = 2 and n = 6 (see p. 144 in [15], problems 197.1 and 197.5), it follows that 6 is the only rational value obtained. in case n is twice an even number, δ(n) is given by δ(n) = 2n tan π n . the fact that tan π n is only rational for n = 4 implies that in this case the only rational value in question is δ(n) = 8. in case n is an odd number, the value δ(n) with respect to the case of having a center of symmetry is given by δ(n) = 2n tan π n cos π 2n . this sequence is monotonically decreasing from 9 to 2π. the proof that δ(n) is monotonically decreasing is not trivial. we include a proof of theorem 2. theorem 2. the sequence δ(n) = 2n tan π n cos π 2n , where n is odd, is monotonically decreasing from 9 to 2π, n = 3, 5, 7, . . .. proof. let g(x) = (tan 2x cos x)/x. we want to show that g′(x) > 0. let g(x) = h(x)/x. then g′(x) = ( xh′(x) − h(x) ) /x2; so g′(x) > 0 iff xh′(x) − h(x) > 0 for 0 < x < π 4 . it suffices to show that (xh′(x) − h(x))′ > 0 for 0 < x < π 4 . we have (xh′ −h)′ = xh′′ + h′ −h′ = xh′′ > 0 iff h′′(x) > 0, h′(x) = 2 sec2 2x cos x− tan 2x sin x, h′′(x) = 8 cos x sin 2x cos3(2x) − 2 sin x cos2(2x) − 2 sin x cos x cos 2x = 2 sin x cos3(2x) ( 8 cos2 x− 2 cos2 x + 2 sin2 x− cos2 x cos2(2x) ) = 2 sin x cos3(2x) ( 8 cos2 x− 2 cos2 x + 2 sin2 x− cos2 x cos2(2x) ) , h′′(x) = 2 sin x cos3(2x) ( 2 + 4 cos2 x− cos2 x cos2(2x) ) = 2 sin x cos3(2x) ( 2 + cos2 x(4 − cos2(2x)) ) > 0 , 24 m. ghandehari, h. martini since |cos x| ≤ 1, and (2 sin x)/ cos3(2x) > 0 for 0 < x < π 4 . thus, h′′(x) > 0 for 0 < x < π 4 , and so g′(x) > 0 for 0 < x < π 4 . the following two theorems give inequalities for self-circumferences of a quadrangle and a trapezoid. p q r s z figure 1 theorem 3. the self-circumference δ+(p,z) of a convex quadrilateral p with respect to the point z of intersection of its diagonals is at least 8, with equality if and only if the quadrilateral is a parallelogram. proof. consider a quadrilateral p with vertices p,q,r,s (see figure 1). let z be the point of intersection of the diagonals. then, by using similar triangles, we get δ+(p,z) = de(z,p) + de(z,r) de(z,r) + de(z,q) + de(z,s) de(z,s) + de(z,p) + de(z,r) de(z,p) + de(z,q) + de(z,s) de(z,s) . thus we have δ+(p,z) = 4 + de(z,p) de(z,r) + de(z,r) de(z,p) + de(z,q) de(z,s) + de(z,s) de(z,q) ≥ 8 , on self-circumferences in minkowski planes 25 where the last inequality follows from the arithmetic-geometric mean inequality. equality holds if and only if de(z,p) = de(z,r),de(z,q) = de(z,s), which implies that p is a parallelogram. theorem 4. the self-circumference δ+(p,o) of a trapezoid p with respect to the midpoint o of one of the diagonals is at least 8, with equality if and only if p is a parallelogram. proof. consider a trapezoid p with vertices p,q,r,s (see figure 2). in the figure, ou is parallel to ps, ov is parallel to pq, ow is parallel to qr, and ox is parallel to rs. by using similar triangles, we get δ+(p,o) = de(p,q) de(ov) + de(q,r) de(ow) + (de(r,s) de(ox) + de(sp) de(ou) , where the equalities de(r,s) = 2de(ov),de(p,q) = 2de(o,x),de(q,r) = 2de(ow), and de(p,s) = 2de(ow) are used. the arithmetic-geometric mean inequality implies that δ+(p,o) ≥ 8, with equality if and only if de(r,s) = de(p,q), yielding again a parallelogram. o p qr s v w x u figure 2 chakerian and talley [3] gave an example of a trapezoid to show that δ+(k,z) and δ−(k,z) do not assume their minimum at the same point, thus answering a question of hammer posed in [8]. 26 m. ghandehari, h. martini 4. curves in the following we use properties of mixed areas of a plane convex body and its polar dual to discuss self-circumferences of certain convex curves. the following theorem shows that the self-circumference of a plane convex body with four-fold symmetry is at least 2π. theorem 5. let k be a plane convex body centered at the origin. assume that r(k,θ) is an equation of the boundary of k in polar coordinates, and assume that r(k,θ) = r(k,θ + π 2 ), 0 ≤ θ ≤ 2π, i.e., k has four-fold symmetry. then its self-circumference satisfies δ(k) ≥ 2π. proof. using the definition given in (8) and four-fold symmetry, we obtain δ(k) = ∫ ds(k,θ) r(k,θ + π 2 ) = ∫ ds(k,θ) r(k,θ) . by the property of the polar dual given in (5) and the properties of mixed areas presented in (6), it follows that δ(k) = ∫ ds(k,θ) r(k,θ) = ∫ h(k∗,θ)ds(k,θ) = 2a(k∗,k) . firey’s result from [4] states that the mixed area of a plane convex body and its polar dual is at least π. it follows that δ(k) is at least 2π. recall that the isoperimetrix t was defined by (7). the following theorem gives the minkowskian length of the boundary of a plane convex body with respect to the isoperimetrix. theorem 6. let k be a plane convex body, and assume that t is the isoperimetrix, that is, the polar dual of k rotated by 90 degrees. then δ+(k,t) = 2a(k), where a(k) is the euclidean area of k. proof. by the definition given in (11) we obtain δ+(k,t) = ∫ ds(k,θ) r(t,θ + π 2 ) . by the definition of the isoperimetrix, r(t,θ + π 2 ) = r(k∗,θ). thus δ+(k,t) = ∫ ds(k,θ) r(k∗,θ) = ∫ h(k,θ)ds(k,θ) = 2a(k) , on self-circumferences in minkowski planes 27 where we have used (4) and (5) giving the euclidean area and the used property of the polar dual. if the boundary of k is a radon curve, then it coincides with its isoperimetrix. thus, the self-circumference of a radon curve is equal to twice its euclidean area. the following theorem refers to the length of the euclidean unit circle with respect to a plane convex body k. theorem 7. let k be a plane convex body, and assume that b is the euclidean unit circle. then the length of b with respect to k equals the euclidean length of the polar dual of k. that is, δ+(b,k) = l(k ∗). proof. by the result of chakerian given in (13) we obtain δ+(b,k) = δ−(k ∗,b∗) = δ−(k ∗,b) . assuming that the polar dual of k is calculated at the center of the euclidean unit circle b, it follows that δ+(b,k) = l(k ∗). in the subcase where k is a square with vertices at (±1, 0), (0,±1), the minkowskian distance is the same as used for the so-called taxicab metric. the polar dual is a square with sides parallel to the axes. thus, the length of a euclidean unit circle in the taxicab metric is the same as the euclidean length of the circumscribed square which is 8, and thus we have finally “squared the circle”. references [1] h. busemann, the foundations of minkowskian geometry, comment. math. helv. 24 (1950), 156-187. [2] g.d. chakerian, mixed areas and the self-circumference of a plane convex body, arch. math. (basel) 34 (1) (1980), 81-83. [3] g.d. chakerian, w.k. talley, some properties of the self-circumference of convex sets, arch. math. (basel) 20 (1969), 431-443. [4] w.j. firey, the mixed area of a convex body and its polar reciprocal, israel j. math. 1 (1963), 201-202. [5] m. ghandehari, r. pfiefer, polygonal circles, mathematics and computer education journal, 29 (2) (1995), 203 – 210. [6] s. go lab, quelques problémes métrique de la géométrie de minkowski, trav. l’acad. mines cracovie 6 (1932), 1-79. [7] b. grünbaum, self-circumference of convex sets, colloq. math. 13 (1964), 55-57. 28 m. ghandehari, h. martini [8] p.c. hammer, unsolved problems, in proc. sympos. pure math. 7 (ed. v. klee), amer. math. soc., providence, r.i., 1963, 498-499. [9] h. martini, a. shcherba, on the self-perimeter of quadrangles for gauges, beitr. algebra geom. 52 (1) (2011), 191-203. [10] h. martini, a. shcherba, on the self-perimeter of pentagonal gauges, aequationes math. 84 (1-2) (2012), 157-183. [11] h. martini, a. shcherba, upper estimates on self-perimeters of unit circles for gauges, colloq. math. 142 (2) (2016), 179-210. [12] h. martini, k.j. swanepoel, antinorms and radon curves, aequationes math. 72 (1-2) (2006), 110-138. [13] h. martini, k.j. swanepoel, g. weiss, the geometry of minkowski spaces – a survey. i, expo. math. 19 (2) (2001), 97-142. [14] h. minkowski, theorie der konvexen körper, insbesondere begründung ihres oberflächenbegriffs, in “gesammelte abhandlungen, vol. 2”, teubner, leipzig-berlin, 1911, 131-229. [15] g. pólya, g. szegö, “problems and theorems in analysis”, vol. 2, springer-verlag, new york-heidelberg, 1976. [16] j.j. schäffer, the self-circumferences of polar convex disks, arch. math. (basel) 24 (1973), 87-90. [17] a.i. shcherba, unit disk of smallest self-perimeter in the minkowski plane, mat. zametki 81 (1) (2007), 125-135 (translation in math. notes 81 (1-2) (2007), 108-116). [18] a.c. thompson, an equiperimetric property of minkowski circles, bull. london math. soc. 7 (3) (1975), 271-272. [19] a.c. thompson, “minkowski geometry”, encyclopedia of mathematics and its applications, 63, cambridge university press, cambridge, 1996. introduction preliminaries polygons curves � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 1 (2019), 123 – 134 doi:10.17398/2605-5686.34.1.123 available online february 8, 2019 characterizations of minimal hypersurfaces immersed in certain warped products eudes l. de lima 1, henrique f. de lima 2,@ , eraldo a. lima jr 3, adriano a. medeiros 3 1 unidade acadêmica de ciências exatas e da natureza universidade federal de campina grande, 58900–000 cajazeiras, paráıba, brazil, eudes.lima@ufcg.edu.br 2 departamento de matemática, universidade federal de campina grande, 58.429–970 campina grande, paráıba, brazil, henrique@mat.ufcg.edu.br 3 departamento de matemática, universidade federal da paráıba, 58.051–900 joão pessoa, paráıba, brazil, eraldo@mat.ufpb.br , adrianoalves@mat.ufpb.br received october 4, 2018 presented by teresa arias-marco accepted january 15, 2019 abstract: our purpose in this paper is to investigate when a complete two-sided hypersurface immersed with constant mean curvature in a killing warped product mn×ρr, whose riemannian base mn has sectional curvature bounded from below and such that the warping function ρ ∈ c∞(m) is supposed to be concave, is minimal (and, in particular, totally geodesic) in the ambient space. our approach is based on the application of the well known generalized maximum principle of omoriyau. key words: killing warped product, constant mean curvature hypersurfaces, minimal hypersurfaces, totally geodesic hypersurfaces. ams subject class. (2010): primary 53c42; secondary 53b30 and 53c50. 1. introduction killing vector fields are important objects which have been widely used in order to understand the geometry of submanifolds and, more particularly, of hypersurfaces immersed in riemannian spaces. into this branch, aĺıas, dajczer and ripoll [1] extended classical bernstein’s theorem [4] to the context of complete minimal surfaces in riemannian spaces of nonnegative ricci curvature carrying a killing vector field. this was done under the assumption that the sign of the angle function between a global gauss mapping and the killing vector field remains unchanged along the surface. afterwards, dajczer, hinojosa and de lira [10] defined a notion of killing graph in a class of riemannian manifolds endowed with a killing vector field and solved the corresponding dirichlet problem for prescribed mean curvature under hypothesis involving @ corresponding author issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.1.123 mailto:eudes.lima@ufcg.edu.br mailto:henrique@mat.ufcg.edu.br mailto:eraldo@mat.ufpb.br mailto:adrianoalves@mat.ufpb.br mailto:henrique@mat.ufcg.edu.br https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 124 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros domain data and the ricci curvature of the ambient space. more recently, dajczer and de lira [7] showed that an entire killing graph of constant mean curvature lying inside a slab must be a totally geodesic slice, under certain restrictions on the curvature of the ambient space. to prove their bernstein type result, they used as main tool the generalized maximum principle of omori [11] and yau [15] for the laplacian in the sense of pigola, rigoli and setti given in [13] (see also [2] for a modern and accessible reference to the generalized maximum principle of omori-yau). when the ambient space is a riemannian product of the type mn × r, it was shown by rosenberg, schulze and spruck [14] that if the ricci curvature of the base mn is nonnegative and its sectional curvature is bounded from below, then any entire minimal graph over mn with nonnegative height function must be a slice. this result extends the celebrated theorem due to bombieri, de giorgi and miranda [5] for entire minimal hypersurfaces in the euclidean space. in [8], the second and third authors jointly with parente studied complete two-sided hypersurfaces immersed in mn × r, whose base is also supposed to have sectional curvature bounded from below. in this setting, they extended a technique developed in [9] obtaining sufficient conditions which assure that such a hypersurface is a slice of the ambient space, provided that its angle function has some suitable behavior. we recall that a hypersurface is said to be two-sided if its normal bundle is trivial, that is, there exists on it a globally defined unit normal vector field. these aforementioned works allow us to discuss a natural question: question. under what reasonable geometric restrictions on a riemannian manifold m n+1 endowed with a killing vector field must a complete two-sided hypersurface σn immersed with constant mean curvature in m n+1 be minimal and, in particular, totally geodesic in this ambient space? as is well known, under suitable assumptions on such a killing vector field, the ambient space m n+1 can be regarded as a killing warped product mn×ρr, for an appropriate n-dimensional riemannian base mn and a certain warping function ρ ∈ c∞(m). assuming that the base mn has sectional curvature bounded from below and supposing that the warping function ρ is concave on mn, our purpose in this paper is just to present satisfactory answers for the above stated question. for this, in order to use the generalized maximum principle of omori-yau, first we establish sufficient conditions to guarantee that the ricci curvature of a complete two-sided hypersurface is bounded from below (see proposition 1). afterwards, in section 4 we state characterizations of minimal hypersurfaces 125 and prove our main results (see theorems 1 and 2, and corollaries 1 and 2). finally, we also discuss the plausibility of the assumptions assumed in our results (see remark 1). 2. killing warped products let m n+1 be an (n + 1)-dimensional riemannian manifold endowed with a killing vector field k. suppose that the distribution d orthogonal to k is of constant rank and integrable. we denote by ψ : mn × i → mn+1 the flow generated by k, where mn is an arbitrarily fixed integral leaf of d labeled as t = 0, which we will suppose to be connected, and i is the maximal interval of definition. without loss of generality, in what follows we will also consider i = r. in this setting, m n+1 can be regard as the killing warped product mn×ρr, that is, the product manifold mn ×r endowed with the warping metric 〈 , 〉 = π∗m (〈 , 〉m ) + (ρ◦πm ) 2π∗r ( dt2 ) , (2.1) where πm and πr denote the canonical projections from m × r onto each factor, 〈 , 〉m is the induced riemannian metric on the base mn and the warping function ρ ∈ c∞ is ρ = |k| > 0, where | · | denotes the norm of a vector field on mn+1. throughout this work, we will deal with hypersurfaces ψ : σn → mn+1 immersed in a killing warped product m n+1 = mn ×ρ r and which are twosided. this condition means that there exists a globally defined unit normal vector field n on σn. let ∇, ∇ and d denote the levi-civita connections in m n+1 , σn and mn, respectively. then, as in [12], the curvature tensor r of the hypersurface σn is given by r(x,y )z = ∇[x,y ] − [∇x,∇y ]z, where [ , ] denotes the lie bracket and x,y,z ∈ x(σ). a well known fact is that the curvature tensor r of the hypersurface σn can be described in terms of the shape operator a and of the curvature tensor r of the ambient space m n+1 = mn ×ρ r by the gauss equation given by r(x,y )z = ( r(x,y )z )> + 〈ax,z〉ay −〈ay,z〉ax, (2.2) 126 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros for every tangent vector fields x,y,z ∈ x(σ), where ( )> denotes the tangential component of a vector field in x(m) along σn. in this paper, we will also consider two particular smooth functions on a connected two-sided hypersurface ψ : σn → mn+1 immersed in a killing warped product m n+1 = mn ×ρ r, namely, the (vertical) height function h = πr ◦ ψ and the angle function θ = 〈n,k〉, where we recall that n denotes the unit normal vector field globally defined on σn. from the decomposition k = k> + θn, it is easy to see that ∇h = 1 ρ2 k> and |∇h|2 = ρ2 − θ2 ρ4 . (2.3) moreover, assuming the constancy of the mean curvature function h = 1 n trace(a), from proposition 2.12 of [3] (see also proposition 6 of [1] or proposition 2.1 of [6]) we have the following formula ∆θ = − ( ric(n,n) + |a|2 ) θ, (2.4) where ric denotes the ricci tensor of m n+1 and |a| stands for the hilbertschmidt norm of the shape operator a of σn. finally, we also recall that it holds the following algebraic relation |a|2 = nh2 + n(n− 1)(h2 −h2), (2.5) where h2 = 2 n(n− 1) s2 is the mean value of the second elementary symmetric function s2 on the eigenvalues of a. 3. auxiliary results in order to prove our main theorems in the next section, we will need use two auxiliary results. the first one is the well known generalized maximum principle of omori [11] and yau [15], which is quoted below (see also [2] for a modern and accessible reference to the generalized maximum principle of omori-yau). lemma 1. let σn be a n-dimensional complete riemannian manifold whose ricci curvature is bounded from below and let u : σn → r be a smooth function satisfying inf σ u > −∞. then, there exists a sequence of points {pk}⊂ σn such that lim k u(pk) = inf σ u, lim k |∇u(pk)| = 0 and lim inf k ∆u(pk) ≥ 0 . characterizations of minimal hypersurfaces 127 the next auxiliary result will give sufficient conditions to guarantee that the ricci curvature of a two-sided hypersurface immersed in a killing warped product m n+1 = mn ×ρ r is bounded from below. in order to prove this result, we will develop some preliminaries computations. let us consider a two-sided hypersurface ψ : σn → mn+1 immersed in killing warped product m n+1 = mn ×ρ r. for vector fields u,v,w tangent to m n+1 , we can write u = u∗ + û, where u∗ and û are the orthogonal projections of u onto tm and tr, respectively. thus, û = 〈u,k〉〈k,k〉 k = 〈u,k〉 ρ2 k, where (as in the previous section) k = ∂t. thus, with a straightforward computation it is not difficult to verify that r(u,v )w = rm (u ∗,v ∗)w∗ − 〈v,k〉 ρ2 r(k,u∗)w∗ + 〈v,k〉〈w,k〉 ρ4 r(u∗,k)k + 〈u,k〉 ρ2 r(k,v ∗)w∗ − 〈u,k〉〈w,k〉 ρ4 r(v ∗,k)k. then, from lemma 7.34 and proposition 7.42 of [12] we get r(u,v )w = rm (u ∗,v ∗)w∗ − 〈v,k〉hessm ρ(u∗,w∗) ρ3 k + 〈v,k〉〈w,k〉〈k,k〉 ρ5 ∇u∗∇(ρ◦πm ) + 〈u,k〉hessm ρ(v ∗,w∗) ρ3 k − 〈u,k〉〈w,k〉〈k,k〉 ρ5 ∇v∗∇(ρ◦πm ), where hessm is the hessian on m n. so, we have that r(u,v )w = rm (u ∗,v ∗)w∗ − 〈v,k〉hessm ρ(u∗,w∗) ρ3 k + 〈v,k〉〈w,k〉 ρ3 du∗dρ + 〈u,k〉hessm ρ(v ∗,w∗) ρ3 k − 〈u,k〉〈w,k〉 ρ3 dv∗dρ. 128 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros in particular, taking a local orthonormal frame {e1, . . . ,en} tangent to σn and x a vector field tangent to σn, we can take u = w = x and v = ei in the last equation to obtain r(x,ei)x = rm (x ∗,e∗i )x ∗ − 〈ei,k〉hessm ρ(x∗,x∗) ρ3 k + 〈ei,k〉〈x,k〉 ρ3 dx∗dρ + 〈x,k〉hessm ρ(e∗i ,x ∗) ρ3 k − 〈x,k〉2 ρ3 de∗i dρ. hence, we conclude that 〈 r(x,ei)x,ei 〉 = 〈rm (x∗,e∗i )x ∗,ei〉− 〈ei,k〉2 ρ3 hessm ρ(x ∗,x∗) + 〈ei,k〉〈x,k〉 ρ3 〈dx∗dρ,ei〉− 〈x,k〉2 ρ3 〈de∗i dρ,ei〉 + 〈ei,k〉〈x,k〉 ρ3 hessm ρ(e ∗ i ,x ∗) = 〈rm (x∗,e∗i )x ∗,e∗i 〉− 〈ei,k〉2 ρ3 hessm ρ(x ∗,x∗) + 〈ei,k〉〈x,k〉 ρ3 hessm ρ(x ∗,e∗i ) + 〈ei,k〉〈x,k〉 ρ3 hessm ρ(x ∗,e∗i ) − 〈x,k〉2 ρ3 hessm (e ∗ i ,e ∗ i ). consequently, we get〈 r(x,ei)x,ei 〉 = km (x ∗,e∗i ) ( 〈x∗,x∗〉〈e∗i ,e ∗ i 〉−〈x ∗,e∗i 〉 2 ) − 〈ei,k〉2 ρ3 hessm ρ(x ∗,x∗) − 〈x,k〉2 ρ3 hessm ρ(e ∗ i ,e ∗ i ) + 2 〈ei,k〉〈x,k〉 ρ3 hessm ρ(x ∗,e∗i ) = km (x ∗,e∗i ) ( 〈x∗,x∗〉〈e∗i ,e ∗ i 〉−〈x ∗,e∗i 〉 2 ) − 1 ρ hessm ρ ( x̃∗i ,x̃ ∗ i ) + 2 ρ hessm ρ ( x̃∗i , ẽ ∗ i ) − 1 ρ hessm ρ ( ẽ∗i , ẽ ∗ i ) , where x̃∗i = 〈ei,k〉 ρ x∗ and ẽ∗i = 〈x,k〉 ρ e∗i . characterizations of minimal hypersurfaces 129 hence,〈 r(x,ei)x,ei 〉 = km (x ∗,e∗i ) ( 〈x∗,x∗〉〈e∗i ,e ∗ i 〉−〈x ∗,e∗i 〉 2 ) − 1 ρ hessm ρ ( x̃∗i − ẽ ∗ i ,x̃ ∗ i − ẽ ∗ i ) . (3.1) therefore, we obtain that n∑ i=1 〈 r(x,ei)x,ei 〉 = n∑ i=1 km (x ∗,e∗i ) ( 〈x∗,x∗〉〈e∗i ,e ∗ i 〉−〈x ∗,e∗i 〉 2 ) − n∑ i=1 1 ρ hessm ρ ( x̃∗i − ẽ ∗ i ,x̃ ∗ i − ẽ ∗ i ) . (3.2) at this point, we recall that a concave function defined on a riemannian manifold mn is a smooth function ρ ∈ c∞(m) whose hessian operator hessmρ is negative semidefinite. now, we are in position to establish the following result: proposition 1. let m n+1 = mn×ρ r be a killing warped product with concave warping function ρ and whose base mn has sectional curvature satisfying km ≥−κ, for some constant κ ≥ 0. let ψ : σn → m n+1 be a two-sided hypersurface with bounded mean curvature h and h2 bounded from below. then, the ricci curvature ric of σn is bounded from below. proof. from the gauss equation (2.2), taking a local orthonormal frame {e1, . . . ,en} tangent to σn, we have that the ricci curvature ric of σn is given by ric(x,x) = n∑ i=1 〈r(x,ei)x,ei〉 + nh〈ax,x〉−〈ax,ax〉, (3.3) for all vector field x tangent to σn. now, we observe that, for each i = 1, . . . ,n, (2.3) implies that 〈x∗,x∗〉〈e∗i ,e ∗ i 〉 = 〈x −〈x,∇h〉k,x −〈x,∇h〉k〉〉〈ei −〈ei,∇h〉k,ei −〈ei,∇h〉k〉 = |x|2 −ρ2|x|2〈ei,∇h〉2 −ρ2〈x,∇h〉2 + ρ4〈x,∇h〉2〈ei,∇h〉2 and 130 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros 〈x∗,e∗i 〉 2 = 〈x −〈x,∇h〉k,ei −〈ei,∇h〉k〉2 = 〈x,ei〉2 − 2ρ2〈x,∇h〉〈x,ei〉〈ei,∇h〉 + ρ4〈x,∇h〉2〈ei,∇h〉2. consequently, we get n∑ i=1 〈x∗,x∗〉〈e∗i ,e ∗ i 〉−〈x ∗,e∗i 〉 2 = (n− 1)|x|2 −ρ2|x|2|∇h|2 − (n− 2)ρ2〈x,∇h〉2 ≤ (n− 1)|x|2. hence, taking into account our constraint on the sectional curvature of mn, we obtain n∑ i=1 km (x ∗,e∗i ) ( 〈x∗,x∗〉〈e∗i ,e ∗ i 〉−〈x ∗,e∗i 〉 2 ) ≥−(n− 1)κ|x|2. (3.4) on the other hand, we have that nh〈ax,x〉−〈ax,ax〉≥−|a|(|nh| + |a|)|x|2 for all tangent vector field x ∈ x(σ). since ρ is concave, it follows from (3.2), (3.3) and (3.4) that ric(x,x) ≥− ( (n− 1)κ + |a|(|nh| + |a|) ) |x|2. therefore, taking into account relation (2.5), our hypothesis on h and h2 assure that the ricci curvature ric of σn is bounded from below. 4. main results in this section, we present our main results concerning the characterization of minimal (and, in particular, totally geodesic) complete two-sided hypersurfaces immersed in a killing warped product. so, we state and prove our first one. theorem 1. let m n+1 = mn ×ρ r be a killing warped product with concave warping function ρ and whose base (not necessarily complete) mn has nonnegative sectional curvature km . let ψ : σ n → mn+1 be a complete two-sided hypersurface with constant mean curvature h and h2 bounded from below. suppose that the angle function θ of σn is bounded away from zero. then, σn is minimal. moreover, if h2 is constant, then σ n is totally geodesic. characterizations of minimal hypersurfaces 131 proof. firstly, since we are assuming that θ is bounded away from zero, for an appropriated choice of n we can suppose that θ > 0 and, consequently, inf σ θ > 0. then, taking into account proposition 1, we can apply lemma 1 to guarantee the existence of a sequence of points {pk}⊂ σn such that lim k θ(pk) = inf σ θ and lim inf k ∆θ(pk) ≥ 0 . on the other hand, from corollary 7.43 of [12] we get ric(n,n) = ric(n∗,n∗) + ric(n⊥,n⊥) (4.1) = ricm (n ∗,n∗) − 1 ρ hessm ρ(n ∗,n∗) −〈n⊥,n⊥〉 ∆mρ ρ = ricm (n ∗,n∗) − 1 ρ hessm ρ(n ∗,n∗) − θ2 ρ3 ∆mρ, where hessm and ∆m are the hessian and the laplacian on m n, respectively. thus, from (2.4) and (4.1) we obtain the following formula ∆θ = − ( ricm (n ∗,n∗) − 1 ρ hessm ρ(n ∗,n∗) − θ2 ρ3 ∆mρ + |a|2 ) θ . (4.2) since ρ is concave, from (4.2) we have that ∆θ ≤− ( ricm (n ∗,n∗) + |a|2 ) θ . (4.3) so, taking into account relation (2.5) jointly with the hypothesis that h is constant and h2 is bounded from below, it follows from (4.3) that 0 ≤ lim inf k ∆θ(pk) ≤− lim k ( ricm (n ∗,n∗) + |a|2 ) θ(pk) ≤− lim k ( ricm (n ∗,n∗) + nh2 ) θ(pk) ≤ 0 . (4.4) consequently, since ricm is nonnegative and inf σ θ > 0, we conclude that h = 0, that is, σn is minimal. finally, assuming that h2 is constant, from relation (2.5) we obtain that |a| is also constant. therefore, returning to (4.4) we get that |a| must be identically zero and, hence, σn is totally geodesic. it is not difficult to verify that from the proof of theorem 1 we obtain the following result: 132 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros corollary 1. let m n+1 = mn ×ρ r be a killing warped product with concave warping function ρ and whose base (not necessarily complete) mn has nonnegative sectional curvature km . let ψ : σ n → mn+1 be a complete two-sided hypersurface with constant mean curvature h and h2 ≥ 0 (not necessarily constant). suppose that the angle function θ of σn is bounded away from zero. then, σn is totally geodesic. proceeding, we consider the case that the sectional curvature of the riemannian base of the ambient space can be negative. in order to obtain our next result we need to assume a suitable constraint on the norm of gradient of the height function. more precisely, we get the following result: theorem 2. let m n+1 = mn ×ρ r be a killing warped product with concave warping function ρ and whose base (not necessarily complete) mn has sectional curvature satisfying km ≥ −κ, for some constant κ > 0. let ψ : σn → mn+1 be a complete two-sided hypersurface with constant mean curvature h and h2 bounded from below. suppose that the angle function θ of σn is bounded away from zero. if the height function of σn satisfies |∇h|2 ≤ α (n− 1)κρ2 |a|2, (4.5) for some constant 0 < α < 1, then σn is minimal. moreover, if h2 is constant, then σn is a slice and mn is complete. proof. as in the the proof of theorem 1, we can choose n such that inf σ θ > 0. then, taking into account our constraint on km , it follows from (4.3) that ∆θ ≤ ( (n− 1)κρ2|∇h|2 −|a|2 ) θ (4.6) using our hypothesis (4.5), from (4.6) we have that ∆θ ≤ (α− 1)|a|2θ . (4.7) thus, in a similar way of the proof of theorem 1, it follows from (4.7) that there exists a sequence of points {pk}⊂ σn such that 0 ≤ lim inf k ∆θ(pk) ≤ lim k ( (α− 1)|a|2θ ) (pk) = (α− 1) inf σ θ lim k |a|2(pk) ≤ 0 . (4.8) hence, from (4.8) we obtain that limk |a|2(pk) = 0. consequently, since h is constant and (2.5) implies that nh2 ≤ |a|2, we conclude that h = 0, characterizations of minimal hypersurfaces 133 that is, σn is minimal. assuming that h2 is constant, using again relation (2.5) we obtain that |a| must be identically zero. therefore, using once more hypothesis (4.5) we get that σn is a slice and, in particular, mn is complete. from the proof of theorem 2 we also get the following consequence: corollary 2. let m n+1 = mn ×ρ r be a killing warped product with concave warping function ρ and whose base (not necessarily complete) mn has sectional curvature satisfying km ≥ −κ, for some constant κ > 0. let ψ : σn → mn+1 be a complete two-sided hypersurface with constant mean curvature h and h2 ≥ 0 (not necessarily constant). suppose that the angle function θ of σn is bounded away from zero. if the height function of σn satisfies condition (4.5), then σn is a slice and mn is complete. we close our paper discussing the plausibility of the assumptions assumed in theorems 1 and 2. remark 1. we observe that theorem 1 does not hold when the base of the ambient space has negative sectional curvature and that hypothesis (4.5) in theorem 2 cannot be extended for α = 1. indeed, let h2 = {(x,y) ∈ r2 : y > 0} be the 2-dimensional hyperbolic space endowed with its canonical complete metric 〈 , 〉h2 = 1 y2 ( dx2 + dy2 ) and let u : h2 → r be the smooth function given by u(x,y) = a ln y, where a ∈ r\{0}. let us consider the entire vertical graph σ(u) = {(x,y,u(x,y)) : y > 0}⊂ h2 ×r. according to example 10 in [8], σ(u) has constant mean curvature h = a 2(1 + a2)1/2 and h2 = 0. moreover, its angle function is given by θ = 1 (1 + a2)1/2 > 0 and its height function h satisfies |∇h|2 = |du|2h2 1 + |du|2h2 = a2 1 + a2 = |a|2. 134 e.l. de lima, h.f. de lima, e.a. lima jr, a.a. medeiros acknowledgements the authors would like to thank the referee for his/her valuable suggestions and useful comments which improved the paper. the second author is partially supported by cnpq, brazil, grant 303977/2015-9. references [1] l.j. aĺıas, m. dajczer , j.b. ripoll, a bernstein-type theorem for riemannian manifolds with a killing field, ann. glob. anal. geom. 31 (2007), 363 – 373. [2] l.j. aĺıas, p. mastrolia, m. rigoli, “ maximum principles and geometric applications ”, springer monographs in mathematics, new york, 2016. [3] j.l.m. barbosa, m. do carmo, j. eschenburg, stability of hypersurfaces with constant mean curvature, math. z. 197 (1988), 123 – 138. [4] s. bernstein, sur les surfaces définies au moyen de leur courboure moyenne ou totale, ann. ec. norm. sup. 27 (1910), 233 – 256. [5] e. bombieri, e. de giorgi, m. miranda, una maggiorazione a priori relativa alle ipersuperfici minimali non parametriche, arch. ration. mech. anal. 32 (1969), 255 – 267. [6] a. caminha, h.f. de lima, complete vertical graphs with constant mean curvature in semi-riemannian warped products, bull. belgian math. soc. simon stevin 16 (2009), 91 – 105. [7] m. dajczer, j.h. de lira, entire bounded constant mean curvature killing graphs, j. math. pures appl. 103 (2015), 219 – 227. [8] h.f. de lima, e.a. lima jr., u.l. parente, hypersurfaces with prescribed angle function, pacific j. math. 269 (2014), 393 – 406. [9] h.f. de lima, u.l. parente, a bernstein type theorem in r×hn, bull. brazilian math. soc. 43 (2012), 17 – 26. [10] m. dajczer, p. hinojosa, j.h. de lira, killing graphs with prescribed mean curvature, calc. var. pde 33 (2008), 231 – 248. [11] h. omori, isometric immersions of riemannian manifolds, j. math. soc. japan 19 (1967), 205 – 214. [12] b. o’neill, “ semi-riemannian geometry with applications to relativity ”, academic press, london, 1983. [13] s. pigola, m. rigoli, a.g. setti, maximum principles on riemannian manifolds and applications, mem. american math. soc. 174, number 822, 2005. [14] h. rosenberg, f. schulze, j. spruck, the half-space property and entire positive minimal graphs in m × r, j. diff. geom. 95 (2013), 321 – 336. [15] s.t. yau, harmonic functions on complete riemannian manifolds, comm. pure appl. math. 28 (1975), 201 – 228. introduction killing warped products auxiliary results main results � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 1 (2022), 75 – 90 doi:10.17398/2605-5686.37.1.75 available online november 17, 2021 on isolated points of the approximate point spectrum of a closed linear relation m. lajnef, m. mnif department of mathematics, faculty of sciences of sfax, university of sfax route de soukra km 3.5, bp 1171, 3000 sfax, tunisia maliklajnaf@gmail.com , maher.mnif@gmail.com received june 28, 2021 presented by m. mbekhta accepted september 30, 2021 abstract: we investigate in this paper the isolated points of the approximate point spectrum of a closed linear relation acting on a complex banach space by using the concepts of quasinilpotent part and the analytic core of a linear relation. key words: linear relation, isolated point of the approximate point spectrum, analytic core, quasinilpotent part. msc (2020): 47a06, 47a10. 1. introduction and preliminaries throughout this paper, (x,‖.‖) will denote a complex banach space. in 2008, gonzález et al. [8] have shown that if 0 is isolated in the approximate point spectrum of a bounded operator t, then the quasinilpotent part h0(t) and the analytic core k(t) of t are closed, h0(t)∩k(t) = {0}, h0(t)⊕k(t) is closed and there exists λ0 6= 0 such that h0(t) ⊕k(t) = k(t −λ0i) = ⋂∞ n=0 im(t −λ0i)n. in recent years, the study of isolated spectral points of a multivalued linear operator (linear relation) has generated a great deal of research attention. it was proved in [9] that for a closed and bounded linear relation t such that 0 is a point of its spectrum, we have the equivalence: 0 is isolated in the spectrum of t ⇐⇒ { h0(t) and k(t) are closed and x = h0(t) ⊕k(t). the previous studies on isolated spectral points in the two cases of linear operators and relations and their extensions motivate us to focus on establishing issn: 0213-8743 (print), 2605-5686 (online) c© the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.37.1.75 mailto:maliklajnaf@gmail.com mailto:maher.mnif@gmail.com https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 76 m. lajnef, m. mnif some necessary conditions for which a point of the approximate point spectrum of a closed linear relation be isolated. by the way, this work could be considered as an extension of the study carried out for the case of operators since it covers the case of closed operators which are not necessary bounded. the importance of the investigation of linear relations is shown by the examples of issues in the study of some cauchy problems associated with parabolic type equations in banach spaces [6]. thus, the generalization of the existing results for bounded operators to the general setting of closed linear relations seems to appear quite naturally. we recall now some basic definitions and properties which are needed in the sequel. a linear relation (or a multivalued linear operator) in a banach space x, t : x → x, is a mapping from a subspace d(t), called the domain of t into the set of nonempty subsets of x verifying t(α1x + α2y) = α1t(x) + α2t(y) for all non zero scalars α1,α2 and vectors x and y ∈ d(t). we denote by lr(x) the class of all linear relations in x. a linear relation t ∈ lr(x) is completely determined by its graph defined by g(t) := {(x,y) ∈ x ×x : x ∈ d(t),y ∈ tx}. let t ∈lr(x). the inverse of t is the relation t−1 given by g(t−1) := {(u,v) ∈ x×x : (v,u) ∈ g(t)}. we say that t is closed if its graph is a closed subspace of x × x, open if γ(t) > 0, where γ(t) =   ∞ if d(t) ⊆ ker(t), inf { ‖tx‖ d(x, ker(t)) : x ∈ d(t)\ker(t) } otherwise. the set of all closed linear relations is denoted by cr(x). we say that t is continuous if the operator qtt is continuous when qt denoted the quotient map from x onto x t(0) . in such a case the norm of t is defined by ‖t‖ := ‖qtt‖. we say that t is bounded if it is continuous and everywhere defined. the set of all bounded and closed linear relations acting between two banach spaces x and y is denoted by bcr(x,y ). if x = y , we write bcr(x,x) := bcr(x). the subspaces ker(t) := t−1(0) and im(t) := t(d(t)) are called respectively the null space and the range space of t. we say that t is surjective if t(d(t)) = x and injective if ker(t) = {0}. note that t is an operator if and only if t(0) = {0}. in addition, the generalized range of t is defined by r∞(t) := ⋂ n∈n im(tn). isolated points of the approximate point spectrum 77 for linear relations s, t ∈ lr(x) the relations s + t, st and s+̂t are defined respectively by s + t := {(x,y + z) : (x,y) ∈ g(s) and (x,z) ∈ g(t)}, st := {(x,z) : (x,y) ∈ g(t) and (y,z) ∈ g(s) for some y ∈ x}, s+̂t := {(x + u,y + v) : (x,y) ∈ g(s) and (u,v) ∈ g(t)}. this last sum is direct when g(s) ∩g(t) = {(0, 0)}. in such case, we write s⊕t. we denote by b(x,y ) the banach algebra of all bounded operators on x and y . if x = y , we write b(x,x) := b(x). recall that a linear relation t is regular if im(t) is closed and ker(tn) ⊆ im(tm) for all n, m ∈ n. the class of all regular linear relations in x will be denoted by r(x). in what follows we write reg(t) := {λ ∈ c : t −λi is regular}. for r > 0 we denote d(0,r) := {λ ∈ c : 0 ≤ |λ| < r} and d∗(0,r) := d(0,r)\{0}. now, we essentially aim to define and study some basic tools of the spectral theory. given a closed linear relation t. for λ ∈ c, we denote by rλ(t) = (λi −t)−1 the resolvent of t at λ. the resolvent set of t is the set defined by ρ(t) = {λ ∈ c : (λi −t)−1 is everywhere defined and single valued}. we say that t is invertible if 0 ∈ ρ(t). the spectrum of t is the set σ(t) = c\ρ(t). furthermore, we say that t is bounded below if there exists some δ > 0 such that δ‖x‖≤‖tx‖ for every x ∈ d(t). the approximate point spectrum of t is defined by σap(t) = {λ ∈ c : t −λi is not bounded below}. let us state now a useful lemma that we use below. lemma 1.1. let t ∈cr(x). then we have the following assertions. (i) if t is surjective then there exists � > 0 such that t −λi is surjective for every |λ| < �. (ii) if t is bounded below then there exists � > 0 such that t−λi is bounded below for every |λ| < �. 78 m. lajnef, m. mnif proof. the proof is similar to the proof of [10, lemma 15]. the structure of this paper is as follows. in section 2, we are mainly interested in studying the quasinilpotent part and the analytic core of a closed linear relation. most properties of these latter subspaces are also gathered. the stated results generalize the concepts of quasinilpotent part and the analytic core recently introduced in [12] to the setting of closed not necessary bounded linear relations. section 3 begins by a generalization to the case of closed linear relations of [9, theorem 3.1] stated above. after that, we develop a significant quantity of interesting technical lemmas. in particular, we set out the concepts of regular linear relations and gap of two subspaces. this leads us to find some necessary conditions for which a point of the approximate point spectrum be isolated. 2. quasinilpotent part and analytic core of a closed linear relation let t ∈cr(x). consider the graph norm ‖.‖t on d(t) defined by ‖x‖t := ‖x‖ + ‖tx‖. in what follows xt denotes the space d(t) endowed with the graph norm. observe that xt is a banach space (since qtt is a closed operator). consider the relation t̃ defined by t̃ : xt −→ x x 7−→ tx. evidently, t̃ is closed and d(t̃) = d(t). then, by virtue of [5, ii.5.1] we get that t̃ ∈bcr(xt ,x). remark 2.1. note that qt t̃ : xt → xt(0) is bounded. moreover, for all x ∈ d(t), ‖qtitx‖ = d(x,t(0)) ≤ ‖x‖ ≤ ‖x‖t , then, qtit is bounded, where it : xt −→ x x 7−→ x. now, let’s further extend the concepts of quasinilpotent part and the analytic core developed in [11, 12] to the case of closed not necessary bounded linear relations. isolated points of the approximate point spectrum 79 definition 2.1. let t ∈cr(x). (i) the quasinilpotent part of t, denoted by h0(t), is the set of all x ∈ d(t) for which there exists a sequence (xn)n ⊆ d(t) satisfying x0 = x, xn+1 ∈ txn for all n ∈ n and ‖xn‖ 1 n t → 0. (ii) the analytic core of t, denoted by k(t), is defined as the set of all x ∈ x for which there exist c > 0 and a sequence (xn)n∈n satisfying x0 = x and for all n ≥ 0, xn+1 ∈ d(t), xn ∈ txn+1 and d(xn, ker(t) ∩t(0)) ≤ cnd(x, ker(t) ∩t(0)). in the next lemma, we collect some elementary properties of h0(t) and k(t). lemma 2.1. let t ∈cr(x). then the following statements hold. (i) if f is a closed subspace of x such that t(f∩d(t)) ⊆ f , then h0(t)∩ f = h0(t|f). (ii) for λ 6= 0, h0(t) ⊆ (λi −t)h0(t) (the closure of (λi − t)h0(t) in x). (iii) t(d(t) ∩k(t)) = k(t). (iv) if f is a closed subspace of x such that t(d(t) ∩f) = f, then f ⊆ k(t). (v) if t ∈r(x), then k(t) = r∞(t) and it is closed. proof. (i) let x ∈ h0(t|f). then, by definition 2.1, there exists (xn)n ⊆ f ∩d(t) such that • x0 = x, • xn+1 ∈ t|fxn, • ‖xn‖ 1 n t |f → 0. but t|fxn = txn and ‖xn‖t |f = ‖xn‖t , then x ∈ h0(t) ∩f. conversely, assume that x ∈ h0(t)∩f. then there exists (xn)n ⊆ d(t) such that 80 m. lajnef, m. mnif • x0 = x ∈ f ∩d(t), • xn+1 ∈ txn, • ‖xn‖ 1 n t → 0. since t(f ∩ d(t)) ⊆ f, then (xn) ⊆ f and so, xn+1 ∈ t |fxn and ‖xn‖t |f = ‖xn‖t . consequently, x ∈ h0(t|f). (ii) let x ∈ h0(t). then, there exists (xn)n such that x0 = x, xn+1 ∈ txn and ‖xn‖ 1 n t −−−→n→∞ 0. let yn = ∑n k=0 xk λk+1 . as, for all n ∈ n, xn ∈ h0(t), then yn ∈ h0(t) and we have (λi −t)yn = x− xn+1 λn+1 + t(0). therefore, x− xn+1 λn+1 ∈ (λi −t)yn. whence, x− xn+1 λn+1 ∈ (λi −t)h0(t). using the fact that ‖xn‖ 1 n ≤‖xn‖ 1 n t → 0, one can deduce that xn+1 λn+1 −−−→ n→∞ 0, which implies that x ∈ (λ−t)h0(t). thus, h0(t) ⊆ (λi −t)h0(t). (iii) let prove the first inclusion t(d(t)∩k(t)) ⊆ k(t). let y ∈ t(d(t)∩ k(t)). then, there exists x ∈ d(t) ∩ k(t) such that y ∈ tx. since x ∈ k(t) then there exist δ > 0 and a sequence (xn)n such that • x0 = x, • for all n ≥ 0, xn+1 ∈ d(t) and xn ∈ txn+1, • d(xn, ker(t) ∩t(0)) ≤ δnd(x, ker(t) ∩t(0)) for all n ∈ n. let (yn)n be the sequence defined by yn+1 = xn for all n ∈ n and y0 = y. since xn−1 ∈ txn, then yn ∈ tyn+1 for all n ≥ 1. on the other hand, since y1 = x0 = x and y ∈ tx, then y0 ∈ ty1. we need only to prove that d(yn,t(0)∩ker(t) ≤ δ′nd(y,t(0)∩ker(t)) for some δ′ > 0. trivially, if y ∈ t(0) ∩ ker(t) there is nothing to prove. if not we get isolated points of the approximate point spectrum 81 that d(yn,t(0) ∩ ker(t) ≤ δ′nd(y,t(0) ∩ ker(t)) with δ′ > 0 be such that δ′ = max { δ , d(x,t(0) ∩ ker(t)) d(y,t(0) ∩ ker(t)) } . thus, y ∈ k(t). for the reverse inclusion, let x ∈ k(t). then there exists δ > 0 and a sequence (un)n such that • u0 = x, • for all n ≥ 0, un+1 ∈ d(t) and un ∈ tun+1, • d(un, ker(t) ∩t(0)) ≤ δnd(x, ker(t) ∩t(0)) for all n ∈ n. since x ∈ tu1 then, in order to show that k(t) ⊆ t(d(t) ∩ k(t)), it is sufficient to prove that u1 ∈ k(t). if u1 ∈ t(0) ∩ ker(t), then there is nothing to prove. if not, let (wn)n be the sequence such that wn = un+1. we have wn = un+1 ∈ tun+2 = twn+1. furthermore, d(wn, ker(t)∩t(0)) = d(un+1, ker(t)∩t(0)) ≤ δ′nd(u1, ker(t)∩t(0)), where δ′ = δ2 d(x,ker(t)∩t(0)) d(u1,ker(t)∩t(0)) . hence, u1 ∈ k(t). (iv) first, we claim that f ∩ d(t) is closed in xt . indeed, let (xn)n ⊆ f ∩d(t) be such that xn xt−−−→ n→∞ x. trivially, x ∈ d(t). on the other hand, we have ‖xn − x‖t −−−→ n→∞ 0. as f is closed in x, then x ∈ f. hence, f ∩d(t) is closed in xt , as claimed. recall that the relation t̃ is closed. let us consider t0 : d(t) ∩f → f, the restriction of t̃. we have g(t0) = g(t̃)∩((d(t̃)∩f)×f) is closed in xt ×x. then, t0 is closed. we have, by hypothesis, im(t0) = f then, by the open mapping theorem [5, theorem iii.4.2], we deduce that t0 is open. thus, there exists a constant γ > 0 such that for all x ∈ d(t0) = d(t) ∩f, dt (x, ker(t0)) ≤ γ‖t0x‖, where dt (x,g) := infα∈g‖x − α‖t . as, for all x ∈ d(t0) and α ∈ ker(t0), ‖x − α‖ ≤ ‖x − α‖t , then d(x, ker(t0)) ≤ dt (x, ker(t0)). hence, d(x, ker(t0)) ≤ γ‖t0x‖. (2.1) now, consider � > 0 and let u ∈ f. then, there exists x ∈ d(t) ∩ f such that u ∈ tx. by (2.1) there exists y ∈ ker(t0) ⊆ ker(t) such that 82 m. lajnef, m. mnif ‖x − y‖ ≤ (γ + �)d(u,t(0)). take u1 = x − y ∈ d(t) ∩ f. we have u ∈ t(u1) and d(u1,t(0) ∩ ker(t)) ≤ (γ + �)d(u,t(0) ∩ ker(t)). continuing in the same manner, we build a sequence (un)n such that u0 = u, for all n ≥ 0, un+1 ∈ d(t) ∩f, un ∈ tun+1 and d(un,t(0) ∩ ker(t)) ≤ (γ + �)nd(u,t(0) ∩ ker(t)). hence, u ∈ k(t). thus, f ⊆ k(t). (v) as t is regular then, by [2, proposition 2.5] and [1, lemma 20], we get that r∞(t) is closed and t(r∞(t) ∩ d(t)) = r∞(t). if follows from (iv) that r∞(t) ⊆ k(t). on the other hand, it is clear that k(t) ⊆ r∞(t). so, k(t) = r∞(t) which is closed, as desired. 3. isolated point of the approximate point spectrum the main objective of this section is to give necessary conditions to ensure that the approximate point spectrum of a closed linear relation t does not cluster at a point λ. to do this, we begin with a generalization to the case of closed linear relations of a theorem stated in [9] dealing with the characterization of isolated points of the spectrum of a bounded closed linear relation. for this, we need the following technical lemma. lemma 3.1. let t ∈cr(x) and x ∈ x. then, r̃.(t)x : ρ(t) −→ xt µ 7−→ r̃µ(t)x = rµ(t)x := (µi −t)−1x is analytic. proof. let λ ∈ ρ(t). by virtue of [5, corollary vi.1.9], we get that if |λ−µ| < ‖rλ(t)‖−1, then rµ(t) = ∑∞ n=0 rλ(t) n+1(µ−λ)n, which implies that r̃µ(t)x = ∑∞ n=0 r̃λ(t) n+1x(µ − λ)n. it was like proving that ∑∞ n=0 r̃λ(t) n+1x(µ − λ)n is convergent on xt . observe that isolated points of the approximate point spectrum 83 ‖r̃λ(t)n+1x‖t = ‖r̃λ(t)n+1x‖ + ‖tr̃λ(t)n+1x‖. moreover, we have ‖tr̃λ(t)n+1x‖ = ‖qttr̃λ(t)n+1x‖ = ‖qt (t −λi + λi)r̃λ(t)n+1x‖ = ‖qt [(t −λi)r̃λ(t)n+1x + λr̃λ(t)n+1x]‖ = ‖qtrλ(t)nx + λqtrλ(t)n+1x‖ ≤‖qtrλ(t)nx‖ + |λ|‖qtrλ(t)n+1x‖ ≤‖rλ(t)nx‖ + |λ|‖rλ(t)n+1x‖. then, ‖r̃λ(t)n+1x‖t ≤ (1 + |λ|)‖rλ(t)n+1x‖ + ‖r̃λ(t)nx‖. since∑ n≥0 rλ(t) n+1x(µ−λ)n, ∑ n≥0 rλ(t) nx(µ−λ)n are absolutely convergent in x, then ∑ n≥0 ‖r̃λ(t) nx‖t |µ−λ|n is convergent. therefore, ∑ n≥0 r̃λ(t) n+1x(µ−λ)n is convergent on xt , as required. theorem 3.1. let t ∈cr(x) and let λ0 ∈ σ(t). then k(λ0i−t) and h0(λ0i −t) are closed and x = h0(λ0i −t) ⊕k(λ0i −t) if and only if λ0 is an isolated point of σ(t). proof. recall that the quasinilpotent part of a closed linear relation is a subspace of xt . then, proceeding as in the proof of [9, theorem 3.1], and taking into account lemma 3.1, we obtain that h0(t) = im(b1) and k(t) = ker(b1) where b1 is the bounded projection defined as follows: b1 := 1 2πi ∮ γλ0 rλ(t)dλ, where γλ0 is a simple closed curve around λ0 such that the closure of the region bounded by γλ0 and containing λ0 intersects σ(t) only at λ0. now, we develop some supplementary technical lemmas that enable us to provide some necessary conditions for which a point of the approximate point spectrum be isolated. the stated notations and terminology are essentially adhered to [8]. using the proof of [1, theorem 23] and lemma 2.1 (v), we get the following lemma. lemma 3.2. let t ∈r(x) and λ ∈ k. then: 84 m. lajnef, m. mnif (i) γ(λi −t) ≥ γ(t) − 3|λ|. (ii) k(t) ⊆ k(λi −t) for all 0 < |λ| < γ(t) 3 . the following lemma gives an analytic core stability result for a regular linear relation. lemma 3.3. let t ∈r(x) and λ ∈ k. then there exists ν > 0 such that k(t) = k(t −λi) for all |λ| < ν. proof. let |µ0| < 17γ(t). then, by virtue of [1, theorem 23], we get that t −µ0i ∈r(x). according to lemma 3.2 (ii) we get that k(t −µ0i) ⊆ k(t −µ0i −λi) whenever |λ| < γ(t −µ0i) 4 . (3.1) as |µ0| < 17γ(t) then, by lemma 3.2 (i), we get that γ(t −µ0i) > 7|µ0|− 3|µ0| > 4|µ0|. hence, for λ = −µ0 in (3.1), we get that k(t − µ0i) ⊆ k(t). using again lemma 3.2, we get that k(t) ⊆ k(t −µ0i). this completes the proof. now, we recall some useful tools for the sequel. definition 3.1. let m and n be two closed subspaces of x. then the gap between m and n is defined by g(m,n) = max(δ(m,n)),δ(n,m)), with δ(m,n) = sup{d(x,n) : x ∈ m and‖x‖ = 1)}. the following lemma is fundamental to the proof of the main result of this section. lemma 3.4. let t ∈cr(x), ω be a connected component of reg(t) and λ0 ∈ reg(t). then: (i) the mapping λ → ker(t −λi) is continuous at λ0 in the gap metric. (ii) im(t −λ0i) is closed, im(t −λi) is closed in a neighborhood of λ0 and the mapping λ → im(t −λi) is continuous at λ0 in the gap metric. isolated points of the approximate point spectrum 85 (iii) ⋂ i≥0 im(t−λii) = ⋂ i>0 im(t−λii), with (λi)i>0 a sequence of distinct points of ω which converges to λ0. (iv) k(t−λ0i) = ⋂ i>0 im(t−λii) with (λi)i>0 a sequence of distinct points of ω which converges to λ0. proof. (i) and (ii) are proved in [3, theorem 3.3]. (iii) let 0 6= x0 ∈ ⋂ i>0 im(t −λii). then, for all i > 0, x0 ‖x0‖ ∈ im(t −λii). first, we note that d ( x0 ‖x0‖ , im(t −λ0i) ) = 1 ‖x0‖ d(x0, im(t −λ0i)). from the last equality and the definitions mentioned above, we deduce that g(im(t −λii), im(t −λ0i)) ≥ δ(im(t −λii), im(t −λ0i)) ≥ sup{d(x, im(t −λ0i) : x ∈ im(t −λii),‖x‖ = 1} ≥ d(x, im(t −λ0i)), ∀x ∈ im(t −λii),‖x‖ = 1 ≥ d ( x0 ‖x0‖ , im(t −λ0i) ) ≥ 1 ‖x0‖ d(x0, im(t −λ0i)). hence, d(x0, im(t −λ0i)) ≤ g(im(t −λii), im(t −λ0i))‖x0‖. this implies, by the use of (ii), that x0 ∈ im(t −λ0i) = im(t −λ0i). thus, x0 ∈ ⋂ i≥0 im(a−λii), as required. (iv) using lemma 3.3 and lemma 2.1 (v), we get that for all i ≥ 0, k(t −λ0i) = k(t −λii) = ⋂ n≥0 im(t −λii)n ⊆ im(t −λii). then, the lefthanded side is contained in the righthanded side. so, it is sufficient to show that ⋂ i>0 im(t − λii) ⊆ k(t − λ0i). to do this, let x ∈ ⋂ i>0 im(t − λii). then, by (iii), x ∈ ⋂ i≥0 im(t − λii). whence, x ∈ im(t − λ0i) and so, there exists y ∈ d(t) such that x ∈ (t − λ0i)y. which implies that for all i ≥ 1, x + (λ0 − λi)y ∈ 86 m. lajnef, m. mnif (t − λii)y. on the other hand, we have for all i ≥ 1, x ∈ im(t − λii). therefore, (λ0 − λi)y ∈ ⋂ i>0 im(t − λii). as λi 6= λ0, then y ∈ ⋂ i>0 im(t −λii). consequently, for all x ∈ ⋂ i>0 im(t −λii), there exists y ∈ ⋂ i>0 im(t −λii)∩d(t) such that x ∈ (t −λ0i)y. whence, (t−λ0i) (⋂ i>0 im(t −λii) ∩d(t) ) ⊇ ⋂ i>0 im(t−λii). on the other hand, it is clear to see that (t − λ0i) (⋂ i>0 im(t −λii) ∩d(t) ) ⊆⋂ i>0 im(t −λii). thus, (t −λ0i) (⋂ i>0 im(t −λii) ∩d(t) ) = ⋂ i>0 im(t −λii). using lemma 2.1 (iv), we get the desired inclusion. with all these auxiliary results behind us, we can now state our main result of this section. we establish a number of important necessary conditions for a point in the approximate point spectrum to be isolated. theorem 3.2. let t ∈ cr(x) and let 0 be an isolated point of σap(t). then: (i) h0(t) and k(t) are closed. (ii) h0(t) ∩k(t) = {0}. (iii) h0(t) ⊕k(t) is closed and there exists λ0 such that h0(t) ⊕k(t) = k(t −λ0i) = r∞(t −λ0i). proof. the proof follows the approach taken in [8, proposition 9] established in the setting of bounded operators. we divide the proof into two cases. first case: assume that t is surjective. then, by lemma 1.1, there exists θ > 0 such that t −λi is bijective for all λ ∈ d∗(0,θ). therefore, by virtue of theorem 3.1, we get h0(t) and k(t) are closed and x = h0(t) ⊕k(t). second case: assume that t is not surjective. since 0 is isolated in σap(t), then there exists µ > 0 such that t−λi is bounded below for each 0 < |λ| < µ. using lemma 3.3, the map λ → k(t − λi) is locally constant on d∗(0,µ). which implies that the map λ → k(t − λi) is constant on d∗(0,µ). now, fix λ0 ∈ d∗(0,µ). then, by virtue of lemma 2.1 (v), we get that k(t −λ0i) = r∞(t −λ0i) := x0 and it is closed. moreover, we have t(r∞(t − λ0i) ∩ d(t)) = (t − λ0i + λ0i)(r∞(t − λ0i) ∩ d(t)) ⊆ r∞(t − λ0i) + λ0r∞(t − λ0i) ⊆ r∞(t − λ0i). let t0 : isolated points of the approximate point spectrum 87 x0 ∩d(t) → x0 be the restriction of t̃, where t̃ is the relation defined by t̃ : xt → x,x 7→ tx. now, we divide the remaining proof into four steps. first step: show that k(t) = k(t0). let x ∈ k(t). we claim that there exist δ > 0 and a sequence (xn)n such that • x0 = x, • xn+1 ∈ d(t) and xn ∈ txn+1 for all n ≥ 0, • ‖xn‖t ≤ δn‖x‖ for all n ≥ 1. indeed, if x ∈ k(t), then there exist c > 0 and a sequence (yn)n such that • y0 = x, • for all n ≥ 0, yn+1 ∈ d(t) and yn ∈ tyn+1, • d(yn, ker(t) ∩t(0)) ≤ cnd(x, ker(t) ∩t(0)). let d > c. then, for all n ≥ 1 there exists αn ∈ t(0) ∩ ker(t) ⊆ d(t) such that ‖yn −αn‖ ≤ dn‖x‖. let (xn)n be the sequence defined by xn+1 = yn+1 − αn+1 for all n ≥ 0 and x0 = x. then, for all n ≥ 0, xn+1 ∈ d(t), xn ∈ txn+1 and ‖xn‖ ≤ dn‖x‖. on the other hand, we have ‖xn‖t = ‖xn‖+‖qttxn‖ = ‖xn‖+‖qt (xn−1)‖. then, ‖xn‖t = ‖xn‖+d(xn−1,t(0)), which implies that ‖xn‖t ≤ dn‖x‖ + ‖xn−1‖≤ (dn + dn−1)‖x‖. consequently, there exists δ > 0 such that ‖xn‖t ≤ δn‖x‖, as claimed. let g be the analytic function g : d(λ0, 1 d ) → xt defined by g(λ) = ∑∞ n=0 xn+1(λ−λ0)n. using remark 2.1, we get for all λ ∈ d(λ0, 1d), qtt (∑ n≥0 (λ−λ0)nxn+1 ) = qt ∑ n≥0 (λ−λ0)nxn. which implies that t (∑ n≥0 (λ−λ0)nxn+1 ) − ∑ n≥0 (λ−λ0)nxn ⊆ t(0). 88 m. lajnef, m. mnif thus, t (∑ n≥0(λ−λ0) nxn+1 ) = ∑ n≥0(λ−λ0) nxn + t(0). whence, (t − (λ−λ0)i) ∑ n≥0 (λ−λ0)nxn+1 = ∑ n≥0 (λ−λ0)nxn + t(0) − ∑ n≥1 xn(λ−λ0)n. therefore, (t − (λ−λ0)i)g(λ) = x + t(0), for each |λ−λ0| < 1 d . (3.2) particularly, x ∈ ⋂ λ∈d(λ0, 1d ) im(t − λi). hence, it follows from lemma 3.4 that x ∈ k(t −λ0i), which means that k(t) ⊆ x0. moreover, let � > 0 be such that � < inf( 1 d ,µ). then, by (3.2), we have for each 0 < |λ − λ0| < �, g(λ) + ker(t −(λ−λ0)i) = (t −(λ−λ0)i)−1x + (t −(λ−λ0)i)−1(t −(λ− λ0)i)(0). then, g(λ) = (t − (λ − λ0)i)−1x ∈ x0 for each 0 < |λ − λ0| < � and, by continuity, we get that x1 = g(λ0) ∈ x0. now, let h be the analytic function h : d(λ0, 1 d ) → xt defined by h(λ) = ∑∞ n=0 xn+2(λ−λ0)n. arguing as in (3.2), we get that (t − (λ−λ0)i)h(λ) = x1 + t(0), for each |λ−λ0| < 1 d . moreover, there exists � > 0 such that for each 0 < |λ − λ0| < �, h(λ) = (t −(λ−λ0)i)−1x1 ∈ x0 and, by continuity, we get that x2 = h(λ0) ∈ x0. in a similar way, we prove that xn ∈ x0 for all n ≥ 1. consequently, x ∈ k(t0) and hence, k(t) ⊆ k(t0). observe that k(t0) ⊆ k(t). then, we obtain k(t) = k(t0). second step: show that h0(t) = h0(t0). we claim that h0(t) ⊆ x0. indeed, according to lemma 2.1 (ii), we get that h0(t) ⊆ im(λi −t) = im(λi −t) for each 0 < |λ| < µ. which implies that h0(t) ⊆ (λi −t)im(λi −t) for each 0 < |λ| < µ. therefore, h0(t) ⊆ im(λi −t)2. as the power of a bounded below linear relation is also a bounded below linear relation then, for each λ ∈ d∗(0,µ), im(λi−t)2 is closed and hence, h0(t) ⊆ im(λi−t)2 for each λ ∈ d∗(0,µ). by repeating this process we get that h0(t) ⊆ r∞(λi −t) = x0. isolated points of the approximate point spectrum 89 therefore, it follows from lemma 2.1 (i) that h0(t) = h0(t0). third step: show that 0 is isolated in σ(t0). it is easy to see that t0−λi is injective for each 0 < |λ| < µ. furthermore, by virtue of lemma 2.1 (iii) we get (t0−λi)(x0∩d(t)) = (t−λi)(x0∩d(t)) = (t−λi)(k(t−λ0i))∩d(t)) = (t −λi)(k(t −λi)∩d(t)) = k(t −λi) = k(t −λ0i) = x0. then, t0−λi is surjective whenever 0 < |λ| < µ and so, t0 − λi is bijective for each 0 < |λ| < µ. hence, 0 is isolated in σ(t0). last step: show that h0(t) and k(t) are closed and x0 = h0(t)⊕k(t). using the third step and theorem 3.1, we get that k(t0) and h0(t0) are closed in x0 and so in x and x0 = h0(t0) ⊕ k(t0). but, we have, by the first step and the second step, that k(t) = k(t0) and h0(t) = h0(t0) then we get the desired result. remark 3.1. at this point, a natural question arises: are necessary conditions given in theorem 3.2 also sufficient? the answer to this question remains open. however, it is known that, in the particular case of bounded operators, the conditions (i) and (ii) of theorem 3.2 are not sufficient to conclude that 0 is isolated in σap(t) (see the remark in [8, page 4]). remark 3.2. it is worthy to point out that the investigation of the quasinilpotent part h0(t) and the analytic core k(t) of a linear relation t is convenient in the study of the spectral properties of relations. however, it is sometimes difficult to find them explicitly. theorem 3.2 gives an alternative way to study the properties of these two subspaces without computing them. example 3.1. let consider the separable hilbert space l2(n) and let then (en)n≥0 be her canonical basis. for k ∈ n∗ fixed, we define the bounded and closed linear relation t in l2(n) by: t((x0,x1, . . .)) =(x1 + . . . + xk,x2 + . . . + xk, . . . ,xk, 0, 0, 0,xk+1,xk+2, . . .) + 〈ek+1〉. we claim that 0 is an isolated point of σap(t). indeed, if we set n = 〈(en)kn=0〉 and m = 〈(en)n≥k+1〉, then we have m⊕n = l2(n) and t = tn ⊕tm , where tn is the bounded nilpotent operator on n of degree k + 1 represented by the matrix   0 1 . . . 1 ... 0 . . . ... ... ... . . . 1 0 0 . . . 0   90 m. lajnef, m. mnif and tm is the linear relation defined on m by tm = s −1 g sd, whether sg and sd are the left and right shift operators on m. it is clear that tm is a bounded below linear relation and hence, by [7, theorem 3.10], we deduce that t is a left drazin invertible linear relation. therefore, it follows from [4, theorem 4.1] that 0 is isolated in σap(t). thus, by virtue of theorem 3.2, we get thath0(t) and k(t) are closed, h0(t) ∩k(t) = {0}, h0(t) ⊕k(t) is closed and there exists λ0 such that h0(t) ⊕k(t) = k(t −λ0i) = r∞(t −λ0i). references [1] t. álvarez, on regular linear relations, acta math. sin. (engl. ser.) 28 (1) (2012), 183 – 194. [2] t. álvarez, m. benharrat, relationship between the kato spectrum and the goldberg spectrum of a linear relation, mediterr. j. math. 13 (1) (2016), 365 – 378. [3] t. álvarez, a. sandovici, regular linear relations on banach spaces, banach j. math. anal. 15 (1) (2021), paper no. 4, 26 pp. [4] y. chamkha, m. kammoun, on perturbation of drazin invertible linear relations, to appear in ukrainian math. j. [5] r. cross, “ multivalued linear operators ”, monographs and textbooks in pure and applied mathematics 213, marcel dekker, inc., new york, 1998. [6] a. favini, a. yagi, multivalued linear operators and degenerate evolution equations, anna. mat. pura appl. (4) 163 (1993), 353 – 384. [7] a. ghorbel, m. mnif, drazin inverse of multivalued operators and its applications, monatsh. math. 189 (2019), 273 – 293. [8] m. gonzález, m. mbekhta, m. oudghiri, on the isolated points of the surjective spectrum of a bounded operator, proc. amer. math. soc. 136 (10) (2008), 3521 – 3528. [9] m. lajnef, m. mnif, isolated spectral points of a linear relation, monatsh. math. 191 (2020), 595 – 614. [10] m. lajnef, m. mnif, on generalized drazin invertible linear relations, rocky mountain j. math. 50 (4) (2020), 1387 – 1408. [11] m. mbekhta, généralisation de la décomposition de kato aux opérateurs paranormaux et spectraux, glasgow math. j. 29 (1987), 159 – 175. [12] m. mnif, a.a. ouled-hmed, analytic core and quasi-nilpotent part of linear relations in banach spaces, filomat 32 (7) (2018), 2499 – 2515. introduction and preliminaries quasinilpotent part and analytic core of a closed linear relation isolated point of the approximate point spectrum � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 36, num. 1 (2021), 1 – 24 doi:10.17398/2605-5686.36.1.1 available online may 7, 2021 structure and bimodules of simple hom-alternative algebras s. attan département de mathématiques, université d’abomey-calavi 01 bp 4521, cotonou 01, bénin syltane2010@yahoo.fr received june 11, 2020 presented by rosa m. navarro accepted march 03, 2021 abstract: this paper is mainly devoted to the study of the structure of hom-alternative algebras. equivalent conditions for hom-alternative algebras being solvable, simple and semi-simple are provided. moreover, some results about hom-alternative bimodule are found. key words: bimodules, solvable, simple, hom-alternative algebras. msc (2020): 13b10, 13d20, 17a30, 17d15. 1. introduction hom-algebras are new classes of algebras which have been studied extensively in the literature during the last decade. they are algebras where the identities defining the structure are twisted by a homomorphism and began with hom-lie algebras [6, 9, 10, 11], motivated by quasi-deformations of lie algebras of vector fields, in particular q-deformations of witt and virasoro algebras. hom-associative algebras were introduced in [14] while hom-alternative and hom-jordan algebras are introduced in [13, 20] and deformations of homalternative and hom-malcev algebras are studied in [5]. questions on the structure of simple algebras in this or that variety are one of the main questions in the theory of rings. this question, for alternative algebras, has been studied by many authors. it turns out that the only simple alternative algebras which are not associative are 8-dimensional algebras over their centers which are generalizations of the original algebra of cayley numbers [15, 22, 23]. hence, all semisimple alternative algebras are known. similarly, it is relevant to study simple hom-algebras in hom-algebras theory. in [4], the authors gave a classification theorem about multiplicative simple hom-lie algebras. inspired by this study, a classification of multiplicative simple hom-jordan algebras is obtained in [18]. issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.36.1.1 mailto:syltane2010@yahoo.fr https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 2 s. attan representations (or bimodules) and deformations are important tools in most parts of mathematics and physics. by means of the representation theory, we would be more aware of the corresponding algebras. the study of bimodules of jordan algebras was initiated by n. jacobson [7]. subsequently the alternative case was considered by schafer [16]. similarly, it is very important to study representations of hom-algebras. fortunately, representations of hom-lie algebras were introduced and studied in [17], see also [1, 3]. later the one of hom-jordan and hom-alternative is presented in [2] where some useful results are obtained. moreover representations of simple hom-lie algebras [4, 12] and the one of simple hom-jordan algebras [18] are introduced and studied in detail. in this paper, based on [18] and on [21], we will study structure and bimodules over simple hom-alternative algebras. the paper is organized as follows: in section two, we give the basics about hom-alternative algebras and provide some new properties. section three deals with the study of simple and solvable hom-alternative algebras. some useful results are obtained (see lemma 3.8 and theorem 3.9). we also mainly prove relevant theorems which are about solvability, simplicity and semi-simplicity of hom-alternative algebras (see theorem 3.14, theorem 3.16 and theorem 3.20). in section four, we prove theorem 4.5 which is very important. it deals with the relationship between bimodules over homalternative algebras of alternative type and the ones over their induced alternative algebras. moreover, some relevant propositions about bimodules over hom-alternative algebras are also displayed as applications of theorem 4.5. all vector spaces are assumed to be over a fixed ground field k of characteristic 0. 2. preliminaries we recall some basic notions, introduced in [6, 14, 19] related to homalgebras. for the map µ : a⊗2 → a, we will write sometimes µ(a ⊗ b) as µ(a,b) or ab for a,b ∈ a. definition 2.1. a hom-module is a pair (m,αm ) consisting of a k-module m and a linear self-map αm : m → m. a morphism f : (m,αm ) → (n,αn ) of hom-modules is a linear map f : m → n such that f ◦αm = αn ◦f. definition 2.2. ([14, 19]) a hom-algebra is a triple (a,µ,α) in which (a,α) is a hom-module, µ : a⊗2 → a is a linear map. the hom-algebra structure and bimodules 3 (a,µ,α) is said to be multiplicative if α ◦ µ = µ ◦ α⊗2 (multiplicativity). a morphism f : (a,µa,αa) → (b,µb,αb) of hom-algebras is a morphism of the underlying hom-modules such that f ◦µa = µb ◦f⊗2. in this paper, we will only consider multiplicative hom-algebras. definition 2.3. let (a,µ,α) be a hom-algebra and λ ∈ k. let r be a linear map satisfying µ(r(x),r(y)) = r ( µ(r(x),y) + µ(x,r(y)) + λµ(x,y) ) , ∀x,y ∈ a. (1) then, r is called a rota-baxter operator of weight λ and (a,µ,α,r) is called a rota-baxter hom-algebra of weight λ. definition 2.4. let (a,µ,α) be a hom-algebra. 1. the hom-associator of a is the linear map asa : a ⊗3 → a defined as asa = µ◦(µ⊗α−α⊗µ). a multiplicative hom-algebra (a,µ,α) is said to be hom-associative algebra if asa = 0. 2. a hom-alternative algebra [13] is a multiplicative hom-algebra (a,µ,α) that satisfies both asa(x,x,y) = 0 (left hom-alternativity), (2) asa(x,y,y) = 0 (right hom-alternativity) (3) for all x,y ∈ a. 3. let (a,µ,α) be a hom-alternative algebra; a hom-subalgebra of (a,µ,α) is a linear subspace h of a, which is closed for the multiplication µ and invariant by α, that is, µ(x,y) ∈ h and α(x) ∈ h for all x,y ∈ h. if furthermore µ(a,b) ∈ h and µ(b,a) ∈ h for all (a,b) ∈ a×h, then h is called a two-sided hom-ideal of a. example 2.5. the octonions algebra o, also called cayley octaves or cayley algebra is 8-dimensional, with a basis (e0,e1,e2,e3,e4,e5,e6,e7), where e0 is the identity for the multiplication. this algebra is twisted into the eightdimensional hom-alternative algebra oα = (o,µ1,α) [20] with the same basis (e0,e1,e2,e3,e4,e5,e6,e7) where α(e0) = e0, α(e1) = e5, α(e2) = e6, α(e3) = e7, α(e4) = e1, α(e5) = e2, α(e6) = e3, α(e7) = e4, 4 s. attan and the multiplication table is: µ1 e0 e1 e2 e3 e4 e5 e6 e7 e0 e0 e5 e6 e7 e1 e2 e3 e4 e1 e5 −e0 e1 e4 −e6 e3 −e2 −e7 e2 e6 −e1 −e0 e2 e5 −e7 e4 −e3 e3 e7 −e4 −e2 −e0 e3 e6 −e1 e5 e4 e1 e6 −e5 −e3 −e0 e4 e7 −e2 e5 e2 −e3 e7 −e6 −e4 −e0 e5 e1 e6 e3 e2 −e4 e1 −e7 −e5 −e0 e6 e7 e4 e7 e3 −e5 e2 −e1 −e6 −e0 and into the eight-dimensional hom-alternative algebra oβ = (o,µ2,β) [13] with the same basis (e0,e1,e2,e3,e4,e5,e6,e7) where β(ei) = ei for all i ∈ {0, 4, 5, 7} and β(ei) = −ei for all i ∈ {1, 2, 3, 6}. the multiplication table is: µ2 e0 e1 e2 e3 e4 e5 e6 e7 e0 e0 −e1 −e2 −e3 e4 e5 −e6 e7 e1 −e1 −e0 e4 e7 e2 −e6 −e5 e3 e2 −e2 −e4 −e0 e5 −e1 e3 e7 e6 e3 −e3 −e7 −e5 −e0 −e6 −e2 −e4 −e1 e4 e4 −e2 e1 e6 −e0 e7 −e3 −e5 e5 e5 e6 −e3 e2 −e7 −e0 −e1 e4 e6 −e6 e5 −e7 e4 e3 e1 −e0 −e2 e7 e7 −e3 −e6 e1 e5 −e4 e2 −e0 nor oα, neither oβ, are alternative algebras. moreover, both α and β are automorphisms of o. similarly as in [13], it easy to prove the following: proposition 2.6. let (a,µ,α) be a hom-alternative algebra and β : a → a be a morphism of (a,µ,α). then (a,β◦µ,β◦α) is a hom-alternative algebra. in particular, if (a,µ) is an alternative algebra and β is a morphism of (a,µ), then (a,β ◦µ,β) is a hom-alternative algebra [13]. definition 2.7. let (a,µ,α) be a hom-alternative algebra. if there is an alternative algebra (a,µ′) such that µ = α ◦ µ′, we say that (a,µ,α) is alternative-type and (a,µ′) is its compatible alternative algebra or the untwist of (a,µ,α). structure and bimodules 5 it is noticed in [13] that a hom-alternative algebra with an invertible twisting map has the compatible alternative algebra. more precisely, we get: corollary 2.8. ([13]) let (a,µ,α) be a hom-alternative algebra where α is invertible then (a,µ′ = α−1 ◦ µ) is an alternative algebra and α is an automorphism with respect to µ′. hence (a,µ,α) is alternative-type and (a,µ′ = α−1 ◦µ) is its compatible alternative algebra. proposition 2.9. let (a1,µ1,α1) and (a2,µ2,α2) be hom-alternative algebras and ϕ : (a1,µ1,α1) → (a2,µ2,α2) be an invertible morphism of hom-algebras. if (a1,µ1,α1) is alternative-type and (a1,µ ′ 1) is its compatible alternative algebra then (a2,µ2,α2) is alternative-type with compatible alternative algebra (a2,µ ′ 2 = ϕ◦µ ′ 1◦(ϕ −1⊗ϕ−1)) such that ϕ : (a1,µ′1) → (a2,µ ′ 2) is an algebra morphism. proof. first, let us prove that (a2,µ ′ 2) is an alternative algebra such that µ′2 = ϕ◦µ ′ 1 ◦ (ϕ −1 ⊗ϕ−1). denote by as1 and as2 the associators of (a1,µ1) and (a2,µ ′ 2) respectively. then as2(u,u,v) = µ ′ 2(µ ′ 2(u,u),v) −µ ′ 2(u,µ ′ 2(u,v)) = ϕ◦µ′1 ( ϕ−1 ◦ϕ◦µ′1(ϕ −1(u),ϕ−1(u)),ϕ−1(v) ) −ϕ◦µ′1 ( ϕ−1(u),ϕ−1 ◦ϕ◦µ′1(ϕ −1(u),ϕ−1(v)) ) = ϕ◦µ′1 ( µ′1(ϕ −1(u),ϕ−1(u)),ϕ−1(v) ) −ϕ◦µ′1 ( ϕ−1(u),µ′1(ϕ −1(u),ϕ−1(v)) ) = ϕ◦ as1(ϕ−1(u),ϕ−1(u),ϕ−1(v)) = ϕ(0) = 0. similarly, we prove that as2(u,v,v) = 0. hence (a,µ ′ 2) is an alternative algebra. next, we have α2 ◦ϕ = ϕ◦α1 and ϕ defines µ2 by µ2 ◦ϕ⊗2 = ϕ◦µ1 since ϕ is a morphism from (a1,µ1,α1) to (a2,µ2,α2), i.e., µ2 = ϕ◦µ1 ◦ (ϕ−1 ⊗ϕ−1) = ϕ◦α1 ◦µ′1 ◦ (ϕ −1 ⊗ϕ−1) = α2 ◦ϕ◦µ′1 ◦ (ϕ −1 ⊗ϕ−1). hence, let take µ′2 = ϕ◦µ ′ 1 ◦ (ϕ −1 ⊗ϕ−1). finally, the fact that ϕ : (a1,µ ′ 1) → (a2,µ ′ 2) is an algebra morphism follows from the definition of µ′2. 6 s. attan the following characterization was given for hom-lie algebras in [17] and hom-associative algebras in [21]. proposition 2.10. given two hom-alternative algebras (a,µa,α) and (b,µb,β), there is a hom-alternative algebra (a ⊕ b,µa⊕b,α + β), where the bilinear map µa⊕b : (a⊕b)×2 → (a⊕b) is given by µa⊕b(a1 + b1,a2 + b2) = µa(a1,a2) + µb(b1,b2) ∀a1,a2 ∈ a, ∀b1,b2 ∈ b, and the linear map (α + β) : (a⊕b) → (a⊕b) is given by (α + β)(a + b) = (α(a) + β(b)) ∀ (a,b) ∈ a×b. (4) proof. first, (α + β) is multiplicative with respect to µa⊕b. indeed, (α + β)◦(µa⊕b)(a1 + b1,a2 + b2) = (α + β)(µa(a1,a2) + µb(b1,b2)) = α◦µa(a1,a2) + β ◦µb(b1,b2) = µa(α(a1),α(a2)) + µb(β(b1),β(b2) = µa⊕b(α(a1) + β(b1),α(a2) + β(b2)) = µa⊕b((α + β)(a1 + b1), (α + β)(a2 + b2)). secondly we prove the left hom-alternativity (2) for a⊕b as follows asa⊕b(a1 + b1,a1 + b1,a2 + b2) = µa⊕b ( µa⊕b(a1 + b1,a1 + b1), (α + β)(a2 + b2) ) −µa⊕b ( (α + β)(a1 + b1),µa⊕b(a1 + b1,a2 + b2) ) = µa⊕b(µa(a1,a1) + µb(b1,b1),α(a2) + β(b2) ) −µa⊕b ( α(a1) + β(b1),µa(a1,a2) + µ(b1,b2) ) = µa(µa(a1,a1),α(a2)) + µb(µb(b1,b1),β(b2)) −µa(α(a1),µa(a1,a2)) −µb(β(b1),µb(b1,b2)) = asa(a1,a1,a2) + asb(b1,b1,b2) = 0. similarly, we prove the right hom-alternativity (3) for a ⊕ b. hence (a ⊕ b,µa⊕b,α + β) is a hom-alternative algebra. structure and bimodules 7 proposition 2.11. let (a,µa,α) and (b,µb,β) be two hom-alternative algebras and ϕ : a → b be a linear map. denote by γϕ ⊂ a⊕b the graph of ϕ. then ϕ is a morphism from the hom-alternative algebra (a,µa,α) to the hom-alternative algebra (b,µb,β) if and only if its graph γϕ is a homsubalgebra of (a⊕b,µa⊕b,α + β). proof. let φ : (a,µa,α) → (b,µb,β) be a morphism of hom-alternative algebras. then we have for all u,v ∈ a, µa⊕b ( (u,ϕ(u)), (v,ϕ(v)) ) = ( µa(u,v),µb(ϕ(u),ϕ(v)) ) = ( µa(u,v),ϕ(µa(u,v)) ) . thus the graph γϕ is closed under the multiplication µa⊕b. furthermore since ϕ◦α = β ◦ϕ, we have (α⊕β)(u,ϕ(u)) = (α(u),β ◦ϕ(u)) = (α(u),ϕ◦α(u)), which implies that γϕ is closed under α⊕β. thus γϕ is a hom-subalgebra of (a⊕b,µa⊕b,α⊕β). conversely, if the graph γϕ ⊂ a⊕b is a hom-subalgebra of (a⊕b,µa⊕b, α⊕β), then we have µa⊕b ( (u,ϕ(u)), (v,ϕ(v)) ) = ( µa(u,v),µb(ϕ(u),ϕ(v)) ) ∈ γϕ, which implies that µb(ϕ(u),ϕ(v)) = ϕ(µa(u,v)). furthermore, (α⊕β)(γϕ) ⊂ γϕ implies (α⊕β)(u,ϕ(u)) = (α(u),β ◦ϕ(u)) ∈ γϕ, which is equivalent to the condition β ◦φ(u) = φ◦α(u), i.e., β ◦ϕ = ϕ◦α. therefore, ϕ is a morphism of hom-alternative algebras. proposition 2.12. let (a,µ,α,r) be a rota-baxter hom-alternative algebra of weight 0 such that r commutes with α. define a new multiplication on a by µr(x,y) = µ(r(x),y) + µ(x,r(y)) for any x,y ∈ a. then ar = (a,µr,α) is a hom-alternative algebra. proof. the multiplicativity of α with respect to µr follows from the one of α with respect to µ and the hypothesis r ◦ α = α ◦ r. to prove the left hom-alternative identity, let pick x,y ∈ a. then, 8 s. attan asr(x,x,y) = µr(µr(x,x),α(y)) −µr(α(x),µr(x,y)) = µr(µ(r(x),x) + µ(x,r(x)),α(y)) −µr ( α(x),µ(r(x),y) + µ(x,r(y)) ) = µ(r(µ(r(x),x) + µ(x,r(x))),α(y)) + µ(µ(r(x),x) + µ(x,r(x)),r◦α(y)) −µ ( r◦α(x),µ(r(x),y) + µ(x,r(y)) ) −µ ( α(x),r ( µ(r(x),y) + µ(x,r(y)) )) = µ(µ(r(x),r(x)),α(y)) + µ(µ(r(x),x),α◦r(y)) + µ(µ(x,r(x)),α◦r(y)) −µ(α◦r(x),µ(r(x),y)) −µ ( α◦r(x),µ(x,r(y)) ) −µ ( α(x),µ(r(x),r(y)) ) (using (1) with λ = 0 and α◦r = r◦α) = asa(r(x),r(x),y) + asa(r(x),x,r(y)) + asa(x,r(x),r(y)) = 0 + 0 = 0 (by (2) in a). hence, we get (2) for ar. similarly, we prove (3) for ar and therefore, the conclusion follows. 3. structures of hom-alternative algebras in this section, we study simple and solvable hom-alternative algebras. this study is inspired by the study given in [18] and [21]. we discuss necessary and sufficient conditions for hom-alternative algebras to be solvable, simple and semi-simple. in the classical case, we already know that every simple alternative algebra is either an associative algebra or a cayley-dickson algebra over its center [15, 22]. as it turns out, there is one non-associative simple alternative algebra. recall that the cayley-dickson algebras are a sequence a0, a1, . . . of non-associative r-algebras with involution. the first few are familiar: a0 = r, a1 = c, a2 = h (the quaternions) and a3 = o (the octonions). each algebra an is constructed from the previous one an−1 by a doubling procedure. the first three cayley-dickson algebras are associative and o is the only one non-associative alternative cayley-dickson algebra. alternativity fails in the higher cayley-dickson algebras. basing on this fact, we will give in this section an example of non hom-associative simple homalternative algebra. structure and bimodules 9 definition 3.1. let (a,µ,α) be a hom-alternative algebra. define its derived sequences as follows: a(0) = a, a(1) = µ(a,a), a(2) = µ ( a(1),a(1) ) , . . . , a(k) = µ ( a(k−1),a(k−1) ) , . . . . the following elementary result will be very useful. lemma 3.2. let (a,µ,α) be a hom-alternative algebra and k ∈ n. then · · · ⊆ a(k+1) ⊆ a(k) ⊆ a(k−1) ⊆ ··· ⊆ a(2) ⊆ a(1) ⊆ a and a(k) is a two-sided hom-ideal of (a,µ,α). proof. first, it is clear that a(1) ⊆ a(0) = a. next let k ∈ n and assume that a(k) ⊆ a(k−1). then, we have a(k+1) = µ ( a(k),a(k) ) ⊆ µ ( a(k−1),a(k−1) ) = a(k). to prove that a(k) is a two-sided hom-ideal of (a,µ,α) for every k ∈ n, by the inclusion condition, it suffices to prove the case n = 1. thanks to a(1) ⊆ a, we have first α(a(1)) = α ( µ(a,a) ) = µ ( α(a),α(a) ) ⊆ µ(a,a) = a(1), and next µ ( a(1),a ) ⊆ µ(a,a) = a(1) and µ ( a,a(1) ) ⊆ µ(a,a) = a(1). thus a(1) is a two-sided hom-ideal of (a,µ,α). definition 3.3. let (a,µ,α) be a hom-alternative algebra; (a,µ,α) is said to be solvable if there exists n ∈ n∗ such that a(n) = {0}. we get the following example of solvable and non solvable hom-alternative algebras respectively. example 3.4. (a) consider the 3-dimensional hom-alternative algebra (a,µ,α) with basis (e1,e2,e3) where µ(e2,e2) = e1,µ(e2,e3) = e1,µ(e3,e2) = e1,µ(e3,e3) = e1,µ(e3,e2) = e1 and α(e1) = e1, α(e2) = e1 + e2, α(e3) = e3. actually, (a,µ,α) is a hom-associative algebra (see [21], theorem 3.6, 10 s. attan hom-algebra a37). then a (1) is a one-dimensional hom-alternative algebra generated by (e1) defined as follows: µ(e1,e1) = 0 and α(e1) = e1. it follows that a(2) = {0} and therefore (a,µ,α) is solvable. (b) consider in the example 2.5, the hom-alternative algebras oα and oβ. from their multiplication tables, we get (oα)(1) = oα and (oβ)(1) = oβ. hence for every k ∈ n we have (oα)(k) = oα and (oβ)(k) = oβ. it follows that neither oα and nor oβ are solvable. definition 3.5. let (a,µ,α) (α 6= 0) be a non trivial hom-alternative algebra. 1. (a,µ,α) is said to be a simple hom-alternative algebra if a(1) 6= {0} and it has no proper two-sided hom-ideal. 2. (a,µ,α) is said to be a semi-simple hom-alternative algebra if a = a1 ⊕a2 ⊕a3 ⊕···⊕ap where ai (1 ≤ i ≤ p) are simple two-sided hom-ideals of (a,µ,α). let give the following example of non-simple hom-alternative algebra. example 3.6. consider the three-dimensional hom-alternative algebra (a,µ,α) over k with basis (e1,e2,e3) defined by µ(e1,e1) = e1, µ(e2,e2) = e2, µ(e3,e3) = e1, µ(e1,e3) = µ(e3,e1) = −e3 and α(e1) = e1, α(e3) = −e3. actually, (a,µ,α) is a hom-associative algebra (see [21], theorem 3.12, homalgebra a′ 3 3). consider the subspace i = span(e1,e3) of a. then one can observe that (i,µ,α) is a proper two-sided hom-ideal of (a,µ,α). hence the hom-alternative (a,µ,α) is not simple. we have the following elementary result which will be used in next sections. proposition 3.7. let (a,µ,α) be a simple hom-alternative algebra. then, for each k ∈ n, a(k) = a. proof. thanks to the definition of a(k), it suffices to prove that a(1) = a. by the simplicity of (a,µ,α), we have a(1) 6= {0}. moreover by lemma 3.2, a(1) is a two-sided hom-ideal of (a,µ,α) which has no proper two-sided hom-ideal, then a(1) = a. the following lemma is useful for next results. structure and bimodules 11 lemma 3.8. let (a,µ,α) be a hom-alternative algebra. then, (ker(α), µ,α) is a two-sided hom-ideal of (a,µ,α). proof. obvious, α(x) = 0 ∈ ker(α) for all x ∈ ker(α). next, let x,z ∈ a and y ∈ ker(α). then α(µ(x,y)) = µ(α(x),α(y)) = µ(α(x), 0) = 0 and α(µ(y,z)) = µ(α(y),α(z)) = µ(0,α(z)) = 0. thus µ(x,y) ∈ ker(α) and µ(y,z) ∈ ker(α) and it follows that (ker(α),µ,α) is a two-sided hom-ideal. proposition 3.9. let (a,µ,α) be a finite dimensional simple homalternative algebra. then the hom-alternative algebra is alternative-type and α is an automorphism of both (a,µ,α) and its induced algebra. proof. by lemma 3.8, ker(α) is a two-sided hom-ideal of the simple homalternative algebra (a,µ,α). therefore ker(α) = {0} or ker(α) = a. since the hom-alternative algebra is non trivial, it follows that ker(α) = {0} and α is an automorphism. thus, a is alternative-type (see corollary 2.8). let (a,µ′ = α−1 ◦µ) be the induced (the compatible) alternative algebra of the simple hom-alternative algebra (a,µ,α). we have α◦µ′ = α◦α−1 ◦µ = α−1 ◦µ◦α⊗2 = µ′ ◦α⊗2, i.e., α is both automorphism of (a,µ′) and (a,µ,α). as in hom-associative algebras case [21], by the above proposition, there exists an induced alternative algebra of any simple hom-alternative algebra (a,µ,α) and α is an automorphism of the induced alternative algebra. moreover, their products are mutually determined. theorem 3.10. two finite dimensional simple hom-alternative algebras (a1,µ1,α) and (a2,µ2,β) are isomorphic if and only if there exists an alternative algebra isomorphism ϕ : a1 → a2 (between their induced alternative algebras) which renders conjugate the two alternative algebra automorphisms α and β that is ϕ◦α = β ◦ϕ. proof. since (a1,µ1,α) and (a2,µ2,β) are finite dimensional simple homalternative algebras, they are alternative-type. let (a1,µ ′ 1) and (a2,µ ′ 2) be their induced alternative algebras respectively. if ϕ : (a1,µ1,α) → (a2,µ2,β) 12 s. attan is an isomorphism of hom-alternative algebras, then ϕ ◦ α = β ◦ ϕ, thus β−1 ◦ϕ = ϕ◦α−1. moreover, ϕ◦µ′1 = ϕ◦α −1 ◦α◦µ′1 = ϕ◦α −1 ◦µ1 = β−1 ◦ϕ◦µ1 = β−1 ◦µ2 ◦ϕ⊗2 = µ′2 ◦ϕ ⊗2. so, ϕ is an isomorphism between the induced hom-alternative algebras. on the other hand, if there exists an isomorphism between the induced alternative algebras satisfying ϕ◦α = β ◦ϕ, then ϕ◦µ1 = ϕ◦α◦µ′1 = β ◦ϕ◦µ ′ 1 = β ◦µ ′ 2 ◦ϕ ⊗2 = µ2 ◦ϕ⊗2. let recall the following: proposition 3.11. ([2]) let (a,µ,α) be a hom-alternative algebra and i be a two-sided hom-ideal of (a,µ,α). then (a/i,µ̄, ᾱ) is a hom-alternative algebra where µ̄(x̄, ȳ) = µ(x,y) and ᾱ(x̄) = ¯α(x) for all x̄, ȳ ∈ a/i. proof. first, note that the multiplicativity of µ̄ with respect to ᾱ follows from the one of µ with respect to α. next, pick x̄, ȳ ∈ a/i. then the left hom-alternativity (2) in (a/i,µ̄, ᾱ) is proved as follows asa/i(x̄, x̄, ȳ) = ū ( ū(x̄, x̄), ᾱ(ȳ) ) − µ̄ ( ᾱ(x̄), µ̄(x̄, ȳ) ) = µ ( µ(x,x)α(y) ) −µ ( α(x),µ(x,y) ) = asa(x,x,y)) = 0̄. hence we get (2) for (a/i,µ̄, ᾱ). similarly, we get (3) and therefore (a/i,µ̄, ᾱ) is a hom-alternative algebra. corollary 3.12. let (a,µ,α) be a finite dimensional hom-alternative algebra such that α2 = α. then, (a/ ker(α), µ̄, ᾱ) is a hom-alternative algebra of alternative-type. proof. it is clear that (a/ ker(α), µ̄, ᾱ) is a hom-alternative algebra by proposition 3.11 since by lemma 3.8, ker(α) is a two-sided hom-ideal of a. • if α is invertible, i.e., ker(α) = {0}, then (a/ ker(α), µ̄, ᾱ) = (a,µ,α) and (a/ ker(α), µ̄, ᾱ) is a hom-alternative algebra of alternative-type (see corollary 2.8). • if α is not invertible, then ker(α) 6= {0}. therefore we have to show that ᾱ is invertible on the hom-alternative algebra (a/ ker(α), µ̄, ᾱ). assume structure and bimodules 13 that x̄ ∈ ker(ᾱ). then ¯α(x) = ᾱ(x̄) = 0̄, i.e., α(x) ∈ ker(α). since α2 = α, we have α(x) = α2(x) = α(α(x)) = 0, which means that x ∈ ker(α), i.e., x̄ = 0̄. if follows that ᾱ is invertible and thanks to the corollary 2.8, the hom-alternative algebra (a/ ker(α), µ̄, ᾱ) is alternative-type. corollary 3.13. let (a,µ,α) be a hom-alternative algebra such that α is invertible. then, (a/ ker(α), µ̄, ᾱ) is a hom-alternative algebra of alternative-type. theorem 3.14. let (a,µ,α) be a hom-alternative algebra such that α is invertible. then (a,µ,α) is solvable if and only if its induced alternative algebra (a,µ′) is solvable. proof. let (a,µ,α) be a hom-alternative algebra such that α is invertible. denote the derived sequences of (a,µ′) and (a,µ,α) by a(i) and a (i) α (i = 1, 2, · · ·) respectively. suppose that (a,µ′) is solvable. then there exists p ∈ n∗ such that a(p) = {0}. note that a(1)α = µ(a,a) = α◦µ ′(a,a) = α ( a(1) ) , a(2)α = µ ( a(1)α ,a (1) α ) = µ ( α ( a(1) ) ,α ( a(1) )) = α2 ◦µ′ ( a(1),a(1) ) = α2 ( a(2) ) , so by induction a (p) α = α p ( a(p) ) . it follows that a (p) α = {0}, which means that (a,µ,α) is solvable. on the other hand, assume that (a,µ,α) is solvable. then there exists q ∈ n∗ such that a(q)α = {0}. by the above proof, we get αq ( a(q) ) = a (q) α . note that α q is invertible since α is, then a(q) = {0} that is (a,µ′) is solvable. lemma 3.15. ([18]) let a be an algebra over a field k that has the unique decomposition of direct sum of simple ideals a = ⊕si=1ai where the ai are not isomorphic to each other and α ∈ aut(a). then α(ai) = ai (i = 1, 2, · · · ,s). 14 s. attan theorem 3.16. (i) let (a,µ,α) be a finite dimensional simple homalternative algebra. then its induced alternative algebra (a,µ′) is semisimple. moreover, (a,µ′) can be decomposed into direct sum of isomorphic simple ideals. in addition, α acts simply transitively on simple ideals of the induced alternative algebra. (ii) let (a,µ′) be a simple alternative algebra and α ∈ aut(a). then (a,µ = α◦µ′,α) is a simple hom-alternative algebra. proof. (i) thanks to corollary 2.8, α is both automorphism with respect to µ′ and µ. assume that a1 is the maximal solvable two-sided ideal of (a,µ ′). then there exists p ∈ n∗ such that a(p)1 = {0}. since µ′ ( a,α(a1) ) = µ′ ( α(a),α(a1) ) = α ( µ′(a,a1) ) ⊆ α(a1), µ′ ( α(a1),a ) = µ′ ( α(a1),α(a) ) = α ( µ′(a1,a) ) ⊆ α(a1),( α(a1) )(p) = α ( a (p) 1 ) = {0}, we obtain that α(a1) is also a solvable two-sided ideal of (a,µ ′). then α(a1) ⊆ a1. moreover µ(a,a1) = α ( µ′(a,a1) ) ⊆ α(a1) ⊆ a1, µ(a1,a) = α ( µ′(a1,a) ) ⊆ α(a1) ⊆ a1. it follows that a1 is a two-sided hom-ideal of (a,µ,α) and we get that a1 = {0} or a1 = a since (a,µ,α) is simple. if a1 = a, thanks to the proof of theorem 3.14, we obtain a(p)α = α p ( a(p) ) = αp ( a (p) 1 ) = {0}. on the other hand, by the simplicity of (a,µ,α), we have by proposition 3.7, a (p) α = a, which is a contradiction. it follows that a1 = {0} and thus, (a,µ′) is semi-simple. now, by the semi-simplicity of (a,µ′), we have a = ⊕si=1ai where for all i ∈ {1, . . . ,s}, ai is a simple two-sided ideal of (a,µ′). since there may be isomorphic alternative algebras among a1, . . . ,as, we can rewrite a as follows: a = a11 ⊕a12 ⊕···⊕a1m1 ⊕a21 ⊕a22 ⊕··· ⊕a2m2 ⊕···⊕at1 ⊕at2 ⊕···⊕atmt, structure and bimodules 15 where (aij,µ ′) ∼= (aik,µ′), 1 ≤ j,k ≤ mi, i = 1, 2, . . . , t. thanks to lemma 3.15, we have α ( ai1 ⊕ai2 ⊕···⊕aimi ) = ai1 ⊕ai2 ⊕···⊕aimi. then µ ( ai1 ⊕ai2 ⊕···⊕aimi,a ) = α ( µ′ ( ai1 ⊕ai2 ⊕···⊕aimi,a )) ⊆ α ( ai1 ⊕ai2 ⊕···⊕aimi ) = ai1 ⊕ai2 ⊕···⊕aimi and µ ( a,ai1 ⊕ai2 ⊕···⊕aimi ) = α ( µ′ ( a,ai1 ⊕ai2 ⊕···⊕aimi )) ⊆ α ( ai1 ⊕ai2 ⊕···⊕aimi ) = ai1 ⊕ai2 ⊕···⊕aimi. it follows that ai1⊕ai2⊕···⊕aimi are two-sided hom-ideals of (a,µ,α). then, the simplicity of (a,µ,α) implies ai1 ⊕ ai2 ⊕ ··· ⊕ aimi = {0} or ai1 ⊕ai2 ⊕···⊕aimi = a. therefore, all but one of ai1 ⊕ai2 ⊕···⊕aimi must be equal to a. without lost of generality, assume that a = a11 ⊕a12 ⊕···⊕a1m1. if m1 = 1 then (a,µ ′) is simple. else, α(a1p) = a1l (1 ≤ l 6= p ≤ m1) since, if α(a1p) = a1p (1 ≤ p ≤ m1), then, a1p would be a non trivial two-sided hom-ideal of (a,µ,α) which contradicts the simplicity of (a,µ,α). in addition, it is clear that a11 ⊕α(a11) ⊕α2(a11) ⊕···αm1−1(a11) is a two-sided hom-ideal of (a,µ,α). therefore, a = a11 ⊕α(a11) ⊕α2(a11) ⊕···⊕αm1−1(a11). in other words, α acts simply transitively on simple ideals of the induced alternative algebra. 16 s. attan (ii) by proposition 2.6, it is clear that (a,µ,α) is a hom-alternative algebra. assume that a1 is a non-trivial two-sided hom-ideal of (a,µ,α, then we get µ′(a,a1) = α −1(µ(a,a1)) ⊆ α−1(a1) ⊆ a1, µ′(a1,a) = α −1(µ(a1,a)) ⊆ α−1(a1) ⊆ a1. it follows that a1 is a non trivial two-sided ideal of (a,µ ′), contradiction. it follows that (a,µ,α) has no trivial ideals. if µ(a,a) = {0}, then, µ′(a,a) = α−1 ( µ(a,a) ) = {0}, which is in contradicts the fact that (a,µ′) is simple. it follows that (a,µ,α) is simple. next, we will give an example of non hom-associative simple hom-alternative algebra using some results about cayley-dickson algebras. example 3.17. consider hom-alternative algebras oα = (o,µ1,α) and oβ = (o,µ2,β) (see example 2.5) obtained from the octonions algebra o which is an eight dimensional simple non-associative alternative algebra. since α ∈ aut(o) and β ∈ aut(o), thanks to theorem 3.16 (ii), both oα and oβ are eight dimensional simple hom-alternative algebras. proposition 3.18. the eight dimensional simple hom-alternative algebras oα and oβ are not isomorphic. proof. suppose that the simple hom-alternative algebras oα and oβ are isomorphic. then by theorem 3.10, there exists an alternative algebra isomorphism ϕ : o → o (between their induced alternative algebras) such that α ◦ ϕ = ϕ ◦ β. this means by the definition of β that for all i ∈ {1, 2, 3, 6}, α(ϕ(ei)) = −ϕ(ei), i.e., λ = −1 is an eigenvalue of α (contradiction since the characteristic polynomial of α is x8 −x7 −x + 1). remark 3.19. we know in the classical case that there exists only one example of a nonassociative simple alternative algebra that is, the cayleydickson algebra over its center [15, 22]. in the hom-algebra setting, proposition 3.18, clearly proves that there is more than one simple non homassociative hom-alternative algebras. structure and bimodules 17 theorem 3.20. (i) let (a,µ,α) be a finite dimensional semi-simple hom-alternative algebra. then (a,µ,α) is alternative-type and its induced alternative algebra (a,µ′) is also semi-simple. (ii) let (a,µ′) be a semi-simple alternative algebra such that a has a decomposition a = ⊕siai where ai (1 ≤ i ≤ s) are simple two-sided ideal of (a,µ′). moreover let α ∈ aut(a) satisfying α(ai) = ai (1 ≤ i ≤ s). then (a,µ = α ◦ µ′,α) is a semi-simple hom-alternative algebra and has the unique decomposition. proof. (i) suppose that (a,µ,α) is a finite dimensional semi-simple homalternative algebra.then a has the decomposition a = ⊕siai where ai (1 ≤ i ≤ s) are simple two-sided hom-ideal of (a,µ,α). then (ai,µ,α|ai ) (1 ≤ i ≤ s) are simple finite dimensional hom-alternative algebras. according to the proof of proposition 3.9, α|ai is invertible and therefore α is invertible. thus thanks to corollary 2.8, the hom-alternative algebra (a,µ,α) is alternativetype and its induced alternative algebra is (a,µ′) with µ′ = α−1 ◦µ. on the other hand, by the proof of theorem 3.16 (ii), ai (1 ≤ i ≤ s) are two-sided ideal of (a,µ′). moreover (ai,µ ′|ai ) (1 ≤ i ≤ s) are induced alternative algebra of finite dimensional simple hom-alternative algebras (ai,µ,α|ai ) (1 ≤ i ≤ s) respectively. thanks to theorem 3.16 (i), (ai,µ′) are semi-simple alternative algebras and can be decomposed into direct sum of isomorphic simple two-sided ideals ai = ai1 ⊕ ai2 ⊕ ···⊕ aimi . it follows that (a,µ ′) is semi-simple and has the decomposition of direct sum of simple two-sided ideals a = a11 ⊕a12 ⊕···⊕a1m1 ⊕a21 ⊕a22 ⊕··· ⊕a2m2 ⊕···⊕as1 ⊕as2 ⊕···⊕asms. (ii) we know by proposition 2.6 that (a,µ,α) is a hom-alternative algebra. next, for all 1 ≤ i ≤ s, the condition α(ai) = ai implies µ(ai,a) = α ( µ′(ai,a) ) ⊆ α(ai) = ai, µ(a,ai) = α ( µ′(a,ai) ) ⊆ α(ai) = ai. it follows that ai are two-sided hom-ideals of (a,µ,α). if there exits non trivial two-sided hom-ideal ai0 of (ai,µ,α), then we have µ(ai0,a) = µ ( ai0,a1 ⊕a2 ⊕···⊕as ) = µ(ai0,ai) ⊆ ai0, µ(a,ai0 ) = µ ( a1 ⊕a2 ⊕···⊕as,ai0 ) = µ(ai0,ai) ⊆ ai0. 18 s. attan it follows that ai0 is a non trivial two-sided hom-ideal of (a,µ,α). thanks to the proof of theorem 3.16 (ii), ai0 is also a non trivial two-sided ideal of (a,µ′). hence, ai0 is also a non trivial two-sided ideal of (ai,µ ′) which is a contradiction. it follows that ai (i = 1, . . . ,s) are simple two-sided homideals of (a,µ,α) and therefore (a,µ,α) is semi-simple and has the unique decomposition. proposition 3.21. let (a,µ,α) be a hom-alternative algebra such that α2 = α. then (a,µ,α) is isomorphic to the decomposition of direct sum of hom-alternative algebras, i.e., a ∼= (a/ ker(α)) ⊕ ker(α). proof. it is clear that (ker(α),µ,α) is a hom-alternative algebra and thanks to proposition 3.11 since ker(α) is a two-sided hom-ideal of (a,µ,α), the quotient hom-algebra (a/ ker(α), µ̄, ᾱ) is a hom-alternative algebra. now set a1 = (a/ ker(α)) ⊕ ker(α) and define µ1 : a×21 → a1 and α1 : a1 → a1 by µ1 ( (x̄,h), (ȳ,k) ) := ( µ(x,y),µ(h,k) ) , α1 ( (x̄,h) ) := ( α(x), 0 ) . then one can show that (a1,µ1,α1) is a hom-alternative algebra which is a decomposition of direct sum of hom-alternative algebras. next, let y ∈ ker(α) ∩ im(α). then, there exists x ∈ a such that y = α(x). moreover, we have 0 = α(y) = α2(x) = α(x) = y. it follows that ker(α) ∩ im(α) = {0}, and then a = ker(α) ⊕ im(α) since for any x ∈ a we have x = (x − α(x)) + α(x) with x − α(x) ∈ ker(α) and α(x) ∈ im(α). now, let show that (im(α),µ,α) ∼= ( v/ ker(α), µ̄, ᾱ ) . note that, it is clear that ( im(α),µ,α ) is a hom-alternative algebra. define ϕ : v/ ker(α) → im(α) by ϕ(x̄) = α(x) for all x̄ ∈ a/ ker(α). clearly, ϕ is bijective and for all x̄, ȳ ∈ a/ ker(α), we have ϕ ( µ̄(x̄, ȳ) ) = ϕ ( µ(x,u) ) = α ( µ(x,y) ) = µ ( α(x),α(y) ) = µ ( ϕ(x̄),ϕ(ȳ) ) , ϕ ( ᾱ(x̄) ) = ϕ ( α(x) ) = α2(x) = α ( ϕ(x̄) ) , i.e., ϕ ◦ µ̄ = µ̄ ◦ ϕ⊗2 and ϕ ◦ ᾱ = α ◦ ϕ. it follows that (im(α),µ,α) ∼=( v/ ker(α), µ̄, ᾱ ) and therefore v = ker(α) ⊕ im(α) ∼= (a/ ker(α)) ⊕ ker(α). structure and bimodules 19 4. bimodules over simple hom-alternative algebras in this section, we mainly study bimodules over hom-alternative algebras of alternative-type. we give a theorem about the relationship between bimodules over hom-alternative algebras of alternative-type and the ones over their induced alternative algebras. moreover, some relevant propositions about bimodules over hom-alternative algebras are also displayed. as a consequence, interesting results about bimodules over finite dimensional simple hom-alternative algebras are given. definition 4.1. let a′ = (a,µ) be any algebra and v be a k-module. 1. a left (resp. right) structure map on v is a morphism δl : a⊗v → v , a⊗v 7−→ a ·v (resp. δr : v ⊗a → v , v ⊗a 7−→ v ·a) of hom-modules. 2. let δl and δr be structure maps on v . then the module associator of v is a trilinear map ( , , )a,v defined as: ( , , )a,v ◦ idv⊗a⊗a = δr ◦ (δr ⊗ ida) −δl ◦ (idv ⊗µ), ( , , )a,v ◦ ida⊗v⊗a = δr ◦ (δl ⊗ ida) −δl ◦ (ida ⊗δr), ( , , )a,v ◦ ida⊗a⊗v = δl ◦ (µ⊗ idv ) −δl ◦ (ida ⊗δl). a bimodule over alternative algebras is given in [8, 16]. definition 4.2. ([8, 16]) let a′ = (a,µ) be an alternative algebra. an alternative a′-bimodule is a k-module v that comes equipped with a (left) structure map δl : a⊗v → v (δl(a⊗v) = a ·v) and a (right) structure map δr : v ⊗a → v (δr(v ⊗a) = v ·a) such that the following equalities holds: (a,v,b)a,v = −(v,a,b)a,v = (b,a,v)a,v = −(a,b,v)a,v . (5) the notion of alternative bimodules has been extended to the homalternative bimodules. more precisely, we get definition 4.3. ([2]) let a = (a,µ,αa) be a hom-alternative algebra. a hom-alternative a-bimodule is a hom-module (v,αv ) that comes equipped with a (left) structure map ρl : a⊗v → v (ρl(a⊗v) = a ·v) and a (right) structure map ρr : v ⊗ a → v (ρr(v ⊗ a) = v · a) such that the following equalities asa,v (a,v,b) = −asa,v (v,a,b) = asa,v (b,a,v) = −asa,v (a,b,v) (6) 20 s. attan hold for all a,b,c ∈ a and v ∈ v , where asa,v is the module hom-associator of the hom-module (v,αv ) defined by asa,v ◦idv⊗a⊗a = ρr ◦ (ρr ⊗αa) −ρl ◦ (αv ⊗µ), asa,v ◦ida⊗v⊗a = ρr ◦ (ρl ⊗αa) −ρl ◦ (αa ⊗ρr), asa,v ◦ida⊗a⊗v = ρl ◦ (µ⊗αv ) −ρl ◦ (αa ⊗ρl). remark 4.4. if αa = ida and αv = idv , the notion of hom-alternative bimodule is reduced to the one of alternative bimodule. theorem 4.5. (i) let a = (a,µ,αa) be a hom-alternative algebra of alternative-type with a′ = (a,µ′) its induced alternative algebra and (v,αv ) be a hom-alternative a-bimodule with the structure maps ρl and ρr such that αv is invertible. then v is an alternative a′-bimodule with the structures maps δl = α −1 v ◦ρl and δr = α −1 v ◦ρr. (ii) let a′ = (a,µ′) be an alternative algebra and a = (a,µ = αa ◦µ′,αa) a corresponding hom-alternative algebra. let v be an alternative a′bimodule with the structure maps δl and δr and αv ∈ end(v ) such that αv ◦δl = δl◦(αa⊗αv ) and αv ◦δr = δr ◦(αv ⊗αa). then (v,αv ) is a hom-alternative a-bimodule with the structures maps ρl = αv ◦δl and ρr = αv ◦δr where αa is a morphism of (a,µ′). proof. (i) let (v,αv ) be an alternative a-bimodule with the structure maps ρl and ρr such that αv is invertible. note that the structure maps δl = α −1 v ◦ρl and δr = α −1 v ◦ρr satisfy conditions αv ◦ δl = δl ◦ (αa ⊗αv ) and αv ◦ δr = δr ◦ (αv ⊗αa). (7) therefore, asa,v ◦ida⊗v⊗a = ρr ◦ (ρl ⊗αa) −ρl ◦ (αa ⊗ρr) = αv ◦ δr ◦ (αv ◦δl ⊗αa) −αv ◦ δl ◦ (αa ⊗αv ◦ δr) = α2v ◦ δr ◦ (δl ⊗ ida) −α 2 v ◦ δl ◦ (ida ⊗δr) (by (7)) = α2v ◦ ( , , )a,v ◦ ida⊗v⊗a. thus, ( , , )a,v ◦ ida⊗v⊗a = (α2v ) −1 ◦ asa,v ◦ida⊗v⊗a. similarly, we get ( , , )a,v ◦ ida⊗a⊗v = (α2v ) −1 ◦ asa,v ◦ida⊗a⊗v , ( , , )a,v ◦ idv⊗a⊗a = (α2v ) −1 ◦ asa,v ◦idv⊗a⊗a. structure and bimodules 21 finally, condition (5) follows from condition (6). (ii) similar to (i). corollary 4.6. let a = (a,µ,αa) be a finite dimensional simple homalternative algebra and (v,αv ) be a hom-alternative a-bimodule with the structure maps ρl and ρr such that αv is invertible. then v is an alternative a′-bimodule with the structures maps δl = α−1v ◦ρl and δr = α −1 v ◦ρr where a′ = (a,µ′) is the induced alternative algebra. definition 4.7. let a = (a,µ,αa) be a hom-alternative algebra and (v,αv ) be a hom-alternative a-bimodule with the structure ρl and ρr. 1. a subspace v0 of v is called an a-subbimodule of (v,αv ) if αv (v0) ⊆ v0, ρl(a⊗v0) ⊆ v0 and ρr(v0 ⊗a) ⊆ v0. 2. the hom-alternative a-module (v,αv ) is said to be irreducible if it has no non trivial a-subbimodules and completely reducible if v = v1 ⊕v2 ⊕···⊕vs where vi (1 ≤ i ≤ s) are irreducible a-subbimodules of (v,αv ). proposition 4.8. let a = (a,µ,αa) be a hom-alternative algebra and (v,αv ) a hom-alternative a-bimodule with the structure maps ρl and ρr. then ker(αv ) is an a-subbimodule of (v,αv ). moreover if αa is surjective then im(αv ) is an a-subbimodule of (v,αv ) and we have the isomorphism of a-bimodules ᾱv : v/ ker(αv ) → im(αv ). proof. obvious, we have αv (ker(αv )) ⊆ ker(αv ). next, let (v,a) ∈ ker(αv ) ×a. then we get αv (ρl(a⊗v)) = ρl(αa(a) ⊗αv (v)) = 0, αv (ρr(v ⊗a)) = ρr(αv (v) ⊗αa(a)) = 0 since αv (v) = 0. therefore ker(αv ) is an a-subbimodule of (v,αv ). similarly, it is obvious that αv (im(αv)) ⊆ im(αv). let (v,a) ∈ im(αv)×a. then there exits v′ ∈ v and if αa is surjective a′ ∈ a such that v = αv (v′) and a = αa(a ′). therefore ρl(a⊗v) = ρl ( αa(a ′) ⊗αv (v′) ) = αv ( ρl(a ′ ⊗v′) ) ∈ im(αv ), ρr(v ⊗a) = ρr ( αv (v ′) ⊗αa(a′) ) = αv ( ρr(v ′ ⊗a′) ) ∈ im(αv ). thus im(αv ) is an a-subbimodule of (v,αv ). 22 s. attan finally, if define the map ᾱv : v/ ker(αv ) → im(αv ) by ᾱv (v̄) = αv (v), then it is easy to prove that ᾱv is an isomorphism. corollary 4.9. let a = (a,µ,αa) be a hom-alternative algebra and (v,αv ) be a finite dimensional irreducible hom-alternative a-bimodule. then αv is invertible. proposition 4.10. let a = (a,µ,αa) be a hom-alternative algebra of alternative-type and (v,αv ) a hom-alternative a-bimodule with the structure maps ρl and ρr such that αv is invertible. if the alternative a′-bimodule v over the induced alternative algebra a′ = (a,µ′) with the structures maps δl = α −1 v ◦ ρl and δr = α −1 v ◦ ρr is irreducible, then the hom-alternative a-bimodule (v,αv ) is also irreducible. proof. assume that the hom-alternative a-bimodule (v,αv ) is reducible. then there exists a non trivial subspace v0 such that (v0,αv |v0 ) is an a-subbimodule of (v,αv ). therefore αv (v0) ⊆ v0, ρl(a ⊗ v)) ∈ v0 and ρr(v ⊗ a) ∈ v0 for all (a,v) ∈ a × v0. hence δl(a ⊗ v)) ∈ α−1v (v0) = v0 and δr(v ⊗ a) ∈ α−1v (v0) = v0. thus v0 is a non trivial a ′-subbimodule of v , contradiction. it follows that (v,αv ) is an irreducible hom-alternative a-bimodule. corollary 4.11. let a = (a,µ,αa) be a finite dimensional simple homalternative algebra and (v,αv ) a hom-alternative a-bimodule with the structure maps ρl and ρr such that αv is invertible. if the alternative a′-bimodule v over the induced alternative algebra a′ = (a,µ′) with the structures maps δl = α −1 v ◦ ρl and δr = α −1 v ◦ ρr is irreducible, then the hom-alternative a-bimodule (v,αv ) is also irreducible. let recall the following result from [16] which is very useful for the next result. theorem 4.12. ([16]) let a′ = (a,µ′) be a semi-simple alternative algebra. then any representation (s,t) of a′ is completely reducible. since the notion of completely reducible representation of a′, is equivalent to the notion of a completely reducible alternative a′-bimodule, thanks to theorem 3.20 (i) and theorem 4.5 (i), we get the following important result. corollary 4.13. let a = (a,µ,α) be a finite dimensional semi-simple hom-alternative algebra. then any hom-alternative a-bimodule (v,αv ) such that αv is invertible, is completely reducible. structure and bimodules 23 example 4.14. consider the following hom-algebra a = (a,µ,α) where the non-zero products are given by µ(e1,e1) = e1, µ(e2,e2) = e2, µ(e3,e3) = e1, µ(e1,e3) = µ(e3,e1) = −e3 and α(e1) = e1, α(e3) = −e3. then a is a semi-simple hom-alternative algebra with the decomposition a1 ⊕a2 where a1 and a2 are simple two sided-hom-ideals generated by (e2) and (e1,e3) respectively. therefore, any hom-alternative a-bimodule (v,αv ) such that αv is invertible, is completely reducible. acknowledgements the author would like to thank the anonymous reviewers for their valuable comments which improved the paper. references [1] b. agrebaoui, k. benali, a. makhlouf, representations of simple hom-lie algebras, j. lie theory 29 (4) (2019), 1119 – 1135. 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[18] c. yao, y. ma, l. chen, structures, classifications and bimodules of multiplicative simple hom-jordan algebras, arxiv:1906.04561v1. [19] d. yau, hom-algebras as deformations and homology, arxiv:0712.3515v1. [20] d. yau, hom-maltsev, hom-alternative and hom-jordan algebras, int. electon. j. algebra 11 (2012), 177 – 217. [21] a. zahari, a. makhlouf structure and classification of hom-associative algebras, acta comment. univ. tartu. math. 24 (1) (2020), 79 – 102. [22] m. zorn, theorie der alternativen ringe, abh. math. sem. univ. hamburg 8 (1) (1931), 123 – 147. [23] m. zorn, alternativkörper und quadratische systeme, abh. math. sem. univ. hamburg 9 (1) (1933), 395 – 402. https://arxiv.org/abs/1906.04561v1 https://arxiv.org/abs/0712.3515v1 introduction preliminaries structures of hom-alternative algebras bimodules over simple hom-alternative algebras e extracta mathematicae vol. 33, núm. 2, 209 – 218 (2018) moving weyl’s theorem from f(t ) to t m. febronio rodŕıguez 1, b.p. duggal2, s.v. djordjević 1 1 buap, facultad de ciencias f́ısico-matemáticas rı́o verde y av. san claudio, san manuel, puebla, pue. 72570, mexico 2 8 redwood grove, northfield avenue, london w5 4sz, england, u.k. mbfebronio222@hotmail.com , bpduggal@yahoo.co.uk , slavdj@fcfm.buap.mx presented by manuel gonzález received may 25, 2017 abstract: schmoeger has shown that if weyl’s theorem holds for an isoloid banach space operator t ∈ b(x) with stable index, then it holds for f(t) whenever f ∈ holo σ(t) is a function holomorphic on some neighbourhood of the spectrum of t. in this note we establish a converse. key words: weyl’s theorem, browder’s theorem, svep. ams subject class. (2000): 47a10, 47a11. 1. introduction recall that an operator t ∈ b(x) has finite ascent if there is p ∈ n for which t −p(0) = t −∞(0) = ∞∪ n=1 t −n(0) ; if in particular t −p(0) = t −w(0) = { x ∈ x : ||t nx|| 1 n → 0 } (transfinite kernel) we shall say that t has finite hyperascent. the transfinite range of an operator is defined by t w(x) =  x ∈ x : there exists k > 0 and a sequence {xn} ⊂ x such that tx1 = x , txn+1 = xn and ∥xn∥ ≤ kn∥x∥ for all positive integers n   . if t ∈ b(x) has finite ascent, then, in particular, it has the “single valued extension property” (svep) at zero, which says [6] that the only holomorphic function f for which (t − z)f(z) = 0 for all z in a neighborhood of zero is 209 210 m.f. rodŕıguez, b.p. duggal, s.v. djordjević the zero function. equivalently, [6], 0 is not in the “local point spectrum” π left loc (t). for t ∈ b(x), let n(t)(= t −1(0)) and r(t)(= t(x)) denote, respectively, the null space and the range of the mapping t. let α(t) = dim n(t) and β(t) = dim x/r(t), if theses spaces are finite dimensional, otherwise let α(t) = ∞ and β(t) = ∞. if the range r(t) of t ∈ b(x) is closed and α(t) < ∞ (respectively, β(t) < ∞), then t is said to be an upper semi-fredholm (respectively, a lower semi-fredholm) operator and we denote t ∈ φ+(x) (respectively t ∈ φ−(x)). if t ∈ φ−(x) ∪ φ+(x) then t is called a semi-fredholm operator (in notation t ∈ φ±(x)) and for t ∈ φ−(x) ∩ φ+(x) we say that t is a fredholm operator (in notation t ∈ φ(x)). for t ∈ φ±(x), the index of t is defined by ind(t) = α(t) − β(t). t has stable index if ind(t − µi) is either ≥ 0 or ≤ 0 (exclusive or) for all µ not in the fredholm spectrum σe(t) of t ; t is isoloid if the isolated points of the spectrum of t are eigenvalues of t. denote with π00(t) the set of isolated eigenvalues of t of finite geometric multiplicity, i.e. π00(t) = {λ ∈ iso σ(t) : 0 < α(t − λ) < ∞}. similarly, with π0(t) we denote the set of all isolated eigenvalues of t of finite algebraic multiplicity (poles of t). obviously, π0(t) ⊆ π00(t). schmoeger, [8], has shown that “if weyl’s theorem holds for an isoloid operator t with stable index”, then it holds for f(t) whenever f ∈ holo σ(t) (or f ∈ holoc σ(t)), the set of all non-trivial holomorphic function on some neighborhood of the spectrum of t (or all function from holo σ(t) that are not constant on connected component). in this note we address the converse problem. specifically, we will give conditions under which “browder’s theorem” (respectively “finite hyperascent property”) is transmitted back from f(t) to t . 2. browder’s theorem recall that t is polaroid if every isolated point λ of the spectrum of t, λ ∈ iso σ(t), is a pole of the resolvent of t. the browder spectrum σb(t) and the weyl spectrum σw(t) of t ∈ b(x) are the sets σb(t) = {λ ∈ σ(t) : t − λ /∈ φ(x) or asc(t −λ) ̸= des(t −λ)} and σw(t) = {λ ∈ σ(t) : t −λ /∈ φ0(x)}, where φ0(x) denotes the set of all fredholm operators with index zero. the essential (fredholm) spectrum is the set σe(t) = {λ ∈ σ(t) : t −λ /∈ φ(x)}. moving weyl’s theorem from f(t) to t 211 here des(t) denotes the descent of t, the smallest positive integer n such that r(t n) = r(t n+1) (if no such n exists, then des(t) = ∞), asc(t) denotes the ascent of t, the smallest positive integer n such that t −n(0) = t −(n+1)(0) (if no such n exists, then asc(t) = ∞). browder’s theorem holds for t if and only if (1) σb(t) ⊆ σw(t), equivalently, [4, theorem 8.3.1], if and only if t has svep on σ(t) \ σw(t), equivalently, [6], if and only if (2) π left loc (t) ⊆ σw(t). remark 2.1. (i) let t ∈ b(x) and f ∈ holo σ(t). then browder’s theorem for f(t) implies the spectral mapping theorem for weyl spectrum: σw(f(t)) = σb(f(t)) = f(σb(t)) ⊇ f(σw(t)). since opposite inclusion always holds, we have σw(f(t)) = f(σw(t)). (ii) the svep property on σ(t) \ σw(t) guarantees us even more: browder’s theorem for f(t) for every f ∈ holoc σ(t). really, let f ∈ holoc σ(t) and f(λ0) ∈ σ(f(t)) \ σw(f(t)). then there is an r ∈ n, a polynomial h and g ∈ holo σ(t) (with no zero in σ(t)) such that f(z) − f(λ0) ≡ (z − λ0)rh(z)g(z) with h(λ0) ̸= 0 and h(λ0) ̸∈ g(σ(t)). it follows f(t) − f(λ0) = (t − λ0)rh(t)g(t) ∈ φ0(x), with 0 /∈ σ(h(t)g(t)) and, consequently, λ0 /∈ σw(t). hence, t has svep at λ0 and, by [1, theorem 2.39], f(t) has svep at f(λ0) that implies browder’s theorem for f(t). (iii) in the case of f ∈ holo σ(t) we need little more, svep at all λ ∈ σ(t) \ σe(t) or injectivity of f. hence, for f ∈ holo σ(t), the passage of browder’s theorem from t to f(t), is not a major problem. more interesting question is how to pass browder’s theorem from f(t) to t. in general, the svep and browder’s theorem do not move from f(t) to t . to see this, it is enough consider an operator t without svep on σ(t) \ σw(t) (in this case no browder’s theorem for t) and f ≡ c ∈ holo σ(t) (for more details see [5, p. 227]). 212 m.f. rodŕıguez, b.p. duggal, s.v. djordjević given t ∈ b(x) and f ∈ holo σ(t), then we define the set (3) af(t) = {λ /∈ σw(t) : f(λ) ∈ σw(f(t))} and we say that t has the property sf if (sf) t has svep at every λ ∈ af(t). theorem 2.2. let t ∈ b(x) be such that browder’s theorem holds for f(t). (i) if f ∈ holo σ(t) and t has the property (sf), then browder’s theorem holds for t. (ii) if f ∈ holoc σ(t), then browder’s theorem holds for g(t), for any g ∈ holoc σ(t). proof. let λ ∈ σ(t) \ σw(t). case i: if f(λ) ∈ f(σ(t)) \ f(σw(t)), then browder’s theorem for f(t) and, consequently, the spectral mapping theorem for weyl spectrum of t, guarantees us that f(λ) is an isolated point of σ(f(t)) (matter of fact it is a pole). then λ is an isolated point of the spectrum of t that implies svep property for t at λ. case ii: let f(λ) ∈ f(σw(t))(= σw(f(t))). (i) then, by property (sf), t has svep at λ. (ii) if f(λ) ∈ σw(f(t)), then by injectivity of f ∈ holoc σ(t), we have that λ ∈ σw(t), which is a contradiction to our assumption. hence, t has svep at all λ /∈ σw(t), and browder’s theorem holds for t . moreover, for any g ∈ holoc σ(t), by remark 2.1 (ii), browder’s theorem holds for g(t), for every g ∈ holoc σ(t). the similar behavior we have in the situation of more general versions of browder’s theorem: the g-browder, a-browder or s-browder theorems. we say that t ∈ b(x) obeys g-browder’s theorem if σbb(t) ⊆ σbw(t) ,(4) a-browder’s theorem if σab(t) ⊆ σaw(t) ,(5) s-browder’s theorem if σsb(t) ⊆ σsw(t) ,(6) where moving weyl’s theorem from f(t) to t 213 σbb(t) = {λ ∈ σ(t) : t − λ is not b-fredholm or asc(t − λ) ̸= des(t − λ)}, σbw(t) = {λ ∈ σ(t) : t − λ is not b-fredholm or ind(t − λ) ̸= 0}, σab(t) = {λ ∈ σa(t) : t − λ /∈ φ+(t) or asc(t − λ) = ∞}, σaw(t) = {λ ∈ σa(t) : t − λ /∈ φ+(t) or ind(t − λ) > 0}, σsb(t) = {λ ∈ σs(t) : t − λ /∈ φ−(t) or des(t − λ) = ∞}, σsw(t) = {λ ∈ σs(t) : t − λ /∈ φ−(t) or ind(t − λ) < 0}. note that t ∈ b(x) is a b-fredholm operator if for some integer n the range space r(t n) is closed and tn = t|r(t n) is a fredholm operator. in this case tm is a fredholm operator and ind(tm) = ind(tn) for each m ≥ n. this enables us to define the index of a b-fredholm operator t as the index of the fredholm operator tn where n is any integer such that r(t n) is closed and such that tn is a fredholm operator. let ∗ ∈ {g, a, s}. it is known that ∗-browder’s theorem holds for t if t has svep at all points λ /∈ σ∗w(t) and that ∗-browder’s theorem implies browder’s theorem (matter of fact, g-browder’s theorem is equivalent to browder’s theorem). moreover, if t has svep at all points λ /∈ σ∗w(t), then the spectral mapping theorem holds for σ∗w(t) and the functions from holoc σ(t). (for more details see [4]). let t ∈ b(x), then for any f ∈ holo σ(t) and ∗ ∈ {g, a, s}, we define the sets a∗f(t) = {λ /∈ σ∗w(t) : f(λ) ∈ σ∗w(f(t))}, and the property s∗f (s∗f) t has svep at every λ ∈ a ∗ f(t), then we have next theorem: theorem 2.3. let t ∈ b(x) and f ∈ holo σ(t). if ∗-browder’s theorem holds for f(t) and t has the property s∗f, then ∗-browder’s theorem holds for t. moreover, ∗-browder’s theorem holds for t if and only if it is holds for g(t), for any g ∈ holoc σ(t). 3. weyl’s theorem if browder’s theorem holds for some t ∈ b(x) together with π0(t) = π00(t), then we say that t satisfies weyl’s theorem. svep alone is not enough 214 m.f. rodŕıguez, b.p. duggal, s.v. djordjević for t to satisfy weyl’s theorem: consider, for example, the quasinilpotent operator q ∈ b(ℓ2), q(x1, x2, x3, . . . ) = (x22 , x3 3 , . . . ). a necessary and sufficient condition for t to satisfy weyl’s theorem is that t satisfies browder’s theorem and, for every λ ∈ π00(t), t − λ has finite hyperascent. furthermore, if t is polaroid and has svep, then both f(t) and f(t ∗) satisfy weyl’s theorem for every f ∈ holoc σ(t) [3]. for moving weyl’s theorem from f(t) to t we need a variant of (3). let t ∈ b(x) and f ∈ holo σ(t), then we define the set (7) πf(t) = {λ ∈ π00(t) : f(λ) ∈ σw(f(t))} and we say that t has the property πf if (πf) t − λ has a finite hyperascent, for every λ ∈ πf(t). remark 3.1. (i) let t ∈ b(x) and {λ1, λ2, . . . , λn} ⊂ c be a finite set of distinct complex numbers. then, for any polynomial p(λ) = ∏n i=1(λi − λ) mi we have p(t)−1(0) = n⊕ i=1 (t − λi)−mi(0). moreover, if p(λ0) ̸= 0, for some complex number λ0, then (t − λ0)−w(0) ∩ p(t)−1(0) = {0}. (ii) if t, s ∈ b(x) is a pair of commuting operators, then t −w(0) ⊆ (ts)−w(0). moreover, if s is an invertible operator, then t −w(0) = (ts)−w(0). theorem 3.2. let t ∈ b(x) and f ∈ holoc σ(t). if weyl’s theorem holds for f(t) and t has the property πf, then weyl’s theorem holds for t. proof. by theorem 2.2, browder’s theorem holds for t, hence we have to show that t − λ has a finite hyperascent, for every λ ∈ π00(t) (see [4, theorem 8.4.5 (vi)]). let λ0 ∈ π00(t); then f(λ0) ∈ f(σ(t)) = σ(f(t)) = σw(f(t)) ∪ π00(f(t)). case i: f(λ0) ∈ π0(f(t)). since weyl’s theorem holds for f(t), f(t) − f(λ0) has a finite hyperascent, i.e., there exists a positive integer p ∈ n such that (f(t) − f(λ0))−w(0) = (f(t) − f(λ0))−p(0). since λ ∈ iso σ(t), x splits into the direct sum of the transfinite kernel (t − λi)−w(0) and the transfinite range (t − λi)wx, both hyperinvariant moving weyl’s theorem from f(t) to t 215 under t . if we write s0 and s1 for the restriction of s ∈ comm(t) to the kernel and the range respectively, then σ(s0) = {λ0} ̸⊆ σ(s1). let r ∈ n, the polynomial h and g ∈ holoc σ(t) (with no zero in σ(t)) be such that f(z) − f(λ0) ≡ (z − λ0)rh(z)g(z) with h(λ0) ̸= 0 and h(λ0) ̸∈ g(σ(t)). it follows f(t) − f(λ0) = (t − λ0)rh(t)g(t) with 0 /∈ σ(g(t)). then (8) (t − λ0)−1(0) ⊆ (t − λ0)−r(0) ⊆ (f(t) − f(λ0))−1(0) and, by remark 3.1 (i), (f(t) − f(λ0))−1(0) = ((t − λ0)rh(t))−1(0) = (t − λ0)−r(0) ⊕ h(t)−1(0). (9) since f(t0) − f(λ0) has hyperascent ≤ p, by remark 3.1 (ii), we have (t − λ0)−w(0) ⊆ (f(t) − f(λ0))−w(0) = (f(t) − f(λ0))−p(0) = (t − λ0)−pr(0) ⊕ h(t)−p(0). again, by (t − λ0)−w(0) ∩ h(t)−p(0) = {0} and remark 3.1, we have (t − λ0)−w(0) ⊆ (t − λ0)−pr(0). since the opposite inclusion is always valid, we have that t − λ0 has finite hyperascent. case ii: if f(λ0) ∈ f(σw(f(t)), since t has a property πf, follows that t − λ0 has finite hyperascent. remark 3.3. a slight modification of the proofs of theorem 2.2 and theorem 3.2 give us the conditions for moving weyl’s theorem form f(t), f ∈ holo σ(t), to t. for this, beside the property πf, we need to suppose that f is an injective function. 216 m.f. rodŕıguez, b.p. duggal, s.v. djordjević if we replace the condition πf with stronger condition (10) λ ∈ iso σ(t) =⇒ f(t) − f(λ) has finite hyperascent, then slight modification of part of the proof of theorem 3.2 give us that t is polaroid. in this case we can extend weyl’s theorem on g(t), for all g ∈ holoc σ(t). theorem 3.4. let t ∈ b(x) and f ∈ holoc σ(t) such that weyl’s theorem holds for f(t) and t has property (10). then weyl’s theorem holds for g(t) and g(t ∗) for all g ∈ holoc σ(t). proof. in view of the hypothesis theorem 2.2 implies that t (so also, t ∗) satisfies browder’s theorem. recall now from theorem 3.2 that if f(t) satisfies condition (10), then t is polaroid (which, in turn, implies that t ∗ is polaroid); hence t and t ∗ satisfy weyl’s theorem. browder’s theorem for t implies that t has svep at points in σ(t) \ σw(t) and by [3, theorem 2.4] g(t) and g(t ∗) satisfy weyl’s theorem for every g ∈ holoc σ(t). 4. applications a banach space operator t ∈ b(x) is hereditarily polaroid, t ∈ hp, if every part of t (i.e., its restriction to an invariant subspace) is polaroid. the class of hp operators is large. it contains amongst others the following classes of operators. (we refer the interested reader to [2] for further, but by no means exhaustive, list of hp operators.) (a) h(p) operators (operators t ∈ b(x) such that h0(t − λ) = (t − λ)−p(0) for some integer p = p(λ) ≥ 0 and all complex λ). this class of operators contains next well known classes: (a-i) hilbert space operators t ∈ b(h) which are either hyponormal (|t ∗|2 ≤ |t |2), or p-hyponormal (|t ∗|2p ≤ |t|2p) for some 0 < p ≤ 1 or (p, k)quasihyponormal (t ∗k(|t|2p − |t ∗|2p)t k ≥ 0) for some integer k ≥ 1 and 0 < p ≤ 1. (a-ii) w-hyponormal (|t̃ ∗| ≤ |t | ≤ ˜|t|, where, for the polar decomposition t = u|t| of t, t̃ = |t| 1 2 u|t | 1 2 ). (a-iii) m-hyponormal (||(t − λ)∗||2 ≤ m||t − λ||2 for some m ≥ 1 and all complex λ) or class a (|t|2 ≤ |t 2|). moving weyl’s theorem from f(t) to t 217 (b) paranormal operators t ∈ b(x) (||tx||2 ≤ ||t 2x|| for all unit vectors x ∈ x). (c) totally paranormal operators t ∈ b(x) (||(t − λ)x||2 ≤ ||(t − λ)2x|| for all unit vectors x ∈ x and complex λ). the classes consisting of paranormal operators and h(p) operators are substantial. thus, the classes consisting of hyponormal or p-hyponormal or (p, 1)-quasihyponormal or (1, 1)-quasihyponormal and class a hilbert space operators are proper subclasses of the class of paranormal operators; the class h(p) contains in particular the classes consisting of operators which are either totally paranormal or generalized scalar or subscalar or multipliers of commutative semi-simple banach algebras [1, p. 175]. moving weyl’s theorem from f(t) to t, for f ∈ holoc σ(t) and t ∈ h(p), is possible applying theorem 3.4. this fact is known and we can find more details in [7]. we have: theorem 4.1. if f(t) ∈ h(p) for some t ∈ b(x) and f ∈ holoc σ(t), then t satisfies weyl’s theorem. moreover, g(t) and g(t ∗) satisfy weyl’s theorem for every g ∈ holoc σ(t). more is true. recall, [1, p. 177], that a banach space operator t satisfies a-weyl’s theorem if σa(t)\σaw(t) = πa00(t), where σa(a) is the approximate point spectrum of t, πa00(t) = {λ ∈ iso σa(t) : 0 < dim(t − λi) −1(0) < ∞} and σaw(t) = {λ ∈ σa(t) : t −λ is not lower semi–fredholm or ind(a−λ) ̸≤ 0} is the weyl essential approximate point spectrum of a. if t has svep, then σ(t) = σa(t ∗), σw(t) = σaw(t ∗) and π00(t) = π a 00(t ∗). since t satisfies weyl’s theorem if and only if σ(t) \ σw(t) = π00(t) [1, p. 166], we have the following: corollary 4.2. if f(t) ∈ h(p) for some t ∈ b(x) and f ∈ holoc σ(t), then g(t ∗) satisfies a-weyl’s theorem for every g ∈ holoc σ(t). proof. since f(t) has svep implies t has svep implies g(t) has svep [1], σa(g(t ∗)) \ σaw(g(t ∗)) = σ(g(t)) \ σw(g(t)). this, since g(t) satisfies weyl’s theorem (see theorem 4.1), implies σa(g(t ∗) \ σaw(g(t ∗)) = π0(g(t)) = π a 0(g(t ∗)). hp operators have svep [2, theorem 2.8], so that if f(t) ∈ hp, for some f ∈ holoc σ(t), then g(t) and g(t ∗) satisfy browder’s theorem for every g ∈ holo σ(t). however, since isolated points of σ(t) may not survive 218 m.f. rodŕıguez, b.p. duggal, s.v. djordjević passage from σ(t) to σ(f(t)), f ∈ holoc σ(t), hp operators do not in general satisfy condition (5). (there is no such problem with h(p) operators.) now, using that λ ∈ iso σ(t) if and only if f(λ) ∈ iso σ(f(t)), the condition (10) and, that, f(t) has svep implies t has svep, way we have new version of [2, theorem 3.6]. theorem 4.3. suppose that f(t) ∈ hp for some t ∈ b(x) and f ∈ holoc σ(t). if f preserves isolated points of σ(t), then t satisfies weyl’s theorem. moreover, g(t) satisfies weyl’s theorem and g(t ∗) satisfies a-weyl’s theorem for every g ∈ holoc σ(t). references [1] p. aiena, “ fredholm and local spectral theory, with applications to multipliers ”, kluwer academic publishers, dordrecht, 2004. [2] b.p. duggal, hereditarily polaroid operators, svep and weyl’s theorems, j. math. anal. appl. 340 (2008), 366 – 373. [3] b.p. duggal, polaroid operators satisfying weyl’s theorem, linear algebra appl. 414 (2006), 271 – 277. [4] b.p. duggal, svep, browder and weyl theorems, in “ topisc in approximation theory iii ”, dirección de fomento editorial buap, puebla, mexico, 2009, 107 – 146. [5] k.b. laursen, m.m. neumann, “ an introduction to local spectral theory ”, the clarendon press, oxford university press, new york, 2000. [6] r. harte, on local spectral theory, in “ recent advances in operator theory and applications ”, birkhäuser, basel, 2009, 175 – 183. [7] m. oudghiri, weyl’s and browder’s theorems for operators satisfying the svep, studia math. 163 (2004), 85 – 101. [8] c. schmoeger, on operators t such that weyl’s theorem holds for f(t), extracta math. 13 (1998), 27 – 33. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 36, num. 1 (2021), 25 – 50 doi:10.17398/2605-5686.36.1.25 available online june 11, 2021 free (rational) derivation k. schrempf austrian academy of sciences, acoustics research institute wohllebengasse 12-14, 1040 vienna, austria math@versibilitas.at , https://orcid.org/0000-0001-8509-009x received april 15, 2021 presented by manuel saoŕın accepted may 21, 2021 abstract: by representing elements in free fields (over a commutative field and a finite alphabet) using cohn and reutenauer’s linear representations, we provide an algorithmic construction for the (partial) non-commutative (or hausdorff-) derivative and show how it can be applied to the noncommutative version of the newton iteration to find roots of matrix-valued rational equations. key words: hausdorff derivative, free associative algebra, free field, minimal linear representation, admissible linear system, free fractions, chain rule, newton iteration. msc (2020): primary 16k40, 16s85; secondary 68w30, 46g05. introduction working symbolically with matrices requires non-commuting variables and thus non-commutative (nc) rational expressions. although the (algebraic) construction of free fields, that is, universal fields of fractions of free associative algebras, is available due to paul m. cohn since 1970 [8, chapter 7], its practical application in terms of free fractions [29] —building directly on cohn and reutenauer’s linear representations [10]— in computer algebra systems is only at the very beginning. the main difficulty for arithmetic —or rather lexetic from the non-existing greek word λεξητικος (from λεξις for word) as analogon to αριθμητικος (from αριθμος for number)— was the construction of minimal linear representations [32], that is, the normal form of cohn and reutenauer [9]. here we will show that free fractions also provide a framework for “free” derivation, in particular of nc polynomials. the construction we provide generalizes the univariate (commutative) case we are so much used to, for example f = f(x) = x3 + 4x2 + 3x + 5 with f ′ = d dx f(x) = 3x2 + 8x + 3. issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.36.1.25 mailto:math@versibilitas.at https://orcid.org/0000-0001-8509-009x https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 26 k. schrempf the coefficients are from a commutative field k (for example the rational q, the real r, or the complex number field c), the (non-commuting) variables from a (finite) alphabet x , for example x = {x,y,z}. for simplicity we focus here (in this motivation) on the free associative algebra r := k〈x〉, aka “algebra of nc polynomials”, and recall the properties of a (partial) derivation ∂x : r → r (for a fixed x ∈x), namely • ∂x(α) = 0 for α ∈ k, and more general, ∂x(g) = 0 for g ∈ k〈x\{x}〉, • ∂x(x) = 1, and • ∂x(fg) = ∂x(f) g + f ∂x(g). in [27] this is called hausdorff derivative. for a more general (module theoretic) context we refer to [7, section 2.7] or [1]. before we continue we should clarify the wording: to avoid confusion we refer to the linear operator ∂x (for some x ∈ x) as free partial derivation (or just derivation) and call ∂xf ∈ r the (free partial) derivative of f ∈ r, sometimes denoted also as fx or f ′ depending on the context. in the commutative, we usually do not distinguish too much between algebraic and analytic concepts. but in the (free) non-commutative setting, analysis is quite subtle [21]. there are even concepts like “matrix convexity” and symbolic procedures to determine (nc) convexity [4]. however, a systematic treatment of the underlying algebraic tools was not available so far. we are going to close this gap in the following. after a brief description of the setup (in particular that of linear representations of elements in free fields) in section 1, we develop the formalism for free derivations in section 2 with the main result, theorem 2.8. to be able to state a (partial) “free” chain rule (in proposition 3.2) we derive a language for the free composition (and illustrate how to “reverse” it) in section 3. and finally, in section 4, we show how to develop a meta algorithm “nc newton” to find matrix-valued roots of a non-commutative rational equation. notation. the set of the natural numbers is denoted by n = {1, 2, . . .}, that including zero by n0. zero entries in matrices are usually replaced by (lower) dots to emphasize the structure of the non-zero entries unless they result from transformations where there were possibly non-zero entries before. we denote by in the identity matrix (of size n) respectively i if the size is clear from the context. by v> we denote the transpose of a vector v. free (rational) derivation 27 1. getting started we represent elements (in free fields) by admissible linear systems (definition 1.6), which are just a special form of linear representations (definition 1.3) and “general” admissible systems [8, section 7.1]. rational operations (scalar multiplication, addition, multiplication, inverse) can be easily formulated in terms of linear representations [10, section 1]. for the formulation on the level of admissible linear systems and the “minimal” inverse we refer to [30, proposition 1.13] resp. [30, theorem 4.13]. let k be a commutative field, k its algebraic closure and x = {x1,x2, . . . , xd} be a finite (non-empty) alphabet. k〈x〉 denotes the free associative algebra (or free k-algebra) and f = k(〈x〉) its universal field of fractions (or “free field”) [6, 10]. an element in k〈x〉 is called (non-commutative or nc) polynomial. in our examples the alphabet is usually x = {x,y,z}. including the algebra of nc rational series [2] we have the following chain of inclusions: k ( k〈x〉 ( krat〈〈x〉〉 ( k(〈x〉) =: f. definition 1.1. ([8, section 0.1], [10]) given a matrix a ∈ k〈x〉n×n, the inner rank of a is the smallest number k ∈ n such that there exists a factorization a = cd with c ∈ k〈x〉n×k and d ∈ k〈x〉k×n. the matrix a is called full if k = n, non-full otherwise. theorem 1.2. ([8, special case of corollary 7.5.14]) let x be an alphabet and k a commutative field. the free associative algebra r = k〈x〉 has a universal field of fractions f = k(〈x〉) such that every full matrix over r can be inverted over f. remark. non-full matrices become singular under a homomorphism into some field [8, chapter 7]. in general (rings), neither do full matrices need to be invertible, nor do invertible matrices need to be full. an example for the former is the matrix b =   · z −y−z · x y −x ·   over the commutative polynomial ring k[x,y,z] which is not a sylvester domain [5, section 4]. an example for the latter are rings without unbounded generating number (ugn) [8, section 7.3]. 28 k. schrempf definition 1.3. ([9, 10]) let f ∈ f. a linear representation of f is a triple πf = (u,a,v) with u >,v ∈ kn×1, full a = a0⊗1+a1⊗x1 +. . .+ad⊗xd with a` ∈ kn×n for all ` ∈ {0, 1, . . . ,d} and f = ua−1v. the dimension of πf is dim (u,a,v) = n. it is called minimal if a has the smallest possible dimension among all linear representations of f. the “empty” representation π = (, , ) is the minimal one of 0 ∈ f with dim π = 0. let f ∈ f and π be a minimal linear representation of f. then the rank of f is defined as rank f = dim π. definition 1.4. ([9]) let π = (u,a,v) be a linear representation of f ∈ f of dimension n. the families (s1,s2, . . . ,sn) ⊆ f with si = (a−1v)i and (t1, t2, . . . , tn) ⊆ f with tj = (ua−1)j are called left family and right family respectively. l(π) = span{s1,s2, . . . ,sn} and r(π) = span{t1, t2, . . . , tn} denote their linear spans (over k). proposition 1.5. ([9, proposition 4.7]) a representation π = (u,a, v) of an element f ∈ f is minimal if and only if both, the left family and the right family are k-linearly independent. in this case, l(π) and r(π) depend only on f. remark. the left family (a−1v)i (respectively the right family (ua −1)j) and the solution vector s of as = v (respectively t of u = ta) are used synonymously. definition 1.6. ([30]) a linear representation a = (u,a,v) of f ∈ f is called admissible linear system (als) for f, written also as as = v, if u = e1 = [1, 0, . . . , 0]. the element f is then the first component of the (unique) solution vector s. given a linear representation a = (u,a,v) of dimension n of f ∈ f and invertible matrices p,q ∈ kn×n, the transformed paq = (uq,paq,pv) is again a linear representation (of f). if a is an als, the transformation (p,q) is called admissible if the first row of q is e1 = [1, 0, . . . , 0]. 2. free derivation before we define the concrete (partial) derivation and (partial) directional derivation, we start with a (partial) formal derivation on the level of admissible linear systems and show the basic properties with respect to the represented elements, in particular that the (formal) derivation does not depend on the als in corollary 2.3. free (rational) derivation 29 in other words: given some letter x ∈x and an admissible linear system a = (u,a,v), there is an algorithmic point of view in which the (free) derivation ∂x defines the als a′ = ∂xa. (alternatively one can identify x by its index ` ∈{1, 2, . . . ,d} and write a′ = ∂`a.) written in a sloppy way, we show that ∂x(a + b) = ∂xa + ∂xb and ∂x(a·b) = ∂xa·b + a·∂xb, yielding immediately the algebraic point of view (summarized in theorem 2.8) by taking the respective first component of the unique solution vectors f = (sa)1 and g = (sb)1. remark. the following definition is much more general then usually needed. one gets the “classical” (partial) derivation with respect to some letter x = x` ∈x (with ` ∈{1, 2, . . . ,d}) for k = 0 resp. (the empty word) a = 1 ∈x∗. definition 2.1. let a = (u,a,v) be an admissible linear system of dimension n ≥ 1 for some element in the free field f = k(〈x〉) and ` 6= k ∈ {0, 1, . . . ,d}. the als ∂`|ka = ∂`|k(u,a,v) = ([ u · ] , [ a a` ⊗xk · a ] , [ · v ]) (of dimension 2n) is called (partial) formal derivative of a, (with respect to x`,xk ∈{1}∪x). for x,a ∈{1}∪x = {1,x1,x2, . . . ,xd} with x 6= a we write also ∂x|aa, having the indices ` 6= k ∈{0, 1, . . . ,d} of x resp. a in mind. lemma 2.2. let af = (uf,af,vf ) and ag = (ug,ag,vg) be admissible linear systems of dimension dimaf ≥ 1 resp. dimag ≥ 1. fix x,a ∈{1}∪x such that x 6= a. then ∂x|a(af + ag) = ∂x|aaf + ∂x|aag. proof. let ` 6= k ∈ {0, 1, . . . ,d} be the indices of x resp. a. we write a`f for the coefficient matrix a` of af resp. a ` g for a` of ag. taking the sum from [30, proposition 1.13] we have ∂x|a(af + ag) = ([ uf · ] , [ af −afu>fug · ag ] , [ vf vg ]) =  [uf · · ·] ,   af −afu>fug a ` f ⊗a −a ` fu > fug ⊗a · ag . a`g ⊗a · · af −afu>fug · · · ag   ,   · · vf vg     30 k. schrempf =  [uf · · ·] ,   af a ` f ⊗a −afu > fug −a ` fu > fug ⊗a · af · −afu>fug · · ag a`g ⊗a · · · ag   ,   · vf · vg     =  [uf · · ·] ,   af a ` f ⊗a −afu > fug 0 · af · 0 · · ag a`g ⊗a · · · ag   ,   · vf · vg     = ([ uf · ] , [ af a ` f ⊗a · af ] , [ · vf ]) + ([ ug · ] , [ ag a ` g ⊗a · ag ] , [ · vg ]) = ∂x|aaf + ∂x|aag. the two main steps are swapping block rows 2 and 3 and block columns 2 and 3, and eliminating the single non-zero (first) column in −a`fu > fug⊗a and −afu>fug (in block column 4) using the first column in block column 2. corollary 2.3. let f,g ∈ f be given by the admissible linear systems af = (uf,af,vf ) and ag = (ug,ag,vg) of dimensions nf,ng ≥ 1 respectively. fix x,a ∈{1}∪x such that x 6= a. then f = g implies that ∂x|aaf −∂x|aag is an als for 0 ∈ f. definition 2.4. let f ∈ f be given by the als a = (u,a,v) and fix x,a ∈ {1}∪x = {1,x1,x2, . . . ,xd} such that x 6= a. denote by ∂x|af the element defined by the als ∂x|aa. the map ∂x|a : f → f, f 7→ ∂x|af is called (partial) formal derivation, the element ∂x|af (partial) formal derivative of f. corollary 2.5. for each x,a ∈{1}∪x with x 6= a, the formal derivation ∂x|a : f → f is a linear map. now we are almost done. before we show the product rule in the following lemma 2.6, we have a look into the left family of the als of the (formal) derivative of a polynomial. let p = x3 ∈ f (and a = 1). a (minimal) als for ∂x|1p is given by free (rational) derivation 31   1 −x · · 0 −1 · · · 1 −x · · 0 −1 · · · 1 −x · · 0 −1 · · · 1 · · · 0 · · · · 1 −x · · · · · · · 1 −x · · · · · · · 1 −x · · · · · · · 1   s =   0 0 0 0 · · · 1   , s =   3x2 2x 1 0 x3 x2 x 1   . notice that the first four entries in the left family of ∂x|1a are si = ∂x|1si+4. lemma 2.6. (product rule) let f,g ∈ f be given by the admissible linear systems af = (uf,af,vf ) and ag = (ug,ag,vg) of dimension nf,ng ≥ 1 respectively. fix x ∈x and a ∈{1}∪x \{x}. then ∂x|a(fg) = ∂x|af g + f ∂x|ag. proof. let ` 6= k ∈ {0, 1, . . . ,d} be the indices of x resp. a. we write a`f for the coefficient matrix a` of af resp. a ` g for a` of ag. we take the sum and the product from [30, proposition 1.13] and start with the als from the right hand side,  af a ` f ⊗a 0 −afu > fug · · · af −vfug · · · · · ag · · · · · · af −vfug · · · · · ag a`g ⊗a · · · · · ag   s =   · · vg · · vg   , subtract block row 6 from block row 3, add block column 3 to block column 6 and remove block row/column 3 to get the als  af a ` f ⊗a −afu > fug · · · af · · −vfug · · af −vfug · · · · ag a`g ⊗a · · · · ag  s =   · · · · vg   . 32 k. schrempf now we can add block row 3 to block row 1 and eliminate the remaining columns in block (1, 3) by the columns {2, 3, . . . ,nf} from block (1, 1), remove block row/column 3 to get the als  af a ` f ⊗a −vfug · · af · −vfug · · ag a`g ⊗a · · · ag  s =   · · · vg   . swapping block rows 2 and 3 and block columns 2 and 3 yields the als  af −vfug a`f ⊗a 0 · ag · a`g ⊗a · · af −vfug · · · ag  s =   · · · vg   of the left hand side ∂x|0(fg). notice the upper right zero in the system matrix which is because of x 6= 1. definition 2.7. let f ∈ f, x ∈ x and a ∈ x \ {x}. the element ∂xf := ∂x|1f is called partial derivative of f. the element ∂x|af is called (partial) directional derivative of f (with respect to a). theorem 2.8. (free derivation) let x ∈ x . then the (partial) free derivation ∂x : f → f = k(〈x〉) is the unique map with the properties • ∂xh = 0 for all h ∈ k(〈x\{x}〉), • ∂xx = 1, and • ∂x(fg) = ∂xf g + f ∂xg for all f,g ∈ f = k(〈x〉). proof. let h be given by the als a = (u,a,v) and let ` ∈ {1, 2, . . . ,d} such that x = x`. we just need to recall the als for ∂xh,[ a a` ⊗ 1 · a ][ s′ s′′ ] = [ · v ] and observe that a` = 0 and thus as ′ = 0, in particular the first component of s′. therefore ∂xh = 0. for ∂xx = 1 we need to minimize  1 −x · −1 · 1 · · · · 1 −x · · · 1  s =   · · · 1   . free (rational) derivation 33 and the product rule ∂x(fg) = ∂xf g + f ∂xg is due to lemma 2.6. for the uniqueness we assume that there exists another ∂′x : f → f with the same properties. from the product rule we obtain ∂x(xf) = f + x∂xf = f + x∂ ′ xf = ∂ ′ x(xf), that is, x(∂xf −∂′xf) = 0 for all f ∈ f, thus ∂x = ∂′x. corollary 2.9. (hausdorff derivation [27]) let x ∈ x . then ∂xκ = 0 for all κ ∈ k, ∂xy = 0 for all y ∈ x \ {x}, ∂xx = 1, and ∂x(fg) = ∂xf g + f ∂xg for all f,g ∈ k〈x〉. remark. more general [8, theorem 7.5.17]: “any derivation of a sylvester domain extends to a derivation of its universal field of fractions.” recall however that the cyclic derivative is not (from) a derivation [27, section 1]. for a discussion of cyclic derivatives of nc algebraic power series we refer to [24]. proposition 2.10. let f ∈ f, x,y ∈ x and a,b ∈ x \{x,y} with a = b if and only if x = y. then ∂x|a(∂y|bf) = ∂y|b(∂x|af), that is, the (partial) derivations ∂x|a and ∂y|b commute. proof. let l,k ∈{1, 2, . . . ,d} the indices of x resp. y. there is nothing to show for the trivial case x = y, thus we can assume l 6= k. let f be given by the admissible linear system a = (u,a,v). then the “left” als ∂x|a(∂y|ba) is   a ak ⊗ b al ⊗a 0 · a 0 al ⊗a · · a ak ⊗ b · · · a  s =   · · · v   . swapping block rows/columns 2 and 3 yields the “right” als ∂y|b(∂x|aa):  a al ⊗a ak ⊗ b 0 · a 0 ak ⊗ b · · a al ⊗a · · · a  s =   · · · v   . 34 k. schrempf since there is no danger of ambiguity, we can define the (free) “higher” derivative of f ∈ f as ∂wf for each word w in the free monoid x∗ with the “trivial” derivative ∂(1)f = f. let w ∈x∗ and σ(w) denote any permutation of the letters of w. then ∂wf = ∂σ(w)f. the proof for nc formal power series in [23, proposition 1.8] is based on words (monomials), that is, ∂x(∂yw) = ∂y(∂xw) ∈ k〈x〉. recall that one gets the (nc) rational series by intersecting the (nc) series and the free field [26, section 9]: k〈〈x〉〉∩k(〈x〉) = krat〈〈x〉〉. overall, (free) nc derivation does not appear that often in the literature. and when there is some discussion it is (almost) always connected with “not simple” [27, section 1], “complicated” [16, section 14.3], etc. this is however not due to the hausdorff derivation but to the use of (finite) formal series as representation (for nc polynomials). using linear representations in the sense of cohn and reutenauer [9] for elements in the free field f can even reveal additional structure, as indicated in example 2.11 (below). for the somewhat more “complicated” example p̃ = 3cyxb + 3xbyxb + 2cyxax + cybxb− cyaxb− 2xbyxax + 4xbybxb − 3xbyaxb + 3xaxyxb− 3bxbyxb + 6axbyxb + 2xaxyxax + xaxybxb −xaxyaxb− 2bxbyxax− bxbybxb + bxbyaxb + 5axbybxb− 4axbyaxb from [3, section 8.2] we refer to [31, example 3.7]. example 2.11. let p = xyzx. a (minimal) polynomial als for p is   1 −x · · · · 1 −y · · · · 1 −z · · · · 1 −x · · · · 1  s =   · · · · 1   . then ∂xp = xyz + yzx admits a factorization into matrices [31, section 3]: ∂xp = [ x y ][y · · z ][ z x ] . free (rational) derivation 35 a minimal als for ∂xp is  1 −x −y · · · · 1 · −y 0 · · · 1 0 −z · · · · 1 · −z · · · · 1 −x · · · · · 1   s =   · · · · · 1   . example 2.12. we will use the directional derivative later in section 4 for the (nc) newton iteration. for q = x2 we get the “sylvester equation” ∂x|aq = xa + ax which is linear in a. in (the next) section 3 we have a look on the “free” chain rule which will turn out to be very elegant. we avoid the term “function” here since one needs to be careful with respect to evaluation (domain of definition), e.g. f = (xy − yx)−1 is not defined for diagonal matrices. for the efficient evaluation of polynomials (by matrices) one can use horner systems [31]. a generalization to elements in free fields is considered in future work. if there is a “compositional structure” available (in admissible linear systems) it could be used to further optimize evaluation. remark. for details on minimization (of linear representations) we refer to [32]. notice in particular that the construction (of the formal derivative) in definition 2.1 preserves refined pivot blocks. therefore, if a is refined, linear (algebraic) techniques suffice for minimization of ∂xa. last but not least, given the alphabet x = {x1,x2, . . . ,xd}, we can define the “free” (canonical) gradient ∇f = [ ∂1f,∂2f, . . . ,∂df ]> = [ ∂x1f,∂x2f, . . . ,∂xdf ]> ∈ fd for some f ∈ k(〈x〉). cyclic gradients are discussed in [34]. and for a “vector valued” element f = (f1,f2, . . . ,fd) we can define the jacobian matrix j(f) = (∂jfi) d i,j=1 = (∂xjfi) d i,j=1. for a discussion of non-commutative jacobian matrices on the level of nc formal power series we refer to [25]. 36 k. schrempf 3. free composition to be able to formulate a (partial) “free” chain rule in an elegant way, we need a suitable notation. it will turn out that cohn’s admissible systems [8, section 7.1] provide the perfect framework for the “expansion” of letters by elements from another free field. first we recall the “classical” (analytical) chain rule: let x, y and z be (open) sets, f = f(x) differentiable on x, g = g(y) differentiable on y and h = h(x) = g ( f(x) ) . then d dx h = d df g d dx f resp. h′(x) = g′ ( f(x) ) f ′(x). x f // h 88y g // z notation. for a fixed d ∈ n let x = {x1,x2, . . . ,xd}, y = {y1,y2, . . . ,yd} and y′ = {y′1,y ′ 2, . . . ,y ′ d} be pairwise disjoint alphabets, that is, x ∩y = x ∩y′ = y ∩y′ = ∅. by fz we denote the free field k(〈z〉). let a ∈ k〈y〉n×n be a linear full matrix and f = (f1,f2, . . . ,fd) a d-tuple of elements fi ∈ fx . by a ◦ f we denote the (not necessarily full) n×n matrix over fx where each letter yi ∈y is replaced by the corresponding fi ∈ fx , that is, a◦ f = a0 ⊗ 1 + a1 ⊗f1 + . . . + ad ⊗fd ∈ fn×nx . we write a◦af = a◦(af1,af2, . . . ,afd ) for a linearized version induced by the d-tuple of admissible linear systems afi = (ufi,afi,vfi ). now let g ∈ fy be given by the als ag = (ug,ag,vg) and f = (f1,f2, . . . , fd) ∈ fdx such that ag ◦ f is full. then (the unique element) h = g ◦ f ∈ fx is defined by the admissible system ãh = (ug,ag ◦ f,vg) and we write ah = (uh,ah,vh) = ( ug ◦af,ag ◦af,vg ◦af ) =: ag ◦af for a linearized version using “linearization by enlargement” [8, section 5.8]. (for details we refer to the proof of proposition 3.2 below.) fixing some x ∈x , we get the (partial) derivative ∂xh of h via the derivative ∂xah[ ah a x h ⊗ 1 · ah ] s = [ · vh ] . free (rational) derivation 37 ag ∼ g ∈ fy ◦f vvmmm mmm mmm mmm ∂y|y′ ((rr rrr rrr rrr rr ãh ∼ h ∈ fx lin. �� a′g ∼ g′ ∈ fy∪y′ ◦(f,f′) �� ah ∼ h ∈ fx ∂x ((pp ppp ppp ppp pp ã′h ∼ h ′ ∈ fx lin. vvmmm mmm mmm mmm m ∂xah ∼ ∂xh = h′ ∼a′h figure 1: let f = (f1, . . . ,fd) with fi ∈ fx = k(〈x〉) given by the d-tuple of admissible linear systems af = (a1, . . . ,ad) and g ∈ fy given by ag = (ug,ag,vg) such that the system matrix ag remains full when we replace each letter yi ∈ y by the respective element fi, written as ag ◦ f. then h = g ◦ f ∈ fx is defined by the admissible system ãh = (ug,ag ◦ f,vg) which we linearize to obtain an als ah for h before we apply the (partial) derivation ∂x (left path). on the other hand, we can apply the (directional) derivation ∂y|y′ by going over to the free field fy∪y′ = k(〈y∪y′〉) with an extended alphabet (with “placeholders” y′i), yielding g ′ = ∂y1|y′1g+. . .+∂yd|y ′ d g given by some als a′g = (u′g,a′g,v′g). then h′ = g′◦(f,f′) ∈ fx is defined by the admissible system ã′h = ( u′g,a ′ g ◦ (f,f′),v′g ) , where also each letter y′i ∈y ′ is replaced by the respective element f′i = ∂xfi ∈ fx . after linearization we get an als a′h such that ∂xah −a ′ h = 0, that is, ∂xh = h ′ (right path). to give a meaning to the right hand side of ∂xh = ∂x(g◦ f), we introduce the “total” (directional) derivative g′ := ∂y|y′g = ∂y1|y′1g + ∂y2|y ′ 2 g + . . . + ∂yd|y′d g ∈ fy∪y′ given by the als a′g := ∂y|y′ag = ([ ug · ] , [ ag ∑d i=1 a (i) g ⊗y′i · ag ] , [ · vg ]) with letters y′i ∈y ′. using a similar notation for the derivatives f ′ = (f ′1,f ′ 2, . . . ,f ′ d) := (∂xf1,∂xf2, . . . ,∂xfd) = ∂xf, we can write h′ := ∂x(g ◦ f) = g′ ◦ (f,f ′) = ∂y|y′g ◦ (f,f ′) (3.1) 38 k. schrempf given by the admissible system a′g ◦ (f,f ′) = ∂y|y′ag ◦ (f,f ′). after an illustration in the following example, we show in proposition 3.2 that indeed ∂xh = h ′ = ∂x(g ◦ f) = ∂y|∂xg ◦ f using an abbreviation for the right hand side of (3.1). for an overview see figure 1. remark. cohn writes an admissible system a = (u,a,v) as “block” [a,v] with not necessarily scalar column v [8, section 7.1]. a ring homomorphism which preserves fullness of matrices is called honest [8, section 5.4]. remark. it is crucial that ag ◦ f is full to be able to define the composition. on a purely algebraic level this is sufficient to define the (partial) chain rule. for a d-tuple g = (g1,g2, . . . ,gd) given by admissible linear systems ag = (ag1,ag2, . . . ,agd ) with agi = (ugi,agi,vgi ) we need agi ◦ f full for all i ∈{1, 2, . . . ,d}. example 3.1. here we take x = {x,y}, y = {f,p} and abuse notation. let f = (x−1 + y)−1 ∈ fx , p = xy ∈ fx and g = pfp ∈ fy. then h = g ◦ (f,p) ∈ fx is given by the (minimal) als   1 −x · · · · · 1 −y · · · · · 1 −x · · · · y 1 −x · · · · · 1 −y · · · · · 1   s =   · · · · · 1   , and ∂xh = ∂x(pfp) = ∂xpfp + p∂xf p + pf ∂xp by free (rational) derivation 39   1 −x · · · · · −1 · · · · · 1 −y · · · · · · · · · · · 1 −x · · · · · −1 · · · · y 1 −x · · · · · −1 · · · · · 1 −y · · · · · · · · · · · 1 · · · · · · 1 −x · · · · · 1 −y · · · · · 1 −x · · · · y 1 −x · · · · · 1 −y · · · · · 1   s =   · · · · · · · · · · · 1   . (3.2) on the other hand, ∂x(∂fg + ∂pg) = ∂x(∂fg) + ∂x(∂pg) is given by (the admissible system)  1 −p · · · −∂xp · · · 1 −f · · · −∂xf · · · 1 −p · · · −∂xp · · · 1 · · · · · · · · 1 −p · · · · · · · 1 −f · · · · · · · 1 −p · · · · · · · 1   s =   · · · · · · · 1   . the summands ∂xpfp and pf ∂xp are easy to read off in the als (3.2). (alternatively one could add row 8 to row 1 resp. row 11 to row 4.) to read off p∂xf p = −p (x−1 + y)−2 p, we just need to recall the (minimal) als  1 −x · −1 y 1 · · 1 −x y 1  s =   · · · 1   . in other words and with g ∈ fy given by the als ag = (ug,ag,vg) of dimension n: ∂xh is given by the admissible system ∂y|∂xag,[ ag ∂xag · ag ] s = [ · v ] 40 k. schrempf of dimension 2n. notice that ∂xag is understood here in a purely symbolic way, that is, over fy∪y′ with additional letters “∂xy” in y′ for each y ∈y. in this sense ∂y|∂xag is actually linear. proposition 3.2. (free chain rule) let x = {x1,x2, . . . ,xd}, y = {y1,y2, . . . ,yd} and y′ = {y′1,y ′ 2, . . . ,y ′ d} be pairwise disjoint alphabets and fix x ∈x . for g ∈ k(〈y〉) given by the als ag = (ug,ag,vg) and the d-tuple f = (f1,f2, . . . ,fd) ∈ k(〈x〉)d such that ag◦f is full. denote by f ′ the d-duple (∂xf1,∂xf2, . . . ,∂xfd). then ∂x(g ◦ f) = ∂y|y′g ◦ (f,f ′) =: ∂y|∂xg ◦ f. proof. for i ∈ {1, 2, . . . ,d} let fi ∈ fx be given by the admissible linear systems afi = (ufi,afi,vfi ) respectively. in the following we assume —without loss of generality— d = 3, and decompose ag ◦ f into [ a11 a12 a12 a ] of size n with a = α0 + α1f1 + α2f2 + α3f3. a generalization of the “linearization by enlargement” [8, section 5.8] is then to start with the full matrix ag ◦ f ⊕ af1 ⊕ . . . ⊕ afd , add ufia −1 fi from the corresponding block row to row n, and −αia−1fi vfi from the corresponding block column to column n in the upper left block (these transformations preserve fullness):  a11 a12 · · · a21 α0 uf1 uf2 uf3 · −α1vf1 af1 · · · −α2vf2 · af2 · · −α3vf3 · · af3   . a partially linearized system matrix for ∂x(ag ◦ f) is  a11 a12 · · · ∗ ∗ · · · a21 α0 uf1 uf2 uf3 ∗ 0 0 0 0 · −α1vf1 af1 · · · 0 a x f1 ⊗ 1 · · · −α2vf2 · af2 · · 0 · a x f2 ⊗ 1 · · −α3vf3 · · af3 · 0 · · a x f3 ⊗ 1 a11 a12 · · · a21 α0 uf1 uf2 uf3 · −α1vf1 af1 · · · −α2vf2 · af2 · · −α3vf3 · · af3   . (3.3) free (rational) derivation 41 on the other hand, the system matrix of ∂yi|y′i ag is  a11 a12 a yi 11 ⊗y ′ i a yi 12 ⊗y ′ i a21 a a yi 21 ⊗y ′ i αiy ′ i a11 a12 a21 a   , the corresponding partially linearized system matrix of (∂yi|y′i ag) ◦ (f,f ′i) is  a11 a12 · · · a yi 11 ⊗f ′ i a yi 12 ⊗f ′ i · · · a21 α0 uf1 uf2 uf3 a yi 21 ⊗f ′ i αif ′ i · · · · −α1vf1 af1 · · · · · · · · −α2vf2 · af2 · · · · · · · −α3vf3 · · af3 · · · · · a11 a12 · · · a21 α0 uf1 uf2 uf3 · −α1vf1 af1 · · · −α2vf2 · af2 · · −α3vf3 · · af3   . to eliminate the boxed entry αif ′ i we just need to recall the derivative ∂xafi of afi = (ufi,afi,vfi ), the invertible matrix q swaps block columns 1 and 2:  0 ufi ·· afi axfi ⊗ 1 −αivfi · afi  q =  ufi 0 ·afi · axfi ⊗ 1 · −αivfi afi   . thus, after summing up (over i ∈ {1, 2, . . . ,d}), we get the system matrix (3.3) and hence ∂xah = ∂x(ag ◦af ) = ∂y|y′ag ◦af,f′, that is, ∂xh = ∂x(g ◦ f) = ∂y|∂xfg ◦ f. free (non-commutative) decomposition is important in control theory [12, section 6.2.2]: “the authors do not know how to fully implement the decompose operation. finding decompositions by hand can be facilitated with the use of certain type of collect command”. 42 k. schrempf let g = fabf +cf +de and f = xy +z, that is, f solves a riccati equation. given h = g ◦f = (xy + z)ab(xy + z) + c(xy + z) + de by the (minimal) admissible linear system ah = (uh,ah,vh),  1 −x −z · −d −c · · · 1 −y · · · · · · · 1 −a · · · · · · · 1 · −b · · · · · · 1 · · −e · · · · · 1 −x −z · · · · · · 1 −y · · · · · · · 1   s =   · · · · · · · 1   , one can read off f = xy + z directly since ah has the form  1 −f · −d −c · · 1 −a · · · · · 1 · −b · · · · 1 · −e · · · · 1 −f · · · · · 1   s =   · · · · · 1   . so, starting with a minimal als ãh for h one “just” needs to find appropriate (invertible) transformation matrices p,q such that ah = pãhq using (commutative) gröbner bases techniques similar to [10, theorem 4.1] or the refinement of pivot blocks [32, section 3]. although this is quite challenging using brute force methods, in practical examples it is rather easy using free fractions, that is, minimal and refined admissible linear systems, as a “work bench” where one can perform the necessary row and column operations manually. “the challenge to computer algebra is to start with an expanded version of h = g ◦f, which is a mess that you would likely see in a computer algebra session, and to automatically motivate the selection of f.” [12, section 6.2.1] working with free fractions is simple, in particular with nc polynomials. the main difficulty in working with (finite) formal power series is that the number of words can grow exponentially with respect to the rank, the minimal free (rational) derivation 43 dimension of a linear representation [31, table 1]. in this case, for h = g ◦f = xyabxy + xyabz + zabxy + zabz + cxy + cz + de, the main “structure” becomes (almost) visible already after minimization of the corresponding “polynomial” admissible linear system (in upper triangular form with ones in the diagonal). 4. application: newton iteration to compute the third root 3 √ z of a (positive) real number z, say z = 2, we just have to find the roots of the polynomial p = x3 − z, for example by using the newton iteration xk+1 := xk −p(xk)/p′(xk), that —given a “good” starting value x0— yields a (quadratically) convergent sequence x0,x1,x2, . . . such that limk→∞x 3 k = z. using fricas [14], the first 6 iterations for x0 = 1 are k |xk −xk−1| xk 1 3.333·10−1 1.3333333333333333 2 6.944·10−2 1.2638888888888888 3 3.955·10−3 1.259933493449977 4 1.244·10−5 1.2599210500177698 5 1.229·10−10 1.2599210498948732 6 0 1.2599210498948732 detailed discussions are available in every book on numerical analysis, for example [17]. as a starting point towards current research, one could take [33]. now we would like to compute the third root 3 √ z of a real (square) matrix z (with positive eigenvalues). we (still) can use xk+1 := 2 3 xk + 1 3 x−2k z if we choose a starting matrix x0 such that x0z = zx0 because then all xk commute with z [19, section 7.3]. for x0 = i and z =  47 84 5442 116 99 9 33 32   respectively 3√z =  3 2 01 4 3 0 1 2   , some iterations are (with ‖.‖f denoting the frobenius norm) 44 k. schrempf k ‖xk −xk−1‖f ‖xk − 3 √ z‖f 4 9.874 23.679 5 6.511 13.818 6 4.109 7.309 7 2.254 3.200 8 8.295·10−1 9.455·10−1 9 1.140·10−1 1.160·10−1 10 2.044·10−3 2.044·10−3 11 1.115·10−6 6.489·10−7 does this sequence of matrices x0,x1, . . . converge? some more iterations reveal immediately that something goes wrong, visible in table 1. k ‖xk −xk−1‖f ‖xk − 3 √ z‖f ‖xkz −zxk‖f 1 65.836 5.385 2.274·10−13 2 22.279 60.756 7.541·10−13 3 14.845 38.500 1.110·10−12 4 9.874 23.679 2.031·10−12 5 6.511 13.818 7.725·10−12 6 4.109 7.309 7.010·10−11 7 2.254 3.200 9.451·10−10 8 8.295·10−1 9.455·10−1 1.716·10−8 9 1.140·10−1 1.160·10−1 3.548·10−7 10 2.044·10−3 2.044·10−3 7.473·10−6 11 1.115·10−6 6.489·10−7 1.574·10−4 12 1.979·10−5 8.971·10−7 3.314·10−3 13 4.168·10−4 1.890·10−5 6.978·10−2 14 8.777·10−3 3.979·10−4 1.469 15 1.848·10−1 8.379·10−3 3.094·10+1 16 3.892 1.764·10−1 6.515·10+2 17 8.195·10+1 3.715 1.372·10+4 18 1.726·10+3 7.823·10+1 2.889·10+5 table 1: the first 18 newton iterations to find 3 √ z for x0 = i. the values in column 2 (and 3) for the first 11 newton steps seem to indicate convergence. however, the (frobenius) norm of the commutator xkz −zxk (column 4) increases steadily and that causes eventually a diverging sequence (xk). free (rational) derivation 45 the problem is that due to rounding errors (in finite precision arithmetics) commutativity of xk (with z) is lost, that is, ‖xkz −zxk‖f increases with every iteration. this happens even for a starting value close to the solution x0 = 3 √ z +εi. for a detailled discussion of matrix roots and how to overcome such problems we refer to [19, section 7], for further information about numerics to [11]. a classical introduction to matrix functions is [15, chapter v]. for the (matrix) square root and discussions about the stability (of newton’s method) we recommend [18]. remark. to ensure commutation of the iterates xk with z one could use an additional correction step solving the sylvester equation z∆xk−∆xkz = xkz−zxk for an update ∆xk. since this is expensive the benefit of using the “commutative” newton iteration would be lost. for fast solutions of sylvester equations we refer to [20]. although what we are going to introduce now as “non-commutative (nc) newton iteration” is even more expensive, it can be implemented as black box algorithm, that is, without manual computation of the non-commutative derivative and any individual implementation/programming. furthermore, there is absolutely no restriction for the initial iterate. to get the nc newton method for solving f(x) = 0 we just need to “truncate” the hausdorff polarization operator [27, section 4] f(x + b) = f(x) + ∂x|bf(x) + 1 2 ∂x|b ( ∂x|bf(x) ) + · · · as analogon to taylor’s formula. thus, for f(x) = x3 −z we have to solve 0 = x3 −z︸︷︷︸ f + bx2 + xbx + x2b︸︷︷︸ ∂x|bf ∈ r(〈x〉) =: f with respect to b. in terms of (square) matrices x,z,b this is just the generalized sylvester equation bx2 + xbx + x2b = z − x3 which is linear in b [19, section 7.2]. taking a (not necessarily with z commuting) starting matrix x0, we can compute b0 and get x1 := x0 + b0 and iteratively xk+1 := xk + bk. minimal admissible linear systems for f = x3−z and ∂x|bf = bx2+xbx+x2b are given by   1 −x · z · 1 −x · · · 1 −x · · · 1  s =   · · · 1   46 k. schrempf and   1 −x · −b · · · 1 −x · −b · · · 1 · · −b · · · 1 −x · · · · · 1 −x · · · · · 1   s =   · · · · · 1   respectively. a minimal als a = (u,a,v) of dimension n = 6 for g = f + ∂x|bf is immediate:  1 −x · −b · z · 1 −x · −b · · · 1 · · −b−x · · · 1 −x · · · · · 1 −x · · · · · 1   s =   · · · · · 1   . since the system matrix is just a linear matrix pencil a = a1 ⊗ 1 + ab ⊗ b + ax ⊗x + az ⊗z we can plug in m×m matrices using the kronecker product a(b,x,z) = a1 ⊗ i + ab ⊗b + ax ⊗x + az ⊗z. in this case, for v = en = [0, . . . , 0, 1] >, the evaluation of g : mm(r)3 → mm(r) is just the upper right m × m block of a(b,x,z)−1. for how to efficiently evaluate (nc) polynomials by matrices using horner systems we refer to [31]. here, a(b,x,z) is invertible for arbitrary b, x and z. given z and x, to find a b such that g(b,x,z) = 0 we have to look for transformation matrices p = p(tij) and q = q(uij) of the form p =   i · · t1,1 t1,2 0 · i · t2,1 t2,2 0 · · i t3,1 t3,2 0 · · · i · · · · · · i · · · · · · i   , q =   i · · 0 0 0 · i · u2,1 u2,2 u2,3 · · i u3,1 u3,2 u3,3 · · · i · · · · · · i · · · · · · i   such that the upper right block of size 3m×3m of pa(b,x,z)q is zero, that is, we need to solve a linear system of equations (with 6m2 +6m2 +m2 = 13m2 free (rational) derivation 47 unknowns) similar to the word problem [30, theorem 2.4]. table 2 shows the nc newton iterations for the starting matrix x0 =  1 0 20 1 0 0 0 1   (4.1) using fricas [14] and the least squares solver dgels from [22]. k ‖bk‖f ‖xk − 3 √ z‖f ‖xkz −zxk‖f 0 46.877 5.745 113.842 1 16.081 42.298 5552.242 2 10.768 26.374 3659.788 3 7.320 15.912 2395.971 4 4.971 9.358 1534.892 5 2.934 6.122 912.700 6 2.651 4.389 414.201 7 1.380 1.846 86.771 8 3.878·10−1 4.875·10−1 5.638 9 9.378·10−2 1.023·10−1 2.652·10−1 10 8.432·10−3 8.506·10−3 6.737·10−3 11 7.407·10−5 7.408·10−5 1.951·10−5 12 5.895·10−9 5.895·10−9 5.106·10−10 13 1.825·10−14 1.521·10−14 1.491·10−12 table 2: the first 13 (nc) newton iterations to find 3 √ z for x0 from (4.1). in the beginning, the iterates xk do not commute with z (column 4). in general, depending on the initial iterate x0, there is no guarantee that one ends up in some prescribed solution x since there can be several (matrix) roots. for the principal p-th root (and further discussion) we refer to [19, theorem 7.2]. for a discussion of taylor’s theorem (for matrix functions) one should have a look in [13]. the goal of this section was mainly for illustration (of the use of the free derivative). how could we attack analysis of convergence (for special classes of rational functions) in this context? “classical” interval newton is discussed in [28, chapter 6.1]. what’s about “nc interval newton”? the next natural 48 k. schrempf step would be multi-dimensional nc newton, say to find a root g(x,y,z) = 0 for g =  g1g2 g3   =   x 3 + z −d( x(1 −yx) )−1 −e xzx−yz + z2 −f   with (matrix-valued) parameters d,e,f. although very technical, one can set up a “joint” linear system of equations to solve  g1 + ∂x|ag1 g1 + ∂y|bg1 g1 + ∂z|cg1 g2 + ∂x|ag2 g2 + ∂y|bg2 g2 + ∂z|cg2 g3 + ∂x|ag3 g3 + ∂y|bg3 g3 + ∂z|cg3   = 0 for matrices a, b and c using the previous approach to find tuples of “individual” transformation matrices pij and qij to create respective upper right blocks of zeros. the problem however is, that this system is overdetermined in general, and starting arbitrary close to a root leads to a residual that causes divergence. how can one overcome that? acknowledgements i am very grateful to tobias mai for making me aware of cyclic derivatives respectively the work of rota, sagan and stein [27] in may 2017 in graz and for a discussion in october 2018 in saarbrücken. i also use this opportunity to thank soumyashant nayak for continuous “noncommutative” discussions, in particular about free associative algebras and (inner) derivations. and i thank the referees for the feedback and a hint on the literature. references [1] g.m. bergman, w. dicks, on universal derivations, j. algebra 36 (2) (1975), 193 – 211. 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[28] s.m. rump, verification methods: rigorous results using floating-point arithmetic, acta numer. 19 (2010), 287 – 449. [29] k. schrempf, free fractions: an invitation to (applied) free fields, arxiv e-prints, september 2018. version 2, october 2020, http://arxiv.org/ pdf/1809.05425. [30] k. schrempf, linearizing the word problem in (some) free fields, internat. j. algebra comput. 28 (7) (2018), 1209 – 1230. [31] k. schrempf, horner systems: how to efficiently evaluate non-commutative polynomials (by matrices), arxiv e-prints, october 2019. [32] k. schrempf, a standard form in (some) free fields: how to construct minimal linear representations, open math. 18 (1) (2020), 1365 – 1386. [33] d. schleicher, r. stoll, newton’s method in practice: finding all roots of polynomials of degree one million efficiently, theoret. comput. sci. 681 (2017), 146 – 166. [34] d. voiculescu, a note on cyclic gradients, indiana univ. math. j. 49 (3) (2000), 837 – 841. http://arxiv.org/pdf/1809.05425 http://arxiv.org/pdf/1809.05425 getting started free derivation free composition application: newton iteration e extracta mathematicae vol. 33, núm. 2, 219 – 227 (2018) minimal matrix representations of decomposable lie algebras of dimension less than or equal to five ryad ghanam, manoj lamichhane, gerard thompson department of mathematics, virginia commonwealth university in qatar, po box 8095, doha, qatar, raghanam@vcu.edu department of mathematics, university of wisonsin at waukesha, waukesha, wi 53188, u.s.a., manoj.lamichhane@uwc.edu department of mathematics, university of toledo, toledo, oh 43606, u.s.a., gerard.thompson@utoledo.edu presented by consuelo mart́ınez received october 27, 2017 abstract: we obtain minimal dimension matrix representations for each decomposable fivedimensional lie algebra over r and justify in each case that they are minimal. key words: lie algebra, lie group, minimal representation. ams subject class. (2000): 17bxx, 53cxx. 1. introduction given a real lie algebra g of dimension n a well known theorem due to ado asserts that g has a faithful representation as a subalgebra of gl(p, r) for some p. in several recently published papers the authors and others have investigated the problem of finding minimal dimensional representations of indecomposable lie algebras of dimension five and less [7, 5, 2, 3]. in particular, in [3] minimal representations have been found for all the indecomposable algebras of dimension five and less. there remains the question of finding minimal dimensional representations of the decomposable lie algebras of dimension five and less and that is the goal of this paper. it should be regarded as a sequel to [3] and as a result, we now have minimal dimensional representations of all lie algebras of dimension up to and including five. burde [1] has defined an invariant µ(g) of a lie algebra g to be the dimension of its minimal faithful representation. in case an algebra g is decomposable, that is, g = g1 ⊕g2 and one has representations for g1 and g2, then there is an obvious way to construct a representation for g, that is, by taking a block diagonal representation, so that µ(g1 ⊕g2) ≤ µ(g1)+µ(g2). however, it an interesting phenomenon that in some cases there are representations of smaller 219 220 r. ghanam, m. lamichhane, g. thompson size than this diagonal representation, so that µ(g1 ⊕ g2) < µ(g1) + µ(g2) is possible. a particular circumstance in which this latter inequality holds is when g1 = r, as an abelian one-dimensional lie algebra and g2 is indecomposable: then µ(g1 ⊕g2) = µ(g2) and the representation for g1 ⊕g2 is obtained by adding multiples of the identity to the minimal dimensional representation for g2. the principal concern of the present paper is to furnish examples where µ(g1 ⊕g2) < µ(g1)+µ(g2) but that are not obtained by the trivial mechanism of adding multiples of the identity. in this paper we shall use the classification of the low-dimensional indecomposable lie algebras found in [6] which is in turn taken from [4]. such algebras are denoted as am.n where m denotes the dimension of the algebra and n the nth one in the list. we shall need indecomposable algebras of dimension from 2 to 4. in dimension 2, the algebra a2.1 is the unique non-abelian algebra. in dimension 3 there are two simple algebras a3.8 = sl(2, r) and a3.9 = so(3) and the remaining seven are solvable. in dimension 4 there are no simple or semi-simple algebras and all 12 classes of algebra are solvable. an outline of this paper is as follows. in section 2 we list all the decomposable lie algebras of dimension ≤ 5. in section 3 we consider classes of algebras given in section 2, for example abelian lie algebras. to show that µ has a certain value we argue in every case that µ has a certain lower bound; then all that is required is to exhibit a particular representation for which that value of µ is attained. in section 4 we consider algebras of the form a2.1⊕ a3.n, which need slightly lengthier arguments. finally, in section 5 we give a list of representations with at least one for each decomposable lie algebra where the value of µ is attained. occasionally we give more than one such representation if it seems to be of particular interest although we do not consider the difficult issue of the equivalence of different representations. 2. classification of decomposable lie algebras the first step is to determine, as abstract lie algebras, all the decomposable lie algebras of dimension ≤ 5. here is a summary of all such algebras. • dimension 2: abelian r2. • dimension 3: abelian r3: nonabelian r ⊕ a2.1. • dimension 4: abelian r4: nonabelian: r2 ⊕ a2.1; a2.1 ⊕ a2.1;r ⊕ a3.n. • dimension 5: abelian r5: nonabelian r3 ⊕ a2.1; r ⊕ a2.1 ⊕ a2.1; r2⊕ a3.n; a2.1⊕ a3.n; r ⊕ a4.n representations of decomposable lie algebras 221 3. minimal dimension matrix representations now we shall consider each of the algebras above with regard to finding minimal dimension matrix representations. 3.1. abelian subalgebras. first of all for an abelian lie algebra of dimension n we have µ = ⌈2 √ n − 1⌉ [1]. for n = 2, 3, 4, 5 we obtain µ(rn) = 2, 3, 4, 4. representations will be given in section 5. 3.2. algebras that have r as a single summand. in each of the cases r⊕a2.1, r⊕a3.1, . . . , r⊕a3.7, r⊕a4.1, . . . , r⊕a4.12 the second factor is solvable. lie’s theorem guarantees an upper triangular representation for the adjoint representation, at least when we complexify the algebra. in fact for all these algebras representations have been found which are almost upper triangular and of minimal dimension [2, 3]. by “almost upper triangular” we mean that the complexified algebra is upper triangular. as such, we can simply add multiples of the identity so as to accommodate the extra factor of r. in the case of r ⊕ a2.1, equivalently we are simply looking at the space of 2 × 2 upper triangular matrices. in the case of r ⊕ a3.8 we have r ⊕ sl(2, r), which is isomorphic to gl(2, r) and therefore µ = 2. in the case of r ⊕ a3.9 we have r ⊕ so(3) and again we can simply add multiples of the identity to the standard three-dimensional representation of so(3). see section 5 for such representations. 3.3. dimension four non-abelian. consulting the list in section 2, the first classes of algebra for which we have not yet supplied a representation are r2 ⊕ a2.1 and a2.1 ⊕ a2.1. in the first case there is an abelian threedimensional subalgebra and consequently µ ≥ 3; in fact µ = 3 and we give two representations in section 5. for a2.1 ⊕ a2.1 clearly µ > 2 because the only four-dimensional subalgebra with µ = 2 is gl(2, r); however, we can find representations in gl(3, r). 3.4. dimension five non-abelian. 3.4.1. r3 ⊕a2.1. the algebra has a four dimensional abelian subalgebra and as such µ ≥ 4. in fact µ = 4. 3.4.2. r ⊕ a2.1 ⊕ a2.1. for this algebra µ = 3. 222 r. ghanam, m. lamichhane, g. thompson 3.4.3. r2 ⊕ a3.n. for 1 ≤ n ≤ 7 each of the algebras has a four dimensional abelian subalgebra and as such µ ≥ 4. in fact in each of these cases we can simply take a block diagonal representation of each three dimensional representation r ⊕ a3.n with r giving µ = 4. for r2 ⊕ a3.8 we can again take a block diagonal representation of r ⊕ a3.8 ≈ gl(2, r) with r giving µ = 3. finally for r2 ⊕a3.9 the fact that µ ≥ 4 follows from an application of schur’s lemma. again we can take a block diagonal representation. 4. algebras a2.1 ⊕ a3.n it remains to discuss the cases a2.1 ⊕a3.n where 1 ≤ n ≤ 9. for 1 ≤ n ≤ 7 these algebras have a three dimensional abelian subalgebra and as such µ ≥ 3. on the other hand for each a3.n we have µ ≤ 3. hence for a2.1 ⊕ a3.n we must have µ ≤ 5 by taking a block diagonal representation. lemma 4.1. over r every 3×3 matrix is equivalent under change of basis to one of the following: a)  α 0 00 β 0 0 0 γ   , b)  λ 1 00 λ 0 0 0 µ   , c)   α 1 0−1 α 0 0 0 β   , d)  λ 1 00 λ 1 0 0 λ   . lemma 4.2. any abelian three-dimensional subalgebra of gl(3, r) contains a multiple of the identity. proof. we refer to lemma 4.1. in case d) the space of matrices that commutes with the given matrix is of the form [ a b c 0 a b 0 0 a ] . likewise in case c) the space of matrices that commutes with the given matrix is of the form[ a b 0 −b a 0 0 0 c ] and hence the result follows in these two cases. in case b) if λ ̸= µ the centralizer consists of matrices of the form [ a b c 0 a 0 0 0 c ] so if there is a an abelian three-dimensional algebra it will contain i. in case b) if λ = µ the centralizer consists of matrices of the form [ a b c 0 a 0 0 d e ] . now we may assume that e = a or else we would be in a different subcase. given two such commuting matrices by subtracting multiples of the original matrix we may assume that b = 0 and so a linear a combination of them will be a multiple of i. finally, in case a) we can assume that all three matrices in the algebra have real eigenvalues of algebraic multiplicity one; otherwise the algebra would fall into case b), c) or d). thus all three matrices are diagonalizable and representations of decomposable lie algebras 223 since they commute, simultaneously diagonalizable and linearly independent. hence again the algebra will contain i. corollary 4.3. a subalgebra of gl(3, r) cannot have a subalgebra isomorphic to a2.1 ⊕ a3.n where 1 ≤ n ≤ 7. corollary 4.4. a subalgebra of gl(3, r) that contains an abelian threedimensional subalgebra is decomposable as an abstract lie algebra. in view of corollary 4.3 it follows for a2.1 ⊕ a3.n, where 1 ≤ n ≤ 7, that µ ≥ 4. in fact µ = 4. it remains to discuss a2.1⊕a3.8 and a2.1⊕a3.9. in the first case we cannot have µ = 3. if we could then the representation for a3.8 = sl(2, r) would be either the 3×3 irreducible representation, which is excluded because the only matrices that commute with it are multiples of the identity and so a2.1 could not be accommodated; or, the 2 × 2 irreducible representation augmented by an extra column of rows and zeros. however, in this latter case the centralizer of sl(2, r) is two-dimensional abelian and again there would not be room for a2.1. the block diagonal representation of a2.1 and a3.8 is four-dimensional and so µ = 4. in the case of a2.1 ⊕ a3.9 again µ ̸= 3 because of schur’s lemma. we can apply schur’s lemma again to show that µ ̸= 4. in fact one needs to exercise some caution because of the same reason that the 4×4 standard representation of so(4) ≈ so(3) ⊕ so(3); as such one can find alternative 4 × 4 representations for r2 ⊕ a3.9 but not for a2.1 ⊕ a3.9. on the other hand the block diagonal representation of a2.1 and a3.9 is five-dimensional and so µ = 5. 5. the representations 5.1. dimension three. 5.1.1. r3.  x 0 00 y 0 0 0 z    x y z0 x 0 0 0 x   . 5.1.2. r ⊕ a2.1. [ x z 0 y ] . 224 r. ghanam, m. lamichhane, g. thompson 5.2. dimension four. 5.2.1. r4.   w 0 0 0 0 x 0 0 0 0 y 0 0 0 0 z     0 0 w x 0 0 y z 0 0 0 0 0 0 0 0     w 0 x z 0 w 0 y 0 0 w 0 0 0 0 w   . 5.2.2. r2 ⊕ a2.1 : [e1, e2] = e2. x w 00 y 0 0 0 z    αx + z y w0 (α + 1)x + z 0 0 0 αx + z   (α ∈ r) . 5.2.3. a2.1 ⊕ a2.1 : [e1, e2] = e2, [e3, e4] = e4. αx + λz y w0 (α + 1)x + λz 0 0 0 αx + µz   (α, λ, µ ∈ r, λ2 + µ2 ̸= 0) . 5.2.4. r ⊕ a3.k (1 ≤ k ≤ 9). lie brackets are the same as for a3.k (1 ≤ k ≤ 9): use the representations given in [3] and add multiples of the identity. 5.3. dimension five. 5.3.1. r5.   q 0 w x 0 q y z 0 0 q 0 0 0 0 q   . 5.3.2. r3 ⊕ a2.1 : [e1, e2] = e2.  x y 0 0 0 z 0 0 0 0 w 0 0 0 0 q   . representations of decomposable lie algebras 225 5.3.3. r ⊕ a2.1 ⊕ a2.1 : [e2, e3] = e2, [e4, e5] = e4.  x q w 0 y 0 0 0 z   . 5.3.4. r ⊕ a4.8,4.9b(−1 0 and every finite dimensional subspaces e ⊂ x and f ⊂ y ∗ there exists t : e −→ y such that (a) te = e for all e ∈ y ∩e, (b) ||te|| ≤ (1 + �)||e|| for all e ∈ e, (c) φf∗(e) = f∗(te) for all e ∈ e, f∗ ∈ f . clearly the notion of ideal imitates plr. however it lacks two important aspects of x in x∗∗ situation; namely, we do not get almost isometry from the finite dimensional subspace e ⊆ x to y and range(p) is not norming for x, where by norming we mean 1-norming. so following two possible strengthening of notion of ideals were considered. definition 1.2. [9] a subspace y of a banach space x is said to be a strict ideal in x if y is an ideal in x and range(p) is norming for x where p : x∗ −→ x∗ is a norm one projection with ker(p) = y ⊥. definition 1.3. [2] a subspace y of a banach space x is said to be an almost isometric ideal (henceforth ai-ideal) in x if for every � > 0 and every finite-dimensional subspace e ⊂ x there exists t : e −→ y which satisfies condition (a) in theorem 1.1 and (1 − �)||e|| ≤ ||te|| ≤ (1 + �)||e|| for all e ∈ e. on the other hand there is a notion of u-ideal, which is a generalisation of m-ideal and is also a strengthening of plr. definition 1.4. [9] a subspace y of a banach space x is said to be a u-ideal in x if there exists a norm one projection p : x∗ → x∗ with ker(p) = y ⊥ and ‖ix∗ − 2p‖ = 1, where ix∗ denotes the identity operator on x∗. it is straightforward to see that strict ideals are ai-ideals and ai-ideals are of course ideals. the inclusions are strict as shown by examples in [2]. if we view above notion of ideals and its subsequent strengthening as generalisations of x in x∗∗ situation, many isometric properties of x are carried to y and much of the studies in this area are devoted to that. however plr can be viewed simply as: identity operator on x∗∗ is an extension of identity operator on x and x∗ is norming for x. on the same ideal operators and relative godun sets 3 vein, the property of y being an ideal in x may be viewed as there exists t : x → y ∗∗ such that ‖t‖ ≤ 1 and t|y = iy . to see this consider the following. suppose y ⊆ x and t : x → y ∗∗ is such that t|y = iy and ‖t‖ ≤ 1. we will refer t as an ideal operator. given t we define p : x∗ → x∗ as p(x∗) = t∗|y ∗ (x∗|y ) and φ : y ∗ → x∗ by φ(y∗) = t∗|y ∗ (y∗). then p is a norm one projection on x∗ with ker(p) = y ⊥ and φ is a hahn– banach extension operator. in the sequel, we will refer these p and φ as ideal projection and hahn–banach extension operator corresponding to t. we also note that this definition is reversible, in the sense that given an ideal projection p , we may define t as above. same is true for φ. with this view point, in section 2 of this paper we provide characterisations of ideal, strict ideal and ai-ideal in terms of the properties of ideal operator t. in the case of ideal and strict ideal (see proposition 2.1) the results are essentially known. proposition 2.1 has some immediate corollaries. it is straightforward to see that if y is 1-complemented in x then y is an ideal in x (if p : x → x is a projection with ‖p‖ = 1 then p∗ is an ideal projection). for a space y which is 1-complemented in its bidual y ∗∗ we show y is an ideal in any superspace x if and only if y is 1-complemented in x. in particular l1(µ) or any reflexive space is an ideal in a superspace if and only if it is 1-complemented. it also follows that any ideal in l1(µ) is 1-complemented. coming to ai-ideals we show that reflexive spaces with a smooth norm cannot have any proper ai-ideal. situation becomes more interesting for the space c[0, 1], the space of all real-valued continuous functions on [0,1] equipped with the supremum norm. it is known that c[0, 1] is universal for the class of separable banach spaces. we show that any ai-ideal of c[0, 1] inherits the universality property from c[0, 1]. also, any separable ai-ideal in l1(µ), µ non-atomic, is isometric to l1[0, 1]. section 3 of this paper is mainly devoted to the study of properties of ideal operator t. there is a need to exercise some caution while dealing with this operator. it may well be the case that for some ideal operator, y has ‘nice’ property in x but there are other ideal operators for which such properties fail. for example, let y be a u-ideal in y ∗∗. then y is always a strict ideal in y ∗∗ under canonical projection π determined by y ∗∗∗ = y ∗ ⊕ y ⊥. but canonical projection may not satisfy u-ideal condition, namely, ‖i − 2π‖ = 1. so we introduce the notion of a maximal ideal operator (vis-a-vis, maximal ideal projection) and discuss properties of maximal ideal operator. however, 4 s. dutta, c.r. jayanarayanan, d. khurana there are situations where there is only one possible ideal operator. here we introduce relative godun set g(y,x) of x with respect to y . we recall that y ⊆ x is said to have unique ideal property (henceforth uip) in x if there is only one possible norm one projection p on x∗ with ker(p) = y ⊥. similarly y ⊆ x is said to have unique extension property (henceforth uep) in x if there is only one possible t : x → y ∗∗ such that ‖t‖ ≤ 1 and t|y = iy . from the above relation of p and t it is clear that y has uip in x if and only if it has uep in x (and in this case φ is also unique). if y has uep in y ∗∗ then we just say y has uep. uep also provides a sufficient condition for an ai-ideal in a dual space to be a local dual for predual. see [6] and references there in for recent results on local duals. in particular a pertinent question in this area is if a separable banach space with non separable dual always has a separable local dual. we now provide some brief preliminaries needed throughout this paper. to show that finite representability of x in a subspace y with condition (a) of theorem 1.1 is enough to characterise ideals through a global property we use following two lemmas from [12]. lemma 1.5. let e be a finite dimensional banach space and t : e −→ y ∗∗ be a linear map for any banach space y . then there exists a net (tα), tα : e −→ y such that (a) ||tα|| −→ ||t ||, (b) tαe −→ te for all e ∈ t−1(y ), (c) t∗αy ∗ −→ t∗y∗ for all y∗ ∈ y ∗. next result from [12] shows that if we are given with t and (tα) satisfying the conditions of lemma 1.5, then we can modify (tα) so that it satisfies the conditions of the following lemma. lemma 1.6. let e be a finite dimensional banach space and t : e −→ y ∗∗ be a linear map for any banach space y . let f ⊆ y ∗ be a finite dimensional banach space, then there exists a net (tα), tα : e −→ y such that (a) ||tα|| −→ ||t||, (b) tαe = te for all e ∈ t−1(y ), (c) t∗αy ∗ = t∗y∗ for all y∗ ∈ f, (d) t∗αy ∗ −→ t∗y∗ for all y∗ ∈ y ∗. ideal operators and relative godun sets 5 in section 3 we will refer to vn-subspaces (very non-constraint subspaces) and their characterisation done in [3]. we recall definitions of vn-subspaces of a banach space and nicely smooth spaces from [3] and [7] respectively. definition 1.7. let y be a subspace of a banach space x. (a) the ortho-complement o(y,x) of y in x is defined as o(y,x) = {x ∈ x : ‖x−y‖≥‖y‖ for all y ∈ y}. we denote o(x,x∗∗) by o(x). (b) y is said to be a vn-subspace of x if o(y,x) = {0}. (c) x is said to be nicely smooth if it is a vn-subspace of its bidual. in this article, we consider only banach spaces over the real field r and all subspaces we consider are assumed to be closed. 2. generalisation of plr we start with the following proposition which is essentially known. however it provides a way to look ideals and strict ideals through some global property. certain known properties of ideals and strict ideals follow trivially if we take this global view point. proposition 2.1. let x be a banach space and y be a subspace of x. then (a) y is an ideal in x if and only if there exists t : x −→ y ∗∗ such that ||t|| ≤ 1 and t|y = iy . (b) y is a strict ideal in x if and only if there exists an isometry t : x −→ y ∗∗ such that t|y = iy . we now note some immediate corollaries. corollary 2.2. let y be a subspace of a banach space x and y be 1-complemented in its bidual. then y is an ideal in x if and only if y is 1-complemented in x. proof. if y is 1-complemented in x then trivially y is an ideal in x. for the converse consider the map t : x −→ y ∗∗ from proposition 2.1 (a). let p : y ∗∗ −→ y ∗∗ be a norm one projection with range(p) = y . if we take q = pt : x −→ x then ||q|| = ||pt|| ≤ 1, q2 = q and range(q) = y . 6 s. dutta, c.r. jayanarayanan, d. khurana corollary 2.3. (a) if y is isometric to any dual space then y ⊆ x is an ideal if and only if y is 1-complemented in x. in particular no reflexive space can simultaneously be a vn-subspace and an ideal in any superspace. (b) l1(µ) ⊆ x is an ideal in x if and only if l1(µ) is 1-complemented in x. (c) y is an ideal in l1(µ) if and only if y is 1-complemented in l1(µ). (d) no infinite dimensional reflexive space can be an ideal in a space with dunford pettis property (see [4, definition 1.10]). proof. proofs of (a) and (b) follow from corollary 2.2. (c) let y be an ideal in l1(µ). then y ∗ is isometric to a 1-complemented subspace of l∞(µ). thus y is isometric to l1(ν) for some positive measure ν. hence y is 1-complemented in its bidual. then, by corollary 2.2, y is 1-complemented in l1(µ). (d) if y is an infinite dimensional reflexive space and y ⊆ x is an ideal then by corollary 2.2 y is 1-complemented in x. however complemented subspaces of a space with dunford pettis property have dunford pettis property and a reflexive space with dunford pettis property is finite dimensional. the notion of ai-ideals is strictly in between the notions of ideals and strict ideals. in the next theorem we characterise ai-ideals in terms of the operator t defined in theorem 2.1. definition 2.4. let x and z be banach spaces. for � > 0, an operator t : x → z is said to be an �-isometry if ‖t‖‖t−1‖≤ 1 + �. theorem 2.5. let y be a subspace of x. then y is an ai-ideal in x if and only if following condition is satisfied. given a finite dimensional subspace e of x and � > 0, there exists a bounded linear map te : x → y ∗∗ such that te|y = iy and te is an �-isometry on e ∩ (te)−1(y ). proof. let e be a finite dimensional subspace of x. without loss of generality, (by possibly adding an element from y ) we assume e ∩y 6= {0}. now consider a net (eα) of finite dimensional subspaces of x such that eα ⊇ e and the �eα-isometry t̃ �eα eα : eα → y with t̃ �eα eα is identity on e ∩y . let te be a weak∗ limit point of this net in the sense that for all x ∈ x and x∗ ∈ x∗, x∗(tex) = limα x∗(t̃ �eα eα x). let tα = t̃ �eα eα |e and te = te|e. ideal operators and relative godun sets 7 it is straightforward to verify that (a) ||tα|| −→ ||te||, (b) tαy −→ tey for all y ∈ t−1e (y ), (c) t∗αy ∗ −→ t∗ey ∗ for all y∗ ∈ y ∗. thus tα and the operator te satisfy conditions of lemma 1.5. now applying a perturbation argument as in lemma 1.6, given any � > 0 we can find sα satisfying conditions of lemma 1.6 and ||sα − tα|| < � for large dim(eα). hence sα is (�eα + �)-isometry and sα −→ te in the weak∗ topology. thus te is an �-isometry on e ∩t−1(y ). conversely, let e be a finite dimensional subspace of x and there exists an operator te satisfying the condition of the theorem. consider te|e : e → y ∗∗ and apply plr to get the desired operator satisfying the definition of ai-ideal. remark 2.6. following example of ai-ideal in c0 was considered in [2]. let y = {(an) ∈ c0 : a1 = 0}. then t : c0 −→ y ∗∗ in this case is given by t(a) = (0,a2,a3, . . .). hence t can not be extended as an isometry beyond t−1(y ). corollary 2.7. let y ⊆ x be an ai-ideal and y be reflexive. if y has uep in x, then y is isometric to x. in particular, in the following cases y is isometric to x. (a) the norm of x is smooth on y . (b) y is a u-ideal in x [2, theorem 2.3]. proof. given x ∈ x, consider e = span{x} and map te : x −→ y ∗∗ as in theorem 2.5. since y is reflexive and te|y = i|y we get te is onto. thus (te)−1(y ) = x. since y has uep in x, there exists a unique t such that te = t for all e. it follows that the ideal operator t is one-one as well. hence t is an isometry. let µ be a non-atomic σ-finite measure. it is proved in [2] that any copy of `1 in l1(µ) can not be an ai-ideal. however l1(µ) contains a 1-complemented copy of `1. hence a copy of `1 can be an ideal in l1(µ). it follows from corollary 2.7 that any copy of `p in lp(µ) can not be an ai-ideal for 1 < p < ∞. 8 s. dutta, c.r. jayanarayanan, d. khurana we will now prove that ai-ideals of c[0, 1] are universal for separable banach spaces. proposition 2.8. let y be an ai-ideal in c[0, 1]. then y is universal for separable banach spaces. proof. let y be an ai-ideal in c[0, 1]. then y is an l1-predual space and it follows from [2, proposition 3.8] that y inherits daugavet property from c[0, 1]. also from [13, theorem 2.6] it follows that y contains a copy of `1. thus y is an l1-predual with non separable dual and hence y is also universal for separable banach spaces (see [10, theorem 2.3]). we now show that for a non-atomic measure µ, any separable ai-ideal in l1(µ) is isometric to l1[0, 1]. proposition 2.9. let y ⊆ l1(µ) be a separable ai-ideal where µ is a non-atomic probability measure. then y is isometric to l1[0, 1]. proof. let y ⊆ l1(µ) be a separable ai-ideal. then by [2, proposition 3.8], it follows that y inherits daugavet property from l1(µ) and thus y ∗ is non separable. since y is an ideal in l1(µ) we have y ∗ is isometric to a 1-complemented subspace of l∞(µ) and thus y is isometric to l1(ν) space for some measure ν. but y has daugavet property so ν can not have atoms. now since ν is non atomic and y is separable we can conclude that y is isometric to l1[0, 1]. the property of being an ai-ideal is inherited from bidual. proposition 2.10. suppose y ⊆ x and y ⊥⊥ is an ai-ideal in x∗∗. then y is an ai-ideal in x∗∗ and hence in particular in x. proof. we note that the property of being ai-ideal is transitive. since y is always an ai-ideal in y ⊥⊥ and y ⊥⊥ is an ai-ideal in x∗∗, y is ai-ideal in x∗∗ as well. we end this section stating a result which connects ai-ideals in a dual space to local dual of preduals. definition 2.11. [6] a closed subspace z of x∗ is said to be a local dual space of a banach space x if for every � > 0 and every pair of finite dimensional subspaces f of x∗ and g of x, there exists an �-isometry l : f → z satisfying the following conditions. ideal operators and relative godun sets 9 (a) l(f)|g = f|g for all f ∈ f, (b) l(f) = f for f ∈ f ∩z. the following result is simple to observe from above definition. proposition 2.12. let x be a banach space and z be a subspace of x∗. if z is a local dual of x, then z is an ai-ideal in x∗. remark 2.13. let y be an ai-ideal in a banach space x. since x is an ai-ideal in x∗∗, y is an ai-ideal in x∗∗. but y cannot be a local dual space of x∗ as it is not norming for x∗. so the converse of proposition 2.12 need not be true. theorem 2.14. let x be a banach space. let z be an ai-ideal in x∗ with uep and z be norming for x. then z is a local dual space of x. proof. let f and g be finite dimensional subspaces of x∗ and x respectively. also, let � > 0. now let ĝ = span{ĝ|z : g ∈ g}, where ĝ is the canonical image of g in x∗∗. then, by [2, theorem 1.4], there exists a hahn– banach extension operator ϕ : z∗ → x∗∗ and an �-isometry l : f → z such that lf = f for all f ∈ f ∩ z and ϕ(ĝ|z)(f) = (ĝ|z)(lf) = (lf)(g) for all g ∈ g and f ∈ f. now to prove z is a local dual space of x, it is enough to prove that l(f)(g) = f(g) for all f ∈ f and g ∈ g. now let g ∈ g. since z is norming for x, ĝ is a hahn–banach extension of ĝ|z. further, by uep, ĝ is the only hahn–banach extension of ĝ|z. therefore (lf)(g) = ϕ(ĝ|z)(f) = f(g) for all f ∈ f and g ∈ g. hence z is a local dual space of x. 3. properties of ideal operators in this section we explore conditions for the ideal operator t to be unique or one-one. any nicely smooth space has uep (see [3, 7]). however any banach space x is a strict ideal in x∗∗. so for an ideal y in x, to get uniqueness we mostly have to consider y to be a strict ideal in x. as mentioned in the introduction, while considering uep one needs to exercise some caution here. for an ideal y in x we will first make sense of a maximal ideal projection through the use of godun set of x with respect to y . we formulate the following lemma for which the equivalence of first three parts is established in [9, lemma 2.2] and the proof for the fourth part goes verbatim as in the proof of (2) ⇒ (4) and (4) ⇒ (3) in [9, proposition 2.3]. 10 s. dutta, c.r. jayanarayanan, d. khurana lemma 3.1. let y be an ideal in x and t be the corresponding ideal operator. for λ,a ∈ r, the following assertions are equivalent. (a) ‖i −λp‖≤ a. (b) for any � > 0, x ∈ x and convex subset a of y such that tx is in the weak∗ closure of a there exists y ∈ a such that ‖x−λy‖ < a‖x‖ + �. (c) for any x ∈ x there exists a net (yα) ⊆ y such that (yα) converges to tx in the weak∗ topology and lim supα‖x−λyα‖≤ a‖x‖. moreover, if y is a strict ideal in x and t is the corresponding strict ideal operator then above assertions are also equivalent to the following. (d) for � > 0 and any sequence (yn) in by with (yn) converges in the weak ∗ topology to tx for some x ∈ bx, there exist n and u ∈ co{yk}nk=1, t ∈ co{yk}∞k=n+1 such that ‖t−λu‖ < a + �. let π be the canonical projection of x∗∗∗ onto x∗. the godun set g(x) is defined to be g(x) = {λ : ‖i −λπ‖ = 1} (see [9]). for ideal y ⊆ x and ideal projection p we define godun set of x with respect to y and p as gp (y,x) = {λ : ‖i −λp‖ = 1}. then it follows that 0 ∈ gp (y,x) and gp (y,x) is a closed convex subset of [0, 2] and thus itself an interval. our next result is an analogue of [9, lemma 2.5] and has interesting consequences. lemma 3.2. let y be a strict ideal in x and p be the corresponding strict ideal projection. (a) if z is a subspace of x such that y ⊆ z ⊆ x then there exists an ideal projection q on z∗ with ker(q) = y ⊥ such that gp (y,x) ⊆ gq(y,z). (b) if z is a closed subspace of y then y/z is an ideal in x/z and there exists an ideal projection p̃ on (x/z)∗ such that gp (y,x) ⊆ g p̃ (y/z,x/z). proof. (a) let t be the corresponding strict ideal operator from x to y ∗∗. consider tz = t|z : z → y ∗∗. we define q : z∗ → z∗ as q(z∗) = (tz) ∗(z∗|y ). it is straightforward to check that ker(q) = y ⊥ ⊆ z∗. thus q is an ideal projection on z∗. the proof now follows from equivalence of (d) and (a) in lemma 3.1. (b) we again let t be the corresponding strict ideal operator from x to y ∗∗. let q : x → x/z be the quotient map. we define t̃ : x/z → (y/z)∗∗ by ideal operators and relative godun sets 11 t̃(x) = q∗∗(tx), where x is the equivalence class containing x. let p̃ be the ideal projection corresponding to t̃. again a straightforward application of equivalence of (d) and (a) in lemma 3.1 gives the desired conclusion. for an ideal y in x, we define godun set of x with respect to y as g(y,x) = ∪{gp (y,x) : p is an ideal projection}. we now verify that g(y,x) is gp (y,x) for some ideal projection p . in the sequel we will refer such projection as maximal ideal projection and the corresponding t as maximal ideal operator. theorem 3.3. let y be an ideal in x. then there exists an ideal projection p such that g(y,x) = gp (y,x). proof. suppose for all ideal projection p , gp (y,x) = {0}. then g(y,x) = {0} and we choose any p as maximal ideal projection. suppose there exists an ideal projection p and λ ∈ gp (y,x) with λ 6= 0. then we claim that [0, 1] ⊆ gp (y,x). to see this, suppose on the contrary that gp (y,x) ⊆ [0,γ] for some 0 < γ < 1. we choose µ ∈ (0,γ). it is straightforward to verify that γ+µ−γµ ∈ gp (y,x) as well. thus γ+µ−γµ ≤ γ. hence µ(1 −γ) ≤ 0 which is a contradiction. now let us consider λ = sup{µ : µ ∈ g(y,x)}. if λ 6= 0, then by above argument either λ = 1 or 1 < λ ≤ 2. in the case λ = 1, there exists an ideal projection p such that g(y,x) = gp (y,x) = [0, 1]. if λ > 1, then choose a sequence (λn) in g(y,x) such that λn > 1 and λn converges to λ. let pn be an ideal projection corresponding to y with ‖i −λnpn‖ = 1 for all n. since b(x∗) is isometric to the dual of projective tensor product of x∗ and x, there exists a bounded linear map p : x∗ → x∗ and a subsequence (denoted again by (pn)) of (pn) such that for every x ∗ ∈ x∗, pn(x ∗) converges to p(x∗) in the weak∗ topology. since for every x∗ ∈ x∗ and n ∈ n, pn(x∗) is a hahn-banach extension of x∗|y , we can see that p(x∗) is also a hahn-banach extension of x∗|y . thus ker(p) = y ⊥. for any x∗ ∈ x∗, since x∗−p(x∗) ∈ y ⊥ = ker(p), we can see that p(p(x∗)) = p(x∗). hence p is an ideal projection corresponding to y . since, for every x∗ ∈ x∗, x∗ − λnpn(x∗) converges to x∗ − λp(x∗) in the weak∗ topology, we can see that ‖(i − λp)(x∗)‖ ≤ lim infn‖(i − λnpn)(x∗)‖ ≤ ‖x∗‖ for every x∗ ∈ x∗. thus ‖i −λp‖ = 1. hence g(y,x) = gp (y,x) = [0,λ]. remark 3.4. we note that g(y,x) = {0} is possible. if y = `1 and 12 s. dutta, c.r. jayanarayanan, d. khurana x = y ∗∗, then following the same argument used in [9, proposition 2.6] it follows that g(y,x) = {0}. we now show that if y is nicely smooth and y embeds in a superspace x as a strict ideal then strict ideal operator t is unique. proposition 3.5. let y be a nicely smooth banach space and y be a strict ideal in a superspace x. then the strict ideal operator is unique. proof. let t1 and t2 be two strict ideal operator. then for any x ∈ x, ‖t1x−y‖ = ‖t2x−y‖ for all y ∈ y . hence t1x−t2x ∈ o(y ) (see [3]). but since y is nicely smooth, o(y ) = {0} and hence t1x = t2x. we will now give a sufficient condition for a strict ideal y in x to be a vn-subspace of x. we will first provide an analogue of [9, proposition 2.7]. proposition 3.6. let y be a strict ideal in x and p a strict ideal projection for y in x∗. if 1 < λ ≤ 2 and ‖i −λp‖ = a < λ then for any proper subespace m ⊆ x∗, m norming for y , we have m is weak∗ dense in x∗. proof. let ry (m) be the greatest constant r such that supx∗∈sm |x ∗(y)| ≥ r‖y‖ for all y ∈ y . we will first show that for any weak∗ closed subspace m ⊆ x∗, ry (m) ≤ λ−1a. without loss of generality let m = ker(x) for some x ∈ sx. consider the isometry t : x −→ y ∗∗ corresponding to p . then by lemma 3.1 there exists a net {yα} ⊆ y such that yα −→ tx in the weak∗ topology and lim sup‖x−λyα‖≤ a‖x‖. now since t is an isometry we have ‖yα‖ −→ 1. for any x∗ ∈ sm , λ|x∗(yα)| = |x∗(x−λyα)| ≤ ‖x−λyα‖. since supx∗∈sm |x ∗(yα)| ≥ ry (m)‖yα‖ and ‖yα‖ −→ 1 it follows that ry (m) ≤ λ−1a. if there exists 1 < λ ≤ 2 and ‖i − λp‖ = a < λ then it follows that for any weak∗ closed proper subspace m of x∗ which is norming for y we have ry (m) < 1. this contradicts m is norming for y and hence we have the result. theorem 3.7. let y ⊆ x be a strict ideal such that ‖i − λp‖ < λ for some 1 < λ ≤ 2 where p is a strict ideal projection. then y is a vn-subspace of x. in particular a strict u-ideal is always a vn-subspace. ideal operators and relative godun sets 13 proof. let y ⊆ x be a strict ideal such that ‖i −λp‖ < λ for some 1 < λ ≤ 2 where p is a strict ideal projection. then it follows from proposition 3.6 that any norming subspace for y separates points in x. hence y is a vnsubspace of x (see [3]). corollary 3.8. let x be a banach space. then the following assertions are equivalent. (a) x∗ is separable. (b) there exists a renorming of x such that x is nicely smooth, that is x∗ has no proper norming subspace. (c) there exists a renorming of x such that every subspace and quotient of x in the new norm are nicely smooth. proof. (a) ⇒ (b) from [9, theorem 2.9] it follows that given 1 < λ < 2 there exists a renorming of x such that λ ∈ g(x). conclusion follows from theorem 3.7. (b) ⇒ (c) follows from lemma 3.2 and theorem 3.7. (c) ⇒ (b) ⇒ (a) is trivial. corollary 3.9. let y be a strict ideal in x with strict ideal projection p and o(y,x) 6= {0}. then either gp (y,x) = {0} or gp (y,x) = [0, 1] and the later happens only if p is bicontractive. proof. let 0 6= x ∈ o(y,x) and m = ker(x). then m is norming for y . now if we assume that ‖i−λp‖ < λ then ry (m) ≤‖i−λp‖λ−1 < 1. but m is norming for y so it follows that ‖i −λp‖≥ λ and thus gp (y,x) ⊆ [0, 1]. if 1 /∈ gp (y,x) that is ‖i − p‖ > 1 then gp (y,x) = {0}. hence the conclusion follows. we now provide a sufficient condition for x ∈ x to be in o(y,x). proposition 3.10. let y ⊆ x be an ideal and t be the corresponding ideal operator. if tx = 0, then x ∈ o(y,x). consequently, if y is also a vn-subspace of x, then any ideal operator t is one-one. proof. let tx = 0. then px∗(x) = 0 for all x∗ ∈ x∗ where p is the ideal projection corresponding to t . thus range(p) ⊆ ker(x). but range(p) is norming for y , hence ker(x) is norming for y and x ∈ o(y,x). 14 s. dutta, c.r. jayanarayanan, d. khurana we now present an extension of [9, theorem 7.4]. towards this, for an ideal y in x with associated ideal operator t, we define ba(y,x) = {x ∈ x : there exists {yn}⊆ y such that yn −→ tx in the weak∗ topology} and kyu (x) = inf { a : tx = ∑ yn in weak ∗ topology and for any n, ‖ ∑n k=1 θkyk‖≤ a,θk = ±1 } . it follows from [11] that if y does not contain a copy of `1 then ba(y,x) = x. as considered in [9], we will say pair (y,x) has property u if kyu (x) < ∞ for all x ∈ x. in this case by closed graph theorem there exists a constant c such that kyu (x) ≤ c||tx|| for all x ∈ ba(y,x). we denote least constant c by kyu (x). we will need the following lemma. lemma 3.11. let y be a strict ideal in x such that y does not contain a copy of `1 and t be a strict ideal operator. then y is a u-ideal in x if and only if kyu (x) = 1. proof. if y is a u-ideal in x then the result follows from [9, lemma 3.4]. conversely, by following the similar arguments as in [9, lemma 5.3] that in this case ‖i − 2p‖≤ kyu (x) where p is the ideal projection corresponding to the ideal operator t. thus ‖i−2p‖ = 1 and y is a strict u-ideal in x. theorem 3.12. let y be a banach space not containing `1. then the following assertions are equivalent. (a) y is a u-ideal in y ∗∗. (b) whenever y is a strict ideal in x, y is a strict u-ideal in x. (c) whenever y is a strict ideal in x, kyu (x) < 2. proof. (a) ⇒ (b) since y is a strict ideal in x, the ideal operator t is an extension of identity operator on y and we have the results by [9, proposition 3.6]. (b) ⇒ (c) follows from lemma 3.11. (c) ⇒ (a) follows from [9, theorem 7.4] by taking x = y ∗∗. ideal operators and relative godun sets 15 we will now give examples where the ideal operator t is unique/ one-one. we first see how does the ideal operator corresponding to an m-ideal in c(k) behave. it is well-known that m-ideals in c(k) are precisely of the form jd = {f ∈ c(k) : f|d = 0} for some closed subset d of k. proposition 3.13. let d be a closed subset of a compact hausdorff space k. then the following are equivalent. (a) jd is a strict ideal in c(k). (b) jd is a vn-subspace of c(k). (c) k \d = k. proof. (a) ⇐⇒ (b) observe that j∗d is norming if and only if k \d = k. (b) ⇐⇒ (c) follows from standard arguments. example 3.14. since the ideal projection corresponding to an m-ideal is unique, the ideal operator t corresponding to jd is unique. in addition, if k \d = k, then, by proposition 3.13, the unique ideal operator t corresponding to jd is an isometry. we next note that the ideal operators corresponding to c(k,x) and c(k,x∗) are isometries. for a compact hausdorff space k and for any banach space x, let wc(k,x) denote the space of x-valued functions on k that are continuous when x has the weak topology, equipped with the supremum norm. also, w∗c(k,x∗) denotes the space of x∗-valued functions on k that are continuous when x∗ has the weak∗ topology, equipped with the supremum norm. proposition 3.15. let x be a banach space. then c(k,x) is a strict ideal in wc(k,x). moreover, c(k,x∗) is a strict ideal in w∗c(k,x∗). proof. the former conclusion follows from the fact that there exists an isometry from wc(k,x) to c(k,x)∗∗ whose restriction to c(k,x) is the canonical embedding. to prove the later conclusion recall that c(k,x∗) = k(x,c(k)), the space of compact operators from x to c(k) and w∗c(k,x∗) = l(x,c(k)), the space of bounded linear operators from x to c(k). it follows from [8, lemma 2] that if y is a banach space having metric approximation property (in short map), then there exists an isometry from l(x,y ) to k(x,y )∗∗ 16 s. dutta, c.r. jayanarayanan, d. khurana whose restriction to k(x,y ) is the canonical embedding. since c(k) has map, it follows that c(k,x∗) is a strict ideal in w∗c(k,x∗). we know that c(k,x) ⊆ ba(k,x) ⊆ c(k,x)∗∗, where ba(k,x) denotes the class of baire-1 functions from k to x. since c(k,x) is a strict ideal in c(k,x)∗∗, it follows that c(k,x) is also a strict ideal in ba(k,x). so the corresponding ideal operator is an isometry. if x has map, then, by [8, lemma 2], k(x) is a strict ideal in l(x). now it follows from [5] that a reflexive space with compact approximation property has map. hence if x is a reflexive space with compact approximation property such that either weak∗ denting points of bx∗ separates points of x ∗∗ or denting points of bx separates points of x ∗, then k(x) is a strict ideal in l(x) and is also a vn-subspace of l(x) (see [3]). so the ideal operator t is an isometry. acknowledgements the second author is grateful to the nbhm, india for its financial support (no.2/40(2)/2014/r&d-ii/6252). he also likes to thank prof. pradipta bandyopadhyay for the discussion regarding local dual spaces. references [1] t. a. abrahamsen, o. nygaard, on λ-strict ideals in banach spaces, bull. aust. math. soc. 83 (2) (2011), 231 – 240. [2] t. a. abrahamsen, v. lima, o. nygaard, almost isometric ideals in banach spaces, glasg. math. j. 56 (2) (2014), 395 – 407. [3] p. bandyopadhyay, s. basu, s. dutta, b. l. lin, very nonconstrained subspaces of banach spaces, extracta math. 18 (2) (2003), 161 – 185. [4] j. bourgain, “new classes of lp-spaces”, lecture notes in mathematics 889, springer verlang, berlin-new york, 1981. [5] c.-m. cho, w. b. johnson, a characterization of subspaces x of lp for which k(x) is an m-ideal in l(x), proc. amer. math. soc. 93 (3) (1985), 466 – 470. [6] m. gonzález, a. mart́ınez-abejón, local duality for banach spaces, expo. math. 33 (2) (2015), 135 – 183. [7] g. godefroy, p. d. saphar, duality in spaces of operators and smooth norms on banach spaces, illinois j. math. 32 (4) (1988), 672 – 695. [8] j. johnson, remarks on banach spaces of compact operators, j. funct. anal. 32 (3) (1979), 304 – 311. [9] g. godefroy, n. j. kalton, p. d. saphar, unconditional ideals in banach spaces, studia math. 104 (1) (1993), 13 – 59. ideal operators and relative godun sets 17 [10] a. j. lazar, j. lindenstrauss, banach spaces whose duals are l1-spaces and their representing matrices, acta math. 126 (1971), 165 – 193. [11] e. odell, h. p. rosenthal, a double-dual characterization of separable banach spaces containing `1, israel j. math. 20 (3-4) (1975), 375 – 384. [12] e. oja, m. põldvere, principle of local reflexivity revisited, proc. amer. math. soc. 135 (4) (2007), 1081 – 1088 (electronic). [13] d. werner, recent progress on daugavet property, irish math. soc. bull. 46 (2001), 77 – 97. introduction and preliminaries generalisation of plr properties of ideal operators � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 1 (2019), 99 – 122 doi:10.17398/2605-5686.34.1.99 available online march 13, 2019 tetrahedral chains and a curious semigroup ian stewart mathematics institute, university of warwick coventry cv4 7al, united kingdom i.n.stewart@warwick.ac.uk received january 7, 2019 presented by juan a. navarro accepted february 27, 2019 and alexandre turull abstract: in 1957 steinhaus asked for a proof that a chain of identical regular tetrahedra joined face to face cannot be closed. świerczkowski gave a proof in 1959. several other proofs are known, based on showing that the four reflections in planes though the origin parallel to the faces of the tetrahedron generate a group r isomorphic to the free product z2 ∗ z2 ∗ z2 ∗ z2. we relate the reflections to elements of a semigroup of 3 × 3 matrices over the finite field z3, whose structure provides a simple and transparent new proof that r is a free product. we deduce the non-existence of a closed tetrahedral chain, prove that r is dense in the orthogonal group o(3), and show that every r-orbit on the 2-sphere is equidistributed. key words: tetrahedral chain, free product, semigroup, density, equidistribution, spherical harmonic, cayley graph. ams subject class. (2010): 20f55, 20m05, 22e46, 37a30, 42c10, 43a15. 1. introduction in 1957 hugo steinhaus contributed two related questions to the problems section of colloquium mathematicum [21]. in loose translation from the french, they were: p 175. the image of a regular tetrahedron t1 (fixed in euclidean space of 3 dimensions) under reflection in one of its faces gives the tetrahedron t2. iteration gives rise to a sequence of pairwise congruent tetrahedra {tn}. supposing that each face serves as a mirror only once, demonstrate that: (1) m 6= n implies tm 6= tn, (2) whatever the region r may be, there exists a sequence of tetrahedra {tn} such that the set of vertices is dense in r. steinhaus indicated that this problem is from the new scottish book, problem 290 1.iii.1956. the original scottish book was a notebook of open issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.1.99 mailto:i.n.stewart@warwick.ac.uk https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 100 i. stewart mathematical problems compiled by regular visitors to the famous scottish café in what is now lviv, ukraine. an english typescript is available [28], and the original can be viewed online as volume 0 at [30]. when world war ii ended, steinhaus revived the book as the new scottish book, volumes 1 and 2 at [30]. figure 1: closed chains of cubes, octahedra, dodecahedra, and icosahedra. adapted from [10]. as far as we are aware, statement (2) is still open. its analogue for cubes is clearly false, but its analogue for the other regular solids is a plausible conjecture. in particular, their dihedral angles are not rational multiples of π. statement (1) implies that a chain of identical regular tetrahedra, joined face to face, cannot be closed. in contrast, it is easy to find closed chains for the other four regular polyhedra, figure 1. (one key difference is that unlike the tetrahedron, opposite faces of these polyhedra are parallel.) stanis law świerczkowski subsequently proved that no closed chain of tetrahedra exists, aside from trivial examples where consecutive tetrahedra coincide. in [24] he proved that two particular rotations in r3 generate a free group on two generators, and stated as a corollary that this result disproves the existence tetrahedral chains and a curious semigroup 101 of a closed chain of regular tetrahedra. he wrote: “this corollary gives a positive answer to a question of h. steinhaus ... . however we shall not prove it here.” in [25] he completed the proof by explaining the connection with chains of tetrahedra. the rotations concerned are  1 3 −2 √ 2 3 0 2 √ 2 3 1 3 0 0 0 1   ,   1 0 0 0 1 3 −2 √ 2 3 0 2 √ 2 3 1 3   . their axes are at right angles to each other and both are rotations through cos−1( 1 3 ). świerczkowski’s proof that these matrices generate a free group uses induction on a sequence of integers determined by the two matrices, and his main aim is to prove that these are not divisible by 3. in passing, we mention that this group-theoretic result can also be used as the basis of a proof of the famous banach-tarski paradox [27]: a solid ball in r3 can be dissected into finitely many disjoint subsets, which can be fitted together via rigid motions to create two solid balls, each congruent to the original one. the free product group discussed below can also be used in this manner. dekker [8] and mason [19] sketched new proofs that no closed tetrahedral chain exists, based on the idea that the group generated by reflections in the four planes through the origin parallel to the faces of the tetrahedron is isomorphic to a free product z2 ∗z2 ∗z2 ∗z2. (without loss of generality, the barycentre of the first tetrahedron in the chain is the origin. it is important to distinguish these linear reflections from the affine reflections in the faces of the tetrahedron, which do not fix the origin; see subsection 2.2.) tomkowicz and wagon [27, theorem 3.10] represents the four reflections as 4×4 matrices using barycentric coordinates, and analyse an arbitrarily long product of these matrices. the entries of such a product are polynomials, evaluated at ±1 3 , ±2 3 . as in [24], the key step in an inductive proof again involves divisibility by a power of 3. say that a chain of tetrahedra is embedded if distinct tetrahedra are disjoint except for the common face of consecutive members of the chain. all of the above proofs rule out the existence of nontrivial closed chains, embedded or not. these proofs are relatively short and simple, but none is particularly transparent. in section 3 we present a new proof, with a clear storyline that emphasises the role of the integer 3. we use cartesian coordinates, but it is possible to recast the discussion using the more traditional barycentric coordinates. with a convenient choice of the initial tetrahedron in r3, with 102 i. stewart vertices at four corners of the cube [−1, 1]3, the 3×3 matrices representing the four reflections have rational entries with denominator 1 or 3; see section 2. that they generate z2 ∗z2 ∗z2 ∗z2 is a reformulation of the statement that the product of any nontrivial sequence of the four matrices (that is, avoiding consecutive repetitions of the same matrix), other than the empty sequence, can never be the identity. for a contradiction, suppose such a sequence exists. let the group generated by the four reflections be r, which is a subgroup of the orthogonal group o(3) acting on r3. if each matrix is multiplied by 3 it has integer entries, and the corresponding product must be the identity multiplied by 3k where k ≥ 1 is the length of the sequence. these products no longer form a group, but together with the zero matrix they form a semigroup. reducing modulo 3, a nontrivial product of the corresponding matrices over z3 must be the zero matrix. theorem 3.2 proves that the four reflection matrices (mod 3) generate an order-33 semigroup. this contains the zero matrix 0, but we prove that no nontrivial product of its nonzero elements is zero. (as before, ‘nontrivial’ means no generator appears twice consecutively.) indeed, every matrix in the semigroup other than 0 has all entries equal to ±1 (mod 3). this contradiction proves that r is isomorphic to z2 ∗ z2 ∗ z2 ∗ z2. (an isomorphic semigroup can be obtained using barycentric coordinates, and the proof can be also be expressed in that framework.) the non-existence of a nontrivial closed chain of regular tetrahedra follows easily, using essentially the argument of świerczkowski [25], which constructs the chain using successive reflections in faces. the translations in the euclidean group e(3) form a normal subgroup and can be factored out, reducing the geometric features required here to sequences of reflections. the four reflections lie in the quotient group o(3). a sequence of reflections determines a unique chain of face-to-face tetrahedra, and a nontrivial sequence determines a nontrivial chain. if this chain closes up, the corresponding sequence fixes the initial tetrahedron. there is one subtlety, discussed briefly in subsection 3.1: this sequence can fix the tetrahedron setwise rather than pointwise. that is, it belongs to the symmetry group of the tetrahedron, but need not be the identity. there are two ways to deal with this possibility. one is to observe that some power of the sequence must then be the identity (it is also necessary to deal with possible repetitions if the sequence starts and ends with the same reflection: this leads to a shorter chain, but after a series of such cancellations it turns out that the result must be nontrivial if the original chain is). the other, employed here, is to check that the semigroup proof remains valid if tetrahedral chains and a curious semigroup 103 we replace the identity by a symmetry of the tetrahedron, because all such symmetries correspond to integer matrices. after multiplying by 3 and reducing modulo 3, these all become the zero matrix and the same proof works. section 4 adds extra information about this semigroup. in the absence of a closed chain, a natural question, also asked by świerczkowski [26], arises: can almost closed chains be formed, in the sense that the gap between the initial and final tetrahedra can be made as small as we please? this question is connected to steinhaus’s statement (2), but it involves the faces of the tetrahedra, not just individual vertices. elgersma and wagon [9] give an affirmative answer for non-embedded chains, based on kronecker’s theorem [1, 17] that if θ is an irrational multiple of π, the set {eniθ : n ∈ z} is dense in the unit circle s1 ⊆ c. the most interesting case arises when the chain is embedded. elgersma and wagon [10, 11] prove the existence of closed embedded chains with arbitrarily small gaps. their construction begins with a boerdijk-coxeter helix [4, 7], also named the tetrahelix by fuller [12]. this is a linear chain of identical regular tetrahedra, all of whose vertices lie on a cylinder. this chain is generated by periodically repeating reflections in four distinct faces. they then construct a ‘quadrahelix’ by joining four copies of a tetrahelix of length l + 1, overlapping them at a common end tetrahedron at the first and third joins, and attaching them face to face at the second join, so that the overall chain has reflectional symmetry about its midpoint. they prove that if l = q − 1 where p/q is a convergent of the continued fraction of θ = cos−1( 2 3 )/(2π), the quadrahelix has the approximate form of a rhombus, and is almost closed, with the size of the gap tending to zero as q increases. for example when l = 601,944 the gap has size 1.3×10−7. here we prove two related results, which do not prove the existence of almost closed chains but have independent interest. section 5 gives a simple proof that r is dense in o(3). steinhaus’s statement (2) asks for more: the group generated by the affine reflections in the faces of a fixed regular tetrahedron has a dense orbit in r3. taking account of the translations is more difficult, in part because the euclidean group e(3) in r3 is non-compact. our density result is too weak to prove statement (2), and has no obvious consequences for almost-closed chains of tetrahedra, because it factors out translations. finally, in section 6, we use the o(3) density result to prove a stronger theorem: the r-orbit of any point of the unit 2-sphere s2 is equidistributed with respect to normalised lebesgue surface measure on s2, where the density of r is defined using the limit of the proportion of words in the generating 104 i. stewart reflections that lie in a given open subset of s2, as the length of the words tends to infinity. the proof is an adaptation of a method of arnold and krylov [2], and can be viewed in the context of ergodic theory of non-abelian group actions; see gorodnik and nevo [13]. 2. reflections in faces of the tetrahedron elgersma and wagon [9, 10, 11] pose the problem using barycentric coordinates and 4×4 matrices. babiker and janeczko [3] analyse chains of regular tetrahedra using a tensor product representation, combining the translations and reflections, to prove several new results. however, the reflections are again represented using barycentric coordinates. in this paper we use cartesian coordinates and 3 × 3 matrices. we choose a fixed reference tetrahedron ∆ ⊆ r3 with vertices p0 : (1, 1, 1) , p1 : (1,−1,−1) , p2 : (−1, 1,−1) , p3 : (−1,−1, 1) . the barycentre is at the origin. see figure 2. x y z figure 2: the cube [−1, 1]3 and the reference tetrahedron ∆. tetrahedral chains and a curious semigroup 105 2.1. symmetries of the tetrahedron. the eight points (±1,±1,±1) are the vertices of a cube. the symmetry group of the cube consists of all permutations of the coordinates (x,y,z) together with sign changes on any coordinate, so has order 48. the symmetry group sym(∆) of the tetrahedron is the subgroup in which the number of minus signs is even, and has order 24. it is, of course, isomorphic to the symmetric group s4, which permutes the four vertices. lemma 2.1. all matrices in the symmetry group sym(∆) have integer entries. proof. all 3 × 3 matrices in the symmetry group of the cube, hence also of the tetrahedron, are signed permutation matrices, with entries 0,±1. 2.2. the 3×3 matrices. next, we compute the four reflection matrices and associated translations. the faces f [pipjpk] through vertices pi,pj,pk have equations: f[p1p2p3] : x + y + z = −1 , f[p0p1p2] : x + y −z = 1 , f[p0p1p3] : x −y + z = 1 , f[p0p2p3] : −x + y + z = 1 . midpoints of centres of these faces are q0 : (−13,− 1 3 ,−1 3 ) , q1 : ( 1 3 , 1 3 ,−1 3 ) , q2 : ( 1 3 ,−1 3 , 1 3 ) , q3 : (−13, 1 3 , 1 3 ) . let mi be reflection in face i, where 0 ≤ i ≤ 3. then it is the identity on that face, and reverses the line perpendicular to the face at its centre. this line joins the midpoint of the face to the remaining vertex. begin with the plane p = f[p1p2p3]. the perpendicular is (a,a,a) : a ∈ r. a general point y = (u,v,w) = (a,a,a) + z where z = (u −a,v −a,w −a) ∈ p . therefore u + v + w − 3a = −1 106 i. stewart so a = u + v + w + 1 3 . then, solving for z, z =   2 3 u − 1 3 v − 1 3 w − 1 3 2 3 v − 1 3 u − 1 3 w − 1 3 2 3 w − 1 3 u − 1 3 v − 1 3   . this maps via m0 to z − (a,a,a), which is m0y =   1 3 u − 2 3 v − 2 3 w − 2 3 1 3 v − 2 3 u − 2 3 w − 2 3 1 3 w − 2 3 u − 2 3 v − 2 3   = r0y + t0 , where r0 =   1 3 −2 3 −2 3 −2 3 1 3 −2 3 −2 3 −2 3 1 3   , t0 =   − 2 3 −2 3 −2 3   . repeating similar calculations for the other three faces, we obtain: for face f [p0p1p2]: r1 =   1 3 −2 3 2 3 −2 3 1 3 2 3 2 3 2 3 1 3   , t1 =   2 3 2 3 −2 3   . for face f [p0p1p3]: r2 =   1 3 2 3 −2 3 2 3 1 3 2 3 −2 3 2 3 1 3   , t2 =   2 3 −2 3 2 3   . for face f [p0p2p3]: r3 =   1 3 2 3 2 3 2 3 1 3 −2 3 2 3 −2 3 1 3   , t3 =   − 2 3 2 3 2 3   . the ri are orthogonal, and ti points perpendicular to the face concerned. since sym(∆) permutes the faces of ∆ and fixes the origin, the ri are conjugate under symmetries of ∆: σriσ −1 = rσ(i) , σ ∈ sym(∆) ≡ s4 . tetrahedral chains and a curious semigroup 107 2.3. factoring out translations. the construction of chains of tetrahedra is intimately related to the subgroup of o(3) generated by the four reflection matrices, and for many purposes we can ignore the translations. definition 2.2. the group r is the subgroup of o(3) generated by the ri for 0 ≤ j ≤ 3. the corresponding affine maps are miy = riy + ti , 0 ≤ i ≤ 3 . the group e(3) of all rigid motions of r3 is a semidirect product e(3) = r3 n o(3) where r3 is the normal subgroup of translations. the homomorphism e(3) → o(3) that factors out r3 maps (ti,ri) to ri. therefore a product of affine maps mikmik−1 · · ·mi2mi1 is a map of the form rikrik−1 · · ·ri2ri1 + wikik−1...i2i1 (2.1) where wikik−1...i2i1 is a translation determined by the semidirect product structure, which we do not state explicitly. the product of the reflections ri gives the orientation of the image tetrahedron; the translation wikik−1...i2i1 leaves the orientation invariant. the sequence of translations is implicit in the sequence of reflections because exactly one pair of faces matches at each stage. 3. free product structure we now give a simple, structural proof of the main result of dekker [8] and mason [19]: theorem 3.1. the group r is isomorphic to the free product z2 ∗ z2 ∗ z2 ∗z2. our proof is based on a rather curious semigroup, and we discuss this first. to get rid of fractions, define qi = 3ri. we can then reduce modulo 3, to get matrices si = qi (mod 3) , 0 ≤ i ≤ 3 , 108 i. stewart which are: s0 =   1 1 11 1 1 1 1 1   , s1 =   1 1 −11 1 −1 −1 −1 1   , s2 =   1 −1 1−1 1 −1 1 −1 1   , s3 =   1 −1 −1−1 1 1 −1 1 1   . here the ±1 lie in z3, but in fact the calculations reported below also apply in z, except when the zero matrix arises and some entries may be multiples of 3. the twelve products sisj (i 6= j) are: s0s1 =   1 1 −11 1 −1 1 1 −1   , s0s2 =   1 −1 11 −1 1 1 −1 1   , s0s3 =   −1 1 1−1 1 1 −1 1 1   , s1s0 =   1 1 11 1 1 −1 −1 −1   , s1s2 =   −1 1 −1−1 1 −1 1 −1 1   , s1s3 =   1 −1 −11 −1 −1 −1 1 1   , s2s0 =   1 1 1−1 −1 −1 1 1 1   , s2s1 =   −1 −1 11 1 −1 −1 −1 1   , s2s3 =   1 −1 −1−1 1 1 1 −1 −1   , s3s0 =   −1 −1 −11 1 1 1 1 1   , s3s1 =   1 1 −1−1 −1 1 −1 −1 1   , s3s2 =   1 −1 1−1 1 −1 −1 1 −1   . clearly q2i = 9r 2 i = 9i, so s 2 i = 0. observe that the sixteen matrices si and sisj (i 6= j) are distinct, and distinct from their negatives −si and −sisj (i 6= j). let s be the set of these 32 matrices. case-by-case analysis tetrahedral chains and a curious semigroup 109 shows that the si satisfy the following relations: sisjsi = si (i 6= j) , sisjsk = −sisk (i 6= j, i 6= k, j 6= k) . (3.1) (using sym(∆) we can reduce this calculation to the special case i = 0, j = 1, k = 2.) let s be the set of all of the above 32 matrices together with the zero matrix, so that s = { 0, ±si, ±sisj : 0 ≤ i, j ≤ 3, i 6= j } . theorem 3.2. (a) the set s is a semigroup. (b) the product of two nonzero members of s is nonzero, except for the trivial cases sisi = 0 , si(sisj) = 0 , (sisj)sj = 0 , (sisj)(sjsk) = 0 , and similar products involving minus signs. proof. for (a) we must show that all products of nonzero elements of s lie in s. for (b) we must also show these products are nonzero. both follow from a case-by-case check. for products sisj this is clear. products of the form si(sjsk) and (sisj)sk are taken care of directly by the relations (3.1). those relations also imply that when i 6= j, k 6= l we have (sisj).(sksl) =   0 if j = k , si if j 6= k , k = i , −sisl if j 6= k , k 6= i . when j = k the string sisjsksl = sisjsjsl is trivial. note in particular that when i 6= j (sisj) 2 = (sisjsi)sj = sisj which is nonzero, unlike squares of the si. we are now ready to give the: 110 i. stewart proof of theorem 3.1. the four free factors z2 are generated respectively by r0, r1, r2, r3. we claim that the only relations between these generators are r2i = i, where i is the identity. using the relations r2i = i we can write any element γ ∈r in the form γ = rikrik−1 · · ·ri2ri1 where rij 6= rij+1 for all 1 ≤ j ≤ k − 1. we claim this representation as a word is unique. if not, some nontrivial word is equal to the identity i. the corresponding nontrivial word in the qi of length l is equal to 3 li. modulo 3, this word becomes zero. consider the corresponding word in the si, which is also nontrivial: w = siksik−1 · · ·si2si1 . by theorem 3.2, w lies in s\{0}, so all of its entries are ±1 (mod 3). therefore w 6= 0, so no nontrivial word in the ri can be the identity. 3.1. non-existence of a closed chain. it is well known that theorem 3.1 implies the non-existence of a nontrivial closed chain of regular tetrahedra. for completeness, we give a proof. theorem 3.3. no nontrivial closed chain of tetrahedra exists. proof. suppose, for a contradiction, that there is such a chain. consider the corresponding product of reflections ri in r. because each reflection ri fixes the origin, the construction of the chain of tetrahedra corresponding to a given element rjkrjk−1 · · ·rj2rj1 ∈r does not add successive tetrahedra to an otherwise stationary chain. instead, the chain corresponding to rjk−1 · · ·rj2rj1 is reflected by rjk , and then translated by an appropriate amount so that it joins to the corresponding face of the reference tetrahedron ∆. thus the chain at stage k has the structure ∆ → rjk ∆ → rjkrjk−1 ∆ → rjkrjk−1rjk−2 ∆ →··· · · ·→ rjkrjk−1 · · ·rj2rj1 ∆ where the arrow indicates ‘joins at a face’. (an alternative approach, in which each new face is added to a growing but otherwise static chain, is geometrically tetrahedral chains and a curious semigroup 111 more natural but involves conjugates of the ri, so this convention is a little simpler algebraically.) a necessary condition for the chain to close up is then that there some nontrivial product of reflections is a symmetry of the tetrahedron: rjkrjk−1 · · ·rj2rj1 = a where a ∈ sym(∆). then the corresponding nontrivial product in s satisfies sjksjk−1 · · ·sj2sj1 = 3 k+1a = 0 by lemma 2.1. this yields the same contradiction as in the proof of theorem 3.1. 4. properties of the semigroup the semigroup s has a lot of structure, which the calculations do not explain. we briefly investigate some of its features. the results of this section are not used later, but they help to explain some aspects of the structure of s from a different point of view. the ±si are symmetric matrices, whereas the ±sisj are not symmetric. there are 29 = 512 matrices of size 3×3 with entries ±1. the 32 such matrices in s \{0} are distinguished by the following properties: (1) all entries are ±1. (2) the matrix has a repeated row, and the remaining row is either the same as the repeated row or the negative of the repeated row. (3) the same goes for columns. it is easy to prove that for matrices satisfying (1), condition (2) holds if and only if (3) does. we omit the proof. proposition 4.1. the equivalent conditions (1)+(2) or (1)+(3) characterise the 32 nonzero elements of the semigroup s. proof. this follows from the list of elements, but we now give an independent proof avoiding case-by-case checking. we count now many such matrices exist. observe that there are 8 possibilities for the first row r1. the second and third rows r2,r3 are all possible choices of r2 = ±r1, r3 = ±r1, with four choices of the ± signs, so in total there are 8 × 4 = 32 such matrices. in other words, the equivalent conditions (1)+(2) or (1)+(3) characterise the elements of s that are not 0. 112 i. stewart proposition 4.1 lets us give an alternative proof that s is a semigroup, without listing all products: theorem 4.2. the collection of matrices satisfying (1)+(2), together with the zero matrix, is a semigroup. proof. as observed, conditions (1) and (2) also imply (3). redefine s to be the set of matrices satisfying these conditions, together with 0. let a,b ∈ s. clearly 0 = 0.0 = 0.a = a.0. it remains to show that ab ∈ s when a,b 6= 0. permuting rows we can write a = p   xx εx   , ε = ±1 , where p is a permutation matrix, and x = [x,y,z] is a row vector. dually, permuting columns we can write b = [ y y δy ] q, δ = ±1 , where q is a permutation matrix, and y = [u,v,w]t is a column vector. then ab = p   x ·y x ·y δx ·yx ·y x ·y δx ·y εx ·y εx ·y εδx ·y  q. either x ·y = 0 and ab is the zero matrix, or x ·y = ±1 and the matrix in the middle clearly also satisfies (1) and (2). now p and q permute its rows and columns, leaving properties (1) and (2) unchanged. remark 4.3. as stated above, the four matrices si are distinguished from the twelve matrices sisj (i 6= j) by symmetry. for the symmetric matrices si, we have x = y and x.y = x.x = (±1)2 + (±1)2 + (±1)2 ≡ 0 (mod 3). for the asymmetric matrices sisj (i 6= j) this does not happen, and x.y ≡ ±1 (mod 3). this is consistent with the relations s2i = 0 but (sisj) 2 6= 0. the semigroup s exhibits a lot of symmetry. we find its automorphism group. some automorphisms are inherited from the symmetry group σ of the tetrahedron ∆, which has order 24 and is isomorphic to s4. since σ permutes the faces of ∆, it permutes the si by conjugation: si 7→ σsiσ−1. tetrahedral chains and a curious semigroup 113 this action extends to −si since conjugation commutes with the negative of the identity, and hence the action extends uniquely to any element of s. clearly this action defines automorphisms σ̂ of s, given by σ̂(si) = sσ(i) , σ̂(−si) = −sσ(i) , σ ∈ s4 , 0 ≤ i ≤ 3 . (4.1) define ŝ4 = { σ̂ : σ ∈ s4 } which is isomorphic to s4. we now prove that the σ̂ are the only automorphisms. proposition 4.4. the automorphism group of s is the group ŝ4 with action (4.1). proof. suppose that α is an automorphism not in ŝ4. the elements in the subset t = {±si : 1 ≤ i ≤ 4} are the only nonzero elements with square 0, so any automorphism α permutes them. moreover, within this set the annihilator of si is{ u ∈t : usi = 0 = siu } = { si,−si : 1 ≤ i ≤ 4 } . therefore if α maps si to ±sj then it must map −si to ∓sj. composing with a suitable permutation in ŝ4 we can assume α(si) = ±si for all i. since α 6∈ ŝ4, we have α(si) = −si for some i. for all j 6= i the relation sisjsi = si in (3.1) implies that (−si)(α(sj))(−si) = −si, so α(sj) = −sj for all j. but this contradicts the second relation sisjsk = −sisk in (3.1) which applies when all i,j,k are different. thus every automorphism lies in ŝ4. remark 4.5. without using the above result, it is clear that the elements ±si satisfy (±si)2 = 0. the products sisj for i 6= j are idempotent: (sisj) 2 = sisj. the elements −sisj for i 6= j are not idempotent, but their squares are: (−sisj)2 = sisj. thus these three classes are distinguished by simple automorphism-invariant properties. 5. density we now use the classification of closed subgroups of o(3) to prove a density theorem for r. this result is too weak to imply the result of [9] that almost closed non-embedded chains with arbitrarily small gaps exist, because 114 i. stewart it factors out the translations in the affine reflections, but it leads to the equidistribution theorem of section 6 and is of independent interest. here the notations so(2),so(3),o(3) refer to lie groups defined over the real numbers. the full classification is not required explicitly; we just need a simple consequence: lemma 5.1. let g be a closed subgroup of o(3). then one of the following conditions is valid: (a) g is finite. (b) the subgroup of g consisting of elements with determinant +1 is conjugate to so(2). (c) g = o(3) or g = so(3). proof. this can be read off from the classification of closed subgroups of o(3); see for example [14, theorem xiii.9.2]. theorem 5.2. the group r is dense in o(3). proof. let r be the closure of r. this is a closed subgroup of o(3), so lemma 5.1 applies. we consider the three cases in turn. (a) this case does not apply: the group r is infinite since it is a free product. (b) this case also does not apply. since so(2) is abelian, elements of r with determinant 1 commute. in particular r0r1 and r2r3 commute. so r0r1r2r3 = r2r3r0r1 contrary to theorem 3.1. (c) this is the only remaining case. the generators ri do not belong to so(3), so r = o(3). therefore r is dense in o(3). corollary 5.3. the subgroup r2 ⊆ r generated by all products rirj (0 ≤ i 6= j ≤ 3) is dense in so(3). proof. since det ri = −1, the subgroup r2 consists precisely of the elements of r that have determinant 1. therefore r2 = r∩ so(3), which is dense in so(3). tetrahedral chains and a curious semigroup 115 6. equidistribution in this section we prove that the orbit of any point of the unit 2-sphere s2 for the action of the group r is equidistributed in s2, in a sense made precise in definition 6.1 below. this is a natural analogue of the theorem of arnold and krylov [2], and we prove it using similar methods, including simplifications suggested by one reviewer that eliminate the use of spherical harmonics. figure 3: the cayley graph of r (schematic). dots (nodes) indicate group elements, with the identity being at the centre. each edge represents left multiplication by a reflection ri. since r 2 i = i these edges are bidirectional. there are four types of edge, for 0 ≤ i ≤ 3, and each node lies on one edge of each type. the tree structure continues recursively to infinity. it is convenient to motivate the method in terms of the cayley graph c(r) of r, for the generating set {r0,r1,r2,r3}, see [5, 18]. the nodes of c(r) correspond to elements of r. edges (of type i) join node γ to riγ. the graph c(r) is an infinite tree, every node of which has valence 4, indicated schematically in figure 3. right multiplication by an element γ ∈r induces an automorphism of c(r) that preserves edge types, because ri(δγ) = (riδ)γ for all δ ∈ r, so an edge of type i from δ to riδ maps to an edge of type i from δγ to ri(δγ). in particular, c(r) is homogeneous in the sense that 116 i. stewart its automorphism group acts transitively. this is a hint that orbits might be equidistributed. if x ∈ s2, the orbit rx wraps the nodes of c(r) around s2, sending γ ∈r to γx. we can therefore use the structure of c(r) to represent the orbit. the length n of a product of reflections rinrin−1 · · ·ri1 = γ ∈ r corresponds to the length of a path in c(r) from the identity to that element. such paths may intersect themselves, or repeat edges. consider a random walk on the cayley graph of r, where each edge of type i occurs with equal probability 1 4 for 0 ≤ i ≤ 3. after n steps the random walk reaches the group element rinrin−1 · · ·ri1 (6.1) where successive rik are chosen randomly from {0, 1, 2, 3}, each with probability 1 4 . let jn be the set of all index sequences j = j1, . . . ,jn, where 0 ≤ jk ≤ 3 for k = 1, . . . ,n. for j ∈ jn define rj = rjnrjn−1 · · ·rj1 and let the length of the sequence be λ(j ) = n. assume that the random walk starts at the identity of r. let u be an open (or, more generally, measurable) subset of s2. for fixed but arbitrary x ∈ s2, let pn(u) be the probability that after n steps of the random walk the point rinrin−1 · · ·ri1x belongs to u. let µ be normalised surface lebesgue measure on s2, that is, lebesgue measure divided by 4π, so that µ(s2) = 1. heuristically, the orbit of x is equidistributed provided that lim n→∞ pn(u) = µ(u) (6.2) for all u. this motivates the following discussion, leading to the definition of equidistribution that we employ in this paper, definition 6.1. let c(s2) be the space of all continuous maps f : s2 → r with inner product 〈f,g〉 = ∫ s2 fg dµ. tetrahedral chains and a curious semigroup 117 the integral is finite because s2 is compact, and the inner product gives c(s2) a hilbert space structure, inducing the norm ‖f‖ = √ 〈f,f〉 . (6.3) the group o(3) of rotations in r3 has a natural action as isometries (normpreserving maps) of c(s2), defined as follows. suppose that γ ∈ o(3). the natural action of o(3) on r3 leaves s2 invariant, so for each γ ∈ o(3) there is an operator gγ on c(s2) defined by (gγf)(x) = f ( γ−1x ) . (6.4) the inverse ensures that this is a left action: gγδf = gγgδf. since µ is o(3)-invariant, ‖gγf‖ = ‖f‖ (6.5) for all γ ∈ o(3). therefore every gγ is an isometry of c(s2). in particular, there are isometries ri (0 ≤ i ≤ 3) of c(s2) such that (rif)(x) = f(r −1 i x) = f(rix) where the latter equality follows from r2i = i. if we define r = r0 + r1 + r2 + r3 4 then the powers r n correspond bijectively to paths of length n through the cayley graph, weighted by the probability 4−n of each such path. this motivates the following definition: definition 6.1. let x ∈ s2, and let fn(x) = (r n f)(x) = 1 4n ∑ j∈jn f(rj x) be the average value of f evaluated at images of x under elements of r having length n (corresponding to paths of length n in the cayley graph). then the r-orbit of x ∈ s2 is equidistributed if and only if lim n→∞ fn(x) = ∫ s2 f dµ (6.6) for any f ∈c(s2). 118 i. stewart recall that the norm of a linear operator a on a banach space b is defined by ‖a‖ = sup ‖x‖=1 ‖ax‖ (x ∈b) . immediate consequences are: ‖ax‖≤‖a‖‖x‖ , ‖ab‖≤‖a‖‖b‖ . (6.7) as in arnold and krylov [2], we also need the following lemma: lemma 6.2. let v1, . . . ,vs ∈c(s2). suppose that ‖vi‖ 6= 0 for 1 ≤ i ≤ s, and ‖v1 + · · · + vs‖ = ‖v1‖ + · · · + ‖vs‖ . (6.8) then there exists v ∈ c(s2) and positive real numbers ri (1 ≤ i ≤ s) such that vi = riv. proof. the triangle inequality implies that ‖v1 + · · · + vs‖ ≤ ‖v1‖ + · · · + ‖vs‖ for any vi ∈ c(s2). we claim that the conditions of the lemma imply that this is an equality with the stated properties of the vi. to prove the claim, recall that a normed vector space is strictly convex if x,y 6= 0 and ‖x + y‖ = ‖x‖+‖y‖ imply that x = cy for some real constant c > 0. the space c(s2) is a hilbert space, hence strictly convex [16, 6]. for such spaces, equality occurs in the triangle inequality if and only if all vi are multiples of each other by nonnegative real numbers. a simple induction completes the proof. corollary 6.3. if ‖vi‖ = 1 for 1 ≤ i ≤ s, and (6.8) holds, then v1 = · · · = vs . (6.9) we can now state and prove the main theorem of this section: theorem 6.4. if x ∈ s2 then the orbit rx is equidistributed in the sense of (6.6). proof. define a polynomial function in c(s2) to be the restriction to s2 of a polynomial function r3 → r in cartesian coordinates (x,y,z). by the stone-weierstrass theorem [20, 22, 23, 29], polynomial functions are dense tetrahedral chains and a curious semigroup 119 in c(s2) with the topology of uniform convergence. therefore it is enough to prove (6.6) when f is polynomial. this equality is obvious when f is constant, so it suffices to prove that lim(r n f)(x) = 0 for any polynomial function whose integral over s2 is zero. let pl be the vector space of polynomial functions p : s2 → r of degree ≤ l such that ∫ s2 p dµ = 0. since this space is finite-dimensional, any two hausdorff vector topologies coincide, so the topology of pointwise convergence is the same as as that given by the norm (6.3). it is therefore sufficient to show that lim‖rnf‖ = 0 for all f ∈ pl. consider the linear operators r n : pl → pl. we have ‖rif‖ = ‖f‖, since reflections preserve µ. hence ‖r‖≤ 1, so ‖rn‖≤ 1 for any n ∈ n. suppose that some ‖rm‖ = k < 1 where m ≥ 1. then (6.7) implies that ‖rnf‖≤ kbn/mc. therefore limn→∞fn(x) = 0, proving (6.6). otherwise we must have ‖rn‖ = 1 for all n ∈ n. we will show that this cannot occur. for a contradiction, suppose it does. the unit ball of pl is compact, so there exists fn ∈ pl with ‖r n fn‖ = ‖fn‖ = 1; hence also ‖rifn‖ = 1 for all i ≤ n. a subsequence of the fn converges to a polynomial f ∈ pl with ‖r n f‖ = ‖f‖ = 1 for all n. now lemma 6.2, with the vi being all rj f for j ∈ jn, implies that r0f = r1f = r2f = r3f , r0r1f = r0r2f = · · · = r2r3f , . . . and more generally, rj f = rkf for all j ,k ∈ jn . (6.10) let km = { rj : λ(j ) = m } be the set of all words in the ri of length m, allowing consecutive repetitions. since r2i = i, we have k0 ⊆ k2 ⊆ k4 ⊆ ···⊆ k2n ⊆ ··· . applying these group elements to x ∈ s2, k0x ⊆ k2x ⊆ k4x ⊆ ···⊆ k2nx ⊆ ··· . 120 i. stewart therefore the orbit of x under r2 ⊆ so(3) is the union r2x = ⋃ n∈n k2nx. now (6.10) implies that f ( r−1j x ) = f ( r−1k x ) for all rj , rk ∈ k2n and all n ∈ n. thus f is constant on the r2-orbit of x. by corollary 5.3, r2 is dense in so(3), so by continuity f is constant on the so(3)-orbit of x. by definition, elements of pl have zero integral over s2, so this contradicts ‖f‖ = 1. remark 6.5. (a) it is also plausible that r is equidistributed in o(3) with respect to haar measure [15]. equivalently, r2 is equidistributed in so(3). however, we have not sought a proof. 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[30] the lviv scottish book, http://www.math.lviv.ua/szkocka/view.php. http://kielich.amu.edu.pl/stefan_banach/pdf/ks-szkocka/ks-szkocka3ang.pdf http://kielich.amu.edu.pl/stefan_banach/pdf/ks-szkocka/ks-szkocka3ang.pdf http://www.math.lviv.ua/szkocka/view.php introduction reflections in faces of the tetrahedron symmetries of the tetrahedron. the 3 3 matrices. factoring out translations. free product structure non-existence of a closed chain. properties of the semigroup density equidistribution e extracta mathematicae vol. 33, núm. 2, 167 – 189 (2018) some new reverses and refinements of inequalities for relative operator entropy s.s. dragomir 1,2 1 mathematics, college of engineering & science, victoria university po box 14428, melbourne city, mc 8001, australia sever.dragomir@vu.edu.au , http://rgmia.org/dragomir 2 school of computer science & applied mathematics, university of the witwatersrand private bag 3, johannesburg 2050, south africa presented by mostafa mbekhta received february 21, 2018 abstract: in this paper we obtain new inequalities for relative operator entropy s(a|b) in the case of operators satisfying the condition ma ≤ b ≤ ma, with 0 < m < m. key words: inequalities for logarithm, relative operator entropy, operator entropy. ams subject class. (2000): 47a63, 47a30. 1. introduction kamei and fujii [6, 7] defined the relative operator entropy s(a|b), for positive invertible operators a and b, by s(a|b) := a 1 2 ( ln ( a− 1 2 ba− 1 2 )) a 1 2 , (1.1) which is a relative version of the operator entropy considered by nakamuraumegaki [12]. in general, we can define for positive operators a, b s(a|b) := s − lim ε→0+ s ( a + ε1h|b ) if it exists, here 1h is the identity operator. for the entropy function η(t) = −t ln t, the operator entropy has the following expression: η(a) = −a ln a = s(a|1h) ≥ 0 for positive contraction a. this shows that the relative operator entropy (1.1) is a relative version of the operator entropy. 167 168 s.s. dragomir following [8, pp. 149-155], we recall some important properties of relative operator entropy for a and b positive invertible operators: (i) we have the equalities s(a|b) = −a1/2 ( ln a1/2b−1a1/2 ) a1/2 = b1/2η ( b−1/2ab−1/2 ) b1/2. (1.2) (ii) we have the inequalities s(a|b) ≤ a ( ln ∥b∥ − ln a ) and s(a|b) ≤ b − a . (1.3) (iii) for any c, d positive invertible operators we have that s ( a + b|c + d ) ≥ s(a|c) + s(b|d). (iv) if b ≤ c then s(a|b) ≤ s(a|c). (v) if bn ↓ b then s(a|bn) ↓ s(a|b). (vi) for α > 0 we have s(αa|αb) = αs(a|b). (vii) for every operator t we have t ∗s(a|b)t ≤ s ( t ∗at|t ∗bt ) . the relative operator entropy is jointly concave, namely, for any positive invertible operators a, b, c, d we have s ( ta + (1 − t)b | tc + (1 − t)d ) ≥ ts(a|c) + (1 − t)s(b|d) for any t ∈ [0, 1]. for other results on the relative operator entropy see [1, 4, 9, 10, 11, 13]. observe that, if we replace in (1.2) b with a, then we get s(b|a) = a1/2η ( a−1/2ba−1/2 ) a1/2 = a1/2 ( − a−1/2ba−1/2 ln ( a−1/2ba−1/2 )) a1/2, some new reverses and refinements of inequalities 169 therefore we have a1/2 ( a−1/2ba−1/2 ln ( a−1/2ba−1/2 )) a1/2 = −s(b|a) (1.4) for positive invertible operators a and b. it is well know that, in general s(a|b) is not equal to s(b|a). in [15], a. uhlmann has shown that the relative operator entropy s(a|b) can be represented as the strong limit s(a|b) = s − lim t→0 a♯tb − a t , (1.5) where a♯νb := a 1/2 ( a−1/2ba−1/2 )ν a1/2, ν ∈ [0, 1] , is the weighted geometric mean of positive invertible operators a and b. for ν = 1 2 we denote a♯b. this definition of the weighted geometric mean can be extended for any real number ν with ν ̸= 0. for t > 0 and the positive invertible operators a, b we define the tsallis relative operator entropy (see also [3]) by tt(a|b) := a♯tb − a t . the following result providing upper and lower bounds for relative operator entropy in terms of tt(·|·) has been obtained in [6] for 0 < t ≤ 1. however, it hods for any t > 0. theorem 1. let a, b be two positive invertible operators, then for any t > 0 we have tt(a|b)(a♯tb)−1a ≤ s(a|b) ≤ tt(a|b). (1.6) in particular, we have for t = 1 that( 1h − ab−1 ) a ≤ s(a|b) ≤ b − a , [6] (1.7) and for t = 2 that 1 2 ( 1h − ( ab−1 )2) a ≤ s(a|b) ≤ 1 2 ( ba−1b − a ) . (1.8) 170 s.s. dragomir the case t = 1 2 is of interest as well. since in this case we have t1/2(a|b) := 2(a♯b − a) and t1/2(a|b)(a♯1/2b) −1a = 2 ( 1h − a(a♯b)−1 ) a , hence by (1.6) we get 2 ( 1h − a(a♯b)−1 ) a ≤ s(a|b) ≤ 2(a♯b − a) ≤ b − a . (1.9) motivated by the above results, in this paper we obtain new inequalities for the relative operator entropy in the case of operators satisfying the condition ma ≤ b ≤ ma, with 0 < m < m. 2. inequalities for log-function we have: theorem 2. for any a, b > 0 we have the inequalities 1 2b min{a, b} (b − a)2 ≥ ln b − ln a − b − a b ≥ 1 2b max{a, b} (b − a)2 (2.1) and 1 2a min{a, b} (b − a)2 ≥ b − a a − ln b + ln a ≥ 1 2a max{a, b} (b − a)2. (2.2) proof. we have∫ b a b − t t dt = b ∫ b a 1 t dt − ∫ b a dt = b(ln b − ln a) − (b − a) giving that ln b − ln a − b − a b = 1 b ∫ b a b − t t dt (2.3) for any a, b > 0. let b > a > 0, then 1 a ∫ b a (b − t) dt ≥ ∫ b a b − t t dt ≥ 1 b ∫ b a (b − t) dt some new reverses and refinements of inequalities 171 giving that 1 2a (b − a)2 ≥ ∫ b a b − t t dt ≥ 1 2b (b − a)2. (2.4) let a > b > 0, then 1 b ∫ a b (t − b) dt ≥ ∫ b a b − t t dt = ∫ a b t − b t dt ≥ 1 a ∫ a b (t − b) dt giving that 1 2b (b − a)2 ≥ ∫ b a b − t t dt ≥ 1 2a (b − a)2. (2.5) therefore, by (2.4) and (2.5) we get 1 2 min{a, b} (b − a)2 ≥ ∫ b a b − t t dt ≥ 1 2 max{a, b} (b − a)2, for any a, b > 0. by utilising the equality (2.3) we get the desired result (2.1). corollary 1. for any y > 0 we have 1 2y min{1, y} (y − 1)2 ≥ ln y − y − 1 y ≥ 1 2y max{1, y} (y − 1)2, (2.6) 1 2 min{1, y} (y − 1)2 ≥ y − 1 − ln y ≥ 1 2 max{1, y} (y − 1)2. (2.7) remark 1. since for any a, b > 0 we have max{a, b} min{a, b} = ab, then (2.1) and (2.2) can also be written as 1 2a max{a, b} ( b − a b )2 ≥ ln b − ln a − b − a b ≥ 1 2a min{a, b} ( b − a b )2 (2.8) and 1 2b max{a, b} ( b − a a )2 ≥ b − a a − ln b + ln a ≥ 1 2b min{a, b} ( b − a a )2 (2.9) 172 s.s. dragomir for any a, b > 0. the inequalities can also be written as 1 2 max{1, y} ( y − 1 y )2 ≥ ln y − y − 1 y ≥ 1 2 min{1, y} ( y − 1 y )2 (2.10) and 1 2y max{1, y}(y − 1)2 ≥ y − 1 − ln y ≥ 1 2y min{1, y}(y − 1)2, (2.11) for any y > 0. in the recent paper [2] we obtained the following inequalities that provide upper and lower bounds for the quantity ln b − ln a − b−a b : 1 2 (b − a)2 min2{a, b} ≥ b − a a − ln b + ln a ≥ 1 2 (b − a)2 max2{a, b} , (2.12) where a, b > 0 and (b − a)2 ab ≥ b − a a − ln b + ln a (2.13) for any a, b > 0. it is natural to ask, which of the upper bounds for the quantity b − a a − ln b + ln a as provided by (2.2), (2.12) and (2.13) is better? it has been shown in [2] that neither of the upper bounds in (2.12) and (2.13) is always best. consider now the difference d1(a, b) := 1 2a min{a, b} (b − a)2 − 1 2 (b − a)2 min2{a, b} = 1 2 (b − a)2 a min2{a, b} (min{a, b} − a) ≤ 0 , which shows that upper bound in (2.2) is always better than the upper bound in (2.12). some new reverses and refinements of inequalities 173 consider the difference d2(a, b) := 1 2a min{a, b} (b − a)2 − (b − a)2 ab = 1 2ab min{a, b} (b − a)2 ( b − 2 min{a, b} ) , which can take both positive and negative values for a, b > 0, showing that neither of the bounds (2.2) and (2.13) is always best. now, consider the difference d(a, b) := 1 2a max{a, b} (b − a)2 − 1 2 (b − a)2 max2{a, b} = 1 2a max2{a, b} (b − a)2(max{a, b} − a) ≥ 0 , which shows that lower bound in (2.2) is always better than the lower bound in (2.12). corollary 2. if y ∈ [k, k] ⊂ (0, ∞), then we have the local inequalities 1 2 min{1, k} (y − 1)2 y ≥ ln y − y − 1 y ≥ 1 2 max{1, k} (y − 1)2 y , (2.14) 1 2 min{1, k} (y − 1)2 ≥ y − 1 − ln y ≥ 1 2 max{1, k} (y − 1)2, (2.15) 1 2 max{1, k} ( y − 1 y )2 ≥ ln y − y − 1 y ≥ 1 2 min{1, k} ( y − 1 y )2 , (2.16) 1 2 max{1, k} (y − 1)2 y ≥ y − 1 − ln y ≥ 1 2 min{1, k} (y − 1)2 y . (2.17) proof. if y ∈ [k, k] ⊂ (0, ∞), then by analyzing all possible locations of the interval [k, k] and 1 we have min{1, k} ≤ min{1, y} ≤ min{1, k} , max{1, k} ≤ max{1, y} ≤ max{1, k} . 174 s.s. dragomir by using the inequalities (2.6) and (2.7) we have 1 2y min{1, k} (y − 1)2 ≥ 1 2y min{1, y} (y − 1)2 ≥ ln y − y − 1 y ≥ 1 2y max{1, y} (y − 1)2 ≥ 1 2y max{1, k} (y − 1)2 and 1 2 min{1, k} (y − 1)2 ≥ 1 2 min{1, y} (y − 1)2 ≥ y − 1 − ln y ≥ 1 2 max{1, y} (y − 1)2 ≥ 1 2 max{1, k} (y − 1)2 for any y ∈ [k, k], that prove (2.14) and (2.15). the inequalities (2.16) and (2.17) follows by (2.16) and (2.17). if we consider the function f(y) = (y−1)2 y , y > 0, then we observe that f ′(y) = y2 − 1 y2 and f ′′(y) = 2 y3 , which shows that f is strictly decreasing on (0, 1), strictly increasing on [1, ∞) and strictly convex for y > 0. we also have f(1 y ) = f(y) for y > 0. by the properties of f we then have that max y∈[k,k] (y − 1)2 y =   (k−1)2 k if k < 1 , max { (k−1)2 k , (k−1)2 k } if k ≤ 1 ≤ k , (k−1)2 k if 1 < k , (2.18) =: u(k, k) some new reverses and refinements of inequalities 175 and min y∈[k,k] (y − 1)2 y =   (1−k)2 k if k < 1 , 0 if k ≤ 1 ≤ k , (k−1)2 k if 1 < k , (2.19) =: u(k, k) . we can provide now some global bounds as follows. from (2.14) we then get for any y ∈ [k, k] that 1 2 min{1, k} u(k, k) ≥ ln y − y − 1 y ≥ 1 2 max{1, k} u(k, k), (2.20) while from (2.17) we get for any y ∈ [k, k] that 1 2 max{1, k}u(k, k) ≥ y − 1 − ln y ≥ 1 2 min{1, k}u(k, k). (2.21) consider z(k, k) : = max y∈[k,k] (y − 1)2 (2.22) =   (1 − k)2 if k < 1 , max{(1 − k)2, (k − 1)2} if k ≤ 1 ≤ k , (k − 1)2 if 1 < k , and z(k, k) := min y∈[k,k] (y − 1)2 =   (1 − k)2 if k < 1 , 0 if k ≤ 1 ≤ k , (k − 1)2 if 1 < k . (2.23) by making use of (2.15) we get 1 2 min{1, k} z(k, k) ≥ y − 1 − ln y ≥ 1 2 max{1, k} z(k, k), (2.24) for any y ∈ [k, k]. 176 s.s. dragomir consider the function g(y) = ( y−1 y )2 , y > 0, then we observe that g′(y) = 2(y − 1) y2 and g′′(y) = 2(3 − 2y) y4 , which shows that g is strictly decreasing on (0, 1), strictly increasing on [1, ∞) strictly convex for y ∈ (0, 3/2) and strictly concave on (3/2, ∞). consider w(k, k) : = max y∈[k,k] ( y − 1 y )2 (2.25) =   ( 1−k k )2 if k < 1 , max {( 1−k k )2 , ( k−1 k )2} if k ≤ 1 ≤ k ,( k−1 k )2 if 1 < k , and w(k, k) := min y∈[k,k] ( y − 1 y )2 =   ( 1−k k )2 if k < 1 , 0 if k ≤ 1 ≤ k ,( k−1 k )2 if 1 < k . (2.26) then by (2.16) we get 1 2 max{1, k}w(k, k) ≥ ln y − y − 1 y ≥ 1 2 min{1, k}w(k, k) (2.27) for any y ∈ [k, k]. 3. operator inequalities we have the following: lemma 1. let x ∈ [k, k] and t > 0, then we have 1 2 min{1, kt} ( xt − 1 t − 1 − x−t t ) ≥ ln x − 1 − x−t t (3.1) ≥ 1 2 max{1, kt} ( xt − 1 t − 1 − x−t t ) ≥ 0 some new reverses and refinements of inequalities 177 and 1 2 max { 1, kt } t ( 1 − x−t t )2 ≥ ln x − 1 − x−t t (3.2) ≥ 1 2 min { 1, kt } t ( 1 − x−t t )2 ≥ 0 . proof. let y = xt ∈ [ kt, kt ] . by using the inequality (2.14) we have 1 2 min{1, kt} (xt + x−t − 2) ≥ t ln x − xt − 1 xt ≥ 1 2 max{1, kt} ( xt + x−t − 2 ) ≥ 0 that is equivalent to (3.1). from the inequality (2.16) we have for y = xt 1 2 max { 1, kt }( 1 − 2x−t + x−2t ) ≥ t ln x − xt − 1 xt ≥ 1 2 min { 1, kt }( 1 − 2x−t + x−2t ) ≥ 0 that is equivalent to (3.2). we have: theorem 3. let a, b be two positive invertible operators and the constants m > m > 0 with the property that ma ≤ b ≤ ma . (3.3) then for any t > 0 we have 1 2 min{1, mt} tt(a|b) ( a−1 − (a♯tb)−1 ) a ≥ s(a|b) − tt(a|b)(a♯tb)−1a (3.4) ≥ 1 2 max{1, mt} tt(a|b) ( a−1 − (a♯tb)−1 ) a ≥ 0 178 s.s. dragomir and 1 2 max { 1, mt } t ( tt(a|b)(a♯tb)−1 )2 a ≥ s(a|b) − tt(a|b)(a♯tb)−1a (3.5) ≥ 1 2 min { 1, mt } t ( tt(a|b)(a♯tb)−1 )2 a ≥ 0 . proof. since ma ≤ b ≤ ma and a is invertible, then by multiplying both sides with a−1/2 we get m1h ≤ a−1/2ba−1/2 ≤ m. denote x = a−1/2ba−1/2 and by using the functional calculus for x that has its spectrum contained in the interval [m, m] and the inequality (3.1), we get 1 2 min{1, mt} (( a−1/2ba−1/2 )t − 1h t − 1h − ( a−1/2ba−1/2 )−t t ) ≥ ln ( a−1/2ba−1/2 ) − 1h − ( a−1/2ba−1/2 )−t t (3.6) ≥ 1 2 max{1, mt} (( a−1/2ba−1/2 )t − 1h t − 1h − ( a−1/2ba−1/2 )−t t ) ≥ 0 for any t > 0. now, if we multiply both sides of (3.6) by a1/2, then we get 1 2 min{1, mt} a1/2 (( a−1/2ba−1/2 )t − 1h t − 1h − ( a−1/2ba−1/2 )−t t ) a1/2 ≥ a1/2 ( ln ( a−1/2ba−1/2 )) a1/2 − a1/2 1h − ( a−1/2ba−1/2 )−t t a1/2 ≥ 1 2 max{1, mt} a1/2 (( a−1/2ba−1/2 )t − 1h t (3.7) − 1h − ( a−1/2ba−1/2 ) −t t ) a1/2 ≥ 0 for any t > 0. observe that a1/2 ln ( a−1/2ba−1/2 ) a1/2 = s(a|b) , some new reverses and refinements of inequalities 179 a1/2 ( a−1/2ba−1/2 )t − 1 t a1/2 = a♯tb − a t = tt(a|b) , a1/2 1h − ( a−1/2ba−1/2 )−t t a1/2 (3.8) = a1/2 ( a−1/2ba−1/2 )t( a−1/2ba−1/2 )−t − (a−1/2ba−1/2)−t t a1/2 = a1/2 ( a−1/2ba−1/2 )t − 1h t ( a−1/2ba−1/2 )−t a1/2 = a1/2 ( a−1/2ba−1/2 )t − 1h t a1/2a−1/2 ( a−1/2ba−1/2 )−t a−1/2a = tt(a|b)(a♯tb)−1a and then by (3.7) we get 1 2 min{1, mt} tt(a|b) ( 1h − (a♯tb)−1a ) ≥ s(a|b) − tt(a|b)(a♯tb)−1a ≥ 1 2 max{1, mt} tt(a|b) ( 1h − (a♯tb)−1a ) ≥ 0 that is equivalent to (3.4). from the inequality (3.2) we also have 1 2 max{1, mt}t ( 1h − ( a−1/2ba−1/2 )−t t )2 (3.9) ≥ ln ( a−1/2ba−1/2 ) − 1h − ( a−1/2ba−1/2 )−t t ≥ 1 2 min{1, mt}t ( 1h − ( a−1/2ba−1/2 )−t t )2 ≥ 0 . 180 s.s. dragomir now, if we multiply both sides of (3.9) by a1/2, then we get 1 2 max{1, mt}ta1/2 ( 1h − ( a−1/2ba−1/2 )−t t )2 a1/2 (3.10) ≥ a1/2 ( ln ( a−1/2ba−1/2 )) a1/2 − a1/2 1h − ( a−1/2ba−1/2 )−t t a1/2 ≥ 1 2 min{1, mt}ta1/2 ( 1h − ( a−1/2ba−1/2 )−t t )2 a1/2 ≥ 0 . from (3.8) we have, by multiplying both sides by a−1/2, that 1h − ( a−1/2ba−1/2 )−t t = a−1/2tt(a|b)(a♯tb)−1a1/2. then a1/2 ( 1h − ( a−1/2ba−1/2 )−t t )2 a1/2 = a1/2 ( a−1/2tt(a|b)(a♯tb)−1a1/2 )2 a1/2 = a1/2a−1/2tt(a|b)(a♯tb)−1a1/2a−1/2tt(a|b)(a♯tb)−1a1/2a1/2 = tt(a|b)(a♯tb)−1tt(a|b)(a♯tb)−1a = (tt(a|b)(a♯tb)−1)2a , which together with (3.10) produces the desired result (3.5). there are some particular inequalities of interest as follows. for t = 1 we get from (3.4) and (3.5) that 1 2 min{1, m} (b − a) ( a−1 − b−1 ) a ≥ s(a|b) − ( 1h − ab−1 ) a (3.11) ≥ 1 2 max{1, m} (b − a) ( a−1 − b−1 ) a ≥ 0 some new reverses and refinements of inequalities 181 and 1 2 max{1, m} ( 1h − ab−1 )2 a ≥ s(a|b) − ( 1h − ab−1 ) a (3.12) ≥ 1 2 min{1, m} ( 1h − ab−1 )2 a ≥ 0 . for t = 1/2 we get from (3.4) and (3.5) that 1 min{1, √ m} (a♯b − a) ( a−1 − (a♯b)−1 ) a ≥ s(a|b) − 2 ( 1h − a(a♯b)−1 ) a (3.13) ≥ 1 max{1, √ m} (a♯b − a) ( a−1 − (a♯b)−1 ) a ≥ 0 and max{1, √ m} ( 1h − a(a♯b)−1 )2 a ≥ s(a|b) − 2 ( 1h − a(a♯b)−1 ) a (3.14) ≥ min{1, √ m} ( 1h − a(a♯b)−1 )2 a ≥ 0 . for t = 2 we get from (3.4) and (3.5) that 1 4 min{1, m2} ( ba−1b − a )( a−1 − b−1ab−1 ) a ≥ s(a|b) − 1 2 ( 1h − ( ab−1 )2) a (3.15) ≥ 1 4 max{1, m2} ( ba−1b − a )( a−1 − b−1ab−1 ) a ≥ 0 and 1 4 max{1, m2} ( 1h − ( ab−1 )2)2 a ≥ s(a|b) − 1 2 ( 1h − ( ab−1 )2) a (3.16) ≥ 1 4 min{1, m2} ( 1h − ( ab−1 )2)2 a ≥ 0 . we have the following: 182 s.s. dragomir lemma 2. let x ∈ [m, m] and t > 0, then we have 1 2 min{1, mt} t ( xt − 1 t )2 ≥ xt − 1 t − ln x ≥ 1 2 max{1, mt} t ( xt − 1 t )2 (3.17) and 1 2 max{1, mt} ( xt − 1 t − 1 − x−t t ) ≥ xt − 1 t − ln x (3.18) ≥ 1 2 min{1, mt} ( xt − 1 t − 1 − x−t t ) . proof. let y = xt ∈ [mt, mt]. by using the inequality (2.15) we have (3.17) and by (2.17) we have (3.18). we also have: theorem 4. let a, b be two positive invertible operators and the constants m > m > 0 with the property (3.3). then for any t > 0 we have 1 2 min{1, mt} t tt(a|b)a−1tt(a|b) ≥ tt(a|b) − s(a|b) (3.19) ≥ 1 2 max{1, mt} t tt(a|b)a−1tt(a|b) ≥ 0 and 1 2 max{1, mt}tt(a|b) ( 1h − (a♯tb)−1a ) ≥ tt(a|b) − s(a|b) (3.20) ≥ 1 2 min{1, mt} tt(a|b) ( 1h − (a♯tb)−1a ) ≥ 0 . proof. if we use the inequality (3.17) for the selfadjoint operator x = a−1/2ba−1/2 that has its spectrum contained in the interval [m, m], then we some new reverses and refinements of inequalities 183 get 1 2 min{1, mt} t (( a−1/2ba−1/2 )t − 1 t )2 ≥ ( a−1/2ba−1/2 )t − 1 t − ln ( a−1/2ba−1/2 ) ≥ 1 2 max{1, mt} t (( a−1/2ba−1/2 )t − 1 t )2 ≥ 0 for any t > 0. if we multiply both sides of this inequality by a1/2 we get 1 2 min{1, mt} ta1/2 (( a−1/2ba−1/2 )t − 1 t )2 a1/2 (3.21) ≥ a1/2 ( a−1/2ba−1/2 )t − 1 t a1/2 − a1/2 ( ln ( a−1/2ba−1/2 )) a1/2 ≥ 1 2 max{1, mt} ta1/2 (( a−1/2ba−1/2 )t − 1 t )2 a1/2 ≥ 0 for any t > 0. since a1/2 ( a−1/2ba−1/2 )t − 1 t a1/2 = tt(a|b) , then ( a−1/2ba−1/2 )t − 1 t = a−1/2tt(a|b)a−1/2 and a1/2 (( a−1/2ba−1/2 )t − 1 t )2 a1/2 = a1/2a−1/2tt(a|b)a−1/2a−1/2tt(a|b)a−1/2a1/2 = tt(a|b)a−1tt(a|b) for any t > 0. 184 s.s. dragomir by making use of (3.21) we then get (3.19). by using inequality (3.18) we have 1 2 max{1, mt} (( a−1/2ba−1/2 )t − 1 t − 1 − ( a−1/2ba−1/2 )−t t ) ≥ ( a−1/2ba−1/2 )t − 1 t − ln ( a−1/2ba−1/2 ) ≥ 1 2 min{1, mt} (( a−1/2ba−1/2 )t − 1 t − 1 − ( a−1/2ba−1/2 )−t t ) ≥ 0 , for any t > 0. if we multiply both sides of this inequality by a1/2 we get 1 2 max{1, mt}a1/2 (( a−1/2ba−1/2 )t − 1 t − 1 − ( a−1/2ba−1/2 )−t t ) a1/2 ≥ a1/2 ( a−1/2ba−1/2 )t − 1 t a1/2 − a1/2 ( ln ( a−1/2ba−1/2 )) a1/2 ≥ 1 2 min{1, mt}a1/2 (( a−1/2ba−1/2 )t − 1 t − 1 − ( a−1/2ba−1/2 )−t t ) a1/2 ≥ 0 for any t > 0, and the inequality (3.20) is obtained. for t = 1 we get from (3.19) and (3.20) that 1 2 min{1, m} (b − a)a−1(b − a) ≥ b − a − s(a|b) (3.22) ≥ 1 2 max{1, m} (b − a)a−1(b − a) ≥ 0 and 1 2 max{1, m}(b − a) ( 1h − b−1a ) ≥ b − a − s(a|b) (3.23) ≥ 1 2 min{1, m}(b − a) ( 1h − b−1a ) ≥ 0 . some new reverses and refinements of inequalities 185 for t = 1/2 we get from (3.19) and (3.20) that 1 min{1, √ m} (a♯b − a)a−1(a♯b − a) ≥ 2(a♯b − a) − s(a|b) (3.24) ≥ 1 max{1, √ m} (a♯b − a)a−1(a♯b − a) ≥ 0 and max { 1, √ m } (a♯b − a) ( 1h − (a♯b)−1a ) ≥ 2(a♯b − a) − s(a|b) (3.25) ≥ min { 1, √ m } (a♯b − a) ( 1h − (a♯b)−1a ) ≥ 0 . for t = 2 we get from (3.19) and (3.20) that 1 4 min{1, m2} ( ba−1b − a ) a−1 ( ba−1b − a ) ≥ 1 2 ( ba−1b − a ) − s(a|b) (3.26) ≥ 1 4 max{1, m2} ( ba−1b − a ) a−1 ( ba−1b − a ) ≥ 0 and 1 4 max { 1, m2 }( ba−1b − a )( 1h − ( b−1a )2) ≥ 1 2 ( ba−1b − a ) − s(a|b) (3.27) ≥ 1 4 min { 1, m2 }( ba−1b − a )( 1h − ( b−1a )2) ≥ 0 . 4. some global bounds for [m, m] ⊂ (0, ∞) and t > 0 and by the use of (2.18) we define ut(m, m) : = u ( mt, mt ) (4.1) =   (mt−1)2 mt if m < 1 , max { (mt−1)2 mt , (mt−1)2 mt } if m ≤ 1 ≤ m , (mt−1)2 mt if 1 < m , 186 s.s. dragomir and by (2.19) ut(m, m) := u ( mt, mt ) =   (1−mt)2 mt if m < 1 , 0 if m ≤ 1 ≤ m , (mt−1)2 mt if 1 < m . (4.2) by (2.20) and (2.21) we have for y = xt ∈ [ mt, mt ] and t > 0 that 1 2t min{1, mt} ut(m, m) ≥ ln x − 1 − x−t t ≥ 1 2t max{1, mt} ut(m, m) (4.3) and 1 2t max { 1, mt } ut(m, m) ≥ xt − 1 t − ln x ≥ 1 2t min{1, mt}ut(m, m) , (4.4) where x ∈ [m, m] and t > 0. using (2.22) and (2.23) we define zt(m, m) : = z(m t, mt) (4.5) =   ( 1 − mt )2 if m < 1 , max {( 1 − mt )2 , ( mt − 1 )2} if m ≤ 1 ≤ m ,( mt − 1 )2 if 1 < m , and zt(m, m) := z ( mt, mt ) =   ( 1 − mt)2 if m < 1 , 0 if m ≤ 1 ≤ m ,( mt − 1 )2 if 1 < m . (4.6) by (2.24) we have for y = xt ∈ [ mt, mt ] and t > 0 that 1 2t min{1, mt} zt(m, m) ≥ xt − 1 t − ln x ≥ 1 2t max{1, mt} zt(m, m) , (4.7) some new reverses and refinements of inequalities 187 where x ∈ [m, m] and t > 0. utilising (2.25) and (2.26) we can define wt(m, m) : = w ( mt, mt ) (4.8) =   ( 1−mt mt )2 if m < 1 , max {( 1−mt mt )2 , ( mt−1 mt )2} if m ≤ 1 ≤ m ,( mt−1 mt )2 if 1 < m , and wt(m, m) := w(m t, mt) =   ( 1−mt mt )2 if m < 1 , 0 if m ≤ 1 ≤ m ,( mt−1 mt )2 if 1 < m . (4.9) by (2.24) we have for y = xt ∈ [ mt, mt ] and t > 0 that 1 2t max { 1, mt } wt(m, m) ≥ ln x − 1 − x−t t ≥ 1 2t min { 1, mt } wt(m, m) , (4.10) where x ∈ [m, m] and t > 0. theorem 5. let a, b be two positive invertible operators and the constants m > m > 0 with the property (3.3). then for any t > 0 we have 1 2t min{1, mt} ut(m, m)a ≥ s(a|b) − tt(a|b)(a♯tb)−1a ≥ 1 2t max{1, mt} ut(m, m)a , 1 2t max { 1, mt } wt(m, m)a ≥ s(a|b) − tt(a|b)(a♯tb)−1a ≥ 1 2t min { 1, mt } wt(m, m)a , 1 2t min{1, mt} zt(m, m)a ≥ tt(a|b) − s(a|b) ≥ 1 2t max{1, mt} zt(m, m)a 188 s.s. dragomir and 1 2t max { 1, mt } ut(m, m)a ≥ tt(a|b) − s(a|b) ≥ 1 2t min { 1, mt } ut(m, m)a . the proof follows by the inequalities (4.4), (4.5), (4.7) and (4.10) in a similar way as the one from the proof of theorem 3 and we omit the details. for t = 1, t = 1/2 and t = 2 one can obtain some particular inequalities of interest, however the details are not provided here. references [1] s.s. dragomir, some inequalities for relative operator entropy; preprint rgmia res. rep. coll. 18 (2015), art. 145. [http://rgmia.org/papers/v18/v18a145.pdf]. [2] s.s. dragomir, reverses and refinements of several inequalities for relative operator entropy; preprint rgmia res. rep. coll. 19 (2015). [http://rgmia.org/papers/v19/]. [3] s. furuichi, k. yanagi, k. kuriyama, fundamental properties of tsallis relative entropy, j. math. phys. 45 (2004) 4868 – 4877. [4] s. furuichi, precise estimates of bounds on relative operator entropies, math. inequal. appl. 18 (2015), 869 – 877. [5] s. furuichi, n. minculete, alternative reverse inequalities for young’s inequality, j. math. inequal. 5 (2011), number 4, 595 – 600. [6] j.i. fujii, e. kamei, uhlmann’s interpolational method for operator means, math. japon. 34 (4) (1989), 541 – 547. [7] j.i. fujii, e. kamei, relative operator entropy in noncommutative information theory, math. japon. 34 (3) (1989), 341 – 348. [8] t. furuta, j. mićić hot, j. pečarić, y. seo, “ mond-pečarić method in operator inequalities. inequalities for bounded selfadjoint operators on a hilbert space ”, monographs in inequalities 1, element, zagreb, 2005. [9] i.h. kim, operator extension of strong subadditivity of entropy, j. math. phys. 53 (2012), 122204, 3 pp. [10] p. kluza, m. niezgoda, inequalities for relative operator entropies, electron. j. linear algebra 27 (2014), 851 – 864. [11] m.s. moslehian, f. mirzapour, a. morassaei, operator entropy inequalities, colloq. math. 130 (2013), 159 – 168. [12] m. nakamura, h. umegaki, a note on the entropy for operator algebras, proc. japan acad. 37 (1961) 149 – 154. some new reverses and refinements of inequalities 189 [13] i. nikoufar, on operator inequalities of some relative operator entropies, adv. math. 259 (2014), 376 – 383. [14] w. specht, zur theorie der elementaren mittel, math. z. 74 (1960), 91 – 98. [15] a. uhlmann, relative entropy and the wigner-yanase-dyson-lieb concavity in an interpolation theory, comm. math. phys. 54 (1) (1977), 21 – 32. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 2 (2019), 201 – 235 doi:10.17398/2605-5686.34.2.201 available online june 3, 2019 hypo-q-norms on cartesian products of algebras of bounded linear operators on hilbert spaces s.s. dragomir 1,2 1 mathematics, college of engineering & science victoria university, melbourne city 8001, australia 2 dst-nrf centre of excellence in the mathematical and statistical sciences school of computer science & applied mathematics university of the witwatersrand, johannesburg 2050, south africa sever.dragomir@vu.edu.au , http://rgmia.org/dragomir received march 6, 2019 presented by horst martini accepted may 14, 2019 abstract: in this paper we introduce the hypo-q-norms on a cartesian product of algebras of bounded linear operators on hilbert spaces. a representation of these norms in terms of inner products, the equivalence with the q-norms on a cartesian product and some reverse inequalities obtained via the scalar reverses of cauchy-buniakowski-schwarz inequality are also given. several bounds for the norms δp, ϑp and the real norms ηr,p and θr,p are provided as well. key words: hilbert spaces, bounded linear operators, operator norm and numerical radius, n-tuple of operators, operator inequalities. ams subject class. (2010): 46c05, 26d15. 1. introduction in [13], the author has introduced the following norm on the cartesian product b(n)(h) := b(h) × ··· × b(h), where b(h) denotes the banach algebra of all bounded linear operators defined on the complex hilbert space h: ‖(t1, . . . ,tn)‖n,e := sup (λ1,...,λn)∈bn ‖λ1t1 + · · · + λntn‖, (1.1) where (t1, . . . ,tn) ∈ b(n)(h) and bn := { (λ1, . . . ,λn) ∈ cn : ∑n i=1 |λi| 2 ≤ 1 } is the euclidean closed ball in cn. it is clear that ‖·‖n,e is a norm on b(n)(h) and for any (t1, . . . ,tn) ∈ b(n)(h) we have ‖(t1, . . . ,tn)‖n,e = ‖(t∗1 , . . . ,t ∗ n)‖n,e , (1.2) issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.2.201 mailto:sever.dragomir@vu.edu.au http://rgmia.org/dragomir https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 202 s.s. dragomir where t∗i is the adjoint operator of ti, i ∈{1, . . . ,n}. it has been shown in [13] that the following inequality holds true: 1 √ n ∥∥∥∥∥ n∑ j=1 tjt ∗ j ∥∥∥∥∥ 1 2 ≤‖(t1, . . . ,tn)‖n,e ≤ ∥∥∥∥∥ n∑ j=1 tjt ∗ j ∥∥∥∥∥ 1 2 (1.3) for any n-tuple (t1, . . . ,tn) ∈ b(n)(h) and the constants 1√n and 1 are best possible. in the same paper [13] the author has introduced the euclidean operator radius of an n-tuple of operators (t1, . . . ,tn) by wn,e (t1, . . . ,tn) := sup ‖x‖=1 ( n∑ j=1 |〈tjx,x〉|2 )1 2 (1.4) and proved that wn,e (·) is a norm on b(n)(h) and satisfies the double inequality: 1 2 ‖(t1, . . . ,tn)‖n,e ≤ wn,e (t1, . . . ,tn) ≤‖(t1, . . . ,tn)‖n,e (1.5) for each n-tuple (t1, . . . ,tn) ∈ b(n)(h). as pointed out in [13], the euclidean numerical radius also satisfies the double inequality: 1 2 √ n ∥∥∥∥∥ n∑ j=1 tjt ∗ j ∥∥∥∥∥ 1 2 ≤ wn,e (t1, . . . ,tn) ≤ ∥∥∥∥∥ n∑ j=1 tjt ∗ j ∥∥∥∥∥ 1 2 (1.6) for any (t1, . . . ,tn) ∈ b(n)(h) and the constants 12√n and 1 are best possible. now, let (e,‖·‖) be a normed linear space over the complex number field c. on cn endowed with the canonical linear structure we consider a norm ‖ ·‖n. as an example of such norms we should mention the usual p-norms ‖λ‖n,p :=   max { |λ1|, . . . , |λn| } if p = ∞,(∑n k=1 |λk| p )1 p if p ∈ [1,∞). the euclidean norm is obtained for p = 2, i.e., ‖λ‖n,2 := ( n∑ k=1 |λk|2 )1 2 . hypo-q-norms on cartesian products 203 it is well known that on en := e×···×e endowed with the canonical linear structure we can define the following p-norms: ‖x‖n,p :=   max { |x1|, . . . , |xn| } if p = ∞,(∑n k=1 |xk| p )1 p if p ∈ [1,∞). where x = (x1, . . . ,xn) ∈ en. following the paper [5], for a given norm ‖ · ‖n on cn, we define the functional ‖ ·‖h,n : en → [0,∞) by ‖x‖h,n := sup ‖λ‖n≤1 ∥∥∥∥∥ n∑ j=1 λjxj ∥∥∥∥∥, (1.7) where x = (x1, . . . ,xn) ∈ en and λ = (λ1, . . . ,λn) ∈ cn. it is easy to see that [5]: (i) ‖x‖h,n ≥ 0 for any x ∈ en, (ii) ‖x + y‖h,n ≤‖x‖h,n + ‖y‖h,n for any x,y ∈ en, (iii) ‖αx‖h,n = |α|‖x‖h,n for each α ∈ c and x ∈ en, and therefore ‖·‖h,n is a semi-norm on en. this will be called the hypo-seminorm generated by the norm ‖ ·‖n on en. we observe that ‖x‖h,n = 0 if and only if ∑n j=1 λjxj = 0 for any (λ1, . . . ,λn) ∈ b(‖·‖n). if there exists λ01, . . . ,λ 0 n 6= 0 such that (λ01, 0, . . . , 0), (0,λ02, . . . , 0), . . . , (0, 0, . . . ,λ 0 n) ∈ b(‖ · ‖n) then the semi-norm generated by ‖ ·‖n is a norm on en. if p ∈ [1,∞] and we consider the p-norms ‖·‖n,p on cn, then we can define the following hypo-q-norms on en: ‖x‖h,n,q := sup ‖λ‖n,p≤1 ∥∥∥∥∥ n∑ j=1 λjxj ∥∥∥∥∥, (1.8) with q ∈ [1,∞]. if p = 1, then q = ∞; if p = ∞, then q = 1; if p ∈ (1,∞), then 1 p + 1 q = 1. for p = 2, we have the hypo-euclidean norm on en, i.e., ‖x‖h,n,e := sup ‖λ‖n,2≤1 ∥∥∥∥∥ n∑ j=1 λjxj ∥∥∥∥∥. (1.9) 204 s.s. dragomir if we consider now e = b(h) endowed with the operator norm ‖·‖, then we can obtain the following hypo-q-norms on b(n)(h) ‖(t1, . . . ,tn)‖h,n,q := sup ‖λ‖n,p≤1 ∥∥∥∥∥ n∑ j=1 λjtj ∥∥∥∥∥ where p,q ∈ [1,∞], (1.10) with the convention that if p = 1, q = ∞, if p = ∞, q = 1 and if p > 1, then 1 p + 1 q = 1. for p = 2 we obtain the hypo-euclidian norm ‖(·, . . . , ·)‖n,e defined in (1.2). if we consider now e = b(h) endowed with the operator numerical radius w(·), which is a norm on b(h), then we can obtain the following hypo-qnumerical radius of (t1, . . . ,tn) ∈ b(n)(h) defined by wh,n,q(t1, . . . ,tn) := sup ‖λ‖n,p≤1 w ( n∑ j=1 λjtj ) with p,q ∈ [1,∞], (1.11) with the convention that if p = 1, q = ∞, if p = ∞, q = 1 and if p > 1, then 1 p + 1 q = 1. for p = 2 we obtain the hypo-euclidian norm wh,n,e(t1, . . . ,tn) := sup ‖λ‖n,2≤1 w ( n∑ j=1 λjtj ) (1.12) and will show further that it coincides with the euclidean operator radius of an n-tuple of operators (t1, . . . ,tn) defined in (1.4). using the fundamental inequality between the operator norm and numerical radius w(t) ≤‖t‖≤ 2w(t) for t ∈ b(h) we have w ( n∑ j=1 λjtj ) ≤ ∥∥∥∥∥ n∑ j=1 λjtj ∥∥∥∥∥ ≤ 2w ( n∑ j=1 λjtj ) for any (t1, . . . ,tn) ∈ b(n)(h) and any λ = (λ1, . . . ,λn) ∈ cn. by taking the supremum over λ with ‖λ‖n,p ≤ 1 we get wh,n,q(t1, . . . ,tn) ≤‖(t1, . . . ,tn)‖h,n,q ≤ 2wh,n,q(t1, . . . ,tn) (1.13) with the convention that if p = 1, q = ∞, if p = ∞, q = 1 and if p > 1, then 1 p + 1 q = 1. hypo-q-norms on cartesian products 205 for p = q = 2 we recapture the inequality (1.5). in 2012, [8] (see also [9, 10]) the author have introduced the concept of s-q-numerical radius of an n-tuple of operators (t1, . . . ,tn) for q ≥ 1 as ws,q(t1, . . . ,tn) := sup ‖x‖=1 ( n∑ j=1 ∣∣〈tjx,x〉∣∣q )1/q (1.14) and established various inequalities of interest. for more recent results see also [12, 14]. in the same paper [8] we also introduced the concept of s-q-norm of an n-tuple of operators (t1, . . . ,tn) for q ≥ 1 as ‖(t1, . . . ,tn)‖s,q := sup ‖x‖=‖y‖=1 ( n∑ j=1 |〈tjx,y〉|q )1/q . (1.15) in [8], [9] and [10], by utilising kato’s inequality [11] |〈tx,y〉|2 ≤ 〈 |t|2αx,x 〉〈 |t∗|2(1−α)y,y 〉 (1.16) for any x,y ∈ h, α ∈ [0, 1], where “absolute value” operator of a is defined by ‖a‖ := √ a∗a, the authors have obtained several inequalities for the s-qnumerical radius and s-q-norm. in this paper we investigate the connections between these norms and establish some fundamental inequalities of interest in multivariate operator theory. 2. representation results we start with the following lemma: lemma 1. let β = (β1, . . . ,βn) ∈ cn. (i) if p,q > 1 and 1 p + 1 q = 1, then sup ‖α‖n,p≤1 ∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ = ‖β‖n,q . (2.1) in particular, sup ‖α‖n,2≤1 ∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ = ‖β‖n,2. (2.2) 206 s.s. dragomir (ii) we have sup ‖α‖n,∞≤1 ∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ = ‖β‖n,1 and sup‖α‖n,1≤1 ∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ = ‖β‖n,∞. (2.3) proof. (i) using hölder’s discrete inequality for p,q > 1 and 1 p + 1 q = 1 we have ∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ ≤ ( n∑ j=1 |αj|p )1/p( n∑ j=1 |βj|q )1/q , which implies that sup ‖α‖n,p≤1 ∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ ≤‖β‖n,q (2.4) where α = (α1, . . . ,αn) and β = (β1, . . . ,βn) are n-tuples of complex numbers. for (β1, . . . ,βn) 6= 0, consider α = (α1, . . . ,αn) with αj := βj|βj|q−2(∑n k=1 |βk|q )1/p for those j for which βj 6= 0 and αj = 0, for the rest. we observe that∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ = ∣∣∣∣∣ n∑ j=1 βj |βj|q−2(∑n k=1 |βk| q )1/pβj ∣∣∣∣∣ = ∑n j=1 |βj| q(∑n k=1 |βk| q )1/p = ( n∑ j=1 |βj|q )1/q = ‖β‖n,q and ‖α‖pn,p = n∑ j=1 |αj|p = n∑ j=1 ∣∣∣βj |βj|q−2 ∣∣∣p(∑n k=1 |βk|q ) = n∑ j=1 ( |βj|q−1 )p (∑n k=1 |βk| q ) = n∑ j=1 |βj|qp−p(∑n k=1 |βk| q ) = n∑ j=1 |βj|q(∑n k=1 |βk| q ) = 1. therefore, by (2.4) we have the representation (2.1). hypo-q-norms on cartesian products 207 (ii) using the properties of the modulus, we have∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ ≤ maxj∈{1,...,n} |αj| n∑ j=1 |βj| , which implies that sup ‖α‖n,∞≤1 ∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ ≤‖β‖n,1, (2.5) where α = (α1, . . . ,αn) and β = (β1, . . . ,βn). for (β1, . . . ,βn) 6= 0, consider α = (α1, . . . ,αn) with αj := βj |βj| for those j for which βj 6= 0 and αj = 0, for the rest. we have ∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ = ∣∣∣∣∣ n∑ j=1 βj |βj| βj ∣∣∣∣∣ = n∑ j=1 |βj| = ‖β‖n,1 and ‖α‖n,∞ = max j∈{1,...,n} |αj| = max j∈{1,...,n} ∣∣∣∣ βj|βj| ∣∣∣∣ = 1 and by (2.5) we get the first representation in (2.3). moreover, we have∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ ≤ n∑ j=1 |αj| max j∈{1,...,n} |βj| , which implies that sup ‖α‖n,1≤1 ∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ ≤‖β‖n,∞, (2.6) where α = (α1, . . . ,αn) and β = (β1, . . . ,βn). for (β1, . . . ,βn) 6= 0, let j0 ∈{1, . . . ,n} such that ‖β‖∞ = max j∈{1,...,n} |βj| = |βj0| . consider α = (α1, . . . ,αn) with αj0 = βj0 |βj0| and αj = 0 for j 6= j0. for this choice we get n∑ j=1 |αj| = ∣∣βj0∣∣ |βj0| = 1 and ∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ = ∣∣∣∣ βj0|βj0|βj0 ∣∣∣∣ = |βj0| = ‖β‖n,∞ , 208 s.s. dragomir therefore by (2.6) we obtain the second representation in (4). theorem 2. let (t1, . . . ,tn) ∈ b(n)(h) and x,y ∈ h, then for p,q > 1 and 1 p + 1 q = 1 we have sup ‖α‖n,p≤1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,y 〉∣∣∣∣∣ = ( n∑ j=1 |〈tjx,y〉|q )1/q (2.7) and in particular sup ‖α‖n,2≤1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,y 〉∣∣∣∣∣ = ( n∑ j=1 |〈tjx,y〉|2 )1/2 . (2.8) we also have sup ‖α‖n,∞≤1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,y 〉∣∣∣∣∣ = n∑ j=1 |〈tjx,y〉| (2.9) and sup ‖α‖n,1≤1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,y 〉∣∣∣∣∣ = maxj∈{1,...,n}{|〈tjx,y〉|} . (2.10) proof. if we take β = (〈t1x,y〉 , . . . ,〈tnx,y〉) ∈ cn in (2.1), then we get( n∑ j=1 |〈tjx,y〉|q )1/q = ‖β‖n,q = sup ‖α‖p≤1 ∣∣∣∣∣ n∑ j=1 αjβj ∣∣∣∣∣ = sup ‖α‖n,p≤1 ∣∣∣∣∣ n∑ j=1 αj 〈tjx,y〉 ∣∣∣∣∣ = sup‖α‖n,p≤1 ∣∣∣∣∣ 〈 n∑ j=1 αjtjx,y 〉∣∣∣∣∣, which proves (2.7). the equalities (2.9) and (2.10) follow by (2.3). corollary 3. let (t1, . . . ,tn) ∈ b(n)(h) and x ∈ h, then for p,q > 1 and 1 p + 1 q = 1 we have sup ‖α‖n,p≤1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,x 〉∣∣∣∣∣ = ( n∑ j=1 |〈tjx,x〉|q )1/q (2.11) hypo-q-norms on cartesian products 209 and, in particular sup ‖α‖n,2≤1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,x 〉∣∣∣∣∣ = ( n∑ j=1 |〈tjx,x〉|2 )1/2 . (2.12) we also have sup ‖α‖n,∞≤1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,x 〉∣∣∣∣∣ = n∑ j=1 |〈tjx,x〉| (2.13) and sup ‖α‖n,1≤1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,x 〉∣∣∣∣∣ = maxj∈{1,...,n}{|〈tjx,x〉|} . (2.14) corollary 4. let (t1, . . . ,tn) ∈ b(n)(h) and x ∈ h, then for p,q > 1 and 1 p + 1 q = 1 we have sup ‖α‖n,p≤1 ∥∥∥∥∥ n∑ j=1 αjtjx ∥∥∥∥∥ = sup‖y‖=1 ( n∑ j=1 |〈tjx,y〉|q )1/q (2.15) and in particular sup ‖α‖n,2≤1 ∥∥∥∥∥ n∑ j=1 αjtjx ∥∥∥∥∥ = sup‖y‖=1 ( n∑ j=1 |〈tjx,y〉|2 )1/2 . (2.16) we also have sup ‖α‖n,∞≤1 ∥∥∥∥∥ n∑ j=1 αjtjx ∥∥∥∥∥ = sup‖y‖=1 n∑ j=1 |〈tjx,y〉| (2.17) and sup ‖α‖n,1≤1 ∥∥∥∥∥ n∑ j=1 αjtjx ∥∥∥∥∥ = maxj∈{1,...,n}{‖tjx‖}. (2.18) proof. by the properties of inner product, we have for any u ∈ h, u 6= 0 that ‖u‖ = sup ‖y‖=1 |〈u,y〉|. 210 s.s. dragomir let x ∈ h, then by taking the supremum over ‖y‖ = 1 in (2.7) we get for p,q > 1 with 1 p + 1 q = 1 that sup ‖y‖=1 ( n∑ j=1 |〈tjx,y〉|q )1/q = sup ‖y‖=1   sup ‖α‖n,p≤1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,y 〉∣∣∣∣∣   = sup ‖α‖n,p≤1   sup ‖y‖=1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,y 〉∣∣∣∣∣   = sup ‖α‖n,p≤1 ∥∥∥∥∥ ( n∑ j=1 αjtj ) x ∥∥∥∥∥, which proves the equality (2.15). the other equalities can be proved in a similar way by using theorem 2, however the details are omitted. we can state and prove our main result. theorem 5. let (t1, . . . ,tn) ∈ b(n)(h). (i) for q ≥ 1 we have the representation for the hypo-q-norm ‖(t1, . . . ,tn)‖h,n,q = sup ‖x‖=‖y‖=1 ( n∑ j=1 |〈tjx,y〉|q )1/q = ‖(t1, . . . ,tn)‖s,q (2.19) and in particular ‖(t1, . . . ,tn)‖n,e = sup ‖x‖=‖y‖=1 ( n∑ j=1 |〈tjx,y〉|2 )1/2 . (2.20) we also have ‖(t1, . . . ,tn)‖h,n,∞ = max j∈{1,...,n} { ‖tj‖ } . (2.21) hypo-q-norms on cartesian products 211 (ii) for q ≥ 1 we have the representation for the hypo--numerical radius wh,n,q(t1, . . . ,tn) = sup ‖x‖=1 ( n∑ j=1 |〈tjx,x〉|q )1/q = ws,q(t1, . . . ,tn) (2.22) and in particular wn,e (t1, . . . ,tn) := sup ‖x‖=1 ( n∑ j=1 |〈tjx,x〉|2 )1/2 . (2.23) we also have wh,n,∞ (t1, . . . ,tn) = max j∈{1,...,n} { w(tj) } . (2.24) proof. (i) by using the equality (2.15) we have for (t1, . . . ,tn) ∈ b(n)(h) that sup ‖x‖=‖y‖=1 ( n∑ j=1 |〈tjx,y〉|q )1/q = sup ‖x‖=1   sup ‖y‖=1 ( n∑ j=1 |〈tjx,y〉|q )1/q = sup ‖x‖=1   sup ‖α‖n,p≤1 ∥∥∥∥∥ n∑ j=1 αjtjx ∥∥∥∥∥   = sup ‖α‖n,p≤1   sup ‖x‖=1 ∥∥∥∥∥ n∑ j=1 αjtjx ∥∥∥∥∥   = sup ‖α‖n,p≤1 ∥∥∥∥∥ n∑ j=1 αjtj ∥∥∥∥∥ = ‖(t1, . . . ,tn)‖h,n,q, which proves (2.19). the rest is obvious. 212 s.s. dragomir (ii) by using the equality (2.11) we have for (t1, . . . ,tn) ∈ b(n)(h) that sup ‖x‖=1 ( n∑ j=1 |〈tjx,x〉|q )1/q = sup ‖x‖=1   sup ‖α‖n,p≤1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,x 〉∣∣∣∣∣   = sup ‖α‖n,p≤1   sup ‖x‖=1 ∣∣∣∣∣ 〈( n∑ j=1 αjtj ) x,x 〉∣∣∣∣∣   = sup ‖α‖n,p≤1 w ( n∑ j=1 αjtj ) = wh,n,q(t1, . . . ,tn), which proves (2.22). the rest is obvious. remark 6. the case q = 2 was obtained in a different manner in [5] by utilising the rotation-invariant normalised positive borel measure on the unit sphere. we can consider on b(n)(h) the following usual operator and numerical radius q-norms, for q ≥ 1 ‖(t1, . . . ,tn)‖n,q := ( n∑ j=1 ‖tj‖q )1/q , wn,q(t1, . . . ,tn) := ( n∑ j=1 wq(tj) )1/q , where (t1, . . . ,tn) ∈ b(n)(h). for q = ∞ we put ‖(t1, . . . ,tn)‖n,∞ := max j∈{1,...,n} { ‖tj‖ } , wn,∞(t1, . . . ,tn) := max j∈{1,...,n} { w(tj) } . corollary 7. with the assumptions of theorem 5 we have for q ≥ 1 that 1 n1/q ‖(t1, . . . ,tn)‖n,q ≤‖(t1, . . . ,tn)‖h,n,q ≤‖(t1, . . . ,tn)‖n,q (2.25) hypo-q-norms on cartesian products 213 and 1 n1/q wn,q(t1, . . . ,tn) ≤ wh,n,q(t1, . . . ,tn) ≤ wn,q(t1, . . . ,tn) (2.26) for any (t1, . . . ,tn) ∈ b(n)(h). in particular, we have [5] 1 √ n ‖(t1, . . . ,tn)‖n,2 ≤‖(t1, . . . ,tn)‖h,n,e ≤‖(t1, . . . ,tn)‖n,2 (2.27) and 1 √ n wn,2(t1, . . . ,tn) ≤ wh,n,e(t1, . . . ,tn) ≤ wn,2(t1, . . . ,tn) (2.28) for any (t1, . . . ,tn) ∈ b(n)(h). proof. let (t1, . . . ,tn) ∈ b(n)(h) and x,y ∈ h with ‖x‖ = ‖y‖ = 1. then by schwarz’s inequality we have( n∑ j=1 |〈tjx,y〉|q )1/q ≤ ( n∑ j=1 ‖tjx‖q ‖y‖q )1/q = ( n∑ j=1 ‖tjx‖q )1/q . by the operator norm inequality we also have( n∑ j=1 ‖tjx‖q )1/q ≤ ( n∑ j=1 ‖tj‖q ‖x‖q )1/q = ‖(t1, . . . ,tn)‖n,q. therefore ( n∑ j=1 |〈tjx,y〉|q )1/q ≤‖(t1, . . . ,tn)‖n,q and by taking the supremum over ‖x‖ = ‖y‖ = 1 we get the second inequality in (2.25). by the properties of complex numbers, we have max j∈{1,...,n} { |〈tjx,y〉| } ≤ ( n∑ j=1 |〈tjx,y〉|q )1/q x,y ∈ h with ‖x‖ = ‖y‖ = 1. 214 s.s. dragomir by taking the supremum over ‖x‖ = ‖y‖ = 1 we get sup ‖x‖=‖y‖=1 ( max j∈{1,...,n} { |〈tjx,y〉| }) ≤‖(t1, . . . ,tn)‖h,n,q (2.29) and since sup ‖x‖=‖y‖=1 ( max j∈{1,...,n} { |〈tjx,y〉| }) = max j∈{1,...,n} { sup ‖x‖=‖y‖=1 |〈tjx,y〉| } = max j∈{1,...,n} { ‖tj‖ } = ‖(t1, . . . ,tnt)‖n,∞, then by (2.29) we get ‖(t1, . . . ,tn)‖n,∞ ≤‖(t1, . . . ,tn)‖h,n,q (2.30) for any (t1, . . . ,tn) ∈ b(n)(h). since ‖(t1, . . . ,tn)‖n,q : = ( n∑ j=1 ‖tj‖q )1/q ≤ ( n‖(t1, . . . ,tn)‖qn,∞ )1/q = n1/q‖(t1, . . . ,tn)‖n,∞, (2.31) then by (2.30) and (2.31) we get 1 n1/q ‖(t1, . . . ,tn)‖n,q ≤‖(t1, . . . ,tn)‖h,n,q for any (t1, . . . ,tn) ∈ b(n)(h). the inequality (2.26) follows in a similar way and we omit the details. corollary 8. with the assumptions of theorem 5 we have for r ≥ q ≥ 1 that ‖(t1, . . . ,tn)‖h,n,r ≤‖(t1, . . . ,tn)‖h,n,q ≤ n r−q rq ‖(t1, . . . ,tn)‖h,n,r (2.32) and [14] wh,n,r(t1, . . . ,tn) ≤ wh,n,q(t1, . . . ,tn) ≤ n r−q rq wh,n,r(t1, . . . ,tn) (2.33) for any (t1, . . . ,tn) ∈ b(n)(h). hypo-q-norms on cartesian products 215 proof. we use the following elementary inequalities for the nonnegative numbers aj, j = 1, . . . ,n and r ≥ q > 0 (see for instance [14])( n∑ j=1 arj )1/r ≤ ( n∑ j=1 a q j )1/q ≤ n r−q rq ( n∑ j=1 arj )1/r . (2.34) let (t1, . . . ,tn) ∈ b(n)(h) and x,y ∈ h with ‖x‖ = ‖y‖ = 1. then by (2.34) we get( n∑ j=1 |〈tjx,y〉|r )1/r ≤ ( n∑ j=1 |〈tjx,y〉|q )1/q ≤ n r−q rq ( n∑ j=1 |〈tjx,y〉|r )1/r . by taking the supremum over ‖x‖ = ‖y‖ = 1 we get (2.32). the inequality (2.33) follows in a similar way and we omit the details. remark 9. for q ≥ 2 we have by (2.32) and (2.33) ‖(t1, . . . ,tn)‖h,n,q ≤‖(t1, . . . ,tn)‖h,n,e ≤ n q−2 2q ‖(t1, . . . ,tn)‖h,n,q (2.35) and wh,n,q(t1, . . . ,tn) ≤ wh,n,e(t1, . . . ,tn) ≤ n q−2 2q wh,n,q(t1, . . . ,tn) (2.36) and for 1 ≤ q ≤ 2 we have ‖(t1, . . . ,tn)‖h,n,e ≤‖(t1, . . . ,tn)‖h,n,q ≤ n 2−q 2q ‖(t1, . . . ,tn)‖h,n,e (2.37) and wh,n,e(t1, . . . ,tn) ≤ wh,n,e(t1, . . . ,tn) ≤ n 2−q 2q wh,n,e(t1, . . . ,tn) (2.38) for any (t1, . . . ,tn) ∈ b(n)(h). also, if we take q = 1 and r ≥ 1 in (2.32) and (2.33), then we get ‖(t1, . . . ,tn)‖h,n,r ≤‖(t1, . . . ,tn)‖h,n,1 ≤ n r−1 r ‖(t1, . . . ,tn)‖h,n,r (2.39) and wh,n,r(t1, . . . ,tn) ≤ wh,n,1(t1, . . . ,tn) ≤ n r−1 r wh,n,r(t1, . . . ,tn) (2.40) for any (t1, . . . ,tn) ∈ b(n)(h). 216 s.s. dragomir in particular, for r = 2 we get ‖(t1, . . . ,tn)‖h,n,e ≤‖(t1, . . . ,tn)‖h,n,1 ≤ √ n‖(t1, . . . ,tn)‖h,n,e (2.41) and wn,e(t1, . . . ,tn) ≤ wh,n,1(t1, . . . ,tn) ≤ √ nwn,e(t1, . . . ,tn) (2.42) for any (t1, . . . ,tn) ∈ b(n)(h). we have: proposition 10. for any (t1, . . . ,tn) ∈ b(n)(h) and p,q > 1 with 1 p + 1 q = 1, then we have ‖(t1, . . . ,tn)‖h,n,q ≥ 1 n1/p ∥∥∥∥∥ n∑ j=1 tj ∥∥∥∥∥ (2.43) and wh,n,q(t1, . . . ,tn) ≥ 1 n1/p w ( n∑ j=1 tj ) . (2.44) proof. let λj = 1 n1/p for j ∈ {1, . . . ,n}, then ∑n j=1 |λj| p = 1. therefore by (1.8) we get ‖(t1, . . . ,tn)‖h,n,q = sup ‖λ‖n,p≤1 ∥∥∥∥∥ n∑ j=1 λjtj ∥∥∥∥∥ ≥ ∥∥∥∥∥ n∑ j=1 1 n1/p tj ∥∥∥∥∥ = 1n1/p ∥∥∥∥∥ n∑ j=1 tj ∥∥∥∥∥. the inequality (2.44) follows in a similar way. we can also introduce the following norms for (t1, . . . ,tn) ∈ b(n)(h), ‖(t1, . . . ,tn)‖s,n,p := sup ‖x‖=1 ( n∑ j=1 ‖tjx‖p )1/p where p ≥ 1 and ‖(t1, . . . ,tn)‖s,n,∞ := sup ‖x‖=1 ( max j∈{1,...,n} ‖tjx‖ ) = max j∈{1,...,n} {‖tj‖}. the triangle inequality ‖·‖s,n,q follows by minkowski inequality, while the other properties of the norm are obvious. hypo-q-norms on cartesian products 217 proposition 11. let (t1, . . . ,tn) ∈ b(n)(h). (i) we have for p ≥ 1, that ‖(t1, . . . ,tn)‖h,n,p ≤‖(t1, . . . ,tn)‖s,n,p ≤‖(t1, . . . ,tn)‖n,p, (2.45) (ii) for p ≥ 2 we also have ‖(t1, . . . ,tn)‖s,n,p = [ wh,n,p/2 ( |t1|2 , . . . , |tn|2 )]1/2 , (2.46) where the absolute value |t| is defined by |t| := (t∗t)1/2. proof. (i) we have for p ≥ 2 and x,y ∈ h with ‖x‖ = ‖y‖ = 1, that |〈tjx,y〉|p ≤‖tjx‖p‖y‖p = ‖tjx‖p ≤‖tj‖p‖x‖p = ‖tj‖p for j ∈{1, . . . ,n}. this implies n∑ j=1 |〈tjx,y〉|p ≤ n∑ j=1 ‖tjx‖p ≤ n∑ j=1 ‖tj‖p , namely ( n∑ j=1 |〈tjx,y〉|p )1/p ≤ ( n∑ j=1 ‖tjx‖p )1/p ≤ ( n∑ j=1 ‖tj‖p )1/p , (2.47) for any x,y ∈ h with ‖x‖ = ‖y‖ = 1. taking the supremum over ‖x‖ = ‖y‖ = 1 in (2.47), we get the desired result (2.45). 218 s.s. dragomir (ii) we have ‖(t1, . . . ,tn)‖s,n,p = sup ‖x‖=1 ( n∑ j=1 ‖tjx‖p )1/p = sup ‖x‖=1 ( n∑ j=1 ( ‖tjx‖2 )p/2 )1/p = sup ‖x‖=1 ( n∑ j=1 〈tjx,tjx〉p/2 )1/p = sup ‖x‖=1   n∑ j=1 〈 t∗j tjx,x 〉p/21/p = sup ‖x‖=1 ( n∑ j=1 〈 |tj|2x,x 〉p/2 )1/p =   sup ‖x‖=1 ( n∑ j=1 〈 |tj|2x,x 〉p/2 )1/(p/2)1/2 = [ wh,n,p/2 ( |t1|2, . . . , |tn|2 )]1/2 , which proves the equality (2.46). 3. some reverse inequalities recall the following reverse of cauchy-buniakowski-schwarz inequality [2] (see also [3, theorem 5.14]): lemma 12. let a,a ∈ r and z = (z1, . . . ,zn), y = (y1, . . . ,yn) be two sequences of real numbers with the property that: ayj ≤ zj ≤ ayj for each j ∈{1, . . . ,n}. (3.1) then for any w = (w1, . . . ,wn) a sequence of positive real numbers, one has the inequality 0 ≤ n∑ j=1 wjz 2 j n∑ j=1 wjy 2 j − ( n∑ j=1 wjzjyj )2 ≤ 1 4 (a−a)2 ( n∑ j=1 wjy 2 j )2 . (3.2) the constant 1 4 is sharp in (3.2). o. shisha and b. mond obtained in 1967 (see [15]) the following counterparts of (cbs )–inequality (see also [3, theorem 5.20 & 5.21]): hypo-q-norms on cartesian products 219 lemma 13. assume that a = (a1, . . . ,an) and b = (b1, . . . ,bn) are such that there exists a,a,b,b with the property that: 0 ≤ a ≤ aj ≤ a and 0 < b ≤ bj ≤ b for any j ∈{1, . . . ,n}, (3.3) then we have the inequality n∑ j=1 a2j n∑ j=1 b2j − ( n∑ j=1 ajbj )2 ≤ (√ a b − √ a b )2 n∑ j=1 ajbj n∑ j=1 b2j. (3.4) and lemma 14. assume that a, b are nonnegative sequences and there exists γ, γ with the property that 0 ≤ γ ≤ aj bj ≤ γ < ∞ for any j ∈{1, . . . ,n}. (3.5) then we have the inequality 0 ≤ ( n∑ j=1 a2j n∑ j=1 b2j )1 2 − n∑ j=1 ajbj ≤ (γ −γ)2 4 (γ + γ) n∑ j=1 b2j. (3.6) we have: theorem 15. let (t1, . . . ,tn) ∈ b(n)(h). (i) we have 0 ≤‖(t1, . . . ,tn)‖2h,n,e − 1 n ‖(t1, . . . ,tn)‖2h,n,1 ≤ 1 4 n‖(t1, . . . ,tn)‖2n,∞ (3.7) and 0 ≤ w2n,e(t1, . . . ,tn) − 1 n w2h,n,1(t1, . . . ,tn) ≤ 1 4 n‖(t1, . . . ,tn)‖2n,∞. (3.8) 220 s.s. dragomir (ii) we have 0 ≤‖(t1, . . . ,tn)‖2h,n,e − 1 n ‖(t1, . . . ,tn)‖2h,n,1 ≤‖(t1, . . . ,tn)‖n,∞‖(t1, . . . ,tn)‖h,n,1 (3.9) and 0 ≤ w2n,e(t1, . . . ,tn) − 1 n w2h,n,1(t1, . . . ,tn) ≤‖(t1, . . . ,tn)‖n,∞wh,n,1(t1, . . . ,tn). (3.10) (iii) we have 0 ≤‖(t1, . . . ,tn)‖h,n,e − 1 √ n ‖(t1, . . . ,tn)‖h,n,1 ≤ 1 4 √ n‖(t1, . . . ,tn)‖n,∞ (3.11) and 0 ≤ wn,e(t1, . . . ,tn) − 1 √ n wh,n,1(t1, . . . ,tn) ≤ 1 4 √ n‖(t1, . . . ,tn)‖n,∞. (3.12) proof. (i) let (t1, . . . ,tn) ∈ b(n)(h) and put r = max j∈{1,...,n} { ‖tj‖ } = ‖(t1, . . . ,tn)‖n,∞. if x,y ∈ h, with ‖x‖ = ‖y‖ = 1 then |〈tjx,y〉| ≤ ‖tjx‖ ≤ ‖tj‖ ≤ r for any j ∈{1, . . . ,n}. if we write the inequality (3.2) for zj = |〈tjx,y〉|, wj = yj = 1, a = r and a = 0, we get 0 ≤ n n∑ j=1 |〈tjx,y〉|2 − ( n∑ j=1 |〈tjx,y〉| )2 ≤ 1 4 n2r2 for any x,y ∈ h, with ‖x‖ = ‖y‖ = 1. hypo-q-norms on cartesian products 221 this implies that n∑ j=1 |〈tjx,y〉|2 ≤ 1 n ( n∑ j=1 |〈tjx,y〉| )2 + 1 4 nr2 (3.13) for any x,y ∈ h, with ‖x‖ = ‖y‖ = 1 and, in particular n∑ j=1 |〈tjx,x〉|2 ≤ 1 n ( n∑ j=1 |〈tjx,x〉| )2 + 1 4 nr2 (3.14) for any x ∈ h, with ‖x‖ = 1. taking the supremum over ‖x‖ = ‖y‖ = 1 in (3.13) and ‖x‖ = 1 in (3.14), then we get (3.7) and (3.8). (ii) let (t1, . . . ,tn) ∈ b(n)(h). if we write the inequality (3.4) for aj = |〈tjx,y〉|, bj = 1, b = b = 1, a = 0 and a = r, then we get 0 ≤ n n∑ j=1 |〈tjx,y〉|2 − ( n∑ j=1 |〈tjx,y〉| )2 ≤ nr n∑ j=1 |〈tjx,y〉| , for any x,y ∈ h, with ‖x‖ = ‖y‖ = 1. this implies that n∑ j=1 |〈tjx,y〉|2 ≤ 1 n ( n∑ j=1 |〈tjx,y〉| )2 + r n∑ j=1 |〈tjx,y〉| , (3.15) for any x,y ∈ h, with ‖x‖ = ‖y‖ = 1 and, in particular n∑ j=1 |〈tjx,x〉|2 ≤ 1 n ( n∑ j=1 |〈tjx,x〉| )2 + r n∑ j=1 |〈tjx,x〉| , (3.16) for any x ∈ h with ‖x‖ = 1. taking the supremum over ‖x‖ = ‖y‖ = 1 in (3.15) and ‖x‖ = 1 in (3.16), then we get (3.9) and (3.10). (iii) if we write the inequality (3.6) for aj = |〈tjx,y〉|, bj = 1, b = b = 1, γ = 0 and γ = r we have 0 ≤ ( n n∑ j=1 |〈tjx,y〉|2 )1 2 − n∑ j=1 |〈tjx,y〉| ≤ 1 4 nr, 222 s.s. dragomir for any x,y ∈ h, with ‖x‖ = ‖y‖ = 1. this implies that( n∑ j=1 |〈tjx,y〉|2 )1 2 ≤ 1 √ n n∑ j=1 |〈tjx,y〉| + 1 4 √ nr, (3.17) for any x,y ∈ h, with ‖x‖ = ‖y‖ = 1 and, in particular( n∑ j=1 |〈tjx,x〉|2 )1 2 ≤ 1 √ n n∑ j=1 |〈tjx,x〉| + 1 4 √ nr, (3.18) for any x ∈ h with ‖x‖ = 1. taking the supremum over ‖x‖ = ‖y‖ = 1 in (3.17) and ‖x‖ = 1 in (3.18), then we get (3.11) and (3.12). before we proceed with establishing some reverse inequalities for the hypoeuclidean numerical radius, we recall some reverse results of the cauchybunyakovsky-schwarz inequality for complex numbers as follows: if γ, γ ∈ c and αj ∈ c, j ∈{1, . . . ,n} with the property that 0 ≤ re [(γ −αj) (αj −γ)] (3.19) = (re γ − re αj) (re αj − re γ) + (im γ − im αj) (im αj − im γ) or, equivalently, ∣∣∣∣αj − γ + γ2 ∣∣∣∣ ≤ 12 |γ −γ| (3.20) for each j ∈{1, . . . ,n}, then (see for instance [4, p. 9]) n n∑ j=1 |αj|2 − ∣∣∣∣∣ n∑ j=1 αj ∣∣∣∣∣ 2 ≤ 1 4 n2 |γ −γ|2 . (3.21) in addition, if re ( γγ̄ ) > 0, then (see for example [4, p. 26]): n n∑ j=1 |αj|2 ≤ 1 4 { re [( γ + γ )∑n j=1 αj ]}2 re (γγ) ≤ 1 4 |γ + γ|2 re (γγ) ∣∣∣∣∣ n∑ j=1 αj ∣∣∣∣∣ 2 . (3.22) hypo-q-norms on cartesian products 223 also, if γ 6= −γ, then (see for instance [4, p. 32]): ( n n∑ j=1 |αj|2 )1 2 − ∣∣∣∣∣ n∑ j=1 αj ∣∣∣∣∣ ≤ 14n|γ −γ| 2 |γ + γ| . (3.23) finally, from [7] we can also state that n n∑ j=1 |αj|2 − ∣∣∣∣∣ n∑ j=1 αj ∣∣∣∣∣ 2 ≤ n [ |γ + γ|− 2 √ re (γγ) ]∣∣∣∣∣ n∑ j=1 αj ∣∣∣∣∣, (3.24) provided re (γγ) > 0. we notice that a simple sufficient condition for (3.19) to hold is that re γ ≥ re αj ≥ re γ and im γ ≥ im αj ≥ im γ (3.25) for each j ∈{1, . . . ,n}. theorem 16. let (t1, . . . ,tn) ∈ b(n)(h) and γ, γ ∈ c with γ 6= γ. assume that w ( tj − γ + γ 2 i ) ≤ 1 2 |γ −γ| for any j ∈{1, . . . ,n}. (3.26) (i) we have w2h,n,e(t1, . . . ,tn) ≤ 1 n w2 ( n∑ j=1 tj ) + 1 4 n|γ −γ|2. (3.27) (ii) if re (γγ) > 0, then wh,n,e(t1, . . . ,tn) ≤ 1 2 √ n |γ + γ|√ (γγ) w ( n∑ j=1 tj ) (3.28) and w2h,n,e(t1, . . . ,tn) (3.29) ≤  1 n w ( n∑ j=1 tj ) + [ |γ + γ|− 2 √ (γγ) ] ·w ( n∑ j=1 tj ) . 224 s.s. dragomir (iii) if γ 6= −γ, then wh,n,e(t1, . . . ,tn) ≤ 1 √ n  w ( n∑ j=1 tj ) + 1 4 |γ −γ|2 |γ + γ|   . (3.30) proof. let x ∈ h with ‖x‖ = 1 and (t1, . . . ,tn) ∈ b(n)(h) with the property (3.26). by taking αj = 〈tjx,x〉 we have∣∣∣∣αj − γ + γ2 ∣∣∣∣ = ∣∣∣∣〈tjx,x〉− γ + γ2 〈x,x〉 ∣∣∣∣ = ∣∣∣∣ 〈( tj − γ + γ 2 i ) x,x 〉∣∣∣∣ ≤ sup ‖x‖=1 ∣∣∣∣ 〈( tj − γ + γ 2 i ) x,x 〉∣∣∣∣ = w ( tj − γ + γ 2 ) ≤ 1 2 |γ −γ| for any j ∈{1, . . . ,n}. (i) by using the inequality (3.21), we have n∑ j=1 |〈tjx,x〉|2 ≤ 1 n ∣∣∣∣∣ n∑ j=1 〈tjx,x〉 ∣∣∣∣∣ 2 + 1 4 n|γ −γ|2 = 1 n ∣∣∣∣∣ 〈 n∑ j=1 tjx,x 〉∣∣∣∣∣ 2 + 1 4 n |γ −γ|2 (3.31) for any x ∈ h with ‖x‖ = 1. by taking the supremum over ‖x‖ = 1 in (3.31) we get sup ‖x‖=1 ( n∑ j=1 |〈tjx,x〉|2 ) ≤ 1 n sup ‖x‖=1 ∣∣∣∣∣ 〈 n∑ j=1 tjx,x 〉∣∣∣∣∣ 2 + 1 4 n|γ −γ|2 = 1 n w2 ( n∑ j=1 tj ) + 1 4 n|γ −γ|2, which proves (3.27). hypo-q-norms on cartesian products 225 (ii) if re (γγ) > 0, then by (3.22) we have for αj = 〈tjx,x〉, j ∈{1, . . . ,n} that n∑ j=1 |〈tjx,x〉|2 ≤ 1 4n |γ + γ|2 re(γγ) ∣∣∣∣∣ n∑ j=1 〈tjx,x〉 ∣∣∣∣∣ 2 = 1 4n |γ + γ|2 re (γγ) ∣∣∣∣∣ 〈 n∑ j=1 tjx,x 〉∣∣∣∣∣ 2 (3.32) for any x ∈ h with ‖x‖ = 1. on taking the supremum over ‖x‖ = 1 in (3.32) we get (3.32). also, by (3.24) we get n∑ j=1 |〈tjx,x〉|2 ≤ 1 n ∣∣∣∣∣ n∑ j=1 〈tjx,x〉 ∣∣∣∣∣ 2 + [ |γ + γ|− 2 √ re (γγ) ]∣∣∣∣∣ n∑ j=1 〈tjx,x〉 ∣∣∣∣∣, for any x ∈ h with ‖x‖ = 1. by taking the supremum over ‖x‖ = 1 in this inequality, we have sup ‖x‖=1 n∑ j=1 |〈tjx,x〉|2 ≤ sup ‖x‖=1  1 n ∣∣∣∣∣ n∑ j=1 〈tjx,x〉 ∣∣∣∣∣ 2 + [ |γ + γ|− 2 √ (γγ) ]∣∣∣∣∣ n∑ j=1 〈tjx,x〉 ∣∣∣∣∣   ≤ 1 n sup ‖x‖=1 ∣∣∣∣∣ 〈 n∑ j=1 tjx,x 〉∣∣∣∣∣ 2 + [ |γ + γ|− 2 √ (γγ) ] sup ‖x‖=1 ∣∣∣∣∣ 〈 n∑ j=1 tjx,x 〉∣∣∣∣∣ = 1 n w2 ( n∑ j=1 tj ) + [ |γ + γ|− 2 √ (γγ) ] w ( n∑ j=1 tj ) , which proves (3.29). 226 s.s. dragomir (iii) by the inequality (3.23) we have( n∑ j=1 |〈tjx,x〉|2 )1 2 ≤ 1 √ n  ∣∣∣∣∣ n∑ j=1 〈tjx,x〉 ∣∣∣∣∣ + 14 |γ −γ| 2 |γ + γ|   = 1 √ n  ∣∣∣∣∣ 〈 n∑ j=1 tjx,x 〉∣∣∣∣∣ + 14 |γ −γ| 2 |γ + γ|   for any x ∈ h with ‖x‖ = 1. by taking the supremum over ‖x‖= 1 in this inequality, we get (3.30). remark 17. by the use of the elementary inequality w (t) ≤ ‖t‖ that holds for any t ∈ b(h), a sufficient condition for (3.26) to hold is that∥∥∥∥tj − γ + γ2 ∥∥∥∥ ≤ 12 |γ −γ| for any j ∈{1, . . . ,n}. (3.33) 4. inequalities for δp and ϑp norms for t ∈ b(h) and p ≥ 1 we can consider the functionals δp(t) := sup ‖x‖=‖y‖=1 ( |〈tx,y〉|p + |〈t∗x,y〉|p )1/p = ‖(t,t∗)‖h,2,p (4.1) and ϑp(t) := sup ‖x‖=1 ( ‖tx‖p + ‖t∗x‖p )1/p = ‖(t,t∗)‖s,2,p . (4.2) it is easy to see that both δp and ϑp are norms on b(h). the case p = 2 for the norm δ := δ2 was considered and studied in [5]. observe that, for any t ∈ b(h) and p ≥ 1, we have wh,2,p((t,t ∗)) = sup ‖x‖=1 ( |〈tx,x〉|p + |〈t∗x,x〉|p )1/p = sup ‖x‖=1 ( |〈tx,x〉|p + |〈tx,x〉|p )1/p = 21/p sup ‖x‖=1 |〈tx,x〉| = 21/pw(t). (4.3) using the inequality (1.13) we have 21/pw(t) ≤ δp(t) ≤ 21+1/pw(t) (4.4) hypo-q-norms on cartesian products 227 for any t ∈ b(h) and p ≥ 1. for p = 2, we get √ 2w(t) ≤ δ(t) ≤ √ 8w(t) (4.5) while for p = 1 we get 2w(t) ≤ δ1(t) ≤ 4w(t) (4.6) for any t ∈ b(h). we have for any t ∈ b(h) and p ≥ 1 that ‖(t,t∗)‖2,p = ( ‖t‖p + ‖t∗‖p )1/p = 21/p‖t‖ and by (2.25) we get ‖t‖≤ δp(t) ≤ 21/p‖t‖ (4.7) for any t ∈ b(h) and p ≥ 1. for p = 2, we get ‖t‖≤ δ(t) ≤ √ 2‖t‖ (4.8) while for p = 1 we get ‖t‖≤ δ1 (t) ≤ 2‖t‖ (4.9) for any t ∈ b(h). from (2.32) we get for r ≥ q ≥ 1 that δr(t) ≤ δq(t) ≤ 2 r−q rq δr(t) (4.10) for any t ∈ b(h). for any t ∈ b(h) and p,q > 1 with 1 p + 1 q = 1, then by (2.43) we have δq(t) ≥ 1 21/p ‖t + t∗‖. (4.11) in particular, for p = q = 2 we get δ(t) ≥ √ 2 2 ‖t + t∗‖, (4.12) for any t ∈ b(h). by using the inequality (2.45) we get δp(t) ≤ ϑp(t) ≤ 21/p‖t‖ (4.13) 228 s.s. dragomir for any t ∈ b(h) and p ≥ 1. for p = 1 we get δ1(t) ≤ ϑ1(t) ≤ 2‖t‖ (4.14) for any t ∈ b(h). for p ≥ 2, by employing the equality (2.46) we get ϑp(t) = [ wh,2,p/2 ( |t|2 , |t∗|2 )]1/2 = [ 22/pw ( |t |2 )]1/2 = 21/p‖t‖ (4.15) for any t ∈ b(h). on utilising (3.7), (3.9) and (3.11) we get 0 ≤ δ2(t) − 1 2 δ21(t) ≤ 1 2 ‖t‖2, (4.16) 0 ≤ δ2(t) − 1 2 δ21(t) ≤‖t‖δ1(t) (4.17) and 0 ≤ δ(t) − 1 √ 2 δ1(t) ≤ √ 2 4 ‖t‖ (4.18) for any t ∈ b(h). observe, by (4.3) we have that wh,2,e((t,t ∗)) = √ 2w(t), for any t ∈ b(h). assume that t ∈ b(h) and γ, γ ∈ c with γ 6= γ such that w ( t − γ + γ 2 i ) , w ( t∗ − γ + γ 2 i ) ≤ 1 2 |γ −γ|, (4.19) then by (3.27) we get w2(t) ≤‖re (t)‖2 + 1 4 |γ −γ|2, (4.20) where re (t) := t+t ∗ 2 . if re (γγ) > 0, then by (3.28) and (3.29) w(t) ≤ 1 2 |γ + γ|√ re (γγ) ‖re (t)‖ (4.21) hypo-q-norms on cartesian products 229 and w2(t) ≤ [ ‖re (t)‖ + [ |γ + γ|− 2 √ (γγ) ]] ‖re (t)‖. (4.22) if γ 6= −γ, then by (3.30) we get w(t) ≤‖re (t)‖ + 1 8 |γ −γ|2 |γ + γ| . (4.23) due to the fact that w(a) = w(a∗) for any a ∈ b(h), the condition (4.19) can be simplified as follows. if m,m are real numbers with m > m and if w ( t − m + m 2 i ) ≤ 1 2 (m −m), then w2(t) ≤‖re (t)‖2 + 1 4 (m −m)2. (4.24) if m > 0, then w(t) ≤ 1 2 m + m √ mm ‖re (t)‖ (4.25) and w2(t) ≤ [ ‖re (t)‖ + (√ m − √ m )2] ‖re (t)‖. (4.26) if m 6= −m, then w(t) ≤‖re (t)‖ + 1 8 (m −m)2 m + m . (4.27) for other numerical radius and norm inequalities, the interested reader may also consult [1] and [6] and compare these results. the details are not provided here. 5. inequalities for real norms if x is a complex linear space, then the functional ‖·‖ is a real norm, if the homogeneity property in the definition of the norms is satisfied only for real numbers, namely we have ‖αx‖ = |α|‖x‖ for any α ∈ r and x ∈ x. 230 s.s. dragomir for instance if we consider the complex linear space of complex numbers c then the functionals |z|p := ( |re (z)|p + |im (z)|p )1/p , p ≥ 1, |z|∞ := max{|re (z)|, |im (z)|}, p = ∞, are real norms on c. for t ∈ b(h) we consider the cartesian decomposition t = re (t) + i im (t) where the selfadjoint operators re (t) and im (t) are uniquely defined by re (t) = t + t∗ 2 and im (t) = t −t∗ 2i . we can introduce the following functionals ‖t‖r,p := ( ‖re (t)‖p + ‖im (t)‖p )1/p , p ≥ 1, and ‖t‖r,∞ := max { ‖re (t)‖,‖im (t)‖ } , p = ∞, where ‖·‖ is the usual operator norm on b(h). the definition can be extended for any other norms on b(h) or its subspaces. using the properties of the norm ‖·‖ and the minkowski’s inequality ( |a + b|p + |c + d|p )1/p ≤ ( |a|p + |c|p )1/p + ( |b|p + |d|p )1/p for p ≥ 1 and a,b,c,d ∈ c, we observe that ‖ · ‖r,p, p ∈ [1,∞] is a real norm on b(h). for p ≥ 1 and t ∈ b we can introduce the following functionals ηr,p(t) : = sup ‖x‖=‖y‖=1 ( |re〈tx,y〉|p + |im〈tx,y〉|p )1/p = sup ‖x‖=‖y‖=1 ( |〈re tx,y〉|p + |〈im tx,y〉|p )1/p = ‖(re t, im t)‖h,2,p , hypo-q-norms on cartesian products 231 θr,p(t) : = sup ‖x‖=1 ( |re〈tx,x〉|p + |im〈tx,x〉|p )1/p = sup ‖x‖=1 ( |〈re tx,x〉|p + |〈im tx,x〉|p )1/p = wh,2,p(re t, im t) and κr,p(t) := sup ‖x‖=1 ( ‖re tx‖p + ‖im tx‖p )1/p = ‖(re t, im t)‖s,2,p . the case p = 2 is of interest since for t ∈ b(h) we have ηr,2(t) : = sup ‖x‖=‖y‖=1 ( |re〈tx,y〉|2 + |im〈tx,y〉|2 )1/2 = sup ‖x‖=‖y‖=1 |〈tx,y〉| = ‖t‖ , θr,2(t) : = sup ‖x‖=1 ( |re〈tx,x〉|2 + |im〈tx,x〉|2 )1/2 = sup ‖x‖=1 |〈tx,x〉| = w(t) and κr,2(t) := sup ‖x‖=1 ( ‖re tx‖2 + ‖im tx‖2 )1/2 = sup ‖x‖=1 (〈 (re t)2x,x 〉 + 〈 (im t)2x,x 〉)1/2 = sup ‖x‖=1 (〈[ (re t) 2 + (im t) 2 ] x,x 〉)1/2 = ∥∥∥(re t)2 + (im t)2∥∥∥1/2 = ∥∥∥∥∥|t| 2 + |t∗|2 2 ∥∥∥∥∥ 1/2 . 232 s.s. dragomir for p = ∞ we have ηr,∞(t) : = sup ‖x‖=‖y‖=1 ( max { |re〈tx,y〉| , |im〈tx,y〉| }) = max { sup ‖x‖=‖y‖=1 |〈re tx,y〉| , sup ‖x‖=‖y‖=1 |〈im tx,y〉| } = max { ‖re t‖,‖im t‖ } , and in a similar way θr,∞(t) = κr,∞(t) = max { ‖re t‖,‖im t‖ } = ‖t‖r,∞ . the functionals ηr,p, θr,p and κr,p with p ∈ [1,∞] are real norms on b(h). we have ηr,p (t) = sup ‖x‖=‖y‖=1 ( |re〈tx,y〉|p + |im〈tx,y〉|p )1/p ≤ ( sup ‖x‖=‖y‖=1 |re〈tx,y〉|p + sup ‖x‖=‖y‖=1 |im〈tx,y〉|p )1/p = (‖re (t)‖p + ‖im (t)‖p)1/p = ‖t‖r,p and ‖t‖r,∞ = sup ‖x‖=‖y‖=1 ( max{|re〈tx,y〉|, |im〈tx,y〉|} ) ≤ sup ‖x‖=‖y‖=1 ( |re〈tx,y〉|p + |im〈tx,y〉|p )1/p = ηr,p(t) for any p ≥ 1 and t ∈ b(h). in a similar way we have ‖t‖r,∞ ≤ θr,p(t) ≤‖t‖r,p and ‖t‖r,∞ ≤ κr,p(t) ≤‖t‖r,p for any p ≥ 1 and t ∈ b(h). hypo-q-norms on cartesian products 233 if we write the inequality (1.13) for n = 2, t1 = re t and t2 = im t then we get θr,p(t) ≤ ηr,p(t) ≤ 2θr,p(t) (5.1) for any p ≥ 1 and t ∈ b(h). using the inequalities (2.25) and (2.26) for n = 2, t1 = re t and t2 = im t then we get 1 21/p ‖t‖r,p ≤ ηr,p(t) ≤‖t‖r,p (5.2) and 1 21/p ‖t‖r,p ≤ θr,p(t) ≤‖t‖r,p (5.3) for any p ≥ 1 and t ∈ b(h). if we use the inequalities (2.32) and (2.33) for n = 2, t1 = re t and t2 = im t then we get for t ≥ p ≥ 1 that ηr,t(t) ≤ ηr,p(t) ≤ 2 t−p tp ηr,t(t) (5.4) and θr,t(t) ≤ θr,p(t) ≤ 2 t−p tp θr,t(t) (5.5) for any t ∈ b(h). for p = 1 we have the functionals ηr,1(t) = sup ‖x‖=‖y‖=1 ( |〈re tx,y〉| + |〈im tx,y〉| ) = ‖(re t, im t)‖h,2,1 , θr,1(t) : = sup ‖x‖=1 ( |〈re tx,x〉| + |〈im tx,x〉| ) = wh,2,1(re t, im t) and κr,1(t) : = sup ‖x‖=1 ( ‖re tx‖ + ‖im tx‖ ) = ‖(re t, im t)‖s,2,1 . by utilising the inequalities (3.7), (3.9) and (3.11) for n = 2, t1 = re t and t2 = im t, then 0 ≤‖t‖2 − 1 2 η2r,1(t) ≤ 1 2 ( max{‖re t‖,‖im t‖} )2 , (5.6) 0 ≤‖t‖2 − 1 2 η2r,1(t) ≤ max { ‖re t‖,‖im t‖ } ηr,1(t) (5.7) 234 s.s. dragomir and 0 ≤‖t‖− √ 2 2 ηr,1(t) ≤ √ 2 4 max{‖re t‖,‖im t‖} (5.8) for any t ∈ b(h). also, by utilising the inequalities (3.8), (3.10) and (3.12) for n = 2, t1 = re t and t2 = im t, then 0 ≤ w2(t) − 1 2 θ2r,1(t) ≤ 1 2 ( max{‖re t‖ ,‖im t‖} )2 , (5.9) 0 ≤ w2(t) − 1 2 θ2r,1(t) ≤ max { ‖re t‖,‖im t‖ } θr,1(t) (5.10) and 0 ≤ w(t) − √ 2 2 θr,1(t) ≤ √ 2 4 max { ‖re t‖,‖im t‖ } (5.11) for any t ∈ b(h). if m,m are real numbers with m > m and if∥∥∥∥ re t − m + m2 i ∥∥∥∥, ∥∥∥∥ im t − m + m2 i ∥∥∥∥ ≤ 12 (m −m), (5.12) then by (3.27) we get w2(t) ≤ 1 2 ‖re t + im t‖2 + 1 2 (m −m)2 . (5.13) if m > 0, then (3.28) and (3.29) we have w(t) ≤ 1 2 √ 2 m + m √ mm ‖re t + im t‖ (5.14) and w2(t) ≤ [ 1 2 ‖re t + im t‖ + (√ m − √ m )2 ] ‖re t + im t‖. (5.15) if m 6= −m, then by (3.30) we get w(t) ≤ 1 √ 2 ( ‖re t + im t‖ + 1 4 (m −m)2 m + m ) . (5.16) finally, we observe that a simple sufficient condition for (5.12) to hold, is that mi ≤ re t, im t ≤ mi in the operator order of b(h). hypo-q-norms on cartesian products 235 acknowledgements the author would like to thank the anonymous referee for valuable suggestions that have been implemented in the final version of the paper. references [1] m. bakherad, k. shebrawi, upper bounds for numerical radius inequalities involving off-diagonal operator matrices, ann. funct. anal. 9 (3) (2018), 297 – 309. [2] s.s. dragomir, a counterpart of schwarz’s inequality in inner product spaces, east asian math. j. 20 (1) (2004), 1 – 10, preprint rgmia res. rep. coll. 6 (2003), http://rgmia.org/papers/v6e/csiips.pdf. 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[15] o. shisha, b. mond, bounds on differences of means, in “ inequalities ”, academic press inc., new york, 1967, 293 – 308. http://rgmia.org/papers/v6e/csiips.pdf https://www.emis.de/journals/jipam/article301.html?sid=301 https://www.emis.de/journals/jipam/article854.html?sid=854 introduction representation results some reverse inequalities inequalities for p and p norms inequalities for real norms e extracta mathematicae vol. 33, núm. 2, 149 – 165 (2018) three-operator problems in banach spaces jesús m.f. castillo, muneo chō, manuel gonzález ∗ departamento de matemáticas, universidad de extremadura, e-06006 badajoz, spain castillo@unex.es department of mathematics, kanagawa university, hiratsuka 259 − 1293, japan chiyom 01@kanagawa-u.ac.jp departamento de matemáticas, universidad de cantabria, e-39071 santander, spain manuel.gonzalez@unican.es received may 31, 2018 abstract: we study the analogue of 3-space problems for classes of operators acting on banach spaces. we show examples of classes of operators having or failing the 3-operator property, and give several methods to obtain classes with this property. key words: three-space property, extending operators, lifting operators, semigroup, operator ideal. ams subject class. (2000): 46b03, 46b08, 46b10. 1. introduction the 3-space problem for a class or property p of banach spaces is the question of whether a banach space x has p provided that a certain subspace y and the corresponding quotient space x/y have p. if so, it is then said that p is a 3-space property. for instance, reflexivity, separability or having density character ℵ are 3-space properties. three-space problems have been a popular topic of research because a positive answer for p yields a technique to get spaces with p; while a negative answer necessarily provides a new insight into banach space constructions. moreover, to get either positive results or counterexamples more often than not requires a blend of different techniques. the monograph [12] contains a thorough, not-too-outdated, treatment of the 3-space problem in banach spaces. ∗the research of the first author was supported in part by project ib16056 de la junta de extremadura; that of the first and third authors was supported in part by mineco (spain), project mtm2016-76958. this paper benefited from a stay in 2016 of castillo and gonzález at kanagawa university invited by prof. cho. 149 150 j.m.f. castillo, m. chō, m. gonzález in this paper we consider the analogue of 3-space problem for operators by means of what we will call the 3-operator property (see definition 1), which was introduced in [29]. we also consider some weak versions of the 3-operator property that were introduced in [13]. some of them can be enjoyed by an operator ideal a and maintain close connections with the fact that the space ideal of a enjoys the 3-space property. we first observe that no (nontrivial) operator ideal can enjoy the 3-operator property, and so we turn our attention to other classes, most remarkably semigroups. our main result (theorem 1) gives a characterization of the semigroups that satisfy the 3-operator property. as a consequence we show in corollary 1 that several semigroups considered in [13] have the 3-operator property. we also describe some categorical methods to obtain new classes with the 3-operator property from a class that have that property. finally we consider some operator ideals related with the separably injective spaces studied in [5] that provide examples satisfying or failing some 3-operator-like properties considered in the paper. 2. three-operator properties a class a of operators is called an operator ideal if it contains the class f of finite-rank operators, is closed under addition, and the composition of an element of a with any operator is in a. typical examples of operator ideals are the classes l of all operators, k of compact operators and w of weakly compact operators. a class s of operators is a semigroup if it contains the bijective operators, it is stable under composition, and given s ∈ l(u, x) and t ∈ l(v, y ), s, t ∈ s ⇐⇒ s ⊕ t ∈ s, where s ⊕ t : u ⊕ v → x ⊕ y is defined by s ⊕ t (u, v) = (su, tv). the notion of operator ideal was thoroughly studied by pietsch [24] (see [25] for details) while the notion of semigroup was considered in [1] and [18, chapter 6]. an operator ideal a has associated a class of banach spaces sp(a) = {x : idx ∈ a}, where idx is the identity in x, which is called the space ideal of a [24]. space ideals are, according to [24], classes of banach spaces containing the finite dimensional spaces and stable under products and complemented subspaces. observe that x ∈ sp(a) if and only if l(x, x) = a(x, x). similarly, a semigroup s has associated a class of spaces ker(s) = {x : 0x ∈ s}, and x ∈ ker(s) if and only if l(x) = s(x). we will see in proposition 1 that, for an operator ideal a, sp(a) coincides with the kernel of the semithree-operator problems in banach spaces 151 groups a+ and a− associated to a. given a class of operators a, we denote ad = {t ∈ l : t ∗ ∈ a}, the dual class of a. note that if a is an operator ideal or a semigroup then so does ad [24, theorem 4.4.2], [18, proposition 6.1.4]. the homological notation will be useful in this paper: recall that an exact sequence 0 −−−−→ y i−−−−→ x q −−−−→ z −−−−→ 0 (2.1) of banach spaces and (linear continuous) operators is a diagram in which the kernel of each arrow coincides with the image of the preceding one. the open mapping theorem yields that y is isomorphic to a subspace of x such that x/i(y ) is isomorphic to z. consequently, p is a 3-space property if whenever one has a short exact sequence (2.1) in which the spaces y, z have p then also x has p. still according to [24], given a space ideal a, the class of all operators that factorize through a space in a form an operator ideal op(a). thus, if the spaces with p form a space ideal (or even if not with some ad hoc amendments), p is a 3-space property if given a commutative diagram 0 −−−−→ y −−−−→ x −−−−→ z −−−−→ 0 idy y idxy yidz 0 −−−−→ y −−−−→ x −−−−→ z −−−−→ 0 (2.2) with exact rows, if idy , idz ∈ op(p) then also idx ∈ op(p). consequently, the following concept makes sense. definition 1. a class a of operators is said to have the 3-operator property if given a commutative diagram with exact rows 0 −−−−→ y −−−−→ x −−−−→ z −−−−→ 0 α y βy yγ 0 −−−−→ y ′ −−−−→ x′ −−−−→ z′ −−−−→ 0 (2.3) if α, γ ∈ a then also β ∈ a. this notion was introduced by zeng and zhong in [29], where they prove that the classes of upper and lower semi-fredholm operators satisfy it (a new proof will be given in corollary 3), and zeng proved in [28] that some classes of operators defined in terms of spectral properties satisfy the 3-operator property. 152 j.m.f. castillo, m. chō, m. gonzález unfortunately, l is the only operator ideal enjoying the 3-operator property: the diagram 0 −−−−→ 0 0−−−−→ x id−−−−→ x −−−−→ 0 0 y yid y0 0 −−−−→ x −−−−→ id x −−−−→ 0 0 −−−−→ 0 shows that an operator ideal satisfying the 3-operator property contains the identity of every banach space. nevertheless, classes of operators with the 3-operator property do exist: example 1. the classical 3-lemma from homological algebra [12, p. 3] shows that the following classes of operators have the 3-space property: (a) the class inj of injective operators. (b) the class dens of dense range operators. (c) the class emb of (into) embedding operators. (d) the class surj of surjective operators. (e) the class iso of bijective operators. two “3-operator-like” properties were introduced in [13, definitions 2 and 5 and propositions 10 and 16]. definition 2. let a be a class of operators. (a) we say that a satisfies the 3s− property if given a push-out diagram 0 −−−−→ y j −−−−→ x q −−−−→ z −−−−→ 0 α y βy ∥∥∥ 0 −−−−→ y ′ j ′ −−−−→ po q ′ −−−−→ z −−−−→ 0 (2.4) with exact rows then α, q ∈ a ⇒ β ∈ a. (b) we say that a satisfies the 3s+ property if given a pull-back diagram 0 −−−−→ y j ′ −−−−→ pb q ′ −−−−→ z′ −−−−→ 0∥∥∥ βy yγ 0 −−−−→ y j −−−−→ x q −−−−→ z −−−−→ 0 (2.5) with exact rows then γ, j ∈ a ⇒ β ∈ a. three-operator problems in banach spaces 153 while l is the only operator ideal that satisfies the 3-operator property, some operator ideals satisfy the 3s− and 3s+ properties: (a) the operator ideals of strictly singular and strictly cosingular operators enjoy respectively properties 3s− and 3s+. this fact is essentially proved in [14, proposition 3.2], and explicitly in [10, lemma 8] and [13, proposition 11]. (b) the operator ideal of p-converging operators cp studied in [11] satisfies the 3s− and 3s+ properties for 1 ≤ p ≤ ∞ [13, propositions 13 and 17]. note that c∞ = c, the completely continuous operators, and c1 = u, the unconditionally converging operators. (c) let k w, r, u, c and wc denote the operator ideals of compact, weakly compact, rosenthal, unconditionally convergent, completely continuous, and weakly completely continuous operators (see [24] for their definitions), and let a be one of these operator ideals. then a satisfies the 3s+ property and its dual ad satisfies the 3s− property [13, propositions 15 and 9]. the following result will be useful later: lemma 1. if a has the property 3s− then ad has the property 3s+; and similarly, if a has the property 3s+ then ad has the property 3s−. proof. for the proof of the first result, it is enough to observe that 0 −−−−→ z∗ q∗ −−−−→ x∗ j∗ −−−−→ y ∗ −−−−→ 0 γ∗ y β∗y ∥∥∥ 0 −−−−→ z′∗ −−−−→ pb∗ −−−−→ y ∗ −−−−→ 0 which is the conjugate of diagram (2.5), is also a push-out diagram like (2.4). indeed, pb is the pull-back of γ and q, and pb∗ can be identified with the push-out of γ∗ and q∗. we refer to [13, section 2] for the details. the proof of the second result is similar. those properties are interesting for the study of 3-space problems. indeed, as it was shown in [13, proposition 18], if an operator ideal a satisfies one of the properties 3s− or 3s+ then the space ideal sp(a) satisfies the 3-space property. 154 j.m.f. castillo, m. chō, m. gonzález for applications of the 3s+ and 3s− properties to the analysis of commutative diagrams of operators we refer to [13, propositions 14 and 19] and [14, proposition 3.3]. 3. semigroups of operators the classes of operators appearing in example 1 satisfy the definition of semigroup presented at the beginning of section 2. this notion of semigroup if closely related with that of operator ideal. it was proved in [1] and [18, chapter 6] that every operator ideal a has associated two semigroups a+ and a− defined as follows. definition 3. let a be an operator ideal and let t ∈ l(x, y ). (a) t ∈ a+ if for every s ∈ l(z, x), ts ∈ a implies s ∈ a. (b) t ∈ a− if for every s ∈ l(y, z), st ∈ a implies s ∈ a. as a direct consequence of the definitions we obtain the following equalities. proposition 1. for every operator ideal a, sp(a) = ker(a+) = ker(a−). the following two 3-operator-like properties for semigroups were introduced in [13, definition 1] in a slightly different way. definition 4. let a be a class of operators. we say that: (a) a satisfies the 3 po property if given a push-out diagram like (2.4), if α ∈ a then also β ∈ a. (b) a satisfies the 3 pb property if given a pull-back diagram like (2.5), if γ ∈ a then also β ∈ a. as in the case of the 3-operator property, an operator ideal satisfying the 3 po property or the the 3 pb property contains the identity of every banach space, hence it is l. for the 3 po property, note that we can construct diagrams like (2.4) with z arbitrary; and we can similarly argue for the 3 pb property. three-operator problems in banach spaces 155 proposition 2. let a be an operator ideal. (a) if a is injective then a+ satisfies the 3 pb property. (b) if a is surjective then a− satisfies the 3 po property. (c) if a+ satisfies the 3 po property or a− satisfies the 3 pb property then sp(a) has the 3-space property. proof. (a) suppose a is injective and γ in diagram (2.5) belongs to a+. we have to show that β ∈ a+. let s : w → pb such that βs ∈ a. from γ ∈ a+ and qβs = γq′s ∈ a, we get q′s ∈ a. the map j : pb → x ⊕ z′ given by jv = (βv, q′v) is an (into) embedding, and βs, q′s ∈ a implies js ∈ a; hence s ∈ a because a is injective. (b) suppose a is surjective and α in diagram (2.4) belongs to ∈ a−. we have to show that β ∈ a−. let s : po → w such that sβ ∈ a. from α ∈ a− and sβj = sj ′α ∈ a, we get sj ′ ∈ a. the map q : y ′ ⊕ x → po given by q(y ′, x) = j ′y ′ + βx is surjective, and sβ, sj ′ ∈ a implies sq ∈ a; hence s ∈ a because a is surjective. (c) see [13, proposition 7]. clearly the 3 po property implies the 3s+, the 3 pb property implies the 3s− property, and lemma 1 has its counterpart admitting a similar proof: lemma 2. if a has the 3 pb property then ad has the 3 po property; and similarly, if a has the 3 po property then ad has the 3 pb property. the following result is useful to understand the 3-operator property. theorem 1. let a be a class of operators stable by composition and containing the bijective operators. then a has the 3-operator property if and only if it has properties 3 po and 3 pb. proof. the direct implication is trivial, because each identity is in a. for the converse implication, given the commutative diagram in definition 1 0 −−−−→ y j −−−−→ x q −−−−→ z −−−−→ 0 α y βy yγ 0 −−−−→ y ′ −−−−→ j ′ x′ −−−−→ q ′ z′ −−−−→ 0 (3.1) 156 j.m.f. castillo, m. chō, m. gonzález we consider the push-out diagram of α and j: 0 −−−−→ y j −−−−→ x q −−−−→ z −−−−→ 0 α y αy ∥∥∥ 0 −−−−→ y ′ −−−−→ j po −−−−→ q z −−−−→ 0 where po = (y ′ × x)/∆ with ∆ = {(αy, −jy) : y ∈ y }, jy ′ = (y ′, 0) + ∆, αx = (0, x) + ∆ and q ( (y ′, 0) + ∆ ) = qx. moreover we consider the pull-back diagram of γ and q′: 0 −−−−→ y ′ j ′ −−−−→ pb q ′ −−−−→ z −−−−→ 0∥∥∥ γy γy 0 −−−−→ y ′ −−−−→ j ′ x′ −−−−→ q ′ z′ −−−−→ 0 where pb = { (x ′, z) ∈ x′ × z : q′x ′ = γz } , j ′y′ = (j ′y ′, 0), γ(x ′, z) = x ′ and q′(x ′, z) = z. let us show that the map ψ : po → pb defined by ψ ( (y ′, x) + ∆ ) = (j ′y ′ + βx, qx) is a bijective operator such that β = γ ψ α. we do it in several steps: (1) ψ takes values in pb: q′(j ′y ′ + βx) = q′βx = γqx. (2) ψ is well-defined since (y ′, x) ∈ ∆ implies y ′ = αy and x = −jy for some y ∈ y . then j ′y ′ + βx = j′αy − βjy = 0 and qx = −qjy = 0. (3) ψ is bijective because the diagram 0 −−−−→ y ′ j −−−−→ po q −−−−→ z −−−−→ 0∥∥∥ yψ ∥∥∥ 0 −−−−→ y ′ −−−−→ j ′ pb −−−−→ q ′ z −−−−→ 0 three-operator problems in banach spaces 157 is commutative. indeed, q′ψ ( (y ′, x) + ∆ ) = q′(j ′y ′ + βx, qx) = qx = q ( (y ′, x) + ∆ ) and ψjy ′ = ψ ( (y ′, 0) + ∆ ) = (j ′y ′, 0) = j ′y ′. (4) for every x ∈ x, γ ψ α x = γ ψ ( (0, x) + ∆ ) = γ(βx, qx) = βx. to conclude the proof, since a has properties 3 po and 3 pb, α ∈ a implies α ∈ a and γ ∈ a implies γ ∈ a, hence β = γ ψ α ∈ a. several examples of semigroups satisfying the 3 po or the 3 pb property were given in [13]. proposition 3. ([13, theorem 1]). let a denote one of the operator ideals k w, r, u, c or wc. then the semigroup a+ satisfies the 3 po property and the semigroup (ad)− satisfies the 3 pb property. note that k = kd by schauder’s theorem and w = wd by gantmacher’s theorem. as a consequence, corollary 1. let a denote one of the operator ideals k w, r, u, c or wc. then the semigroups a+ and (ad)− satisfy the 3-operator property. proof. the operator ideal a is injective, hence the semigroup a+ satisfies the 3 pb property by proposition 2. since a+ also satisfies the 3 po property (proposition 3), it has the 3-operator property by theorem 1. similarly, since ad is surjective and (ad)− satisfies the 3 pb property, (ad)− has the 3-operator property. since cp is injective (hence cdp is surjective), (cp)+ satisfies the 3 pb property and (cdp )− the 3 po property. however the following questions remain open: (a) do the semigroups (cp)+ and (cdp )− satisfy the the 3-operator property? (b) does (cp)+ satisfy the 3 po property, or (cdp )− satisfy the 3 pb property? for the obtention of lifting result for sequences when the quotient map belongs to one of the semigroups w+, r+, c+ or wc+ we refer to [19]. 158 j.m.f. castillo, m. chō, m. gonzález 4. categorical methods to obtain 3-operator classes a functor f acting on the category of banach spaces is said to be exact if it transforms exact sequences into exact sequences. examples of exact functors can be found in [12, section 2.2] and include the following ones: • the duality functor, f(x) = x∗ and f(t ) = t ∗ (as well as the biduality functor, α-transfinite dual, etc). • the residual functor, f(x) = x∗∗/x and f(t ) = t ∗∗/t , where if t : x → y then t ∗∗/t : x∗∗/x → y ∗∗/y is the operator induced by t ∗∗. • the ultrapower functor, f(x) = xu and f(t ) = tu, where u is a non-trivial ultrafilter. • the ultra-residual functor, f(x) = xu/x and f(t ) = tu/t . • the c(k, ·) functor, k a compact space: c(k, x) is the banach space of all continuous functions f : k → x, and for each t : x → y , the operator c(k, t ) : c(k, x) → c(k, y ) is defined by c(k, t )(f)(k) = t (f(k)). one obviously has: proposition 4. let a be a class of operators with the 3-operator property and let f be an exact functor. then the class f−1(a) = {t : f(t ) ∈ a} has the 3-operator property. let us show that nontrivial results can be obtained with this method. it was proved in [20] that w+ coincides with the class of tauberian operators introduced in [22] and w− coincides with the class of cotauberian operators introduced in [27]. since (see [18]), w+ = {t : t ∗∗/t ∈ inj} and w− = {t : t ∗∗/t ∈ dens} one has corollary 2. the semigroups w+ of tauberian and w− of cotauberian operators have the 3-operator property. three-operator problems in banach spaces 159 it was proved in [3] that (w+)dd = {t : t ∗∗ ∈ w+} is properly contained in w+. it follows from proposition 4 and corollary 2 that (w+)dd has the 3-operator property. in [21] (see also [4]) the operators s that can be represented as t ∗∗/t for some operator t are studied. recall that many banach spaces (such as separable of weakly compactly generated [8]) are linearly isometric to some x∗∗/x. the operators t such that t ∗∗/t ∈ emb are called strongly tauberian in [26]. in a similar way as in the case of tauberian operators, it can be shown that the class of strongly tauberian operators has the 3-operator property. the semigroup k+ coincides with the class of upper semi-fredholm operators (operators with closed range and finite dimensional kernel) while k− coincides with the class of lower semi-fredholm operators (operators with closed and finite codimensional range) (see [17]). since k+ = {t : tu/t ∈ inj} = {t : tu/t ∈ emb} k− = {t : tu/t ∈ dens} = {t : tu/t ∈ surj} one has corollary 3. the semigroups k+ and k− have the 3-operator property. as in corollaries 2 and 3, it is possible to show other classes with the 3-operator property by applying exact functors to known classes that satisfy that property. it would be interesting to identify some of them with known classes of operators. 5. separable injectivity revisited the density character of a banach space x, dens(x), is the smallest cardinal ℵ for which x has a subset of cardinality ℵ spanning a dense subspace. definition 5. let ℵ be a cardinal. a banach space x is said to be ℵ-injective if every operator t : y → x admits an extension t : e → x to a superspace e ⊃ y whenever dens(e) < ℵ. the space x is said to be universally ℵ-injective if every operator t : y → x admits an extension t : e → x to a superspace e ⊃ y whenever dens(y ) < ℵ . 160 j.m.f. castillo, m. chō, m. gonzález the separably injective and universally separably injective banach spaces (corresponding to the choice ℵ = ℵ1) have been recently studied in the monograph [5]. we present now an operator approach to those ideas. definition 6. let ℵ be a cardinal and let t : x → y be an operator. (i) t ∈ e0(ℵ) if for every super-space e ⊃ x with dens(e/x) < ℵ there exists an extension t̂ : e → y . (ii) t ∈ e(ℵ) if for every subspace m ⊂ x with dens(m) < ℵ and every superspace e ⊃ m, the restriction t|m : m → y admits an extension t̂|m : e → y . proposition 5. the classes e0(ℵ) and e(ℵ) are operator ideals. proof. it is obvious that e0(ℵ) + e0(ℵ) ⊂ e0(ℵ), e(ℵ) + e(ℵ) ⊂ e(ℵ), and both classes contain the finite rank operators. we prove that given t ∈ e0(ℵ)(x, x′), r ∈ l(x′, y ′) and s ∈ l(y, x) one gets rts ∈ e0(ℵ)(y, y ′). let e be a superspace of y such that dens (e/y ) < ℵ and consider the diagram 0 −−−−→ y −−−−→ e −−−−→ e/y −−−−→ 0 s y ys ∥∥∥ 0 −−−−→ x −−−−→ po −−−−→ e/y −−−−→ 0 t y x′ r y x′ since t ∈ e0, it admits an extension t : po → x′, thus the operator rts admits the extension rts. the proof for e(ℵ) is analogous. our interest in these uncommon operator ideals appears explained in the next result. proposition 6. let ℵ be a cardinal. (a) x ∈ sp(e0(ℵ)) if and only if x is ℵ-injective. (b) x ∈ sp(e(ℵ)) if and only if x is universally ℵ-injective. three-operator problems in banach spaces 161 proof. it is clear that idx ∈ e0(ℵ) if and only if x is complemented in every superspace e so that dens (e/x) < ℵ; namely, x is ℵ-injective. it is also clear that if x is universally ℵ-injective then idx ∈ e(ℵ). to prove the converse, let y be a banach space with dens(y ) < ℵ which is a subspace of a space e, and let t : y → x be an operator. consider the diagram 0 −−−−→ y −−−−→ e −−−−→ e/y −−−−→ 0 t y yt′ ∥∥∥ 0 −−−−→ t(y ) −−−−→ po −−−−→ e/y −−−−→ 0 i y x since dens(t(y )) ≤ dens(y ), the canonical inclusion i can be extended to po, therefore t can be extended to e, which shows that x is universally ℵ-injective. it is therefore clear that e0(ℵ) (resp. e(ℵ)) are non-injective operator ideals containing (although probably different from) the class of all operators that factorize through an ℵ-injective (resp. universally ℵ-injective) space. they are likely not to be surjective either. more ad hoc versions of these operator ideals have been introduced and studied by domański [16]: he fixes a banach space z and considers the ideas ez of those operators t : x → y such that for every super-space e ⊃ x with e/x ≃ z there exists an extension t̂ : e → y . therefore, it will turn out (see [5] for details) that e0(ℵ) = ∪ z ez when the intersection runs over all spaces z with density character < ℵ. one has proposition 7. the class e0(ℵ) satisfies the 3s− property. proof. we consider a push-out diagram 0 −−−−→ y −−−−→ x q −−−−→ z −−−−→ 0 α y βy ∥∥∥ 0 −−−−→ y ′ −−−−→ po −−−−→ z −−−−→ 0 with q, α ∈ e0(ℵ), and we have to show that β ∈ e0(ℵ). let x′ be a superspace of x so that dens (x′/x) < ℵ. our goal is to extend β to an operator b : x′ → po. 162 j.m.f. castillo, m. chō, m. gonzález let q : x′ → z be an extension of q. the commutative diagram 0 0y y 0 −−−−→ y −−−−→ x q −−−−→ z −−−−→ 0y y ∥∥∥ 0 −−−−→ ker q −−−−→ x′ q −−−−→ z −−−−→ 0y y ker q/y x′/xy y 0 0 shows that ker q is a superspace of y with dens(ker q/y ) < ℵ. let a : ker q → y ′ be an extension of α. given x ′ ∈ x′ pick x ∈ x so that qx ′ = qx and define b : x′ → po by means of b(x ′) = ( a(x ′ − x), x ) + ∆. to check it is well defined, observe that if q(x ′) = q(x) = q(w) then( a(x ′ − x), x ) − ( a(x ′ − w), w ) = (a(w − x), x − w) ∈ ∆ since x − w ∈ y . the operator b is continuous since ∥bx ′∥ = inf y∈y ∥∥(a(x ′ − x), x) − (ay, −y)∥∥ = inf y∈y ∥∥(a(x ′ − x − y), x + y)∥∥ ≤ inf y∈y ∥∥a(x ′ − x − y)∥∥ + ∥x + y∥ ≤ ∥ax ′∥ + 2∥qx∥ ≤ ∥a∥∥x′∥ + 2∥qx ′∥ ≤ ( ∥a∥ + 2∥q∥ ) ∥x ′∥. finally, b is an extension of β since when x ∈ x one has b(x) = ( a(x − x), x ) + ∆ = (0, x) + ∆ = β(x). three-operator problems in banach spaces 163 we do not know however if the operator ideal e0 satisfies the 3s+. thus, by the result [13, proposition 18] above mentioned, we get that the class sp(e0(ℵ)) of ℵ-injective banach spaces has the 3-space property. this fact is well-known (see [6, 5]). the homological argument can be found in [9]: every property having the form ext(·, x) = 0 or ext(x, ·) = 0 is a 3-space property. a forerunner can be found in [15]. by lemma 1, the class e0(ℵ)d has property 3s+, from where it follows also that sp(e0(ℵ)d) has the 3-space property, something we already knew since it is a standard fact that if sp(a) has the 3-space property then also sp(ad) has the 3-space property. moreover, to identify the class sp(e0(ℵ)d) is easy and, surprisingly, the class turns out to be independent of ℵ: proposition 8. for every ℵ, sp(e0(ℵ)d) is the class of l1-spaces. proof. recall [5] that ℵ-injective spaces are l∞-spaces and that dual l∞spaces are injective. thus, x∗ is ℵ-injective if and only if it is an l∞-space, which occurs if and only if x is an l1-space. curiously enough, if one defines the apparently dual classes l(ℵ) of operators that can be lifted to any superquotient having kernel with density character strictly lesser than ℵ the identification of sp(l(ℵ)) is not so simple. indeed, observe that a banach space in sp(l(ℵ)) must be ℵ-projective, with the obvious meaning that ext(x, y ) = 0 for every banach space y with dens(y ) < ℵ. it is clear that a separable separably projective must be ℓ1. it is also clear that an ℵ-projective space is such that any ℵ-projective space must be an l1-space with the schur property (see [5] for details). there are however uncountably many non-mutually isomorphic l1-subspaces of ℓ1. it is likely that that the answer to the following question be positive. question 1. is a separably projective space projective? equivalently [24, theorem c.3.8], is a separably projective space isomorphic to ℓ1(i) for some set i? turning to the main topic of this paper, observe that the case of the ideal e(ℵ) is quite different form that of e0(ℵ) since, surprisingly, one has lemma 3. e(ℵ) does not enjoy either 3s− or 3s+. proof. otherwise, the associated space class sp(e0(ℵ)) of universally ℵinjective spaces would enjoy the 3-space property, something that, under ch, has been shown to be false in [7]. 164 j.m.f. castillo, m. chō, m. gonzález references [1] p. aiena, m. gonzález, a. mart́ınez-abejón, operator semigroups in banach space theory, boll. unione mat. ital. sez. b artic. ric. mat. 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[27] d.g. tacon, generalized semi-fredholm transformations, j. austral. math. soc. 34 (1) (1983), 60 – 70. [28] q. zeng, five short lemmas in banach spaces, carpathian j. math. 32 (1) (2016), 131 – 140. [29] q. zeng, h. zhong, three-space theorem for semi-fredholmness, arch. math. (basel) 100 (1) (2013), 55 – 61. e extracta mathematicae vol. 33, núm. 2, 127 – 143 (2018) exposed polynomials of p ( 2r2 h(1 2 ) ) sung guen kim ∗ department of mathematics, kyungpook national university daegu 702 − 701, south korea sgk317@knu.ac.kr presented by ricardo garćıa received february 2, 2018 abstract: we show that every extreme polynomials of p ( 2r2 h( 1 2 ) ) is exposed. key words: the krein-milman theorem, extreme polynomials, exposed polynomials, the plane with a hexagonal norm. ams subject class. (2000): 46a22. 1. introduction according to the krein-milman theorem, every nonempty convex set in a banach space is fully described by the set of its extreme points. let n ∈ n. we write be for the closed unit ball of a real banach space e and the dual space of e is denoted by e∗. we recall that if x ∈ be is said to be an extreme point of be if y, z ∈ be and x = λy + (1 − λ)z for some 0 < λ < 1 implies that x = y = z. x ∈ be is called an exposed point of be if there is an f ∈ e∗ so that f(x) = 1 = ∥f∥ and f(y) < 1 for every y ∈ be \ {x}. it is easy to see that every exposed point of be is an extreme point. we denote by extbe and expbe the sets of extreme and exposed points of be, respectively. we denote by l(ne) the banach space of all continuous n-linear forms on e endowed with the norm ∥t∥ = sup∥xk∥=1 |t(x1, . . . , xn)|. a n-linear form t is symmetric if t(x1, . . . , xn) = t ( xσ(1), . . . , xσ(n) ) for every permutation σ on {1, 2, . . . , n}. we denote by ls(ne) the banach space of all continuous symmetric n-linear forms on e. a mapping p : e → r is a continuous n-homogeneous polynomial if there exists a unique t ∈ ls(ne) such that p(x) = t(x, . . . , x) for every x ∈ e. in this case it is convenient to write t = p̌ . we denote by p(ne) the banach space of all continuous n-homogeneous polynomials from e into r endowed with the norm ∥p∥ = sup∥x∥=1 |p(x)|. note that the spaces l(ne), ∗ this research was supported by the basic science research program through the national research foundation of korea(nrf) funded by the ministry of education, science and technology (2013r1a1a2057788). 127 128 s. g. kim ls(ne), p(ne) are very different from a geometric point of view. in particular, for integral multilinear forms and integral polynomials one has ([2], [9], [42]) extbli(ne) = {ϕ1ϕ2 · · · ϕn : ϕi ∈ extbe∗} , extbpi(ne) = {±ϕ n : ϕ ∈ e∗, ∥ϕ∥ = 1}, where li(ne) and pi(ne) are the spaces of integral n-linear forms and integral n-homogeneous polynomials on e, respectively. for more details about the theory of multilinear mappings and polynomials on a banach space, we refer to [10]. let us say about the stories of the classification problems of extbx and expbx if x = p(ne). choi et al. ([4], [5]) initiated the classification problems and classified extbx if x = p ( 2l2p ) for p = 1, 2, where l2p = r 2 with the lp-norm. b. grecu [14] classified extbx if x = p ( 2l2p ) for 1 < p < 2 or 2 < p < ∞. kim [18] classified expbx if x = p ( 2l2p ) for 1 ≤ p ≤ ∞. kim et al. [34] showed that every extreme 2-homogeneous polynomials on a real separable hilbert space is also exposed. kim ([20], [26]) characterized extbx and expbx for x = p ( 2d∗(1, w) 2 ) , where d∗(1, w) 2 = r2 with the octagonal norm ∥(x, y)∥d∗ = max { |x|, |y|, |x|+|y| 1+w : 0 < w < 1 } . he showed [26] that extbp(2d∗(1,w)2) ̸= expbp(2d∗(1,w)2). in [31], kim classified extbx and using the classification of extbx, kim computed the polarization and unconditional constants of the space x if x = p ( 2r2 h( 1 2 ) ) , where r2 h(w) denotes the space r2 endowed with the hexagonal norm ∥(x, y)∥h(w) := max{|y|, |x| + (1 − w)|y|}. we refer to ([1]–[9], [11]–[43]) and references therein for some recent work about extremal properties of multilinear mappings and homogeneous polynomials on some classical banach spaces. we will denote by t((x1, y1), (x2, y2)) = ax1x2 +by1y2 +c(x1y2 +x2y1) and p(x, y) = ax2 + by2 + cxy a symmetric bilinear form and a 2-homogeneous polynomial on a real banach space of dimension 2, respectively. recently, kim [31] classified the extreme points of the unit ball of p ( 2r2 h( 1 2 ) ) as follows: extb p ( 2r2 h( 1 2 ) ) ={± y2, ±(x2 + 1 4 y2 ± xy ) , ± ( x2 + 3 4 y2 ) , ± [ x2 + ( c2 4 − 1 ) y2 ± cxy ] , ± [ cx2 + ( c+4 √ 1−c 4 − 1 ) y2 ± ( c + 2 √ 1 − c ) xy ] (0 ≤ c ≤ 1) } . exposed polynomials of p ( 2r2 h( 1 2 ) ) 129 in this paper, we show that that every extreme polynomials of p ( 2r2 h( 1 2 ) ) is exposed. 2. results theorem 2.1. ([31]) let p(x, y) = ax2 + by2 + cxy ∈ p ( 2r2 h( 1 2 ) ) with a ≥ 0, c ≥ 0 and a2 + b2 + c2 ̸= 0. then: case 1 : c < a. if a ≤ 4b, then ∥p∥ = max { a, b, ∣∣1 4 a + b ∣∣ + 1 2 c, 4ab−c 2 4a , 4ab−c 2 2c+a+4b , 4ab−c 2 |2c−a−4b| } = max { a, b, ∣∣1 4 a + b ∣∣ + 1 2 c } . if a > 4b, then ∥p∥ = max { a, |b|, ∣∣1 4 a + b ∣∣ + 1 2 c, |c2−4ab| 4a } . case 2 : c ≥ a. if a ≤ 4b, then ∥p∥ = max { a, b, ∣∣1 4 a + b ∣∣ + 1 2 c, |c2−4ab| 2c+a+4b } . if a > 4b, then ∥p∥ = max { a, |b|, ∣∣1 4 a + b ∣∣ + 1 2 c, c 2−4ab 2c−a−4b } . theorem 2.2. ([31]) extb p ( 2r2 h( 1 2 ) ) ={± y2, ±(x2 + 1 4 y2 ± xy ) , ± ( x2 + 3 4 y2 ) , ± [ x2 + ( c2 4 − 1 ) y2 ± cxy ] , ± [ cx2 + ( c+4 √ 1−c 4 − 1 ) y2 ± ( c + 2 √ 1 − c ) xy ] (0 ≤ c ≤ 1) } . theorem 2.3. let f ∈ p ( 2r2 h( 1 2 ) )∗ with α = f(x2), β = f(y2), γ = f(xy). then ∥f∥ = sup { |β|, ∣∣α + 1 4 β ∣∣ + |γ|, ∣∣α + 3 4 β ∣∣, ∣∣α + (c2 4 − 1 ) β ∣∣ + c|γ|,∣∣cα + (c+4√1−c 4 − 1 ) β ∣∣ + (c + 2√1 − c)|γ| (0 ≤ c ≤ 1)}. proof. it follows from theorem 2.2 and the fact that ∥f∥ = sup { |f(p)| : p ∈ extb p ( 2r2 h( 1 2 ) )}. 130 s. g. kim note that if ∥f∥ = 1, then |α| ≤ 1, |β| ≤ 1, |γ| ≤ 1 2 . we are in a position to show the main result of this paper. theorem 2.4. expb p ( 2r2 h( 1 2 ) ) = extb p ( 2r2 h( 1 2 ) ). proof. let (0 ≤ c ≤ 1) p1(x, y) = y 2 , p +2 (x, y) = x 2 + 1 4 y2 + xy , p −2 (x, y) = x 2 + 1 4 y2 − xy , p3(x, y) = x 2 + 3 4 y2 , p +4,c(x, y) = x 2 + ( c2 4 − 1 ) y2 + cxy , p −4,c(x, y) = x 2 + ( c2 4 − 1 ) y2 − cxy , p +5,c(x, y) = cx 2 + ( c+4 √ 1−c 4 − 1 ) y2 + (c + 2 √ 1 − c)xy , p −5,c(x, y) = cx 2 + ( c+4 √ 1−c 4 − 1 ) y2 − (c + 2 √ 1 − c)xy . claim 1: p1 = y 2 ∈ expb p ( 2r2 h( 1 2 ) ). let f ∈ p ( 2r2 h( 1 2 ) )∗ be such that α = 1 5 , β = 1 , γ = 0 . indeed, f(p1) = 1 , |f(p ±2 )| = 9 20 , |f(p3)| = 19 20 . (*) note that for all 0 ≤ c ≤ 1, |f(p ±4,c)| = 4 5 − c2 4 ≤ 4 5 , (**) |f(p ±5,c)| = | √ 1 − c + 9c 20 − 1| ≤ 11 20 . (***) exposed polynomials of p ( 2r2 h( 1 2 ) ) 131 hence, by theorem 2.3, 1 = ∥f∥. we will show that f exposes p1. let q(x, y) = ax2 + by2 + cxy ∈ p ( 2r2 h( 1 2 ) ) such that 1 = ∥q∥ = f(q). we will show that q = p1. since p ( 2r2 h( 1 2 ) ) is a finite dimensional banach space with dimension 3, by the krein-milman theorem, b p ( 2r2 h( 1 2 ) ) is the closed convex hull of extb p ( 2r2 h( 1 2 ) ). then, q(x, y) = up1(x, y) + v +p +2 (x, y) + v −p −2 (x, y) + tp3(x, y) + ∞∑ n=1 λ+n p + 4,c+n (x, y) + ∞∑ n=1 λ−n p − 4,c−n (x, y) + ∞∑ m=1 δ+mp + 5,a+m (x, y) + ∞∑ m=1 δ−mp − 5,a−m (x, y) , for some u, v±, t, λ±n , δ ± m, ∈ r (n, m ∈ n) with 0 ≤ c±n , a±m ≤ 1 and |u| + |v+| + |v−| + |t| + ∞∑ n=1 |λ+n | + ∞∑ n=1 |λ−n | + ∞∑ m=1 |δ+m| + ∞∑ m=1 |δ−m| = 1 . we will show that v± = t = λ±n = δ ± m = 0 for every n, m ∈ n. subclaim: v± = t = 0. assume that v+ ̸= 0. it follows that 1 = f(q) = uf(p1) + v +f(p +2 ) + v −f(p −2 ) + tf(p3) + ∞∑ n=1 λ+n f(p + 4,cn ) + ∞∑ n=1 λ−n f(p − 4,cn ) + ∞∑ m=1 δ+mf(p + 5,am ) + ∞∑ m=1 δ−mf(p − 5,am ) ≤ |u| + |v+||f(p +2 )| + |v −||f(p −2 )| + |t||f(p3)| + ∞∑ n=1 |λ+n ||f(p + 4,cn )| + ∞∑ n=1 |λ−n ||f(p − 4,cn )| + ∞∑ m=1 |δ+m||f(p + 5,am )| + ∞∑ m=1 |δ−m||f(p − 5,am )| ≤ |u| + 9 20 |v+| + 9 20 |v−| + 19 20 |t| + 4 5 ∞∑ n=1 |λ+n | + 4 5 ∞∑ n=1 |λ−n | + 11 20 ∞∑ m=1 |δ+m| + 11 20 ∞∑ m=1 |δ−m| (by (*), (**), (***)) 132 s. g. kim < |u| + |v+| + 9 20 |v−| + 19 20 |t| + 4 5 ∞∑ n=1 |λ+n | + 4 5 ∞∑ n=1 |λ−n | + 11 20 ∞∑ m=1 |δ+m| + 11 20 ∞∑ m=1 |δ−m| ≤ |u| + |v+| + |v−| + |t| + ∞∑ n=1 |λ+n | + ∞∑ n=1 |λ−n | + ∞∑ m=1 |δ+m| + ∞∑ m=1 |δ−m| = 1 , which is impossible. therefore, v+ = 0. using a similar argument as above, we have v− = t = 0. subclaim: λ±n = δ ± m = 0 for every n, m ∈ n. assume that λ+n0 ̸= 0 for some n0 ∈ n. it follows that 1 = f(q) = uf(p1) + λ + n0 f(p +4,cn0 ) + ∑ n∈n,n ̸=n0 λ+n f(p + 4,c+n ) + ∞∑ n=1 λ−n f(p − 4,c−n ) + ∞∑ m=1 δ+mf(p + 5,a+m ) + ∞∑ m=1 δ−mf(p − 5,a−m ) ≤ |u| + |λ+n0||f(p + 4,c+n0 )| + ∑ n∈n,n ̸=n0 |λ+n ||f(p + 4,c+n )| + ∞∑ n=1 |λ−n ||f(p − 4,c−n )| + ∞∑ m=1 |δ+m||f(p + 5,a+m )| + ∞∑ m=1 |δ−m||f(p − 5,a−m )| < |u| + |λ+n0| + 4 5 ∑ n∈n,n̸=n0 |λ+n | + 4 5 ∞∑ n=1 |λ−n | + 11 20 ∞∑ m=1 |δ+m| + 11 20 ∞∑ m=1 |δ−m| ≤ |u| + ∞∑ n=1 |λ+n | + ∞∑ n=1 |λ−n | + ∞∑ m=1 |δ+m| + ∞∑ m=1 |δ−m| = 1 , which is impossible. therefore, λ+n = 0 for every n ∈ n. using a similar argument as above, we have λ−n = δ ± m = 0 for every n, m ∈ n. therefore, q(x, y) = up1(x, y). hence u = 1, so q = p1. therefore, f exposes p1. claim 2: p5,0 = 2xy ∈ expbp ( 2r2 h( 1 2 ) ). let f ∈ p ( 2r2 h( 1 2 ) )∗ be such that α = β = 0 , γ = 1 2 . exposed polynomials of p ( 2r2 h( 1 2 ) ) 133 we will show that f exposes p5,0. indeed, f(p5,0) = 1, f(p1) = 0, f(p ± 2 ) = ±1 2 , f(p3) = 0, − 1 2 ≤ f(p ±4,c) = ± c 2 ≤ 1 2 (0 ≤ c ≤ 1) . note that, for 0 < c ≤ 1, −1 < f(p ±5,c) = ± c + 2 √ 1 − c 2 < 1 . (†) hence, by theorem 2.3, 1 = ∥f∥. let q(x, y) = up1(x, y) + v +p +2 (x, y) + v −p −2 (x, y) + tp3(x, y) + ∞∑ n=1 λ+n p + 4,c+n (x, y) + ∞∑ n=1 λ−n p − 4,c−n (x, y) + ∞∑ m=1 δ+mp + 5,a+m (x, y) + ∞∑ m=1 δ−mp − 5,a−m (x, y) , for some u, v±, t, λ±n , δ ± m, ∈ r (n, m ∈ n) with 0 ≤ c±n , a±m ≤ 1 and |u| + |v+| + |v−| + |t| + ∞∑ n=1 |λ+n | + ∞∑ n=1 |λ−n | + ∞∑ m=1 |δ+m| + ∞∑ m=1 |δ−m| = 1 . we will show that v± = t = λ±n = δ ± m = 0 for every n, m ∈ n. subclaim: v+ = 0. assume that v+ ̸= 0. it follows that 1 = f(q) = v+f(p +2 ) + v −f(p −2 ) + ∞∑ n=1 λ+n f(p + 4,c+n ) + ∞∑ n=1 λ−n f(p − 4,c−n ) + ∞∑ m=1 δ+mf(p + 5,a+m ) + ∞∑ m=1 δ−mf(p − 5,a−m ) < |v+| + 1 2 |v−| + ∞∑ n=1 |λ+n ||f(p + 4,c+n )| + ∞∑ n=1 |λ−n ||f(p − 4,c−n )| + ∞∑ m=1 |δ+m||f(p + 5,a+m )| + ∞∑ m=1 |δ−m||f(p − 5,a−m )| ≤ |v+| + |v−| + ∞∑ n=1 |λ+n | + ∞∑ n=1 |λ−n | + ∞∑ m=1 |δ+m| + ∞∑ m=1 |δ−m| ≤ 1 , 134 s. g. kim which is impossible. therefore, v+ = 0. using a similar argument as claim 1, we have v− = λ±n = 0 for every n ∈ n. hence, q(x, y) = up1(x, y) + tp3(x, y) + ∞∑ m=1 δ+mp + 5,a+m (x, y) + ∞∑ m=1 δ−mp − 5,a−m (x, y) . it follows that 1 = f(q) = ∞∑ m=1 δ+mf(p + 5,a+m ) + ∞∑ m=1 δ−mf(p − 5,a−m ) ≤ ∞∑ m=1 |δ+m||f(p + 5,a+m )| + ∞∑ m=1 |δ−m||f(p − 5,a−m )| ≤ ∞∑ m=1 |δ+m| + ∞∑ m=1 |δ−m| ≤ 1 , which shows that f(p + 5,a+m ) = f(p − 5,a−m ) = ∞∑ m=1 |δ+m| + ∞∑ m=1 |δ−m| = 1 , u = t = 0 for all m ∈ n . by (†), p ± 5,a±m = p5,0 for every m ∈ n and ∑∞ m=1 δ + m+ ∑∞ m=1 δ − m = 1. therefore, q = p5,0. hence, f exposes p5,0. claim 3: p +2 = x 2 + 1 4 y2 + xy ∈ expb p ( 2r2 h( 1 2 ) ). let f ∈ p ( 2r2 h( 1 2 ) )∗ be such that α = 1 2 = β , γ = 3 8 . we will show that f exposes p2. indeed, f(p + 2 ) = 1, f(p − 2 ) = 1 4 , f(p1) = 1 2 , f(p ±3 ) = 7 8 . by some calculation, we have |f(p ±4,c)| ≤ 1 2 , |f(p ±5,c)| ≤ 57 64 for 0 ≤ c ≤ 1 . hence, by theorem 2.3, 1 = ∥f∥. by similar arguments as claims 1 and 2, f exposes p +2 . obviously, p − 2 ∈ expbp ( 2r2 h( 1 2 ) ). claim 4: p +4,0 = x 2 − y2 ∈ expb p ( 2r2 h( 1 2 ) ). exposed polynomials of p ( 2r2 h( 1 2 ) ) 135 let f ∈ p ( 2r2 h( 1 2 ) )∗ be such that α = 1 2 = −β , γ = 0 . we will show that f exposes p4,0. indeed, f(p +4,0) = 1 , |f(p1)| = 1 2 , |f(p ±2 )| = 3 8 , |f(p3)| = 1 8 . note that |f(p ±4,c)| = 1 − c2 8 < 1 for 0 < c ≤ 1 . note that, for 0 ≤ c ≤ 1, |f(p ±5,c)| = 3c + 4 − 4 √ 1 − c 8 ≤ 7 8 . hence, by theorem 2.3, 1 = ∥f∥. by similar arguments as claims 1 and 2, f exposes p +4,0. claim 5: p3 = x 2 + 3 4 y2 ∈ expb p ( 2r2 h( 1 2 ) ). let f ∈ p ( 2r2 h( 1 2 ) )∗ be such that α = 5 8 , β = 1 2 , γ = 0 . we will show that f exposes p3. indeed, f(p3) = 1 , |f(p1)| = 1 2 , |f(p ±2 )| = 3 4 . note that |f(p ±4,c)| ≤ 1 4 , |f(p ±5,c)| ≤ 1 3 for 0 ≤ c ≤ 1 . hence, by theorem 2.3, 1 = ∥f∥. by similar arguments as claims 1 and 2, f exposes p3. claim 6: p +5,1 = x 2 − 3 4 y2 + xy ∈ expb p ( 2r2 h( 1 2 ) ). let f ∈ p ( 2r2 h( 1 2 ) )∗ be such that α = 11 16 , β = − 1 4 , γ = 1 8 . 136 s. g. kim we will show that f exposes p +5,1. indeed, f(p +5,1) = 1 , |f(p1)| = 1 4 , |f(p ±2 )| ≤ 3 4 , |f(p3)| = 1 2 . note that 3 4 ≤ f(p ±4,c) < 1 , − 1 4 ≤ f(p ±5,c) < 1 for 0 ≤ c < 1 . hence, by theorem 2.3, 1 = ∥f∥. by similar arguments as claims 1 and 2, f exposes p +5,1. obviously, p − 5,1 ∈ expbp ( 2r2 h( 1 2 ) ). claim 7: p +4,c = x 2 + (c 2 4 − 1)y2 + cxy ∈ expb p ( 2r2 h( 1 2 ) ) for 0 < c < 1. let f ∈ p ( 2r2 h( 1 2 ) )∗ be such that α = 3 4 − c2 16 , β = − 1 4 , γ = c 8 . indeed, f(p +4,c) = 1 , 3 4 ≤ f(p −4,c) = 1 − c2 4 < 1 , |f(p1)| = 1 4 , 1 2 ≤ f(p ±2 ) ≤ 3 4 , 1 2 ≤ f(p3) < 9 16 . (*) note that for every t ∈ [0, 1] with t ̸= c, f(p +4,t) = − 1 16 t2 + c 8 t + ( 1 − c2 16 ) and f(p −4,t) = − 1 16 t2 − c 8 t + ( 1 − c2 16 ) . hence, we have, for every t ∈ [0, 1] with t ̸= c, 1 < min { 1 − c2 16 , 1 − (1 − c)2 16 } ≤ f(p +4,t) < 1 (**) and −1 < 1 − (1 + c)2 16 ≤ f(p −4,t) ≤ 1 − c2 16 < 1 . exposed polynomials of p ( 2r2 h( 1 2 ) ) 137 note that, for every t ∈ [0, 1], f(p +5,t) = ( −c2 + 2c + 11 16 ) t + ( c − 1 4 )√ 1 − t + 1 4 and f(p −5,t) = ( −c2 − 2c + 11 16 ) t + ( c + 1 4 )√ 1 − t + 1 4 . hence, we have that, for every t ∈ [0, 1], −1 < c 4 ≤ f(p +5,t) ≤ −c2 + 2c + 15 16 < 1 (***) and −1 < c + 2 4 ≤ f(p −5,t) ≤ −c2 − 2c + 15 16 < 1 . hence, by theorem 2.3, 1 = ∥f∥. we will show that f exposes p +4,c. let q(x, y) = ax2 + by2 + cxy ∈ p ( 2r2 h( 1 2 ) ) such that 1 = ∥q∥ = f(q). we will show that q = p +4,c. by the krein-milman theorem, q(x, y) = up1(x, y) + v +p +2 (x, y) + v −p −2 (x, y) + tp3(x, y) + ∞∑ n=1 λ+n p + 4,c+n (x, y) + ∞∑ n=1 λ−n p − 4,c−n (x, y) + ∞∑ m=1 δ+mp + 5,a+m (x, y) + ∞∑ m=1 δ−mp − 5,a−m (x, y) , for some u, v±, t, λ±n , δ ± m, ∈ r (n, m ∈ n) with 0 ≤ c±n , a±m ≤ 1 and |u| + |v+| + |v−| + |t| + ∞∑ n=1 |λ+n | + ∞∑ n=1 |λ−n | + ∞∑ m=1 |δ+m| + ∞∑ m=1 |δ−m| = 1 . we will show that u = v± = t = λ−n = δ ± m = 0 for every n, m ∈ n. assume 138 s. g. kim that δ+m0 ̸= 0 for some m0 ∈ n. it follows that 1 = f(q) = uf(p1) + v +f(p +2 ) + v −f(p −2 ) + tf(p3) + ∞∑ n=1 λ+n f(p + 4,c+n ) + ∞∑ n=1 λ−n f(p − 4,c−n ) + ∞∑ m=1 δ+mf(p + 5,a+m ) + ∞∑ m=1 δ−mf(p − 5,a−m ) < 1 4 |u| + 3 4 |v+| + 3 4 |v−| + 9 16 |t| + ∞∑ n=1 |λ+n | + ∞∑ n=1 |λ−n | + |δ + m0 | + ∑ m̸=m0 |δ+m| + ∞∑ m=1 |δ−m| (by (*), (**), (***)) ≤ 1 , which is impossible. therefore, δ+m = 0 for every m ∈ n. using a similar argument as above, we have u = v± = t = λ−n = 0. therefore, q(x, y) = ∞∑ n=1 λ+n p + 4,c+n (x, y) . we will show that if c+n0 ̸= c for some n0 ∈ n, then λ + n0 = 0. assume that λ+n0 ̸= 0. it follows that 1 = f(q) = λ+n0f(p + 4,c+n0 ) + ∑ n̸=n0 λ+n f(p + 4,c+n ) < |λ+n0| + ∑ n ̸=n0 |λ+n | = 1 , which is impossible. therefore, λ+n = 0 for every n ∈ n. therefore, q(x, y) = ( ∑ c+n =c λ+n ) p +4,c(x, y) = p + 4,c(x, y) . therefore, f exposes p +4,c. obviously, p − 4,c ∈ expbp ( 2r2 h( 1 2 ) ) for 0 < c ≤ 1. claim 8: p +5,c = cx 2 + ( c+4 √ 1−c 4 −1 ) y2 +(c+2 √ 1 − c)xy ∈ expb p ( 2r2 h( 1 2 ) ) for 0 < c < 1. exposed polynomials of p ( 2r2 h( 1 2 ) ) 139 let f ∈ p ( 2r2 h( 1 2 ) )∗ be such that α = 1 2 ( 1 − c + 4 √ 1 − c 4 ) , β = − c 2 , γ = c + 2 √ 1 − c 4 . note that 0 ≤ α < 3 8 , − 1 2 < β ≤ 0 , 1 4 < γ ≤ 1 2 . we will show that f exposes p +5,c. indeed, f(p +5,c) = 1 , |f(p1)| < 1 2 , 0 < f(p +2 ) < 1 2 , − 1 < f(p −2 ) < − 1 8 , − 1 8 ≤ f(p3) < 0 . (*) note that for every t ∈ [0, 1], f(p +4,t) = − c 8 t2 + ( c + 2 √ 1 − c 4 ) t + 1 2 + 3c 8 − √ 1 − c 2 and f(p −4,t) = − c 8 t2 − ( c + 2 √ 1 − c 4 ) t + 1 2 + 3c 8 − √ 1 − c 2 . hence, we have for every t ∈ [0, 1], − 1 < 1 2 + 3c 8 − √ 1 − c 2 ≤ f(p +4,t) ≤ c + 1 2 < 1 , (**) − 1 < 1 2 − √ 1 − c ≤ f(p −4,t) ≤ 1 2 + 3c 8 − √ 1 − c 2 < 1 . note that for every t ∈ [0, 1] with t ̸= c, f(p +5,t) = 1 2 t + √ 1 − c √ 1 − t + c 2 and f(p −5,t) = ( 1 − c − √ 1 − c 2 ) t − (c + √ 1 − c) √ 1 − t + c 2 . hence, we have for every t ∈ [0, 1] with t ̸= c, − 1 < min { c 2 + √ 1 − c, c + 1 2 } ≤ f(p +5,t) < 1 , (***) − 1 < − ( c 2 + √ 1 − c ) ≤ f(p −5,t) ≤ 1 2 − √ 1 − c < 1 . 140 s. g. kim hence, by theorem 2.3, 1 = ∥f∥. let q(x, y) = ax2 + by2 + cxy in p ( 2r2 h( 1 2 ) ) such that 1 = ∥q∥ = f(q). by the krein-milman theorem, q(x, y) = up1(x, y) + v +p +2 (x, y) + v −p −2 (x, y) + tp3(x, y) + ∞∑ n=1 λ+n p + 4,c+n (x, y) + ∞∑ n=1 λ−n p − 4,c−n (x, y) + ∞∑ m=1 δ+mp + 5,a+m (x, y) + ∞∑ m=1 δ−mp − 5,a−m (x, y) , for some u, v±, t, λ±n , δ ± m, ∈ r (n, m ∈ n) with 0 ≤ c±n , a±m ≤ 1 and |u| + |v+| + |v−| + |t| + ∞∑ n=1 |λ+n | + ∞∑ n=1 |λ−n | + ∞∑ m=1 |δ+m| + ∞∑ m=1 |δ−m| = 1 . we will show that u = v± = t = λ±n = δ − m = 0 for every n, m ∈ n. assume that λn0 ̸= 0 for some n0 ∈ n. it follows that 1 = f(q) = uf(p1) + v +f(p +2 ) + v −f(p −2 ) + tf(p3) + ∞∑ n=1 λ+n f(p + 4,c+n ) + ∞∑ n=1 λ−n f(p − 4,c−n ) + ∞∑ m=1 δ+mf(p + 5,a+m ) + ∞∑ m=1 δ−mf(p − 5,a−m ) < 1 2 |u| + 1 2 |v+| + 1 2 |v−| + 1 2 |t| + |λ+n0| + ∑ n̸=n0 |λ+n | + ∞∑ m=1 |δ+m| + ∞∑ m=1 |δ−m| ≤ 1 (by (*), (**), (***)) , which is impossible. therefore, λ+n = 0 for every n ∈ n. using a similar argument as above, we have u = v± = t = λ−n = δ − m = 0 for every n, m ∈ n. therefore, q(x, y) = ∞∑ m=1 δ+mp + 5,a+m (x, y) . we will show that if a+m0 ̸= c for some m0 ∈ n, then δ + m0 = 0. assume that exposed polynomials of p ( 2r2 h( 1 2 ) ) 141 δ+m0 ̸= 0. it follows that 1 = f(q) = δ+m0f(p + 5,a+m0 ) + ∑ m ̸=m0 δ+mf(p + 5,am ) < |δ+m0| + ∑ m ̸=m0 |δ+m| = 1 which is impossible. therefore, δ+m0 = 0. therefore, q(x, y) = ( ∑ am=a δ+m ) p +5,c(x, y) = p + 5,c(x, y) . therefore, f exposes p +5,c. obviously, p − 5,c ∈ expbp ( 2r2 h( 1 2 ) ) for 0 < c < 1. therefore, we complete the proof. references [1] r.m. aron, m. klimek, supremum norms for quadratic polynomials, arch. math. 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[43] r.a. ryan, b. turett, geometry of spaces of polynomials, j. math. anal. appl. 221 (1998), 698 – 711. e extracta mathematicae vol. 33, núm. 1, 1 – 10 (2018) a generalization of the hyers -ulam-aoki type stability of some banach lattice -valued functional equation nutefe kwami agbeko, patŕıcia szokol institute of mathematics, university of miskolc, 3515 miskolc-egyetemváros, hungary matagbek@uni-miskolc.hu institute of mathematics, mta-de research group “equations, functions and curves”, hungarian academy of sciences and university of debrecen, p. o. box 12, 4010 debrecen, hungary szokol.patricia@inf.unideb.hu presented by pier l. papini received april 30, 2017 abstract: we obtained a generalization of the stability of some banach lattice-valued functional equation with the addition replaced in the cauchy functional equation by lattice operations and their combinations. key words: banach lattices, hyers-ulam-aoki type of stability. ams subject class. (2010): 39b82, 46b30, 46b42. 1. introduction all along (x , ∧x , ∨x ) will stand for a normed riesz space and (y, ∧y, ∨y) for a banach lattice with x + and y+ their respective positive cones. let us pose the following problem. problem 1. given three numbers ε, p, q ∈ (0, ∞), two riesz spaces g1 and g2 with g2 being endowed with a metric d(·, ·), four lattice operations ∆∗g1, ∆ ∗∗ g1 ∈ {∧g1, ∨g1} and ∆ ∗ g2 , ∆∗∗g2 ∈ {∧g2, ∨g2}, does there exist some real number δ > 0 such that, if a mapping f : g1 → g2 satisfies d (( f ( (τq|xt|)∆∗g1(η q|y|) )) ∆∗g2 ( f ( (τq|x|)∆∗∗g1(η q|y|) )) , ( τpf(|x|) ) ∆∗∗g2 ( ηpf(|y|) )) ≤ δ (1.1) for all x, y ∈ g1 and all τ, η ∈ [0, ∞), then an operation-preserving functional t : g1 → g2 exists with the property that d ( t(x), f(x) ) ≤ ε 1 2 n.k. agbeko, p. szokol for all x ∈ g1 and all τ, η ∈ [0, ∞)? if in (1.1) we let τ = η = 1, then the above problem reduces to the problem posed and treated in [5]. the study of functional equations and inequalities in lattice environments is motivated by the fact that many addition-related results or theorems can be extended and can be proved mutatis mutandis. for more references about the earliest extensions of the kind, we would refer the reader to the papers [1, 2, 3, 4]. the main goal of this paper is to show how ulam-hyers-aoki styled version of perturbation (1.1) leads to the unique solution of the functional equation( t ( (τq|x|)∆∗x (η q|y|) )) ∆∗y ( t ( (τq|x|)∆∗∗x (η q|y|) )) = ( τpt(|x|) ) ∆∗∗y ( ηpt(|y|) ) (1.2) for all x, y ∈ x and all τ, η ∈ [0, ∞), where ∆∗x , ∆ ∗∗ x ∈ {∧x , ∨x } and ∆∗y, ∆ ∗∗ y ∈ {∧y, ∨y} are fixed lattice operations. remark 1.1. if we let η = τ and y = x in equation (1.2), then we observe that t (τq |x|) = τpt (|x|) (1.3) for all x ∈ x and all τ ∈ [0, ∞). the results we obtained are straightforward generalizations of agbeko [3, 5, 6] and salahi et al [16]. for an additional reference we would like to mention the paper [7] where we proved separation and stability results for operators mapping a semi-group with values in a riesz lattice. we recall that a functional h : x → y is cone-related if h (x +) = {h (|x|) : x ∈ x} ⊂ y+ (see more about this notion in [3, 4]). some few references about hyers-ulam stability problems and solutions can be found, e.g. in [8, 11, 13, 14, 15]. our theorems will be deduced from the following forti’s result [10]. theorem 1.1. (forti) let (x, d) be a complete metric space and s an appropriate set. assume some functions f : s → x, g : s → s, h : x → x and δ : s → [0, ∞) satisfy the inequality d ( h ( f(g(x)) ) , f(x) ) ≤ δ(x), (1.4) stability of a functional equation on banach lattices 3 for all x ∈ s. if function h both is continuous and satisfies the inequality d ( h (u) , h (v) ) ≤ φ ( d (u, v) ) , u, v ∈ x, (1.5) for a certain non-decreasing subadditive function φ : [0, ∞) → [0, ∞) and the series ∞∑ j=0 φj ( δ ( gj (x) )) (1.6) is convergent for every x ∈ s, then there exists a unique function f : s → x solution of the functional equation h ( f (g (x)) ) = f (x) , x ∈ s, (1.7) and satisfying the following inequality: d ( f (x) , f (x) ) ≤ ∞∑ j=0 φj ( δ ( gj (x) )) . (1.8) the function f is given by f (x) = lim n→∞ hn ( f(gn(x)) ) . (1.9) 2. the main results theorem 2.1. given a pair of real numbers (p, q) ∈ (0, ∞) × (0, ∞), consider a cone-related functional f : x → y for which there are numbers ϑ > 0 and α with qα ∈ (p, ∞) such that∥∥∥f((τq|x|)∆∗x (ηq|y|))∆∗yf((τq|x|)∆∗∗x (ηq|y|)) − ( τpf(|x|) ) ∆∗∗y ( ηpf(|y|) )∥∥∥ ≤ 2(p−1)ϑ(∥x∥α + ∥y∥α) (2.1) for all x, y ∈ x and all τ, η ∈ [0, ∞). then the sequence (2npf (2−nq |x|))n∈n is a cauchy sequence for every x ∈ x . let the functional t : x + → y+ be defined by t (|x|) = lim n→∞ 2npf ( 2−nq |x| ) (2.2) for all x ∈ x . then t both is the unique solution of (1.2) and satisfies inequality ∥∥t (|x|) − f (|x|) ∥∥ ≤ 2pϑ 2qα − 2p ∥x∥α (2.3) for every x ∈ x . 4 n.k. agbeko, p. szokol theorem 2.2. given a pair of real numbers (p, q) ∈ (0, ∞) × (0, ∞), consider a cone-related functional f : x → y for which there are numbers β ∈ [0, ∞), ϑ > 0 and α with qα ∈ (0, p) such that∥∥∥f((τq|x|)∆∗x (ηq|y|))∆∗yf((τq|x|)∆∗∗x (ηq|y|)) − ( τpf(|x|) ) ∆∗∗y ( ηpf(|y|) )∥∥∥ ≤ β + ϑ2−(p+1)(∥x∥α + ∥y∥α) (2.4) for all x, y ∈ x and all τ, η ∈ [0, ∞). then the sequence (2−npf (2nq |x|))n∈n is a cauchy sequence for every fixed x ∈ x . let the functional t : x + → y+ be defined by t (|x|) = lim n→∞ 2−npf ( 2nq|x| ) (2.5) for all x ∈ x . then t both is the unique solution of (1.2) and satisfies inequality ∥∥t(|x|) − f(|x|)∥∥ ≤ β2p 2p − 1 + ϑ∥x∥α2qα 2p − 2qα (2.6) for every x ∈ x . remark 2.1. if the conditions of theorem 2.1 or theorem 2.2 hold true, then f (0) = 0. proof. the proof is similar to its counterpart in [5, 6] under the conditions of theorem 2.1 or theorem 2.2 when β = 0. under the condition of theorem 2.2 with β > 0, we need to prove that f (0) = 0. suppose in the contrary that f (0) > 0 were true. then by letting x = y = 0 and η = τ in (2.4), inequality∥∥f(0) − τpf(0)∥∥ ≤ β follows or, equivalently |τp − 1| ≤ β ∥f (0)∥ < ∞, which, as τ tends to infinity, would lead to an absurdity, indeed. hence the relation f (0) = 0 must be true. remark 2.2. let z be a set closed under the scalar multiplication, i.e. bz ∈ z whenever b ∈ r and z ∈ z. given a number c ∈ r let the function γ : z → z be defined by γ (z) = cz. then γj : z → z the j-th iteration of γ is given by γj (z) = cjz for every natural number j ≥ 2. stability of a functional equation on banach lattices 5 3. proof of the main results we shall use the technique in [5] to prove the main theorems. proof of theorem 2.1. first, if we choose τ = η = 2, y = x and replace x by 2−qx in inequality (2.1) then we obviously have∥∥2pf(2−q|x|) − f(|x|)∥∥ ≤ ϑ2p−qα∥x∥α. (3.1) next, let us define the following functions: g : x + → x +, g (|x|) = 2−q |x| , for all x ∈ x , δ : x + → [0, ∞) , δ (|x|) = ϑ2p−qα ∥x∥α , for all x ∈ x , φ : [0, ∞) → [0, ∞) , φ (t) = 2pt, h : y+ → y+, h (|y|) = 2p |y| , for all y ∈ y, d(·, ·) : y+ × y+ → [0, ∞), d ( |y1|, |y2| ) = ∥∥|y1| − |y2|∥∥, for all y1, y2 ∈ y. we shall verify the fulfilment of all the three conditions of the forti’s theorem as follows. (i) from inequality (3.1) we obviously have d ( h ( f(g(|xt|)) ) , f(|x|) ) = ∥∥h(f(g(|x|))) − f(|x|)∥∥ = ∥∥2pf(2−q|x|) − f(|x|)∥ ≤ ϑ2p−qα ∥∥x∥α = δ(|x|). (ii) d ( h(|y1|), h(|y2|) ) = 2p ∥∥|y1|−|y2|∥∥ = φ(d(|y1| , |y2|)) for all y1, y2 ∈ y. (iii) clearly, on the one hand φ is a non-decreasing subadditive function on the positive half line, and on the other hand by applying remark 2.2 on both the iterations gj and φj of g and φ respectively, one can observe that ∞∑ j=0 φj ( δ(gj(|x|)) ) = ϑ2p−qα∥x∥α ∞∑ j=0 2(p−qα)j = ϑ∥x∥α 2p 2qα − 2p < ∞. then in view of forti’s theorem, sequence (hn(f(gn|x|)))n∈n is a cauchy sequence for every x ∈ x and thus so is (2npf(2−nq|x|))n∈n. furthermore, the mapping (2.2) satisfies inequality (2.3). 6 n.k. agbeko, p. szokol next, we prove that t solves (1.2). in fact, in (2.1) substitute x with 2−nqx also y with 2−nqy, and fix arbitrarily τ, η ∈ [0, ∞). then∥∥∥∥∥f ( (τq|x|)∆∗x (η q|y|) 2nq ) ∆∗yf ( (τq|x|)∆∗∗x (η q|y|) 2nq ) − ( τpf ( |x| 2nq )) ∆∗∗y ( ηpf ( |y| 2nq ))∥∥∥∥∥ ≤ 2(p−1)ϑ (∥∥∥ x 2nq ∥∥∥α + ∥∥∥ y 2nq ∥∥∥α). multiplying both sides of this last inequality by 2np yields 2np ∥∥∥∥∥f ( (τq|x|)∆∗x (η q|y|) 2nq ) ∆∗yf ( (τq|x|)∆∗∗x (η q|y|) 2nq ) − ( τpf ( |x| 2nq )) ∆∗∗y ( ηpf ( |y| 2nq ))∥∥∥∥∥ ≤ ϑ2(1−p) ∥x∥ α + ∥y∥α 2n(qα−p) . (3.2) taking the limit in (3.2) we have via (2.2) that∥∥t((τq|x|)∆∗x (ηq|y|))∆∗yt((τq|x|)∆∗∗x (ηq|y|))−(τpt(|x|))∆∗∗y (ηpt(|y|))∥∥ = 0 for all τ, η ∈ [0, ∞) and all x, y ∈ x , which is equivalent to (1.2). thus t also satisfies (1.3) in remark 1.1. finally we show the uniqueness, using a technique in [16]. in fact, assume that there is another functional s : x → y which satisfies (1.2) and the inequality ∥s(|x|) − f(|x|)∥ ≤ δ2∥x∥α2 for some numbers α2, δ2 ∈ (0, ∞) with qα2 > p, and for all x ∈ x . in (2.3) let δ1 := 2pϑ 2qα−2p , α1 := α and by choosing τ = 2 −n in remark 1.1 one can observe that for all x ∈ x∥∥s(|x|) − t(|x|)∥∥ = 2np∥∥s(2−nq|x|) − t(2−nq|x|)∥∥ ≤ 2np ∥∥f(2−nq|x|) − t(2−nq|x|)∥∥ + 2np ∥∥s(2−nq|x|) − f(2−nq|x|)∥∥ ≤ 2npδ1∥2−nqx∥α1 + 2npδ2∥2−nqx∥α2 = 2(p−qα1)nδ1∥x∥α1 + 2(p−qα2)nδ2∥x∥α2. hence ∥∥s(|x|) − t(|x|)∥∥ ≤ 2(p−qα1)nδ1∥x∥α1 + 2(p−qα2)nδ2∥x∥α2 which, in the limit, yields ∥s(|x|)−t(|x|)∥ = 0 or equivalently s(|x|) = t(|x|) for all x ∈ x . stability of a functional equation on banach lattices 7 this was to be proven. proof of theorem 2.2. first, if we choose τ = η = 2−1, y = x and replace x by 2qx in inequality (2.4) then we obviously have∥∥2−pf(2q|x|) − f(|x|)∥∥ ≤ β + ϑ2qα−p∥x∥α. (3.3) next, let us define the following functions: g : x + → x +, g(|x|) = 2q|x|, for all x ∈ x , δ : x + → [0, ∞), δ(|x|) = β + ϑ2qα−p∥x∥α, for all x ∈ x , φ : [0, ∞) → [0, ∞), φ(t) = 2−pt, h : y+ → y+, h(|y|) = 2−p|y|, for all y ∈ y, d(· , ·) : y+× y+ → [0, ∞), d ( |y1|, |y2| ) = ∥∥|y1| − |y2|∥∥, for all y1, y2 ∈ y. we shall verify the fulfilment of all the three conditions of the forti’s theorem as follows. (i) from inequality (3.3) we obviously have d ( h ( f(g(|x|)) ) , f(|x|) ) = ∥∥h(f(g(|x|))) − f(|x|)∥∥ = ∥∥2−pf(2q|x|) − f(|x|)∥∥ ≤ β + ϑ2qα−p∥x∥α = δ(|x|). (ii) d ( h(|y1|), h(|y2|) ) = 2−p ∥∥|y1|−|y2|∥∥= φ(d(|y1|, |y2|)) for all y1, y2 ∈ y. (iii) clearly, on the one hand φ is a non-decreasing subadditive function on the positive half line, and on the other hand by applying remark 2.2 on both the iterations gj and φj of g and φ respectively, one can observe that ∞∑ j=0 φj ( δ(gj(|x|)) ) = β ∞∑ j=0 2−pj + ϑ2qα−p∥x∥α ∞∑ j=0 2(qα−p)j = β2p 2p − 1 + ϑ∥x∥α2qα 2p − 2qα < ∞. then in view of forti’s theorem, sequence (hn ( f(gn|x|)) ) n∈n is a cauchy sequence for every x ∈ x and thus so is ( 2−npf(2nq|x|) ) n∈n. furthermore, the mapping (2.5) satisfies inequality (2.6). 8 n.k. agbeko, p. szokol next, we prove that t solves (1.2). in fact, in (2.4) substitute x with 2nqx also y with 2nqy, and fix arbitrarily τ, η ∈ [0, ∞). then∥∥∥f(2nq((τq|x|)∆∗x (ηq|y|)))∆∗yf(2nq((τq|x|)∆∗∗x (ηq|y|))) − ( τpf(2nq|x|) ) ∆∗∗y ( ηpf(2nq|y|) )∥∥∥ ≤ β + 2−(p+1)ϑ ( ∥2nqx∥α + ∥2nqy∥α ) . dividing both sides of this last inequality by 2np yields∥∥∥∥∥ f ( 2nq ( (τq|x|)∆∗x (η q|y|) )) ∆∗yf ( 2nq ( (τq|x|)∆∗∗x (η q|y|) )) 2np − ( τpf(2nq|x|) ) ∆∗∗y ( ηpf(2nq|y|) ) 2np ∥∥∥∥∥ ≤ β2−np + 2−(p+1)ϑ(∥x∥α + ∥y∥α)2(qα−p)n. (3.4) taking the limit in (3.4) we have via (2.5) that∥∥∥t((τq|x|)∆∗x (ηq|y|))∆∗yt((τq|x|)∆∗∗x (ηq|y|))−(τpt(|x|))∆∗∗y (ηpt(|y|))∥∥∥ = 0 for all τ, η ∈ [0, ∞) and all x, y ∈ x , which is equivalent to (1.2). thus t satisfies (1.3) in remark 1.1. finally we show the uniqueness, using a technique in [16]. in fact, assume that there is another functional s : x → y which satisfies (1.2) and the inequality ∥s(|x|) − f(|x|)∥ ≤ β2 + δ2∥x∥α2 for some numbers α2, δ2 ∈ (0, ∞), β2 ∈ [0, ∞) with qα2 < p, and for all x ∈ x . in (2.6) let β1 := β2p 2p−1, δ1 := ϑ2qα 2p−2qα , α1 := α and by choosing τ = 2 n in remark 1.1 one can observe that for all x ∈ x∥∥s(|x|) − t(|x|)∥∥ = 2−np∥∥s(2nq|x|) − t(2nq|x|)∥∥ ≤ 2−np ∥∥f(2nq|x|) − t(2nq|x|)∥∥ + 2−np ∥∥s(2nq|x|) − f(2nq|x|)∥∥ ≤ 2−np ( β1 + δ1∥2nqx∥α1 ) + 2−np ( β2 + δ2∥2nqx∥α2 ) = 2−np(β1 + β2) + δ12 (qα1−p)n∥x∥α1 + δ22(qα2−p)n∥x∥α2. hence∥∥s(|x|) − t(|x|)∥∥ ≤ 2−np(β1 + β2) + δ12(qα1−p)n∥x∥α1 + δ22(qα2−p)n∥x∥α2 stability of a functional equation on banach lattices 9 which, in the limit, yields ∥s(|x|)−t(|x|)∥ = 0 or equivalently s(|x|) = t(|x|) for all x ∈ x . this completes the proof. to end our paper we give an example showing that stability fails to occur in general. example 1. fix arbitrarily τ, η ∈ (0, 2) and consider the function f : [0, ∞) → [0, ∞) , f (x) = xα+1, α = p q . since f is increasing the first equality in the chain below is valid, entailing the subsequent relations:∣∣∣f((τqx) ∨ (ηqy)) − (τpf(x)) ∧ (ηpf(y))∣∣∣ = ∣∣(τqx)α+1 ∨ (ηqy)α+1 − (τpxα+1) ∧ (ηpyα+1)∣∣ ≤ (τqx)α+1 ∨ (ηqy)α+1 + (τpxα+1) ∧ (ηpyα+1) ≤ (2qx)α+1 ∨ (2qy)α+1 + (2pxα+1) ∧ (2pyα+1) ≤ 2p+q(xα+1 ∨ yα+1) + 2p+q(xα+1 ∧ yα+1) = 2p+q(xα+1 + yα+1) for all x, y ∈ [0, ∞). now, let t : [0, ∞) → [0, ∞) be a function such that t (µqx) = µpt (x) for all x ∈ [0, ∞) and all µ ∈ [0, ∞). since x = ( x1/q )q , and α is the ratio of p and q, we can then note that t (x) = xαt (1) for every x ∈ [0, ∞). now, sup x∈(0, ∞) |f (x) − t (x)| 2p+qxα+1 = sup x∈(0, ∞) ∣∣∣xα+1 − t((x1q )q)∣∣∣ 2p+qxα+1 = sup x∈(0, ∞) ∣∣xα+1 − xαt (1)∣∣ 2p+qxα+1 = 1 2p+q sup x∈(0, ∞) ∣∣∣∣1 − t (1)x ∣∣∣∣ = ∞. the above example about the lack of stability on the real line in lattice environments is the counterpart of the example given by s. czerwik [9] in the addition environments for quadratic mappings. 10 n.k. agbeko, p. szokol acknowledgements the second author is supported by the hungarian academy of sciences, and otka grant pd124875. references [1] n.k. agbeko, on optimal averages, acta math. hungar. 63 (2) (1994), 133 – 147. 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[16] n. salehi, s.m.s. modarres, stablity of maximum preserving quadratic functional equation in banach lattices, miskolc math. notes 17 (1) (2016), 581 – 589. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae article in press available online july 19, 2023 radon-nikodýmification of arbitrary measure spaces p. bouafia 1 , t. de pauw 2,∗ 1 fédération de mathématiques fr3487, centralesupélec 3 rue joliot curie, 91190 gif-sur-yvette, france 2 université paris cité and sorbonne université, cnrs, imj-prg, f-75013, paris, france philippe.bouafia@centralesupelec.fr , depauw@imj-prg.fr received dec 6, 2022 presented by g. plebanek accepted may 2, 2023 and j. jaramillo abstract: we study measurable spaces equipped with a σ-ideal of negligible sets. we find conditions under which they admit a localizable locally determined version – a kind of fiber space that locally describes their directions – defined by a universal property in an appropriate category that we introduce. these methods allow to promote each measure space (x, a ,µ) to a strictly localizable version (x̂, â , µ̂), so that the dual of l1(x, a ,µ) is l∞(x̂, â , µ̂). corresponding to this duality is a generalized radon-nikodým theorem. we also provide a characterization of the strictly localizable version in special cases that include integral geometric measures, when the negligibles are the purely unrectifiable sets in a given dimension. key words: measurable space with negligibles; radon-nikodým theorem; strictly localizable measure space; integral geometric measure; purely unrectifiable. msc (2020): primary 28a15; secondary 28a75,28a05. contents 1 foreword 2 2 measurable spaces with negligibles 10 3 supremum preserving morphisms 15 4 localizable, 4c and strictly localizable msns 19 5 localizable locally determined msns 28 6 gluing measurable functions 37 7 existence of 4c and lld versions 43 8 strictly localizable version of a measure space 47 9 a directional radon-nikodým theorem 51 10 4c version deduced from a compatible family of lower densities 53 11 applications 60 references 64 ∗ the second author was partially supported by the science and technology commission of shanghai (no. 18dz2271000). issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) mailto:philippe.bouafia@centralesupelec.fr mailto:depauw@imj-prg.fr https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 2 p. bouafia, t. de pauw 1. foreword the radon-nikodým theorem does not hold for every measure space (x, a ,µ). one way to phrase this precisely is to consider the canonical embedding υ : l∞(x, a ,µ) → l1(x, a ,µ)∗. the following hold. (a) υ is injective (this corresponds to the uniqueness almost everywhere of radon-nikodým derivatives) if and only if (x, a ,µ) is semi-finite. (b) υ is surjective (this corresponds to the existence of radon-nikodým derivatives) if and only if the boolean algebra a /nµ,loc is dedekind complete (i.e. order complete as a lattice). while (a) is classical, see e.g. [6, 243g(a)], (b) is recent and due to the second author, see [2, 4.6]. let us recall the relevant definitions. given a measure space (x, a ,µ), we abbreviate a f := a ∩ {a : µ(a) < ∞} and nµ := a ∩ {n : µ(n) = 0}. we say that (x, a ,µ) is semi-finite if every a ∈ a of infinite measure contains some f ∈ a f \ nµ. equivalently, µ(a) = sup{µ(f) : a ⊇ f ∈ a f}. we further define the σ-ideal of locally µ-null sets as follows: nµ,loc := a ∩ { a : a ∩ f ∈ nµ for all f ∈ a f } . it is easy to see [2, 4.4] that (x, a ,µ) is semi-finite if and only if nµ,loc = nµ. thus we obtain the following classical criterion, [6, 243g(b)]. (c) υ is an isometric isomorphism if and only if (x, a ,µ) is semi-finite and the boolean algebra a /nµ is dedekind complete. though semi-finiteness is a natural property, caratheodory’s method does not always provide it. for instance, the measure spaces (r2, ah 1, h 1) and (r2, b(r2), i 1∞) are not semi-finite – see [8, 439h] and [4, 3.3.20]. here, h 1 is the 1-dimensional hausdorff measure in the euclidean plane [4, 2.10.2], ah 1 is the σ-algebra consisting of h 1-measurable sets in caratheodory’s sense, i 1∞ is a 1-dimensional integral geometric measure [4, 2.10.5(1)], and b(r2) is the σ-algebra whose members are the borel subsets of r2. both h 1(γ) and i 1∞(γ) coincide with the usual euclidean length of γ when this is a lipschitz curve. it is natural to want to associate, with an arbitrary (x, a ,µ), an improved version of itself – in a universal way – ideally one for which the radon-nikodým theorem holds. this is one of our several achievements in this paper. it is not difficult to modify slightly the measure µ, keeping the underlying measurable radon-nikodýmification of arbitrary measure spaces 3 space (x, a ) untouched, in order to make it semi-finite. specifically, letting µsf (a) = sup { µ(a∩f) : f ∈ a f } , for a ∈ a , one checks that (x, a ,µsf ) is semi-finite and that nµsf = nµ,loc. however, it appears to be a more delicate task to modify (x, a ,µ) in a canonical way in order for υ to become surjective. an idea for testing if υ is surjective is as follows. given α ∈ l1(x, a ,µ)∗ we apply the radon-nikodým theorem “locally”, as it is valid on each finite measure subspace (f, af ,µf ), f ∈ a f , i.e. we represent by integration the functional α◦ ιf ∈ l1(f, af ,µf )∗, where ιf : l1(f, af ,µf ) → l1(x, a ,µ) is the obvious map. this produces a family of radon-nikodým derivatives 〈ff〉f∈a f . by the almost everywhere uniqueness of radon-nikodým derivatives in finite measure spaces, this is a compatible family in the sense that f ∩ f ′ ∩ {ff 6= ff ′} ∈ nµ, for every f,f ′ ∈ a f . in order to obtain a globally defined radon-nikodým derivative, one ought to be able to “glue” together the functions of this family. a gluing of 〈ff〉f∈a f is, by definition, an a -measurable function f : x → r such that f ∩ {f 6= ff} ∈ nµ for every f ∈ a f . the question whether such a gluing exists takes us away from the realm of measure spaces, as it rather pertains to measurable spaces with negligibles, abbreviated msns, i.e. triples (x, a , n ) where (x, a ) is a measurable space and n ⊆ a is a σ-ideal. the notion of compatible family 〈fe〉e∈e of ae-measurable functions e → r subordinated to an arbitrary collection e ⊆ a readily makes sense in this more general setting, as does the notion of gluing of such a compatible family. that each compatible family of partially defined measurable functions admits a gluing is equivalent to the boolean algebra a /n being dedekind complete. in this case we say that (x, a , n ) is localizable. equivalently, (x, a , n ) is localizable if and only if every collection e ⊆ a admits an n -essential supremum (see 3.3 for a definition), which corresponds to taking an actual supremum in the boolean algebra a /n . for a proof of these classical equivalences, see e.g. [2, 3.13]. it will be convenient to call n -generating a collection e ⊆ a that admits x as an n -essential supremum. for instance, one easily checks that if (x, a ,µ) is a semi-finite measure space, then a f is nµ-generating. let (x, a , n ) be a localizable msn, e ⊆ a be n -generating, and 〈fe〉e∈e be a compatible family of partially defined measurable functions. the problem of gluing this compatible family in our setting is reminiscent of the fact that, for a topological space x, the functor of continuous functions on open sets is a sheaf. however, unlike in the case of continuous functions, 4 p. bouafia, t. de pauw in order to define f globally, we ought to make choices on the domains e∩e′, for e,e′ ∈ e , because fe and fe′ do not coincide everywhere there, but merely almost everywhere. in an attempt to avoid the issue, one can replace e with an almost disjointed refinement of itself, say f . by this we mean that each member of f is contained in a member of e , that f is n -generating, and that f ∩f ′ ∈ n whenever f,f ′ ∈ f are distinct. the existence of f follows from zorn’s lemma, see 4.9. still, f ∩f ′ may not be empty whenever f,f ′ ∈ f are distinct and we are again in a position to make choices. a step further along the road would be to produce from f a disjointed family g whose union is conegligible. from classical measure theory, we learn of two situations when this is doable. first, in the presence of a lower density of (x, a , n ) (see 10.1 for a definition), and second when card f 6 c (see the proof of 7.6). in those cases, a gluing exists. in fact, in the context of measure spaces, the existence of a lower density yields a somewhat stronger structure than localizability. in order to state this, we need one more definition. we say that a measure space (x, a ,µ) is locally determined if it is semi-finite and if the following holds: ∀a ⊆ x : [ ∀f ∈ a f : a∩f ∈ a ] ⇒ a ∈ a . a complete locally determined measure space (x, a ,µ) admits a lower density if and only if it is strictly localizable, which means, by definition, that there exists a partition g ⊆ a f of x such that a = p(x) ∩{a : a∩g ∈ a for all g ∈ g} and µ(a) = ∑ g∈g µ(a ∩ g), for a ∈ a . see [7, 341m] for a proof. the existence of a lower density for a strictly localizable measure space follows from the case of finite measure spaces by gluing, and the case of finite measure spaces is a consequence of a martingale convergence theorem. even though the notion of a lower density makes sense for msns, their existence does not hold for even the most natural generalization of finite measure spaces, namely ccc msns (satisfying the countable chain condition, 4.3 and 4.5), see [16]. both notions of localizability (of an msn) and local determination (of a measure space) seem to express in different ways the fact that “there are enough measurable sets”. for instance, one easily checks that an msn (x, a ,{∅}), such that a contains all singletons, is localizable if and only if a = p(x). thus, given an arbitrary msn (x, a , n ), one may naively attempt to “add measurable sets” in a smart way in order to obtain a localizable msn ( x, â , n̂ ) , just radon-nikodýmification of arbitrary measure spaces 5 as many as needed, and that would be a “localizable version” of (x, a , n ). unfortunately, within zfc this cannot always be done while “sticking in the base space x”, as shown by the following, quoted from [2]. theorem. assume that: (1) c ⊆ [0, 1] is some cantor set of hausdorff dimension 0; (2) x = c × [0, 1]; (3) a is a σ-algebra such that b(x) ⊆ a ⊆ p(x); (4) n = nh 1 or n = npu. then (x, a , n ) is consistently not localizable. here, npu consists of those subsets s of x that are purely 1-unrectifiable, i.e. h 1(s ∩ γ) = 0 for every lipschitz (or, for that matter, c1) curve γ ⊆ r2. thus, one may need to also add points to the base space x and, in particular cases such as the one above, we give a very specific way of doing so, in the last section of this paper. in the case of a general measure space (x, a ,µ), we can get a feeling of what needs to be done, when trying to define the gluing of a compatible family 〈ff〉f∈a f . indeed, each x ∈ x may belong to several f ∈ a f and this calls for considering an appropriate quotient of the fiber bundle { (x,f) : x ∈ f ∈ a f } . one of the tasks that we assign ourselves in this paper is to define a general notion of “localization” of an msn and to prove existence results in some cases. since a definition of “localization” will involve a universal property, it is critical to determine which category is appropriate for our purposes. as this offers unexpected surprises, we describe the several steps in some detail. the objects of our first category msn are the saturated msn (x, a , n ), by what we mean that for every n,n ′ ⊆ x, if n ⊆ n ′ and n ′ ∈ n , then n ∈ n . this is in analogy with the notion of a complete measure space. in order to define the morphisms between two objects (x, a , n ) and (y, b, m ), we say that a map f : x → y is [(a , n ), (b, m )]-measurable if f−1(b) ∈ a for every b ∈ b and f−1(m) ∈ n for every m ∈ m . for instance, if x is a polish space and µ is a diffuse probability measure on x, there exists [17, 3.4.23] a borel isomorphism f : x → [0, 1] such that f#µ = l 1, where l 1 is the lebesgue measure, thus f is [(b(x), nµ), (b([0, 1]), nl 1 )]-measurable. we define an equivalence relation for such measurable maps f,f ′ : x → y by saying that f ∼ f ′ if and only if {f 6= f ′} ∈ n . the morphisms in the category msn between the objects (x, a , n ) and (y, b, m ) are the equivalence classes of 6 p. bouafia, t. de pauw [(a , n ), (b, m )]-measurable maps. at this stage, we need to suppose that (x, a , n ) is saturated for the relation of equality almost everywhere to be transitive 2.7. with this assumption, the composition of measurable maps is also compatible with ∼, see 2.8. we let loc be the full subcategory of msn whose objects are the localizable msns. we may be tempted to define the localization of a saturated msn (x, a , n ) as its coreflection (if it exists) along the forgetful functor forget: loc → msn, and the question of existence in general becomes that of the existence of a right adjoint to forget. specifically, we may want to say that a pair [( x̂, â , n̂ ) ,p ] , where ( x̂, â , n̂ ) is saturated localizable msn and p is a morphism x̂ → x, is a localization of (x, a , n ) whenever the following universal property holds. for every pair [(y, b, m ),q], where (y, b, m ) is a saturated localizable msn and q is a morphism y → x, there exists a unique morphism r: y → x̂ such that q = p◦r. (y, b, m ) ( x̂, â , n̂ ) (x, a , n ) q ∃!r p (5) however, we now illustrate that the notion of morphism defined so far is not yet the appropriate one that we are after. we consider the msn (x, a ,{∅}) where x = r and a is the σ-algebra of lebesgue measurable subsets of r. we recall that we want the localization of (x, a ,{∅}) to be [(x, p(x),{∅}),p] with p induced by the identity idx. assume if possible that this is the case. in the diagram above we consider (y, b, m ) = (x, a , nl 1 ) and q induced by the identity. note that this is, indeed, a localizable msn since it is associated with a σ-finite measure space (see 4.5 and 4.4). thus, there would exist a morphism r in msn such that p ◦ r = q. picking r ∈ r, this implies that x ∩{x : r(x) 6= x} is lebesgue negligible. the measurability of r would then imply that r−1(s) ∈ a for every s ∈ p(x), contradicting the existence of non lebesgue measurable subsets of r. the problem with the example above is that the objects (x, a , nl 1 ) and (x, a ,{∅}) should not be compared, in other words that q should not be a legitimate morphism. we say that a morphism f : (x, a , n ) → (y, b, m ) of the category msn is supremum preserving if the following holds for (one, and therefore every) f ∈ f. if f ⊆ b admits an m -essential supremum s ∈ b, then f−1(s) is an n -essential supremum of f−1(f ). it is easy to see that adding this condition to the definition of morphism rules out the q radon-nikodýmification of arbitrary measure spaces 7 considered in the preceding paragraph. we define the category msnsp to be that whose objects are the saturated msns and whose morphisms are those morphisms of msn that are supremum preserving. we define similarly locsp. we now define the localizable version (if it exists) of a saturated msn with the similar universal property illustrated in (5), except for we now require all morphisms to be in msnsp, i.e. supremum preserving. in other words, it is a coreflection of an object of msnsp along forget: locsp → msnsp. unfortunately, this is not quite yet the right setting. indeed, we show in 4.13 that if x is uncountable and c (x) is the countable-cocountable σ-algebra of x, then [(x, p(x),{∅}),ι] (with ι induced by idx) is not the localizable version of (x, c (x),{∅}). this prompts us to introduce a new category. we say that an object (x, a , n ) of msn is locally determined if for every n -generating collection e ⊆ a the following holds: ∀a ⊆ x : [ ∀e ∈ e : a∩e ∈ a ] ⇒ a ∈ a . in case (x, a , n ) is the msn associated with some complete semi-finite measure space (x, a ,µ), then it is locally determined (in the sense of msns) if and only if (x, a ,µ) is locally determined (in the sense of measure spaces) – see 5.3(f) – even though the latter sounds stronger because we test with any generating family e . we say that an object of msn is lld if it is both localizable and locally determined, and we let lldsp be the corresponding full subcategory of locsp. we now define the lld version of an object of msnsp to be its coreflection (if it exists) along forget: lldsp → msnsp, i.e. it satisfies the corresponding universal property illustrated in (5) with y and x̂ being lld, and the morphisms being supremum preserving. this definition is satisfactory in at least the simplest case, 5.4 : if (x, a ,{∅}) is so that a contains all singletons, then it admits [(x, p(x),{∅}),ι] as its lld version. our general question has now become whether forget : lldsp → msnsp admits a right adjoint. freyd’s adjoint functor theorem [1, 3.3.3] could prove useful, however do not know whether it applies, mostly because we do not know whether coequalizers exist in msnsp. we gather in table 1 the information that we know about limits and colimits in the three categories we introduced. in view of proving some partial existence result for lld versions, we introduce the intermediary notion of a 4c (or cccc) saturated msn, short for coproduct (in msnsp) of ccc saturated msns. it is easy to see that 4c msns are lld, 4.6 and 4.4. the 4c version of an object of msnsp is likewise defined by its universal property in diagram (5), using supremum preserving 8 p. bouafia, t. de pauw morphisms. our main results are about locally ccc msns, i.e. those saturated msns (x, a , n ) such that eccc = a ∩{z : the submsn (z, az, nz) is ccc} is n -generating. a complete semi-finite measure space (x, a ,µ) is clearly locally ccc, since a f is nµ-generating. similarly, one can define the more general locally localizable objects in msnsp. in 4.14, we give an example of an msn which is not even locally localizable. msnsp locsp lldsp equalizers exist if {f = g} ? exist 5.10 is meas. 3.7(c) products (countable) exist 2.13 ? ? coequalizers ? ? see 5.7 ? see 5.7 coproducts exist 3.7(d) exist 4.6 exist 4.6 and 5.3(d) table 1: limits and colimits in the three categories of msns. theorem. let (x, a , n ) be a saturated locally ccc msn. the following hold: (1) (x, a , n ) admits a 4c version, 7.4. (2) if furthermore eccc contains an n -generating subcollection e such that card e 6 c and each (z, az) is countably separated, for z ∈ e , then (x, a , n ) admits an lld version which is also its 4c version. by saying that a measurable space (z, az) is countably separated we mean that az contains a countable subcollection that separates points in z. the 4c version ( x̂, â , n̂ ) is obtained as a coproduct ∐ z∈e (z, az, nz) where e is an n -generating almost disjointed refinement of eccc, whose existence ensues from zorn’s lemma. in order to establish that this, in fact, is also the lld version under the extra assumptions in (2), we need to build an appropriate morphism r in diagram (5), associated with an lld pair [(y, b, m ),q]. it is obtained as a gluing of 〈qz〉z∈e where qz : q−1(z) → x̂ is the obvious map. since e is almost disjointed, 〈qz〉z∈e is compatible and, since (y, b, m ) in diagram (5) is localizable, the only obstruction to gluing is that x̂ is not r. notwithstanding, ( x̂, â ) = ∐ z∈e (z, az) is itself countably separated because card e 6 c, 6.8 so that the local determinacy of (y, b, m ) and the fact radon-nikodýmification of arbitrary measure spaces 9 that q−1(e ) is m -generating (because e is n -generating and q is supremum preserving) provides a gluing r, 6.10. we now explain how this applies to associating, in a canonical way, a strictly localizable measure space with any measure space (x, a ,µ). first, we recall that without changing the base space x we can render the measure space complete and semi-finite. in that case, a f is nµ-generating and witnesses the fact that the saturated msn (x, a , nµ) is locally ccc. by the theorem above, it admits a 4c version [( x̂, â , n̂ ) ,p ] . theorem. let (x, a ,µ) be a complete semi-finite measure space and[( x̂, â , n̂ ) ,p ] its corresponding 4c version. let p ∈ p. there exists a unique (and independent of the choice of p) measure µ̂ defined on â such that p#µ̂ = µ and nµ̂ = n̂ . furthermore ( x̂, â , µ̂ ) is a strictly localizable measure space, and the banach spaces l1(x, a ,µ) and l1 ( x̂, â , µ̂ ) are isometrically isomorphic. of course, the general process for constructing x̂ is non constructive, as it involves the axiom of choice to turn a f into an almost disjointed generating family. this is why, in the last two sections of this paper, we explore a particular case where we are able to describe explicitly x̂ as a quotient of a fiber bundle, all “hands on”. we start with the measure space( rm, b(rm), i k∞ ) where 1 6 k 6 m−1 are integers, b(rm) is the σ-algebra of borel subsets of rm, and i k∞ is the integral geometric measure described in [4, 2.10.5(1)] and [11, 5.14]. note that it is not semi-finite, [4, 3.3.20]. thus, we replace it with its complete semi-finite version ( rm, b̃(rm), ĩ k∞ ) . we let e be the collection of k-dimensional submanifolds m ⊆ rm of class c1 such that φm = h k m is locally finite. it follows from the besicovitch structure theorem [4, 3.3.14] that e is n ĩ k∞ -generating, 11.2(ii). now, for each x ∈ rm we define ex = e ∩{m : x ∈ m} and we define on ex an equivalence relation as follows. we declare that m ∼x m ′ if and only if lim r→0+ h k(m ∩m ′ ∩b(x,r)) α(k)rk = 1. letting [m]x denote the equivalence class of m ∈ ex, we prove 11.2 that underlying set of the 4c, lld, and strictly localizable version of the msn( rm, b̃(rm), n ĩ k∞ ) can be taken to be x̂ = { (x, [m]x) : x ∈ rm and m ∈ ex } . 10 p. bouafia, t. de pauw this leads to an explicit description of the dual of l1 ( rm, b̃(rm), ĩ k∞ ) as l∞ ( x̂, â , n̂ ) . we close this foreword with a comment about the “baire category” counterpart of our work on measure spaces. consider a nonempty topological space x which is either completely metrizable or locally compact hausdorff. we recall that a subset of x is termed meager if it is a countable union of nowhere dense subsets of x (i.e. subsets whose closure has empty interior). meager sets in x clearly form a σ-ideal which we denote by m . furthermore, a ⊆ x is called baire measurable if it is the symmetric difference of an open set and a meager set. letting b be the collection of baire measurable subsets of x, we note that (x, b, m ) is a saturated msn and that, under our assumption on x, x 6∈ m . in case x is polish (i.e. completely metrizable and separable), everything turns out perfect from the point of view of this paper, reflecting the situation of σ-finite measure spaces: this is because (x, b, m ) then satisfies the countable chain condition. it is therefore localizable, by 4.4, and locally determined, by 5.3(c). in the general case, however, we do not know whether our results in section 7 apply to showing that (x, b, m ) admits a localizable version. we are indebted to david fremlin whose point of view on measure theory – generously shared in his immense treatise [5, 6, 7, 8, 9, 10] – influenced our work in this paper. it is the second author’s pleasure to record useful conversations with francis borceux. 2. measurable spaces with negligibles definition 2.1. (σ-algebra) let x be a set. a σ-algebra on x is a set a ⊆ p(x) such that (1) ∅∈ a ; (2) if a ∈ a then x \a ∈ a ; (3) if 〈an〉n∈n is a sequence in a then ⋃ n∈n an ∈ a . if a is a σ-algebra on x then x ∈ a and ⋂ n∈n an ∈ a whenever 〈an〉n∈n is a sequence in a . clearly {∅,x} and p(x) are σ-algebras on x, respectively the coarsest and the finest. if 〈ai〉i∈i is a nonempty family of σ-algebras on x then ⋂ i∈i ai is a σ-algebra on x. thus each e ⊆ p(x) is contained in a coarsest σ-algebra on x which we will denote by σ(e ). if e , a ⊆ p(x) and a = σ(e ) we say that the σ-algebra a is generated by e . clearly, if radon-nikodýmification of arbitrary measure spaces 11 e1 ⊆ e2 ⊆ p(x) then σ(e1) ⊆ σ(e2). a measurable space is a couple (x, a ) where x is a set and a is a σ-algebra on x. in this case, if no confusion is possible we call measurable the members of a . definition 2.2. (measurable maps) let (x, a ) and (y, b) be measurable spaces and f : x → y . we say that f is (a , b)-measurable (or simply measurable if no confusion can occur) if f−1(b) ∈ a whenever b ∈ b. measurable spaces, together with measurable maps, form a well-defined category, as one can check that the composition of two measurable maps is measurable. definition 2.3. (σ-ideal) let (x, a ) be a measurable space. a σ-ideal n of a is a subset of a that satisfies the following requirements: (1) ∅∈ n ; (2) if a ∈ a , n ∈ n and a ⊆ n then a ∈ n ; (3) if 〈nn〉n∈n is a sequence in n , then ⋃ n∈n nn ∈ n . definition 2.4. (measurable space with negligibles) a measurable space with negligibles (abbreviated msn) is a triple (x, a , n ) where (x, a ) is a measurable space and n is a σ-ideal of a . given an msn (x, a , n ), elements belonging to n are referred to as n -negligible sets (simply negligible sets if no confusion can occur). complements of n -negligible sets are called n -conegligible sets (or simply conegligible sets). we can associate to any measure space (x, a ,µ) the msn (x, a , nµ) where nµ is the σ-ideal nµ = a ∩{n : µ(n) = 0}. conversely, any msn (x, a , n ) derives from a measure space: it suffices to consider the measure µ: a → [0,∞] that sends negligible sets to 0 and the remaining sets to ∞. definition 2.5. (saturated msns) an msn (x, a , n ) is called saturated whenever the following property holds: for all n ∈ n , any subset n ′ ⊆ n is a -measurable – therefore, in fact, n ′ ∈ n . this property is of purely technical nature, as an msn (x, a , n ) that does not have it can be turned into a saturated msn (x, ā , n̄ ), by setting: n̄ = p(x) ∩{n̄ : n̄ ⊆ n for some n ∈ n }, ā = p(x) ∩{ā : a ā ∈ n̄ for some a ∈ a } = p(x) ∩{a n̄ : a ∈ a and n̄ ∈ n̄ }. here, denotes the symmetric difference of sets. we call ( x, ā , n̄ ) the saturation of (x, a , n ). in case the original msn corresponds to a measure space 12 p. bouafia, t. de pauw (x, a ,µ), its saturation corresponds to the measure space usually referred to as the completion of (x, a ,µ). we will denote the latter by ( x, ā , µ̄ ) . definition 2.6. let (x, a , n ) and (y, b, m ) be two msns. we say that a map f : x → y is [(a , n ), (b, m )]-measurable (or simply measurable) if (1) f is (a , b)-measurable; (2) f−1(m) ∈ n for every m ∈ m . it is easy to check that measurability in the above sense is preserved by composition. definition 2.7. (morphisms of saturated msns) let (x, a , n ) and (y, b, m ) be two saturated msns. a morphism from (x, a , n ) to (y, b, m ) is an equivalence class of [(a , n ), (b, m )]-measurable maps under the relation ∼ of equality almost everywhere: f ∼ f ′ whenever {f 6= f ′}∈ n . in order to check that this relation is, indeed, transitive, it is important to assume that (x, a , n ) is saturated for otherwise we would not know that {f 6= f ′′} ∈ a when f,f ′′ : x → y are both [(a , n ), (b, m )]-measurable. also, in the special case where x is n -negligible and y = ∅, we follow the convention that there is unique morphism from (x, a , n ) to (∅,{∅},{∅}). lemma 2.8. let (x, a , n ), (y, b, m ) and (z, c , p) be msns and let f,f ′ : x → y and g,g′ : y → z be maps. if (a) f,f ′ are [(a , n ), (b, m )]-measurable; (b) g,g′ are [(b, m ), (c , p)]-measurable; (c) (x, a , n ) is saturated; (d) f ∼ f ′, g ∼ g′, then g ◦f, g′ ◦f ′ are [(a , n ), (c , p)]-measurable and g ◦f ∼ g′ ◦f ′. proof. the first conclusion follows from hypotheses (a) and (b) and paragraph 2.6. the second conclusion is a consequence of {g ◦f 6= g′ ◦f ′}⊆{f 6= f ′}∪f−1({g 6= g′}) and hypotheses (a), (c) and (d). radon-nikodýmification of arbitrary measure spaces 13 definition 2.9. (category msn) thanks to the preceding result, there is a notion of composition for morphisms between saturated msns: if f : (x, a , n ) → (y, b, m ) and g : (y, b, m ) → (z, c , p) are morphisms, we let g ◦ f : (x, a , n ) → (z, c , p) be the equivalence class of g ◦f where f ∈ f and g ∈ g. this allows to define the category msn whose objects are saturated msns and whose morphisms are described in the paragraph 2.7. additionally, we add the convention that, for a negligible saturated msn, i.e. an msn of the form ( x, p(x), p(x) ) , there is a unique morphism from ( x, p(x), p(x) ) to ( ∅,{∅},{∅} ) . this way, negligible saturated msns are isomorphic to one another in the category msn. the categorical point of view is rarely considered in measure theory, mainly due to the lack of a well-behaved notion of morphism between measure spaces. the category msn also appears in the work [14] under the name strictems. we start to investigate the existence of limits and colimits in this category. definition 2.10. (submsn) let (x, a , n ) be an msn and z∈p(x). we define the submsn (z, az, nz), where az := {a ∩ z : a ∈ a } and nz := {n ∩ z : n ∈ n }. note that in the special case where z is a -measurable, we have az = a ∩{a : a ⊆ z} and nz = n ∩{n : n ⊆ z}. the inclusion map ιz : z → x is [(az, nz), (a , n )]-measurable and induces a morphism ιz between (z, az, nz) and (x, a , n ). proposition 2.11. let f,g : (x, a , n ) → (y, b, m ) be a pair of morphisms in the category msn, represented by the maps f ∈ f and g ∈ g, and set z := {f = g}. then the equalizer of f,g is [(z, az, nz),ιz]. proof. as f ◦ ιz = g ◦ ιz, we have clearly f ◦ ιz = g ◦ ιz. let h be any other morphism (t, c , p) → (x, a , n ) that satisfies the relation f ◦ h = g ◦ h and let h ∈ h. then h−1(z) is conegligible in t. up to modifying h, we can suppose that it has values in z. the restriction h′ : t → z of h is [(c , p), (az, nz)]-measurable and we have h = ιz ◦ h′, leading to a factorization h = ιz ◦ h′. this factorization is unique, as any morphism h′ satisfying h = ιz ◦ h′ must derive from a map h′ : t → z that coincides almost everywhere with h. proposition 2.12. the category msn has coproducts. consider a family of saturated msns 〈(xi, ai, ni)〉i∈i. its coproduct is the msn (x, a , n ) whose underlying set is x = ∐ i∈i xi, and whose σ-algebra and σ-ideal are 14 p. bouafia, t. de pauw defined by a = p(x) ∩{a : a∩xi ∈ ai for all i ∈ i}, n = p(x) ∩{n : n ∩xi ∈ ni for all i ∈ i}. for i ∈ i, the canonical morphism ιi : (xi, ai, ni) → (x, a , n ) is the morphism induced by the inclusion map ιi : xi → x. proof. notice that, indeed, (x, a , n ) is a saturated msn. let (y, b, m ) be a saturated msn and 〈fi〉i∈i be a collection of morphisms from (xi, ai, ni) to (y, b, m ), each fi being represented by a measurable map fi. we set f = ∐ i∈i fi, the map such that f ◦ ιi = fi for any i ∈ i. it is clear that f is [(a , n ), (b, m )]-measurable and f ◦ ιi = fi holds for all i ∈ i. we need to show that f is the unique morphism (x, a , n ) → (y, b, m ) with this property. suppose g : (x, a , n ) → (y, b, m ) is another morphism, represented by a measurable map g : x → y , for which g ◦ ιi = fi for all i ∈ i. then f and g coincide almost everywhere on each xi, which implies, due to the choice of n , that f and g are equal almost everywhere and f = g. proposition 2.13. the category msn has countable products. let 〈 (xi, ai, ni) 〉 i∈i be a countable family of saturated msns. its product is the msn (x, a , n ), whose underlying set is the product x = ∏ i∈i xi, whose σ-ideal is n = p(x) ∩ { n : ∃〈ni〉i∈i ∈ ∏ i∈i ni, n ⊆ ⋃ i∈i π−1i (ni) } , where πi : x → xi denotes the projection map, and whose σ-algebra is the saturation of ⊗ i∈i ai: a = { a n : a ∈ ⊗ i∈i ai and n ∈ n } . for i ∈ i, the projection morphism πi : (x, a , n ) → (xi, ai, ni) is the map induced by the projection map πi. proof. first note that, by construction of a and n , the projection maps πi are [(a , n ), (ai, ni)]-measurable. let (y, b, m ) be a saturated msn radon-nikodýmification of arbitrary measure spaces 15 and 〈fi〉i∈i be a collection of morphisms from (y, b, m ) to (xi, ai, ni), each fi being represented by a measurable map fi : y → xi. we define f = ∏ i∈i fi : y → ∏ i∈i xi that assigns y ∈ y to 〈fi(y)〉i∈i. clearly, f is (b, ⊗ i∈i ai)-measurable. moreover, for any negligible set n ∈ n , we can find a sequence 〈ni〉i∈i such that n ⊆ ⋃ i∈i π −1 i (ni). thus f−1(n) ⊆ ⋃ i∈i (πi ◦f)−1(ni) = ⋃ i∈i f−1i (ni). as i is countable and f−1i (ni) ∈ m for all i ∈ i, we find that f −1(n) ∈ m , which entails that the map f is [(b, m ), (a , n )]-measurable. let g : (y, b, m ) → (x, a , n ) be another morphism satisfying the identities πi ◦ g = fi for i ∈ i. let g : y → x be a representative of g. the coordinate functions πi◦g must coincide with fi almost everywhere. as there are only countably many of them, we conclude that f and g are equal almost everywhere, that is, f = g. remark 2.14. in case (xi, ai, ni) are associated with measure spaces (xi, ai,µi), i = 1, 2, the σ-ideal n considered in the above proposition may not coincide with nµ1⊗µ2 . this is the case, for instance, when (xi, ai,µi) = (r, b(r), l 1), i = 1, 2, since the diagonal d = r2 ∩{(x,x) : x ∈ r}∈ b(r)⊗b(r) is l 2-negligible but does not belong to n . 3. supremum preserving morphisms 3.1. (motivation) one of the reasons we were led to introduce msns is that the category of measure spaces and (equivalence classes of) measure preserving measurable maps does not have good properties at all. roughly speaking, this can be attributed to the fact that it has very few arrows. one way to increase their number is to define as morphisms (x, a ,µ) → (y, b,ν) the (a , b)-measurable maps ϕ: x → y such that the pushforward measure ϕ#µ is absolutely continuous with respect to ν. if we drop the measures and retain only which sets have measure zero, we get the notion of [(a , nµ), (b, nν)]measurability of 2.6. however, doing so, we may introduce some “irregular” maps. for example, if a is the σ-algebra of lebesgue measurable sets of the real line, l 1 the lebesgue measure and ν the counting measure on (r, a ) then the identity map induces a morphism (r, a , l 1) → (r, a ,ν). but, l 1 does not really compare to ν, although it is absolutely continuous with respect 16 p. bouafia, t. de pauw to ν. for instance, l 1 has no radon-nikodým density with respect to ν, not even in the generalized sense of section 9. forgetting the measures, the morphism of msns (r, a , nl 1 ) → (r, a ,{∅}) is still somehow inappropriate. to avoid this, we restrict our attention to the supremum preserving morphisms introduced below. this will allow us to define a new category msnsp of saturated msns with supremum preserving morphisms. later, we will be able to define localizable versions of msns and similar notions by means of universal properties to be satisfied in msnsp. definition 3.2. (boolean algebras) many of the properties that we will introduce underneath for msns are related to their boolean algebra, defined in the following way: given an msn (x, a , n ), we observe that the σ-algebra a is a boolean algebra and n is an ideal of a in the ring-theoretic sense; we then associate to (x, a , n ) the quotient boolean algebra a /n . when we restrict our attention to saturated msns, this construction becomes functorial. call bool(x, a , n ) = a /n the boolean algebra of a saturated msn. given a morphism f : (x, a , n ) → (y, b, m ) represented by a measurable map f : x → y , we define bool(f) : bool(y, b, m ) → bool(x, a , n ) that maps the equivalence class of b ∈ b to the equivalence class of f−1(b). this map is well-defined because of the [(a , n ), (b, m )]-measurability of f, it is a morphism of boolean algebras, and it does not depend on the representative of f, as one can easily check. definition 3.3. let (x, a , n ) be an msn and e be a subcollection of a . we say that u ∈ a is an n -essential upper bound of e whenever e \ u ∈ n for all e ∈ e . furthermore, a measurable set s ∈ a is an n -essential supremum of e whenever (1) s is an n -essential upper bound of e ; (2) if s′ is an n -essential upper bound of e , then s \s′ ∈ n . in particular, if s, s′ are both n -essential suprema of e , their symmetric difference s s′ is negligible. in other words, an essential supremum, when it exists, is unique up to negligible sets. in fact, it corresponds to a (unique) supremum in bool(x, a , n ). a collection e ⊆ a that admits x as an n -essential supremum is called n -generating. we will use the following radon-nikodýmification of arbitrary measure spaces 17 repeatedly. if e ⊆ a and s ∈ a is an n -essential supremum of e , then e ∪{x \s} is n -generating. the next ubiquitous lemma expresses that ∩ is distributive over the (partially defined) operation of taking essential suprema. it implies the following fact, which we will use frequently: if e is n -generating and a ∈ a \ n , then e ∩a 6∈ n for some e ∈ e . lemma 3.4. (distributivity lemma) let (x, a , n ) be an msn, e ⊆ a be a collection that has an n -essential supremum s, and c ∈ a . then c ∩s is an n -essential supremum of {c ∩e : e ∈ e}. proof. condition (1) in definition 3.3 is met because c ∩e \c ∩s = c ∩ (e \s) ∈ n for all e ∈ e . as for (2), we let s′ be an n -essential upper bound for {c ∩ e : e ∈ e}. we claim that s′′ := s′ ∪ (x \ c) is an n -essential upper bound for e . indeed, for any e ∈ e , we have e \s′′ = (c ∩e) \s′ ∈ n . it follows that (c ∩s) \s′ = s \s′′ ∈ n . 3.5. note that if (x, a , n ) and (y, b, m ) are msns, f : x → y is [(a , n ), (b, m )]-measurable, e ⊆ b, and s ∈ b is an m -essential upper bound of e , then f−1(s) is an n -essential upper bound of f−1(e ). however, if s is an m -essential supremum of e then f−1(s) may not be an n -essential supremum of f−1(e ). consider, for instance, (x, a , n ) = (r, b(r), nl 1 ), (y, b, m ) = (r, b(r),{∅}), f = idr, and e = {{x} : x ∈ r}. then r is an {∅}-essential supremum of e , ∅ is an nl 1 -essential supremum of e = f−1(e ), and r\∅ 6∈ nl 1 . 3.6. there are several objects that we can call supremum preserving. for saturated msns (x, a , n ) and (y, b, m ), we define • a morphism of boolean algebras ϕ: a → b is called supremum preserving if, for any family e ⊆ a that admits a supremum, the family ϕ(e) admits a supremum and ϕ(sup e) = sup ϕ(e). • a morphism f : (x, a , n ) → (y, b, m ) is called supremum preserving whenever bool(f) is. • an [(a , n ), (b, m )]-measurable map f : x → y is called supremum preserving if, for any collection e ⊆ b with an m -essential supremum s, f−1(s) is an n -essential supremum of f−1(e ) := {f−1(e) : e ∈ e}. 18 p. bouafia, t. de pauw for a morphism f represented by f ∈ f, the supremum preserving characters of f, f and bool(f) are all equivalent. also, the composition of two supremum preserving morphisms is supremum preserving. we call msnsp the subcategory of msn that consists of saturated msns and supremum preserving morphisms. in the next proposition, we gather some basic facts about supremum preserving morphisms and the category msnsp. proposition 3.7. the following hold: (a) two saturated msns are isomorphic in msn if and only if they are isomorphic in msnsp. (b) let (x, a , n ) be a saturated msn and z ∈ a . the morphism ιz : (z, az, nz) → (x, a , n ) induced by the inclusion map ιz : z → x is supremum preserving. (c) let f,g : (x, a , n ) → (y, b, m ) be a pair of morphisms in msnsp, represented by f ∈ f and g ∈ g. if z := {f = g} is a -measurable, then( (z, az, nz),ιz ) is the equalizer of f,g in msnsp. (d) the category msnsp has coproducts, which are preserved by the forgetful functor msnsp → msn. proof. (a) let f be an isomorphism in msn. then bool(f) is an isomorphism of boolean algebras. more specifically, it is an isomorphism of posets and for this reason it preserves suprema. (b) this is the content of lemma 3.4. (c) by proposition 2.11, ((z, az, nz),ιz) is the equalizer of f,g in msn and by (b) the morphism ιz is a morphism of msnsp. let h: (t, c , p) → (x, a , n ) be a supremum preserving morphism that satisfies f ◦ h = g ◦ h. recalling the proof of proposition 2.11, there is a representative h ∈ h with values in z, and its restriction h′ : t → z induces the unique morphism h′ such that h = ιz ◦ h′. the results follows from the fact that h′ is easily checked to be supremum preserving. (d) let 〈(xi, ai, ni)〉i∈i be a family of saturated msns, (x, a , n ) be their coproduct in the category msn, and 〈fi〉i∈i be a family of supremum preserving morphisms from (xi, ai, ni) to a saturated (y, b, m ), each represented by fi : xi → y . we need to show that f := ∐ i∈i fi : x → y is supremum preserving. for this, let e ⊆ b be a collection that has an m -essential supremum s. we observe that f−1(s) = ∐ i∈i f −1 i (s) is an n -essential upper bound of f−1(e ). let u be a second n -essential radon-nikodýmification of arbitrary measure spaces 19 upper bound of f−1(e ). then xi ∩ u is an ni-essential upper bound of {xi ∩f−1(e) : e ∈ e} = f−1i (e ). it follows that f −1 i (s)\(xi ∩u) ∈ ni. as this happens for all i ∈ i, we get that f−1(s) \u ∈ n . 4. localizable, 4c and strictly localizable msns definition 4.1. (localizable msn) an msn (x, a , n ) is localizable whenever each collection e ⊆ a admits an n -essential supremum. equivalently, (x, a , n ) is localizable whenever its boolean algebra a /n is dedekind complete, that is, each subset of a /n has a supremum. originally, localizability was introduced by segal [15] in the context of measure spaces. since then, many minor variations over the definition in that context have been proposed (see [13] for an overview). we will follow the definition in [6, chapter 2]. a measure space (x, a ,µ) is called localizable whenever (1) it is semi-finite, i.e. for all a ∈ a with µ(a) > 0, there is a measurable set a′ ⊆ a such that 0 < µ(a′) < ∞; (2) the underlying msn (x, a , nµ) is localizable. 4.2. (semi-finite measure space) in the definition of localizable measure space, semi-finiteness plays on important rôle. let us rephrase it. given (x, a ,µ) a measure space, we abbreviate a f := a ∩{e : µ(e) < ∞}. we say that n ∈ a is locally µ-negligible whenever n∩e ∈ nµ for every e ∈ a f . we let nµ,loc be the σ-ideal consisting of locally µ-negligible measurable sets. the following are equivalent: (1) nµ = nµ,loc. (2) (x, a ,µ) is semi-finite. (3) a f is nµ-generating. the only non trivial part is (3) ⇒ (1). if n ∈ a then n is an nµ-essential supremum of { n ∩ f : f ∈ a f } , according to the distributivity lemma 3.4. if also n ∈ nµ,loc, then it follows that n ∈ nµ. the notion of locally µ-negligible sets will appear again in 5.2. next we introduce some classes of localizable msns that will appear throughout the paper. 20 p. bouafia, t. de pauw 4.3. (countable chain condition) let (x, a , n ) be an msn. a family e ⊆ a \ n is called almost disjointed whenever e ∩ e′ ∈ n for any pair of distinct e,e′ ∈ e . the msn (x, a , n ) is said to have the countable chain condition (in short: is ccc) whenever an almost disjointed family in a \ n is at most countable. the previous notions have counterparts in the realm of boolean algebras. given a boolean algebra a, a subset e ⊆ a is called disjointed whenever x∧y = 0 for any pair of distinct elements x,y ∈ e. the boolean algebra a has the countable chain condition (or: is ccc) whenever each of its disjointed families is at most countable. of course, an msn (x, a , n ) is ccc if and only if its boolean algebra a /n is. in the following proposition, we show that being ccc is stronger than localizability. it is related to the fact, first established in [18], that a dedekind σ-complete boolean algebra (that is, a boolean algebra where countable collections have suprema) having the countable chain condition is dedekind complete. proposition 4.4. if an msn (x, a , n ) is ccc and e ⊆ a is a collection, then there is a countable subcollection e ′ ⊆ e such that ⋃ e ′ is an n -essential supremum of e . in particular, (x, a , n ) is localizable. proof. suppose the existence of a collection e ⊆ a for which one cannot find a countable subcollection e ′ ⊆ e whose union is an n -essential supremum of e . this assumption allows us to construct transfinitely a sequence 〈eα〉α<ω1 with values in e such that for every α < ω1, one has fα := eα \ ⋃ β<α eβ 6∈ n . but the disjointed family {fα : α < ω1} contradicts the fact that (x, a , n ) is ccc. proposition 4.5. let (x, a ,µ) be a finite measure space. then the space (x, a , nµ) is ccc. proof. let e ⊆ a \ n be an almost disjointed family. for each positive integer n, set en = e ∩{e : µ(e) > n−1}. as µ(x) > µ ( ⋃ en) > n −1 card en, we have that en is finite. consequently, e is at most countable. proposition 4.6. a coproduct ∐ i∈i(xi, ai, ni) of saturated localizable msns is localizable. proof. let e be a collection of measurable sets of some coproduct∐ i∈i(xi, ai, ni). for each i ∈ i, the collection ei := {xi ∩ e : e ∈ e} radon-nikodýmification of arbitrary measure spaces 21 has an ni-essential supremum si ⊆ xi. we then routinely check that s := ∐ i∈i si ∈ a is an n -essential supremum of e . definition 4.7. (stronger notions of localizability) an msn is called strictly localizable if it is isomorphic to a coproduct of the form∐ i∈i(xi, ai, nµi), where (xi, ai,µi) are complete finite measure spaces. examples of strictly localizable msns are provided by msns associated to complete σ-finite measure spaces (x, a ,µ). indeed, denoting 〈xi〉i∈i a countable partition of x into measurable subsets of finite µ measure, one can verify that (x, a , nµ) is isomorphic to ∐ i∈i(xi, axi, nµ xi). likewise, we say that an msn is cccc (abbreviated 4c) whenever it is isomorphic to a coproduct of saturated ccc msns. we have the chain of implications strictly localizable =⇒ 4c =⇒ localizable. the first implication comes from proposition 4.5, the second one from propositions 4.4 and 4.6. examples of non localizable spaces are provided by the next results. lemma 4.8. let (x, a , n ) be a localizable msn, e ⊆ a \n an almost disjointed family. then card(a /n ) > 2card e . proof. consider the application p(e ) → a /n which maps each subcollection e ′ ⊆ e to the equivalence class of its n -essential supremum. we claim that this map is injective. indeed, suppose e ′, e ′′ ⊆ e are distinct. call s′ (resp. s′′) an n -essential supremum of e ′ (resp. e ′′). without loss of generality, there is f ∈ e ′ \ e ′′. by lemma 3.4, f ∩ s′ (resp. f ∩ s′′) is an essential supremum of {f ∩ e : e ∈ e ′} (resp. {f ∩ e : e ∈ e ′′}). we deduce that f \s′ ∈ n and, taking the almost disjointed character of e into account, that f ∩ s′′ ∈ n . this implies that s′ and s′′ do not induce the same equivalence class in a /n . we will use the following many times. lemma 4.9. let (x, a , n ) be an msn and let c ⊆ a be n -generating. there exists e ⊆ a \ n with the following properties. (a) e is almost disjointed. (b) for each e ∈ e , there exists c ∈ c such that e ⊆ c. (c) e is n -generating. 22 p. bouafia, t. de pauw proof. there is no restriction to assume that n 6= a ; in particular, c 6= ∅. consider the set e consisting of those e ⊆ a \ n that satisfy conditions (a) and (b) above, ordered by inclusion. thus, e is nonempty and one readily checks that every chain in e possesses a maximal element. therefore, e admits a maximal element e , according to zorn’s lemma. we ought to show that e is n -generating. if this were not the case, there would exist an n -essential upper bound u ∈ a of e such that x \u 6∈ n . the latter, together with the fact that c is n -generating, implies the existence of c ∈ c such that c ∩ (x \u) 6∈ n . then, e ∪{c ∩ (x \u)} contradicts the maximality of e . proposition 4.10. (zfc + ch) let x be a polish space endowed with its borel σ-algebra b(x) and µ: b(x) → [0,∞] be a semi-finite borel measure. under the continuum hypothesis, one has the following dichotomy: either µ is σ-finite, or the msn (x, b(x), nµ) is not localizable. proof. let e be associated with c := b(x) ∩ {a : µ(a) < ∞} in lemma 4.9. recall 4.2 that c is nµ-generating. if e is countable, then⋃ e is measurable and, accordingly, an nµ-essential upper bound of e . thus x \ ⋃ e ∈ nµ, since e is nµ-generating. we have proven that µ is σ-finite. on the other hand, if e is uncountable, the continuum hypothesis guarantees that it has cardinal greater or equal to c. assume if possible that (x, b(x),µ) is localizable. as the map b(x) → b(x)/nµ is onto, we deduce from lemma 4.8 that card b(x) > 2c > c. however, borel sets are suslin, and suslin sets are continuous images of closed subsets of a particular polish space, the baire space, see e.g [17, 3.3.18]. this gives the upper bound card b(x) 6 c, contradicting the preceding inequality. definition 4.11. (p-version of an msn) let p be a property associated to msns. we suppose that the property p is hereditary: if (x, a , n ) has p then the msns ( z, az, nz ) also has p for all z ∈ a . “being strictly localizable”, “being 4c” or “being localizable” are examples of hereditary properties. let (x, a , n ) be a saturated msn. we define a p-version of (x, a , n ) to be a couple [( x̂, â , n̂ ) ,p ] consisting of a saturated msn ( x̂, â , n̂ ) with the property p and a supremum preserving morphism p: ( x̂, â , n̂ ) → (x, a , n ) satisfying the following property: for any saturated msn (y, b, m ) with the property p and any supremum preserving morphism q: (y, b, m ) → (x, a , n ), there radon-nikodýmification of arbitrary measure spaces 23 is a unique supremum preserving morphism r: (y, b, m ) → ( x̂, â , n̂ ) such that q = p◦r. (y, b, m ) ( x̂, â , n̂ ) (x, a , n ) q ∃!r p by this definition, a p-version must satisfy a universal property, and as such it is unique up to a unique isomorphism of the category msnsp. more specifically, if [( x̂, â , n̂ ) ,p ] and [( x̂′, â ′, n̂ ′ ) ,p′ ] are two p-versions, then we easily check that there is a unique isomorphism r: ( x̂, â , n̂ ) → ( x̂′, â ′, n̂ ′ ) such that p′ ◦r = p. definition 4.12. (atomic msns) one of our motivations in this article is to find a universal construction that transforms an msn into something with better localizability properties. as such, it is wise to first have a look at the not so easy case of msns (x, a , n ) such that all singletons are a -measurable and n = {∅}. we call such msns atomic. in an atomic msn (x, a ,{∅}), it is easy to see that a subset e ⊆ a has an {∅}-essential supremum if and only if ⋃ e ∈ a , in which case ⋃ e is the {∅}-essential supremum. therefore the msn (x, a ,{∅}) is localizable if and only if a = p(x). in other words, the non localizability of (x, a ,{∅}) can only be due to the lack of measurable sets; therefore it seems sensible to ask for (x, p(x),{∅}) to be the “localization” of (x, a ,{∅}). unfortunately, proposition 4.13 gives a negative result. it tells us that the localizable version of an msn, as defined in 4.11, is not the right notion of “localization”. this issue will be addressed in section 5 by introducing a notion of local determination for msns. proposition 4.13. let x be an uncountable set, c (x) be its countablecocountable σ-algebra. let ι : (x, p(x),{∅}) → (x, c (x),{∅}) be the morphism induced by the identity map. then [ (x, p(x),{∅}),ι ] is not a localizable version of (x, c (x),{∅}). proof. that ι is supremum preserving follows from the discussion in paragraph 4.12. assume if possible that ((x, p(x),{∅}),ι) is a localizable version of (x, a ,{∅}). 24 p. bouafia, t. de pauw we will get a contradiction if we manage to build a localizable saturated msn (y, b, m ) and a function q : y → x that is [(b, m ), (c (x),{∅})]measurable, supremum preserving, but not (b, p(x))-measurable. we choose y = x2 ×{0, 1}. for any subset b ⊆ y , we call b[0] and b[1] the subsets defined by b[i] := x2 ∩{(x1,x2) : (x1,x2, i) ∈ b} for i ∈{0, 1} we let b = p(y ) ∩{b : b[0] b[1] is countable}. we claim that b is a σ-algebra of y . the stability of b under countable unions is a consequence of the formula(⋃ n∈n bn ) [0] (⋃ n∈n bn ) [1] ⊆ ⋃ n∈n bn[0] bn[1] that holds for any sequence 〈bn〉n∈n of subsets in y , and we leave the other points to the reader. finally, we define the σ-ideal m := b∩{m : m[0] = ∅}. clearly (y, b, m ) is a saturated msn. let us show that (y, b, m ) is localizable. let e ⊆ b be any collection. we set a := ⋃ e∈e e[0] and s := a ×{0, 1}. the set s is b-measurable, because s[0] = s[1] = a. for any e ∈ e , we have (e\s)[0] = e[0]\s[0] = ∅, meaning that s is an m -essential upper bound of e . denoting by u another essential upper bound of e , then (e \u)[0] = e[0] \u[0] = ∅ for all e ∈ e . it follows that a ⊆ u[0] and (s \ u)[0] = s[0] \ u[0] = ∅. thus, s is an m -essential supremum, as we wanted. now, let σ : x → x be a bijection of x without fixed points. for example, choose a partition x = z∪z′ into subsets z,z′ that have the same cardinality as x, choose a bijection f : z → z′ and set σ so that σ(x) = f(x) for all x ∈ z and σ(x) = f−1(x) for all x ∈ z′. we define the map q : y → x by ∀(x1,x2, i) ∈ y, q(x1,x2, i) = { x2 if i = 1 and x2 = σ(x1), x1 otherwise . first we show that q is [(b, m ), (c (x),{∅})]-measurable. it suffices to show that q−1({x}) ∈ b for all x ∈ x. but we have q−1({x})[0] = {x}×x, q−1({x})[1] = ( {x}× (x \{σ(x)}) ) ∪{(σ−1(x) ,x)}. radon-nikodýmification of arbitrary measure spaces 25 consequently, q−1({x})[0] q−1({x})[1] has only two elements. by the definition of b, this ensures the measurability of q−1({x}). however, we claim that q is not (b, p(x))-measurable. to this end, we will show that q−1(z) 6∈ b. we have q−1(z)[0] = z ×x, q−1(z)[1] = {(x1,x2) : x1 ∈ z,x2 6= σ(x1)}∪{(σ−1(x),x) : x ∈ z}. it follows that q−1(z)[0] q−1(z)[1] ={(x,σ(x)) :x ∈ z}∪{(σ−1(x),x) :x ∈ z} is uncountable. thus, q−1(z) 6∈ b. it only remains to prove that q is supremum preserving. let e ⊆ c (x) be a collection that has an {∅}-essential supremum s. this implies that s = ⋃ e . first suppose that e consists only of singletons. we wish to prove that q−1(s) is an m -essential supremum of q−1(e ) = {q−1{x} : x ∈ s}. of course, q−1(s) is an m -essential upper bound of q−1(e ). let u an arbitrary m -essential upper bound of q−1(e ). for all x ∈ s, we have q−1{x}\u ∈ m , meaning that {x}× x = (q−1{x})[0] ⊆ u[0]. thus s × x ⊆ u[0], which implies (q−1(s) \ u)[0] = q−1(s)[0] \ u[0] = s × x \ u[0] = ∅. it means that q−1(s) \ u ∈ m . thus, we have shown that q−1(s) is an m -essential supremum of e . now we turn to the general case, where e need not consist only of singletons. let e ′ = {{x} : x ∈ e ∈ e}. clearly, e and e ′ have the same {∅}-essential supremum s := ⋃ e = ⋃ e ′. by what precedes, q−1(s) is an m -essential supremum of q−1(e ′) and it is an m -essential upper bound of q−1(e ). an m -essential upper bound u of q−1(e ) is also an upper bound for q−1(e ′), as any member of q−1(e ′) is a subset of a member of q−1(e ). therefore, q−1(s) \ u ∈ m , showing that q−1(s) is an m -essential supremum of q−1(e ). 4.14. (example of an msn with no localizable part) consider an msn of the form (x, p(x), k (x)), where x is a set of cardinality ℵ1 and k (x) is the σ-ideal of countable subsets. there is a bijection ϕ: x → x×x and we can use it to construct an uncountable family of “horizontal lines” hx := ϕ −1(x ×{x}) indexed by x ∈ x witnessing that (x, p(x), k (x)) is not ccc. actually, we can do better and prove that it is not localizable. suppose {hx : x ∈ x} has a k (x)-essential supremum s. for each x ∈ x choose a point px ∈ s∩hx. then it is easy to see that u := s\{px : x ∈ x} is an essential upper bound for the family of horizontal lines, however 26 p. bouafia, t. de pauw s \ u = {px : x ∈ x} is not negligible, contradicting that s is an essential supremum. observe that the msn (x, p(x), k (x)) is isomorphic to all its non negligible submsns. in particular, it has no nontrivial ccc or localizable part, an unpleasant situation that we will rule out in the next paragraph by introducing the notions of “locally localizable” and “locally ccc” msn. we will prove nonetheless that ( x, p(x), k (x) ) has a 4c version, that is disappointingly the trivial msn (∅,{∅},{∅}) (with the only morphism from there to ( x, p(x), k (x) ) ). to establish this fact, one needs to prove that if (y, b, m ) is a 4c msn and f : (y, b, m ) → ( x, p(x), k (x) ) is a supremum preserving morphism, then y ∈ m (actually, the supremum preserving character of f will not be used). we can reduce to the case where (y, b, m ) is ccc. we reproduce an argument due to ulam [19], showing that there is a family 〈an,α〉n∈n,α<ω1 of subsets of x such that • for all n ∈ n, the family 〈an,α〉α<ω1 is disjointed; • for all α < ω1, the union ⋃ n∈n an,α is conegligible (that is, cocountable). any ordinal β < ω1 is countable, so we can select a sequence 〈kα,β〉α<β of distinct integers. let 〈xβ〉β<ω1 be an enumeration of all the elements in x. set an,α := {xβ : β > α and kα,β = n} for every n ∈ n and α < ω1. for distinct α,α′ < ω1, there cannot be some xβ ∈ an,α ∩ an,α′, for otherwise we would have kα,β = kα′,β. in addition, one has ⋃ n∈n an,α = {xβ : β > α} whose complement in x is the countable set {xβ : β 6 α}. now, fix a representative f ∈ f. the family 〈 f−1(an,α) 〉 α<ω1 being disjointed, the set cn := ω1 ∩ { α : f−1(an,α) 6∈ m } is countable for all n ∈ n. hence the existence of some α ∈ ω1 \ ⋃ n∈n cn. now we see that the set f−1 (⋃ n∈n an,α ) is both negligible and conegligible in y , which can happen only if y ∈ m . definition 4.15. let p be a hereditary property associated to msns. we say that an msn (x, a , n ) is locally p whenever one of the following equivalent statements holds: (a) the collection a ∩ { z : (z, az, nz) has the property p } is n -generating; radon-nikodýmification of arbitrary measure spaces 27 (b) for any y ∈ a \n there is z ∈ a \n such that z ⊆ y and (z, az, nz) has the property p. proof. (proof of the equivalence) (a) =⇒ (b) for y ∈ a \n , an application of lemma 3.4 gives that y is an essential supremum of{ y ∩z : z ∈ a and (z, az, nz) has the property p } . therefore, there must be some z ∈ a such that y ∩z 6∈ n and (z, az, nz) has the property p. the subset y ∩z ⊆ y establishes (b). (b) =⇒ (a) clearly, x is an n -essential upper bound of the collection a ∩ {z : (z, az, nz) has the property p}. let s be another upper bound. if x\s were not negligible, (b) gives the existence of some measurable z ∈ a \ n such that z ⊆ x \s and (z, az, nz) has the property p. but z \s = z 6∈ n , which contradicts that s is an essential upper bound. for instance, in a semi-finite measure space (x, a ,µ), any non negligible set a ∈ a \ nµ contains a measurable subset z of nonzero finite measure. by (b), this implies that the associated msn (x, a , nµ) is locally strictly localizable. we conclude this section with an important property of “local isomorphism” that holds for p-versions. proposition 4.16. let (x, a , n ) be a saturated msn and[( x̂, â , n̂ ) ,p ] a p-version of it. fix a representative map p ∈ p. for any f ∈ a , we set f̂ := p−1(f) and we call pf : ( f̂, â f̂ , n̂ f̂ ) → (f, af , nf ) the morphism induced by the restriction pf : f̂ → f of p. (a) [( f̂, â f̂ , n̂ f̂ ) ,pf ] is the p-version of (f, af , nf ); (b) if (f, af , nf ) has the property p, then pf is an isomorphism. proof. (a) since the property p is hereditary, we can assert that (f̂, â f̂ , n̂ f̂ ) has it. we also readily check that pf is supremum preserving. let q: (y, b, m ) → (f, af , nf ) be a supremum preserving morphism starting from a saturated msn with the property p. then ιf ◦q is a supremum preserving morphism ending in (x, a , n ). it has a lifting r: (y, b, m ) → ( x̂, â , n̂ ) . let r ∈ r and q ∈ q be representatives. as p(r(y)) = q(y) for m -almost all y ∈ y , we lose no generality in supposing that r has values in f̂ . calling its restriction r′ : y → f̂ , we see 28 p. bouafia, t. de pauw that pf ◦ r′ and q coincide m -almost everywhere. the induced morphism r′ provides a factorization of q through ( f̂, â f̂ , n̂ f̂ ) . to establish the uniqueness of this factorization, we proceed as follows. for any morphism r′′ such that pf ◦r′′ = q we notice that ιf ◦q = ιf ◦pf ◦r′′ = p◦ ιf̂ ◦r ′′. since this holds for r′ we obtain p ◦ ι f̂ ◦ r′ = p ◦ ι f̂ ◦ r′′ and, by uniqueness of the factorization relative to the universal property of ( x̂, â , n̂ ) , ι f̂ ◦r′ = ι f̂ ◦r′′. thus, r′ and r′′ coincide m -almost everywhere. (b) if (f, af , nf ) has property p, then obviously [ (f, af , nf ), id ] is a second p-version. from the uniqueness of the p-version, we obtain a isomorphism r: (f, af , nf ) → ( f̂, â f̂ , n̂ f̂ ) such that id = pf ◦ r, whence pf = r −1. 5. localizable locally determined msns in order to motivate the main definition in this section, we start with the following result, of which we can think as a way of testing whether an msn has a property p. for instance, if each f ∈ f corresponds to a ccc submsn, then (x, a , n ) is 4c. the difficulty in applying this proposition stems with both hypotheses: conditions (1) and (2) will be turned into a definition in 5.2, whereas condition (3), that f is disjointed rather than merely almost disjointed, calls for techniques that transform almost disjointed generating families (whose existence, in applications, follows from lemma 4.9) into partitions – see the proof of theorem 7.6 in case card e 6 c and the notion of compatible family of densities introduced in section 10. proposition 5.1. let (x, a , n ) be an msn and f ⊆ a . assume that (1) for every a ⊆ x the following holds:[ ∀f ∈ f : a∩f ∈ a ] ⇒ a ∈ a ; (2) for every n ⊆ x the following holds:[ ∀f ∈ f : n ∩f ∈ n ] ⇒ n ∈ n ; (3) f is a partition of x. then the msns (x, a , n ) and ∐ f∈f (f, af , nf ) are isomorphic in msnsp. radon-nikodýmification of arbitrary measure spaces 29 proof. we abbreviate (y, b, m ) for ∐ f∈f (f, af , nf ). since f is a partition of x, there is a canonical bijection ϕ : x → y . its inverse ϕ−1 is [(b, m ), (a , n )]-measurable, by definition of coproduct of msns. we now show that ϕ is [(a , n ), (b, m )]-measurable. given b ∈ b, we note that ϕ−1(b) = ⋃ f∈f b ∩f, whence ϕ −1(b) ∩f = b ∩f ∈ a for every f ∈ f , by definition of b. we infer from hypothesis (1) that ϕ−1(b) ∈ a . let m ∈ m . as above we infer from the definition of m that ϕ−1(m) ∩f ∈ n for every f ∈ f , whence ϕ−1(m) ∈ n , in view of hypothesis (2). in other words, (x, a , n ) and (y, b, m ) are isomorphic in msn. the conclusion follows from proposition 3.7(a). definition 5.2. we borrow the following definition from [6, 211h]. a measure space (x, a ,µ) is locally determined whenever it is semi-finite and, for every subset a ⊆ x,[ ∀e ∈ a f : a∩e ∈ a ] ⇒ a ∈ a , where, as usual, a f = a ∩{e : µ(e) < ∞}. the definition relies on the particular collection a f (which is nµ-generating, recall 4.2). this makes sense because we are dealing with a measure space. it is a rather good surprise that we can define an analogous notion of locally determined msns, by substituting for a f an arbitrary generating collection. namely, a saturated msn (x, a , n ) is called locally determined whenever the following holds. for every n -generating collection e ⊆ a and every a ⊆ x, [ ∀e ∈ e : a∩e ∈ a ] ⇒ a ∈ a . an msn that is both localizable and locally determined is called lld. the following is useful as well. we say that a saturated msn (x, a , n ) has locally determined negligible sets whenever the following holds. for every n -generating collection e ⊆ a and every n ⊆ x,[ ∀e ∈ e : n ∩e ∈ n ] ⇒ n ∈ n . we observe that if (x, a , n ) is locally determined, then it has locally determined negligible sets. indeed, let e ⊆ a and n ⊆ x be as above, we first infer from the local determinacy of (x, a , n ) that n ∈ a and, in turn from the distributivity lemma 3.4, that n is an n -essential supremum of {n ∩e : e ∈ e}. therefore, n ∈ n . 30 p. bouafia, t. de pauw next we prove some elementary properties concerning locally determined msns. in particular, the consistency between both notions of local determination (for complete semi-finite measure spaces and msns) is established in proposition 5.3(f). here, the semi-finiteness property of a measure space is critical as the following example shows. we consider h 1, the 1-dimensional hausdorff measure in r2 and a the σ-algebra consisting of h 1-measurable subsets of r2 in the sense of carathéodory. the following hold: (a) ∀a ⊆ r2 : [ ∀f ∈ a f : a∩f ∈ a ] ⇒ a ∈ a ; (b) the measure space (r2, a , h 1) is not semi-finite; (c) the (saturated) msn (r2, a , nh 1 ) does not have locally determined negligible sets and, in particular, is not locally determined. for (a), see for instance [2, 6.2]. for (b), see [8, 439h]. now (c) follows for example from [2, 4.4]. it follows from 4.2 that a f is not nh 1 -generating. proposition 5.3. the following hold. (a) being locally determined is a hereditary property. (b) being locally determined is a property invariant under isomorphisms in msnsp. (c) a saturated ccc msn is locally determined. (d) a coproduct of locally determined msns is locally determined. (e) a 4c msn is locally determined. (f) a complete semi-finite measure space (x, a ,µ) is locally determined (as a measure space) if and only if the msn (x, a , nµ) is locally determined. proof. (a) let (x, a , n ) be a locally determined msn and z ∈ a . let e be an nz-generating family in the submsn (z, az, nz) and a ⊆ z be such that e ∩a ∈ az for any e ∈ e . the family e ∪{x \z} is n -generating in (x, a , n ) and e ∩a ∈ a for all e ∈ e ∪{x \z}. it follows that a ∈ a . (b) let (x, a , n ) and (y, b, m ) be two saturated msns, f : x → y and g : y → x be two measurable supremum preserving maps that induce reciprocal isomorphisms. assume that (x, a , n ) is locally determined. let e ⊆ b be an m -generating collection and b ⊆ y be such that e ∩ b ∈ b for all e ∈ e . then f−1(e) ∩ f−1(b) = f−1(e ∩ b) ∈ a . as f is supremum preserving, f−1(e ) is n -generating. and as (x, a , n ) is locally radon-nikodýmification of arbitrary measure spaces 31 determined, we infer that f−1(b) ∈ a . therefore g−1(f−1(b)) ∈ b. but b g−1(f−1(b)) ∈ m and as (y, b, m ) is saturated we conclude that b ∈ b. (c) let (x, a , n ) be a saturated ccc msn, e ⊆ a an n -generating family and a ∈ p(x) be such that e ∩a ∈ a for all e ∈ e . by proposition 4.4, there is a countable subset e ′ ⊆ e that is n -generating. then a = ( ⋃ e∈e ′ e ∩a ) ∪ ( a\ ⋃ e ′ ) since x \ ⋃ e ′ is n -negligible and (x, a , n ) is saturated, we infer that a\ ⋃ e ′ is a -measurable. therefore, a ∈ a . (d) let (x, a , n ) be the coproduct of a family 〈(xi, ai, ni)〉i∈i of locally determined msns. it is readily saturated. let e ⊆ a be an n -generating family and a ⊆ x such that e ∩ a ∈ a for all e ∈ e . for all i ∈ i, the family ei := {e∩xi : e ∈ e} is ni-generating in (xi, ai, ni) by lemma 3.4. this observation leads to the fact that a ∩ xi ∈ ai for all i ∈ i, in other words, a ∈ a . (e) this obviously follows from (c) and (d). (f) suppose that the measure space (x, a ,µ) is locally determined. let e ⊆ a be an nµ-generating family and a ⊆ x be such that e ∩a ∈ a for all e ∈ e . let f ∈ a f . by lemma 3.4, the collection {f ∩ e : e ∈ e} is nf -generating in ( f, af , (nµ)f ) and of course f ∩ e ∩ a ∈ af for all e ∈ e . on top of that, ( f, af , (nµ)f ) is a ccc msn by proposition 4.5 and it is saturated. we get from (c) above that a ∩ f is measurable. as this happens for all f ∈ a f , we conclude that a ∈ a . conversely, suppose the msn (x, a , nµ) is locally determined. owing to the semi-finiteness of (x, a ,µ), the collection a f is nµ-generating. then (x, a ,µ) is easily seen to be locally determined: if a ∈ p(x) satisfies a∩f ∈ a for all f ∈ a f , then a ∈ a . 5.4. (lld version of an atomic msn) as a first result, we mention that the lld version of an atomic msn (x, a ,{∅}) is the space [(x, p(x),{∅}),ι], where ι is the morphism induced by the identity map (that ι is supremum preserving follows from 4.12). this amounts to prove that, for any lld msn (y, b, m ), a [(b, m ), (a ,{∅})]-measurable supremum preserving map q : y → x is automatically (b, p(x))-measurable. indeed, let s ∈ p(x). then q−1(s)∩q−1{x} is either q−1{x} or ∅, hence q−1(s) ∩ q−1{x} is b-measurable for every x ∈ x. besides, q is supremum preserving, thus the collection {q−1{x} : x ∈ x} is m -generating. by local determination in (y, b, m ) we conclude that q−1(s) ∈ b. 32 p. bouafia, t. de pauw 5.5. call lldsp the full subcategory of msnsp that consists of lld msns, and consider the forgetful functor forget: lldsp → msnsp. in categorical terms, an lld version is the coreflection of a saturated msn (x, a , n ) along the functor forget, see [1, chapter 3]. in this paper, we do not answer the question whether there exists an lld version for each saturated msn. this is equivalent to the existence of a right adjoint r of forget. as a matter of fact, if such an adjoint exists, there would be a natural transformation ε: forget ◦ r =⇒ idmsnsp such that the pair [ r(x, a , n ),ε(x,a ,n ) ] gives the lld version of any saturated msn (x, a , n ). in search for an abstract proof of the existence of r, one might think of using freyd’s adjunction theorem, [1, theorem 3.3.3]. following this path, one needs to establish (setting aside the solution set condition) that: (a) the category msnsp is cocomplete; (b) the forgetful functor forget preserves small colimits. assertion (a) boils down to showing that msnsp has two types of small colimits: coproducts and coequalizers. the existence of the former is shown in proposition 3.7(d). we do not know whether coequalizers exist in msnsp and it is the main difficulty here. as for (b), which, regarding the existence of lld versions, is a necessary condition even if (a) were to be false, we have already proven in propositions 4.6 and 5.3(d) that coproducts of lld msns are lld. our next goal is proposition 5.7 which states that coequalizers of lld msns are lld. before that, we need to introduce some notation and a lemma. for a saturated msn (x, a , n ) and an arbitrary e ⊆ a , we define ae := p(x) ∩{a : e ∩a ∈ a for all e ∈ e}, ne := p(x) ∩{n : e ∩n ∈ n for all e ∈ e}. it is clear that (x, ae , ne ) is a saturated msn. lemma 5.6. let (x, a , n ) be a localizable saturated msn and e ⊆ a an n -generating family. let ι be the morphism (x, ae , ne ) → (x, a , n ) induced by the identity map on x. then bool(ι) is an isomorphism. in particular, ι is supremum preserving and (x, ae , ne ) is localizable. proof. first we make the following observation, to be used later in the proof: a ∩ ne = n . indeed, if n ∈ a is such that e ∩ n ∈ n for all radon-nikodýmification of arbitrary measure spaces 33 e ∈ e , then by the distributivity lemma 3.4, we conclude that n ∈ n . this proves the inclusion a ∩ ne ⊆ n , the reciprocal being trivial. the identity map ι: x → x is [(ae , ne ), (a , n )]-measurable because a ⊆ ae and n ⊆ ne . let us show that bool(ι) : a /n → ae /ne is injective by inspecting its kernel. let a ∈ a /n be a class represented by a ∈ a such that bool(ι)(a) = 0, in other words a = ι−1(a) ∈ ne . then a ∈ a ∩ ne = n . this means that a = 0 in a /n . therefore, bool(ι) is injective. now let us show that bool(ι) is surjective. to this end, let h ∈ ae /ne be a class represented by h ∈ ae . we ought to prove that h is in the range of bool(ι). set f := {e ∩h : e ∈ e}. note that f ⊆ a . the localizability of (x, a , n ) guarantees that f has an n -essential supremum s ∈ a . in particular, e ∩h \s ∈ n for all e ∈ e , meaning that h \s ∈ ne . we also claim that s\h ∈ ne . indeed, let e0 ∈ e . set s′ := s\(e0∩s\h). we note that s′ ∈ a . for all e ∈ e , we have e ∩h \s′ = e ∩h \s ∈ n , as h \s ∈ ne . this means that s′ is an n -essential upper bound of f . it follows that s \ s′ = e0 ∩ s \ h ∈ n . as e0 ∈ e is arbitrary, we obtain s \h ∈ ne , as required. we proved that h s ∈ ne . calling s the equivalence class of s in a /n , we have that bool(ι)(s) = h. proposition 5.7. consider the following diagram in msnsp, where ((z, c , p),h) is the coequalizer of f,g. (x, a , n ) (y, b, m ) (z, c , p) f g h (a) if (y, b, m ) is localizable, so is (z, c , p). (b) if (y, b, m ) is lld, so is (z, c , p). proof. (a) let us call 2 the special msn ( {0, 1}, p({0, 1}),{∅} ) . first we show the following intermediate result: for any msn (x, a , n ), there is a one-to-one correspondence υx between the boolean algebra bool(x, a , n ) and the set of morphisms hom ( (x, a , n ),2 ) (those are automatically supremum preserving since the boolean algebra of 2 is finite). given a class a ∈ bool(x, a , n ), represented by a set a, the characteristic function 1a : x → {0, 1} induces a morphism 1a which only depends on the equivalence class a. indeed, if a′ is another representative of a, then 1a and 1a′ coincide n -almost everywhere. we set υx(a) := 1a. 34 p. bouafia, t. de pauw this map is surjective because each morphism ϕ: (x, a , n ) → 2 is represented by a map ϕ ∈ ϕ which has the form ϕ = 1ϕ−1({1}). it is injective because if 1a coincides with 1b almost everywhere, for measurable sets a and b, then a and b yield the same equivalence class in bool(x, a , n ). now we turn to the proof of conclusion (a). by naturality of υ, the following diagram is commutative, where hom(h,2), hom(f,2) and hom(g,2) denote the right composition with h, f, and g, respectively: hom((z, c , p),2) hom((y, b, m ),2) hom((x, a , n ),2) bool(z, c , p) bool(y, b, m ) bool(x, a , n ) hom(h,2) hom(f,2) hom(g,2) υz bool(h) υy bool(f) bool(g) υx we show that hom(h,2) is injective. indeed, if ϕ and ψ are such that hom(h,2)(ϕ) = hom(h,2)(ψ) then, upon letting k = ϕ ◦ h = ψ ◦ h, we infer that k ◦ f = k ◦ g. by the universal property of (z, c , p), there exists a unique ` ∈ hom ( (z, c , p),2 ) such that ` ◦ h = k. since ϕ and ψ have the property of `, we conclude that they coincide. similarly, the universal property of coequalizers tells us that the range of hom(h,2) consists of those morphisms k such that hom(f,2)(k) = hom(g,2)(k). on the second line of the diagram, these two observations translate to the fact that bool(h) induces an isomorphism of boolean algebras from bool(z, c , p) onto the boolean subalgebra a := bool(y, b, m ) ∩ { ξ : bool(f)(ξ) = bool(g)(ξ) } . it remains to prove that a is dedekind complete. let e ⊆ a be a collection. it has a supremum s in bool(y, b, m ), as (y, b, m ) is localizable. since f and g are supremum preserving, we have bool(f)(s) = sup bool(f)(e) = sup bool(g)(e) = bool(g)(s). hence s ∈ a and a is dedekind complete. (b) that (z, c , p) is localizable follows from (a). let g ⊆ c be any p-generating family. we wish to prove that cg ⊆ c . if we manage to do so, then (z, c , p) is locally determined, as g is arbitrary. let h be a representative of h. by definition, it is [(b, m ), (c , p)]measurable and supremum preserving. we claim that it is, in fact, [(b, m ), (cg , pg )]-measurable. indeed, let c ∈ cg . for all g ∈ g , we have g ∩ c ∈ c which implies that h−1(g) ∩ h−1(c) = h−1(g ∩ c) ∈ b. radon-nikodýmification of arbitrary measure spaces 35 moreover, h−1(g ) is m -generating, as g is p-generating and h is supremum preserving. thus, since (y, b, m ) is locally determined, we have that h−1(c) ∈ b. next, if p ∈ pg , then h−1(p) ∈ b by what precedes and h−1(p) ∩ h−1(g) = h−1(p ∩ g) ∈ m for all g ∈ g . by the distributivity lemma 3.4, we obtain h−1(p) ∈ m . denote as h′ : (y, b, m ) → (z, cg , pg ) the morphism induced by h, and denote as ι: (z, cg , pg ) → (z, c , g ) the morphism induced by the identity map idz. by lemma 5.6, we have bool(h ′) = bool(h) ◦ bool(ι)−1, which is the composition of two supremum preserving morphisms of boolean algebras. thus, h′ is a supremum preserving as well. also, we recall h ◦ f = h ◦ g. as h and h′ are induced by the same map, we deduce that h′◦f = h′◦g. by the universal property of coequalizers, there is a morphism k: (z, c , p) → (z, cg , pg ) such that h′ = k◦h. (z, cg , pg ) (x, a , n ) (y, b, m ) (z, c , p) ι f g h h′ k hence ι◦ k ◦ h = ι◦ h′ = h = id(z,c ,p) ◦h. the uniqueness in the universal property of equalizers implies that h is an epimorphism. thus ι◦k = id(z,c ,p). a representative k ∈ k must satisfy z = idz(k(z)) = k(z) for p-almost all z ∈ z, i.e. p = z ∩{z : z 6= k(z)} ∈ p. let c ∈ cg . since k is (c , cg )measurable, it follows that k−1(c) ∈ c . since k−1(c) c ⊆ p , we deduce that k−1(c) c ∈ c and, in turn, c ∈ c . 5.8. the last two results of this section show that the category lldsp has better categorical properties than msnsp: it has equalizers in full generality. this result is reminiscent of [6, 214ie]. proposition 5.9. let (x, a , n ) be an lld msn and y ⊆ x any subset. then the submsn (y, ay , ny ) is lld and the canonical morphism ιy : (y, ay , ny ) → (x, a , n ) is supremum preserving. proof. first we show that the map ιy : y → x is supremum preserving. let e ⊆ a and assume s ∈ a is an n -essential supremum of e . the set s ∩ y = ι−1y (s) is an ny -essential upper bound of ι −1 y (e ). let u ∈ ay be an arbitrary ny -essential upper bound of ι −1 y (e ). we ought to show that s∩y \u ∈ ny . for all e ∈ e , one has e∩s∩y \u ⊆ e∩y \u ∈ ny . as 36 p. bouafia, t. de pauw (x, a , n ) is saturated, ny ⊆ n and, also, e ∩s ∩y \u ∈ n . of course (x \s) ∩s ∩y \u = ∅ is also n -negligible. since the family e ∪{x \s} is n -generating and (x, a , n ) is locally determined, we deduce that s∩y \u is a -measurable and, in turn, that it is n -negligible by the distributivity lemma 3.4. the proof that ιy is supremum preserving is complete. since the morphism bool(ιy ) : a /n → ay /ny is onto and supremum preserving, and a /n is dedekind complete, so is ay /ny , meaning that (y, ay , ny ) is localizable. it remains to show that (x, a , n ) is locally determined. we claim the following: if e ⊆ ay is ny -generating and n ∈ p(y ) satisfies e ∩n ∈ ny for all e ∈ e , then n ∈ ny . by definition of ay , any set e ∈ e can be written as e = e′∩y , for some e′ ∈ a , so there is a subset e ′ ⊆ a such that e = ι−1y (e ′). the localizability of (x, a , n ) guarantees the existence of an n -essential supremum s of e ′. for all e′ ∈ e ′ one has e′∩n = e′∩y ∩n ∈ ny ⊆ n , because e′∩y ∈ e . also s∩y = ι−1z (s) is an ny -essential supremum of e = ι −1 z (e ′), by the first paragraph. recalling that e is ny -generating, we find that y \s = y \(s∩y ) ∈ ny . consequently, (x \s) ∩n ⊆ y \s ∈ ny ⊆ n . as (x, a , n ) is saturated, we find that (x \s) ∩n ∈ n . in conclusion, e′∩n ∈ n for any e′ that belongs to the n -generating family e ′ ∪{x \ s}. since (x, a , n ) is locally determined, we infer that n ∈ a and then that n ∈ n by the distributivity lemma 3.4. as n ⊆ y , we conclude that n ∈ ny . now let e ⊆ ay be an ny -generating collection and a ∈ p(y ) be such that e ∩ a ∈ ay for all e ∈ e . we want to prove that a ∈ ay . as (y, ay , ny ) is localizable, {e ∩ a : e ∈ e} has an ny -essential supremum s. this implies that e ∩ a \ s ∈ ny for all e ∈ e . by the claim above, a\s ∈ ny . fix e0 ∈ e . note that e0 ∩ (s \a) = (e0 ∩s) \ (e0 ∩a) ∈ ay . also, e ∩a\ ((s \ (e0 ∩s \a)) = e ∩a∩ ((y \s) ∪ (e0 ∩s \a)) = e ∩a\s ∈ ny , for all e ∈ e . in other words, s \ (e0 ∩ s \ a) is an ny -essential upper bound of {e ∩a : e ∈ e}. as s is an ny -essential supremum of this family, s \ (s \ (e0 ∩s \a)) = e0 ∩s \a ∈ ny . applying again the claim above, we deduce that s \ a ∈ ny from the arbitrariness of e0. summing up, a s ∈ ny . as s ∈ ay we infer that a ∈ ay . the proof that (y, ay , ny ) is locally determined is now complete. radon-nikodýmification of arbitrary measure spaces 37 corollary 5.10. lldsp has equalizers preserved by the forgetful functor lldsp → msn. proof. consider a pair of supremum preserving morphisms f,g : (x, a , n ) → (y, b, m ) in the category lldsp, represented by maps f ∈ f and g ∈ g. let h: (t, c , p) → (x, a , n ) be another supremum preserving morphism in lldsp, such that f ◦h = g ◦h. set z = {f = g}. we know since proposition 2.11 that ( (z, az, nz),ιz ) is the equalizer of the pair f,g in the category msn, so there is a unique morphism h′ : (t, c , p) → (x, a , n ) such that h = ιz ◦ h′. by the proposition 5.9, ιz is supremum-preserving and (z, az, nz) is lld. it remains to prove that h′ is supremum preserving. this follows from the fact that bool(h) = bool(h′) ◦ bool(ιz), where bool(h) is supremum preserving and bool(ιz) is supremum preserving and surjective. 6. gluing measurable functions definition 6.1. let (x, a , n ) be an msn and (y, b) a measurable space. let e ⊆ a be a collection. a family subordinated to e is a family of functions 〈fe〉e∈e such that: (1) fe : e → y is (ae, b)-measurable for every e ∈ e . we further say that 〈fe〉e∈e is compatible whenever (2) for all pairs e,e′ ∈ e one has e ∩e′ ∩{fe 6= fe′}∈ n . a gluing of a compatible family 〈fe〉e∈e subordinated to e is a function f : x → y such that: (3) f is (a , b)-measurable; (4) e ∩{f 6= fe}∈ n for every e ∈ e . in this section, we will be mainly concerned about the existence of gluings, as they will be of use in the construction of the 4c version of a locally ccc msn in section 7. this turns out to depend both on the domain and the target space. in case where (y, b) is the real line equipped with its borel σ-algebra (r, b(r)), we can glue measurable functions together if (x, a , n ) is localizable. in fact, this important property is a characterization of localizability. the interested reader may find a proof of this classical result expressed in the language of msns in [2, proposition 3.13]. only the measurable structure of (r, b(r)) is involved, thus, the result holds in the more general case where 38 p. bouafia, t. de pauw (y, b) is a standard borel space, see [17, chapter 3]. many questions arise when we remove the condition that (y, b) is a standard borel space. in this case, we need some additional assumptions on (x, a , n ). we will focus on two cases: (x, a , n ) is 4c or lld. but first, we prove that a gluing inherits some of the properties of the functions fe. lemma 6.2. let (x, a , n ) be a saturated msn, (y, b, m ) an msn, and e ⊆ a an n -generating collection. we let 〈fe〉e∈e be a compatible family of functions subordinated to e and we assume that: (1) for every e ∈ e , the map fe is [(ae, ne), (b, m )]-measurable; (2) the family 〈fe〉e∈e has a gluing f. then: (a) the gluing f is [(a , n ), (b, m )]-measurable; (b) if fe is supremum preserving, for every e ∈ e , then so is f. proof. we start with the following easy observation. for each e ∈ e and b ∈ b one has f−1e (b) (e ∩f −1(b)) ⊆ e ∩{fe 6= f}∈ n . (a) as the gluing f is (a , b)-measurable by definition, we need only show that f−1(m) ∈ n for m ∈ m . since f−1e (m) is n -negligible, the above observation applied with b = m ensures that e ∩ f−1(m) ∈ n for any e ∈ e . we next use lemma 3.4 to assert that f−1(m) is an n -essential supremum of {e∩f−1(m) : e ∈ e}. this forces f−1(m) to be n -negligible. (b) let f ⊆ b be a collection that admits an m -essential supremum s. since fe is supremum preserving for every e ∈ e , f−1e (s) is an n -essential supremum of {f−1e (f) : f ∈ f} and it ensues from the observation above, applied with b ∈ {s}∪ f , that e ∩f−1(s) is an n -essential supremum of {e ∩f−1(f) : f ∈ f}. therefore, f−1(s) = n ess supe∈e e ∩f −1(s) (by lemma 3.4) = n ess supe∈e ( n ess supf∈f e ∩f −1(f) ) (from what precedes) = n ess supf∈f ( n ess supe∈e e ∩f −1(f) ) = n ess supf∈f f −1(f) (by lemma 3.4). radon-nikodýmification of arbitrary measure spaces 39 proposition 6.3. let (x, a , n ) be a locally determined msn and (y,b) be any nonempty measurable space. let e ⊆ a be an n -generating collection. if a compatible family 〈fe〉e∈e has a gluing, then it is unique up to equality almost everywhere. proof. let f,g : x → y be two gluings of 〈fe〉e∈e . we warn the reader that the measurability of {f 6= g} is not immediate, since the diagonal {(y,y) : y ∈ y} may not be measurable in (y 2, b⊗b). notwithstanding, for all e ∈ e , we have e ∩{f 6= g} ⊆ (e ∩{f 6= fe}) ∪ (e ∩{g 6= fe}). since f,g are gluings and (x, a , n ) is saturated, it follows that e∩{f 6= g}∈ n . this happens for any e in the n -generating set e . by local determination and the distributivity lemma 3.4, {f 6= g}∈ n . proposition 6.4. let (x, a , n ) be a 4c msn and (y, b) be any nonempty measurable space. let e ⊆ a be an n -generating collection. any compatible family 〈fe〉e∈e subordinated to e admits a unique gluing f up to equality n -almost everywhere. proof. first observe that the uniqueness of the gluing up to almost everywhere equality follows from proposition 6.3, as a 4c msn is locally determined by proposition 5.3(e). let us treat the special case where (x, a , n ) is a saturated ccc msn. according to proposition 4.4, we can find a sequence of sets 〈ei〉i∈n in e such that ⋃ i∈n ei provides an n -essential supremum of e . we then define the (a , b)-measurable map f : x → y which, for all i ∈ n, coincides with fi on the set ei \ ⋃ j 3−n0 − ∞∑ n=n0+1 3−n|1cn(y1) −1cn(y2)|> 3 −n0 − 3−n0 2 > 0 thus h(y1) 6= h(y2), which shows that f is injective. (b) ⇒ (c) is obvious. (c) ⇒ (a) let u be a countable basis for the topology of x. if there is an injective measurable map h: (y, b) → (x, b(x)), then h−1(u ) ⊆ b is a countable set that separates points. proposition 6.8. let 〈(yi, bi)〉i∈i be a family of countably separated measurable spaces. if card i 6 c, then ∐ i∈i(yi, bi) is countably separated. proof. for each i ∈ i, there is an injective (bi, b(r))-measurable map hi : yi → r by proposition 6.7. choose an arbitrary injective map g : i → r. radon-nikodýmification of arbitrary measure spaces 41 let h: ∐ i∈i yi → r 2 be the map defined by h(yi) := (hi(yi),g(i)) for all i ∈ i and yi ∈ yi. let b ⊆ r2 be a borel set. then, for any i ∈ i, we have h−1(b) ∩yi = h−1i (r∩{x : (x,g(i)) ∈ b}) = h −1 i (b g(i)). this last set is bi-measurable as hi is measurable and the horizontal section bg(i) is borel. as i is arbitrary, we conclude that h−1(b) is measurable. this means that h is measurable. by proposition 6.7, it follows that ∐ i∈i(yi, bi) is countably separated. remark 6.9. the restriction on the cardinal of i is necessary, since a countably measurable space must have cardinal less or equal than c by proposition 6.7(b). proposition 6.10. let (x, a , n ) be an lld msn and (y, b) be a nonempty countably separated measurable space. let e ⊆ a be n -generating. any compatible family 〈fe〉e∈e subordinated to e admits a gluing, unique up to equality almost everywhere. proof. let h be a measurable injective map (y, b) → (r, b(r)), whose existence follows from proposition 6.7. now, 〈h ◦ fe〉e∈e is still a compatible family of measurable functions, this time with values in (r, b(r)). as (x, a , n ) is localizable, it admits a gluing g : x → r. for all e ∈ e , one has e ∩ g−1(r \ h(y )) ⊆ e ∩{g 6= h ◦ fe}. therefore e ∩ g−1(r \ h(y )) is negligible. this holds for any e in the n -generating set e . by local determination and lemma 3.4, we deduce that g−1(r \h(y )) ∈ n . thus, we lose no generality in supposing, from now on, that g takes values in h(y ). define f := h−1 ◦ g. we claim that f is a gluing. for e ∈ e , we observe that e ∩{f 6= fe} ⊆ e ∩{g 6= h◦fe} ∈ n , since h is injective. therefore, condition (4) of 6.1 is satisfied. also, let b ∈ b, then (e ∩ f−1(b)) f−1e (b) ⊆ e ∩{f 6= fe} ∈ n . since fe is measurable, we have f −1 e (b) ∈ a and, in turn, e ∩f −1(b) ∈ a . since e is arbitrary, we deduce that f−1(b) ∈ a , by local determination, showing that f is measurable. of course, the uniqueness of the gluing is given by proposition 6.3. 6.11. in this paragraph, we exhibit an lld msn (x, a , n ), a measurable space (y, b) and, within this setting, a compatible family of measurable maps that cannot be glued. with regards to proposition 6.4, it is natural to turn towards fremlin’s example in [6, §216e] of a localizable, locally determined but 42 p. bouafia, t. de pauw not strictly localizable1 measure space (x, a ,µ). let us recall its construction. fix a set y with cardinal greater than c and we set x := {0, 1}p(y ). for any y ∈ y , we define xy ∈ x by ∀z ∈ p(y ), xy(z) = { 1 if y ∈ z, 0 if y 6∈ z. let k ⊆ p(p(y )) be the family of countable subsets of p(y ). for any k ∈ k and y ∈ y , we define fy,k := x ∩{x : x(z) = xy(z) for all z ∈ k}. then we define, for all y ∈ y , ay = p(x) ∩ { a : there is k ∈ k such that fy,k ⊆ a or fy,k ⊆ x \a } . let us prove that ay is a σ-algebra. clearly ∅∈ ay and ay is closed under complementations. let 〈an〉n∈n be a sequence in ay. suppose there is some n0 ∈ n and k ∈ k such that fy,k ⊆ an0 . then fy,k ⊆ ⋃ n∈n an, which implies ⋃ n∈n an ∈ ay. suppose on the contrary that for all n ∈ n, there is kn ∈ k such that fy,kn ⊆ x \an. then ⋂ n∈n fy,kn = fy, ⋃ n∈n kn ⊆ x \ ⋃ n∈n an which also gives that ⋃ n∈n an ∈ ay. finally set a := ⋂ y∈y ay and define the measure µ: a → [0,∞] by ∀a ∈ a , µ(a) = card ( y ∩{y : xy ∈ a} ) . for the rest of the discussion, we admit that (x, a ,µ) is complete, localizable, locally determined and not strictly localizable. the proof of the latter relies on a non trivial fact in infinitary combinatorics; we refer to [6, 216e(f)(g)] for more details. the associated msn (x, a , nµ) is saturated, localizable, and it is locally determined, by proposition 5.3(e). define ey = x∩{x : x({y}) = 1} for all y ∈ y . this set is a measurable, because fy,{{y}} = ey (hence ey ∈ ay), and for any z ∈ y \ {y}, we have fz,{{y}} = x \ey (hence ey ∈ az). note that y ∩{z : xz ∈ ey} = {y}. we now choose b to be the countable cocountable σ-algebra of y . for any y ∈ y , we define the measurable map fy : (ey, aey ) → (y, b) that is constant equal to y. we claim that 〈fy〉y∈y is a compatible family of measurable maps subordinated to 〈ey〉y∈y . this ensues from the fact that ey ∩ ez ∈ nµ for 1in the context of measure spaces, we follow the terminology of [6]: (x, a ,µ) is strictly localizable whenever there is a measurable partition 〈xi〉i∈i such that a set a ⊆ x is measurable whenever the sets a∩xi are, and in that case µ(a) = ∑ i∈i µ(a∩xi). radon-nikodýmification of arbitrary measure spaces 43 any distinct y,z ∈ y . assume by contradiction that we can find a gluing f : x → y . we will use the decomposition 〈f−1({y})〉y∈y to show (x, a ,µ) is strictly localizable. let a ⊆ x such that a∩f−1({y}) ∈ a for all y. we want to show that a ∈ a . for y ∈ y , we have • case xy ∈ a: as a ∩ f−1({y}) ∈ ay and xy ∈ a ∩ f−1({y}), there is k ∈ k such that fy,k ⊆ a ∩ f−1({y}) ⊆ a (because xy ∈ fy,k, this is the branch of the dichotomy, in the definition of ay, that occurs). therefore a ∈ ay. • case xy 6∈ a: since xy 6∈ a∩f−1({y}) ∈ ay, we can find k ∈ k such that fy,k ⊆ x\ ( a∩f−1({y}) ) . we deduce that fy,k∩f−1({y}) ⊆ x\a. but xy ∈ f−1({y}) ∈ ay, so there is k′ ∈ k such that fy,k′ ⊆ f−1({y}). whence fy,k∪k′ = fy,k ∩fy,k′ ⊆ fy,k ∩f−1({y}) ⊆ x \a. it follows that a ∈ ay. in any case, we have shown that a ∈ ay. as y ∈ y is arbitrary, a ∈ a . now, one observes that the only z ∈ y such that xz ∈ f−1({y}) is y. therefore µ(a∩f−1({y})) equals 1 if xy ∈ a and 0 otherwise. in consequence, we have µ(a) = ∑ y∈y µ(a∩f −1({y})) as desired. 7. existence of 4c and lld versions theorem 7.1. let (x, a , n ) be a saturated msn and e ⊆ a \n . we suppose that (1) (z, az, nz) is 4c for every z ∈ e ; (2) e is almost disjointed; (3) e is n -generating. then the pair consisting of the msn( x̂, â , n̂ ) = ∐ z∈e (z, az, nz) and the morphism p = ∐ z∈e ιz is the 4c version of (x, a , n ) (as usual ιz is the morphism induced by the inclusion map ιz : z → x). proof. the msn ( x̂, â , n̂ ) is 4c as a coproduct of 4c msns (this is a general fact, in any category, a coproduct of coproducts is a coproduct, see [1, proposition 2.2.3]), and p is supremum preserving, according to 3.7(b) 44 p. bouafia, t. de pauw and (d). observe that each z ∈ e is also a subset of x̂ and we denote by ι̂z : z → x̂ the corresponding inclusion map. let (y, b, m ) be a 4c msn and q: (y, b, m ) → (x, a , n ) be a supremum preserving morphism, represented by q ∈ q. for all z ∈ e , call qz := ι̂z ◦ (q q−1(z)) : q−1(z) → x̂. because q is supremum preserving, y = q−1(x) is an m -essential supremum of the family 〈q−1(z)〉z∈e . the family e being almost disjointed and q being measurable, q−1(z) ∩ q−1(z′) = q−1(z ∩ z′) ∈ m for any distinct z,z′ ∈ e . as a result, the family 〈qz〉z∈e subordinated to 〈q−1(z)〉z∈e is compatible. by proposition 6.4, this family has a gluing r: y → x̂ and, by lemma 6.2, r is [(b, m ), (â , n̂ )]-measurable and supremum preserving. call p := ∐ z∈e ιz. for each z ∈ e , we have q−1(z) ∩{p◦r 6= q}⊆ q−1(z) ∩{r 6= ι̂z ◦ (q q−1(z))}∈ m . the family {q−1(z) : z ∈ e} is m -generating and (y, b, m ) is locally determined, so we conclude that {p◦r 6= q}∈ m , that is, p◦r = q. as for uniqueness, let r be any morphism such that p ◦ r = q, and call r ∈ r one of its representatives. fix z ∈ e . observe that ι̂z(p(z)) = z for all z ∈ z. for m -almost every x ∈ q−1(z), we have p(r(x)) = q(x) which implies that r(x) ∈ z ⊆ x̂. for such an x, we find that r(x) = ι̂z(p(r(x))) = ι̂z(q(x)) = qz(x). hence, r is a gluing of the compatible family 〈qz〉z∈e and we invoke the uniqueness part of proposition 6.4 to conclude. 7.2. consider the following example, taken from [6, 216d]. let x be a set of cardinality greater or equal than ℵ2. for each x,y ∈ x, we define hy = x ×{y} and vx = {x}× x. sets of this form are respectively called horizontal and vertical lines. we define a σ-algebra a of x2 by declaring that a ∈ a iff for all x,y ∈ x, the trace a ∩ hy (resp. a ∩ vx) is either countable or cocountable in hy (resp. vx). also, we define the σ-ideal n of a as follows: n ∈ n if and only if the intersection of n with any line is countable. clearly, (x2, a , n ) is saturated. we assert that it is not localizable. suppose if possible that the family of horizontal lines {hy : y ∈ x} has an n -essential supremum s. then for all y ∈ x, the intersection s ∩ hy is cocountable in hy, that is, ny := x ∩{x : (x,y) 6∈ s} is countable. let z be a subset of x of cardinality ℵ1. then card ⋃ y∈z ny 6 ℵ1, hence the existence of x ∈ x \ ⋃ y∈z ny. radon-nikodýmification of arbitrary measure spaces 45 this implies that vx∩s is not countable, so it is cocountable in vx. however, s \vx is easily checked to be an n -essential upper bound of {hy : y ∈ x}, as hy ∩ vx is negligible for all y. since vx ∩ s = s \ (s \ vx) 6∈ n , we get a contradiction. the family of all lines e := {hy : y ∈ x}∪{vx : x ∈ x} satisfies the three hypotheses of theorem 7.1. applying the theorem, we see that the 4c version of (x2, a , n ) can be described as the coproduct of all lines. doing so, we see that each point (x,y) in the base msn (x2, a , n ) is duplicated in the 4c version: the “fibers” p−1({(x,y)}) contains two elements, which represent the horizontal and vertical directions emanating from the point (x,y). if a given msn has no obvious choice of a family satisfying the conditions of theorem 7.1, we can justify the existence of a 4c version in a non constructive way. lemma 7.3. let (x, a , n ) be a saturated msn and c ⊆ a an n -generating collection such that (z, az, nz) is ccc for all z ∈ c . then we can find a collection e ⊆ a \ n that satisfies conditions (1), (2) and (3) of theorem 7.1 and such that each of its members is a subset of some member of c . moreover, we can suppose card e 6 max{ℵ0, card c}. proof. let e be associated with c in lemma 4.9. it clearly satisfies conditions (1), (2), and (3) of 7.1, since a submsn of a ccc msn is ccc as well. if c is infinite, then for all z ∈ c , call ez := e ∩{z′ : z′ ⊆ z}. then, each ez is at most countable, since it is an almost disjointed family in the ccc msn (z, az, nz). as e = ⋃ z∈c ez, we conclude that card e 6 card c . corollary 7.4. every saturated locally ccc msn admits a 4c version. proof. apply lemma 7.3 to the family c := a ∩{z : (z, az, nz) is ccc} and then theorem 7.1. 7.5. it is worth noticing that all the arguments contained in theorem 7.1 and corollary 7.4 remain valid provided we replace “ccc” by “strictly localizable”, “locally ccc” by “locally strictly localizable”, and “4c” by “strictly localizable”. summing up, a saturated locally strictly localizable msn (x, a , n ) has a strictly localizable version, which is constructed as the coproduct of submsns whose underlying sets belongs to a family e that satisfies hypotheses (2), (3) of theorem 7.1 and (1’) (z, az, nz) is strictly localizable for every z ∈ c . 46 p. bouafia, t. de pauw since (1’) implies (1) we can apply theorem 7.1 again to conclude that the 4c and strictly localizable versions of (x, a , n ) are the same. as for the existence of lld versions, we have a partial result, which applies for most locally ccc msns that one is likely to encounter in analysis. theorem 7.6. let (x, a , n ) be a saturated msn with a collection c ⊆ a such that (1) (z, az, nz) is ccc for all z ∈ c ; (2) c is n -generating; (3) card c 6 c. the following hold: (a) if (x, a , n ) has an lld version, then it coincides with the 4c version. (b) if moreover (z, az) is countably separated for all z ∈ c , then the lld version exists. proof. (a) recall (x, a , n ) has a 4c version, according to corollary 7.4. suppose (x, a , n ) has an lld version [( x̂, â , n̂ ) ,p ] . in view of proposition 5.3(e), conclusion (a) will be established if we prove that ( x̂, â , n̂ ) is 4c. to this end, we need to find a suitable decomposition in x̂. apply lemma 7.3 to get an almost disjointed n -generating family e such that card e 6 c and (z, az, nz) is ccc for all z ∈ e . choose an injection c : e → r : z 7→ cz and p ∈ p. let fz : p−1(z) → r be the constant map equal to cz; it is readily( âp−1(z), b(r) ) -measurable. the family 〈fz〉z∈e is obviously compatible, since e is almost disjointed. as ( x̂, â , n̂ ) is localizable, this family has an (â , b(r))-measurable gluing f : x̂ → r. we now show that 〈f−1{cz}〉z∈e is a partition of x̂ into ccc measurable pieces. since c is injective, the family 〈f−1{cz}〉z∈e is, indeed, a partition of x̂ and, since f is (â , b(r))-measurable, f−1{cz} ∈ â for all z ∈ e . as f is a gluing of 〈fz〉z∈e , we have p−1(z) \f−1{cz} = p−1(z) ∩{f 6= fz}∈ n̂ for all z ∈ e . moreover, since c is injective, for all z′ ∈ e distinct from z, one has p−1(z′) ∩ (f−1{cz}\p−1(z)) ⊆ p−1(z′) ∩{f 6= fz′}∈ n̂ . radon-nikodýmification of arbitrary measure spaces 47 also, p−1(z) ∩ ( f−1{cz} \ p−1(z) ) = ∅ is clearly negligible. recalling that ( x̂, â , n̂ ) is saturated and that p−1(e ) is n̂ -generating (because e is n -generating and p is supremum preserving), one infers from the distributivity lemma 3.4 that f−1{cz}\p−1(z) ∈ n̂ . thus p−1(z) f−1{cz} ∈ n̂ . as p is a local isomorphism, according to propositions 4.16(b) and 5.3(a), ( p−1(z), âp−1(z), n̂p−1(z) ) is ccc. by what precedes, so is ( f−1{cz}, âf−1{cz}, n̂f−1{cz} ) . therefore, the msn (y, b, m ) := ∐ z∈e ( f−1{cz}, âf−1{cz}, n̂f−1{cz} ) is 4c, by definition. it remains to establish that (x̂, â , n̂ ) and (y, b, m ) are isomorphic in msnsp. this is a consequence of proposition 5.1 applied to the measurable partition f = {f−1{cz} : z ∈ e}. recalling that p−1(e ) is n̂ -generating, it ensues from the preceding paragraph that so is f . since (x̂, â , n̂ ) is locally determined (whence, has locally determined negligible sets, recall 5.2), f satisfies hypotheses (1) and (2) of proposition 5.1. (b) apply lemma 7.3 to c and let e be the family thus obtained. by theorem 7.1, the msn ( x̂, â , n̂ ) := ∐ z∈e (z, az, nz) and the morphism p induced by p = ∐ z∈e ιz constitute the 4c version of (x, a , n ). furthermore,( x̂, â ) is countably separated, by proposition 6.8. in order to prove that[( x̂,â, n̂ ) ,p ] is an lld version, we need to adapt the end of the proof of 7.1. let (y, b, m ) be an lld msn and q: (y, b, m ) → (x, a , n ) a supremum preserving morphism represented by q ∈ q. as before, we let ι̂z : z → x̂ be the inclusion map and qz := ι̂z ◦ ( q q−1(z) ) for all z ∈ e . the family 〈qz〉z∈e subordinated to 〈 q−1(z) 〉 z∈e is compatible. this time we use the gluing result 6.10 instead, that provides a gluing r: y → x of 〈qz〉z∈e . we argue as before to show that r induces the unique supremum preserving morphism r: (y, b, m ) → (x, a , n ) such that p◦r = q. 8. strictly localizable version of a measure space lemma 8.1. let (x, a ,µ) be a measure space and e ⊆ a an nµ-generating collection that is closed under finite union. then, for every a ∈ a , we have µ(a) = sup{µ(a∩z) : z ∈ e}. proof. if α := sup{µ(a∩z) : z ∈ e} is infinite, there is nothing to prove. otherwise, select an increasing sequence〈zn〉n∈n such that limn µ(a∩zn) = α. set a′ := a ∩ ⋃ n∈n zn. suppose that µ((a \ a ′) ∩ z) > 0 for some z ∈ e . 48 p. bouafia, t. de pauw then α > µ(a∩ (zn ∪z)) > µ(a∩zn) + µ((a\a′) ∩z). letting n → ∞ gives a contradiction. so we conclude that (a \ a′) ∩ z is negligible for all z ∈ e . with the help of lemma 3.4, we obtain that µ(a\a′) = 0. consequently, µ(a) = µ(a′) = limn→∞µ(a∩zn) = α. definition 8.2. (pushforward of a measure by a morphism) let (x, a , n ) be a saturated msn. a measure µ: a → [0,∞] is absolutely continuous with respect to n whenever n ∈ n implies µ(n) = 0. let q: (x, a , n ) → (y, b, m ) a morphism of saturated msns. we define the pushforward measure q#µ := q#µ, where q is any representative of q. this definition makes sense, because, for all q′ ∈ q and a ∈ a , we have µ(q−1(a) (q′)−1(a)) = 0, owing to the absolute continuity of µ. trivially, q#µ is absolutely continuous with respect to m . definition 8.3. (pre-image measure) let (x, a ,µ) be a complete semi-finite measure space. to simplify the notations, we abbreviate nµ to n . following the discussion in paragraph 4.15, (x, a , n ) is locally ccc. recording corollary 7.4, (x, a , n ) has a 4c version [( x̂, â , n̂ ) ,p ] and we shall show that there is a unique measure µ̂ on (x̂, â ) such that (1) nµ̂ = n̂ ; (2) p#µ̂ = µ. such a measure µ̂ is referred to as the pre-image measure of µ. moreover, we will show that the measure space ( x̂, â , µ̂ ) is strictly localizable; we say that[( x̂, â , µ̂ ) ,p ] is the strictly localizable version of the measure space (x, a ,µ). we start to prove the uniqueness of µ̂. fix a representative p ∈ p. for any f ∈ a we define f̂ := p−1(f) and call pf : f̂ → f the restriction of p, which induces, as usual, a morphism pf : ( f̂, â f̂ , n̂ f̂ ) → ( f, af , nf ) . call a f := a ∩ {f : µ(f) < ∞}. a pre-image measure µ̂ must satisfy pf# (µ̂ f̂) = µ f for every f ∈ a f . but (f, af , nf ) is ccc, so by proposition 4.16, pf is an isomorphism, forcing µ̂ f̂ = (p −1 f )#(µ f) to hold. since p is supremum preserving, the collection { f̂ : f ∈ a f } admits x̂ as an n̂ essential supremum. by (1) and lemma 8.1 we infer that µ̂(a) = sup { µ̂(a∩ f̂) : f ∈ a f } = sup { (p−1f )#(µ f)(a∩ f̂) : f ∈ a f } for all a ∈ â , from which the uniqueness of the pre-image measure follows straightforwardly. radon-nikodýmification of arbitrary measure spaces 49 8.4. to deal with the existence of pre-image measures, we will fix a 4c version, obtained by an application theorem 7.1 to the family a f defined above. as all 4c versions of (x, a , n ) are isomorphic, there is no restriction in considering this special case. henceforth we suppose that ( x̂, â , n̂ ) = ∐ z∈e (z, az, nz), where e ⊆ a f \ n is a collection such that (a), (b) and (c) of theorem 7.1 hold. we now define µ̂ on (x̂, â ) by µ̂ (∐ z∈e az ) := ∑ z∈e µ(az) each az being an arbitrary measurable subset of z. we choose the representative p = ∐ z∈e ιz of p, each ιz : z → x being the inclusion map. obviously, (x̂, â , µ̂) is a strictly localizable measure space and nµ̂ = n̂ , which is condition (1) of paragraph 8.3. the next result gathers some facts about the measure µ̂. in particular, condition (2) of paragraph 8.3 is proven in proposition 8.5(b). proposition 8.5. with the notations of paragraph 8.4: (a) for all a ∈ a , one has µ(a) = ∑ z∈e µ(a∩z). (b) p#µ̂ = µ. (c) for every set a ∈ â with σ-finite µ̂-measure, there is b ∈ a with σ-finite µ-measure such that µ̂(a b̂) = 0, where b̂ := p−1(b). proof. (a) when e ∩ {z : µ(a ∩ z) > 0} is uncountable, the result follows easily, for there is α > 0 such that eα := e ∩{z : µ(a∩z) > α} is infinite. taking a countable subset e ′α ⊆ eα, then µ(a) > µ ( a∩ ⋃ e ′α ) = ∑ z∈e ′α µ(a∩z) = ∞ because e ′α is almost disjointed. on the other hand, suppose e ′ := e ∩{z : µ(a ∩ z) > 0} is countable and set a′ := a\ ⋃ e ′. then µ(a′∩z) = 0 for every z ∈ e . by lemma 3.4, a′ is an n essential supremum of {a′ ∩ z : z ∈ e}, which forces a′ to be negligible. consequently, µ(a) = µ ( a∩ ⋃ e ′ ) = ∑ z∈e ′ µ(a∩z) = ∑ z∈e µ(a∩z) 50 p. bouafia, t. de pauw (b) for any a ∈ a , one has p#µ̂(a) = µ̂(p −1(a)) = µ̂ (∐ z∈e a∩z ) = ∑ z∈e µ(a∩z). we conclude by means of (a). (c) let a ∈ â a set of σ-finite µ̂ measure. writing a = ∐ z∈e az, each az being a measurable subset of z, the set e ′ := e ∩{z : µ(az) > 0} must be countable. define b := ⋃ {az : z ∈ e ′}. we claim that the set a b̂ = ∐ z∈e ( az (b∩z) ) is negligible, or, equivalently, all az (b∩z) are negligible. indeed, for z ∈ e ′, one has az (b ∩z) ⊆ ⋃{ az′ ∩z : z′ ∈ e ′ and z′ 6= z } ∈ n . if z 6∈ e ′, then both az and b ∩z are negligible. proposition 8.6. the banach spaces l1(x, a ,µ) and l1(x̂, â , µ̂) are isometrically isomorphic. proof. for any f,f ′ ∈ f ∈ l1(x, a ,µ) we check that f◦p and f ′◦p coincide almost everywhere. thus, the linear map ϕ: l1(x, a ,µ) → l1(x̂, â , µ̂) which assigns to f the equivalence class of f ◦p is well-defined. furthermore, we have ∫ x |f|dµ = ∫ ∞ 0 µ ( {|f| > t} ) dt = ∫ ∞ 0 p#µ̂ ( {|f| > t} ) dt = ∫ ∞ 0 µ̂ ( {|f ◦p| > t} ) dt = ∫ x̂ |f ◦p|dµ̂, showing that ϕ is an isometry. let us show that ϕ is onto. let f̂ be an integrable function on ( x̂, â , µ̂ ) . as {f̂ 6= 0} has σ-finite µ̂ measure, proposition 8.5(b) provides a set b ∈ a of σ-finite µ measure such that µ̂({f̂ 6= 0} b̂) = 0. but ( b, ab, nb ) is strictly localizable, and by proposition 4.16, the morphism pb : ( b̂, â b̂ , n̂ b̂ ) → (b, ab, nb) induced by the restriction pb : b̂ → b of p is an isomorphism. we choose qb : b → b̂ a representative of p−1b and define the map f : x → r by f(x) := f̂(qb(x)) for x ∈ b and f(x) := 0 otherwise. finally, because {f̂ 6= f ◦p}⊆ ( b̂ ∩{x : qb(p(x)) 6= x} ) ∪ ({f̂ 6= 0}\ b̂), the maps f̂ and f ◦p coincide almost everywhere. radon-nikodýmification of arbitrary measure spaces 51 corollary 8.7. the dual of l1(x, a ,µ) is l∞(x̂, â , µ̂). proof. it follows from proposition 8.6 and the (strict) localizability of (x̂, â , µ̂), see e.g. [6, 243g]. definition 8.8. (semi-finite version) we report on [6, 213x(c)]. let (x, a ,µ) be a measure space. we define a measure µ̌ on a by the formula µ̌(a) = sup { µ(a∩f) : f ∈ a f } , a ∈ a . as usual, a f = a ∩{a : µ(a) < ∞}. the following hold. (1) (x, a , µ̌) is semi-finite. (2) if a ∈ a and µ a is σ-finite, then µ a = µ̌ a. (3) if a ∈ a and µ̌(a) < ∞, then there are f ∈ a f and n ∈ nµ̌ such that a = f ∪n. (4) the banach space l1(x, a ,µ) and l1(x, a , µ̌) are isometrically isomorphic. these all straightforwardly follow from the definition. if we let (x,ã,µ̃) be the completion of (x, a , µ̌), it follows from (4) that l1(x, a ,µ) is isometrically isomorphic to l1(x,ã,µ̃) and, in turn, to l1(x̂, â , µ̂), according to proposition 8.6. in other words, we have associated with each measure space (x, a ,µ) a strictly localizable “version”, and we have identified the dual of l1(x, a ,µ). however, reference to zorn’s lemma in section 7 (by means of lemma 4.9) makes it difficult to understand the corresponding space x̂. this is why we determine x̂ explicitly in sections 10 and 11, in some special cases of interest. 9. a directional radon-nikodým theorem in this section, we prove an extension of the radon-nikodým theorem for measure spaces that are not necessarily localizable, in connection with the duality outlined in corollary 8.7. so to speak, it involves a generalized radon-nikodým density that also depends on the direction: as a function, it is defined on the strictly localizable version. this result is a slight extension of the radon-nikodým theorem that was discovered independently by mcshane [12, theorem 7.1] and zaanen [20]. using fremlin’s version of the radon-nikodým theorem [6, 232e] in the proof 52 p. bouafia, t. de pauw below instead of the standard one (between measure spaces of finite measure), we are able to weaken one of the hypotheses in [12] and ask (2) instead. but the main difference with [12] and [20] is in terms of formulation. in their work, the radon-nikodým density takes the form of a “quasi-function” or a “cross-section”, a notion that is very close to that of a compatible family of measurable functions. theorem 9.1. let (x, a ,µ) be a complete semi-finite measure space and ν a semi-finite measure on (x, a ). we let [( x̂, â , µ̂ ) ,p ] be the strictly localizable version of (x, a ,µ). suppose that (1) ν is absolutely continuous with respect to µ. (2) for all a ∈ a such that ν(a) > 0, there is an a -measurable subset f ⊆ a such that µ(f) < ∞ and ν(f) > 0. then there is a â -measurable function f : x̂ → r+, unique up to equality µ̂-almost everywhere, such that ν = p#(fµ̂). proof. we let a f := {f : ν(f) < ∞}. we claim that this family is nµ-generating. let u ∈ a be an nµ-essential upper bound of a f . then µ(f \ u) = 0 for all f ∈ a f . by absolute continuity, it follows that ν((x \ u) ∩ f) = ν(f \ u) = 0 for all f ∈ a f . however, a f is nν-generating, by semi-finiteness of ν, and a routine application of the distributivity lemma 3.4 shows that ν(x \ u) = 0. hence x \ u ∈ a f and µ(x \u) = µ((x \u) \u) = 0. now, the measure ν f is truly continuous with respect to µ f, for f ∈ a f . indeed, the hypotheses of [6, 232b(b)] are all satisfied. thus we can apply fremlin’s version of the radon-nikodým theorem. it says that ν f has a radon-nikodým density gf : f → r+ with respect to µ f . it is easy to show that, for any f,f ′ ∈ a f , one has µ ( f ∩f ′∩{gf 6= g′f} ) = 0. hence 〈gf〉f∈a f is a compatible family subordinated to a f . fix a representative p ∈ p and set f̂ := p−1(f) and ff := gf ◦ pf for each f ∈ a f . where pf : f̂ → f is the restriction of p. we claim that 〈ff〉f∈a f is a compatible family subordinated to 〈f̂〉f∈a f . indeed, for distinct f,f ′ ∈ a f , we have f̂ ∩ f̂ ′ ∩{ff 6= ff ′} = p−1(f ∩f ′ ∩{gf 6= gf ′}). since p is [(â , nµ̂), (a , nµ)]-measurable, f̂ ∩ f̂ ′ ∩{ff 6= ff ′} ∈ nµ̂. owing to the supremum preserving character of p, the family {f̂ : f ∈ a f} is nµ̂-generating. by proposition 6.4, the family 〈ff〉f∈a f has a gluing radon-nikodýmification of arbitrary measure spaces 53 f : x̂ → r+. for every a ∈ a and f ∈ a f , we have ν(a∩f) = ∫ 1a∩fgfdµ radon-nikodým theorem = ∫ 1a∩fgfdp#µ̂ µ̂ is a pre-image measure = ∫ 1p−1(a∩f)ffdµ̂ = ∫ p−1(a)∩f̂ fdµ̂ f = ff a.e on f̂ = (fµ̂)(p−1(a) ∩ f̂) applying lemma 8.1, we obtain ν(a) = sup { ν(a∩f) : f ∈ a f } . also, if we set z := x̂ ∩{x : f(x) > 0}, then in the submsn ( z, âz, (nµ̂)z ) the family {z ∩ f̂ : f ∈ a f} admits z as an (nµ̂)z-essential supremum, because p◦ ιz is supremum preserving (ιz : z → x̂, being the inclusion map, is supremum preserving, and the composition of supremum preserving maps is supremum preserving). since (nµ̂)z = nfµ̂ z, we can apply lemma 8.1 again and deduce (fµ̂) ( p−1(a) ) = (fµ̂ z) ( p−1(a) ∩z ) = sup { (fµ̂ z)(p−1(a) ∩z ∩ f̂) : f ∈ a f } = sup { (fµ̂)(p−1(a) ∩ f̂) : f ∈ a f } . hence ν(a) = p#(fµ̂)(a). now we prove the uniqueness of f. let f ′ be another density, and suppose µ̂({f ′ > f}) > 0. by semi-finiteness of µ̂ there is a set a ∈ â such that a ⊆ {f ′ > f} and 0 < µ̂(a) < ∞. by proposition 8.5(c), there is b ∈ a such that µ̂(a p−1(b)) = 0. however, we have p#(f ′µ̂)(b) = (f ′µ̂)(a) > (fµ̂)(a) = p#(fµ̂)(b), which is a contradiction. it follows that f ′ 6 f almost everywhere. similarly, we prove the reverse inequality. 10. 4c version deduced from a compatible family of lower densities we devote this section to an explicit construction of the 4c and lld version under some extra assumptions. it will be applied in the next section. 54 p. bouafia, t. de pauw definition 10.1. let (x, a , n ) be an msn. a lower density for (x, a , n ) is a function θ : a → a such that: (1) θ(a) = θ(b) for all a,b ∈ a such that a b ∈ n ; (2) a θ(a) ∈ n for all a ∈ a ; (3) θ(∅) = ∅; (4) θ(a∩b) = θ(a) ∩ θ(b) for all a,b ∈ a . proposition 10.2. let (x, a , n ) be a saturated msn, e ⊆ a , and θ : a → a a lower density. assume that (a) for all z ∈ e , the submsn (z, az, nz) is ccc; (b) e is n -generating; (c) one has (i) ∀a ⊆ x : [ ∀z ∈ e : a∩z ∈ a ] ⇒ a ∈ a ; (ii) ∀n ⊆ x : [ ∀z ∈ e : n ∩z ∈ n ] ⇒ n ∈ n . then (x, a , n ) is 4c. proof. let e1 be associated with e in lemma 4.9. thus, e1 is almost disjointed and n -generating, and (z, az, nz) is ccc for all z ∈ e1. we claim that e may be replaced by e1 in hypothesis (c). let a ∈ p(x) be such that a∩z ∈ a for every z ∈ e1. let z′ ∈ e . define z := {z∩z′ : z ∈ e1 and z∩z′ 6∈ n }. notice that (z′, az′, nz′) is ccc and z is almost disjointed. thus z is countable and ⋃ z is an n -essential supremum of z . besides, by the distributivity lemma 3.4, the family {z∩z′ : z ∈ e1} admits z′ as an n -essential supremum. this family differs from z only by negligible sets. therefore, z′ ⋃ z ∈ n . since (x, a , n ) is saturated, we deduce that (a∩z′) ( a∩ ⋃ z ) ∈ n . thus, one needs to establish that a∩ ⋃ z ∈ a in order to show that a∩z′ ∈ a . this is readily done by observing that a∩ ⋃ z = ⋃{ a∩z ∩z′ : z ∈ e1 and z ∩z′ 6∈ n } is a countable union of measurable sets. we just proved that a∩z′ ∈ a for all z′ ∈ e . by hypothesis (c)(i), a ∈ a . now assume that a ∩ z ∈ n , for each z ∈ e1, and let z′ and z be as above. since z′ is an n -essential supremum of z , it follows from lemma 3.4 that a ∩ z′ is an n -essential radon-nikodýmification of arbitrary measure spaces 55 supremum of {a∩z∩z′ : z ∈ e1 and z∩z′ 6∈ n }. therefore, a∩z′ ∈ n . since z′ is arbitrary, it follows that a ∈ n , by hypothesis (c)(ii). next we define e2 = {θ(z) : z ∈ e1}. the family e2 is disjointed, for θ(z)∩θ(z′) = θ(z ∩z′) = ∅ for any distinct z,z′ ∈ e1, since z ∩z′ ∈ n . we next claim that e (or, for that matter, e1) may be replaced by e2 in hypothesis (c). indeed, for every a ⊆ x and every z ∈ e1, (a∩z) (a∩ θ(z)) ⊆ z θ(z) ∈ n , therefore (i) a ∩ z ∈ a if and only if a ∩ θ(z) ∈ a , and (ii) a ∩ z ∈ n if and only if a ∩ θ(z) ∈ n , since (x, a , n ) is saturated. in particular, letting n := x \∪e2, we infer from (c)(ii) with e replaced by e2 that n ∈ n . finally, the conclusion follows from proposition 5.1 applied to f = e2 ∪{n}. definition 10.3. let (x, a , n ) be a saturated msn and e ⊆ a . a compatible family of lower densities is a family 〈θz〉z∈e such that (1) for all z ∈ e , the map θz : az → az is a lower density for (z, az, nz); (2) for all z,z′ ∈ e and a ⊆ z ∩z′ a measurable set, θz(a) = θz′(a); (3) θz(z) = z for all z ∈ e . condition (3) is merely of technical nature. if a family satisfies only (1) and (2), we can enforce (3) by replacing e with {θz(z) : z ∈ e} and observing that θz : az → az restricts to aθz(z) → aθz(z). this, indeed, follows from the fact that θz(a) ⊆ θz(b), whenever a,b ∈ az and a ⊆ b, and θz ◦ θz = θz, as one easily checks from the definition of lower density. definition 10.4. (germ space) in the sequel, we consider a saturated msn (x, a , n ) that has a compatible family of lower densities 〈θz〉z∈e , where e is a family such that (1) e is n -generating; (2) (z, az, nz) is ccc for each z ∈ e . under these assumptions, we will now construct a new msn ( x̂, â , n̂ ) that we call the germ space of (x, a , n ) associated with e and 〈θz〉z∈e . for every x ∈ x, we set ex := e ∩{z : x ∈ z} and we define the relation ∼x on ex by z ∼x z′ ⇐⇒ x ∈ θz(z∩z′). we claim that it is an equivalence relation. indeed, it is reflexive because of 10.3(3); it is symmetric because of 56 p. bouafia, t. de pauw the set equality θz(z ∩z′) = θz′(z ∩z′) implied by 10.3(2). let us check that it is transitive. for z,z′,z′′ ∈ ex such that z ∼x z′ ∼x z′′, we have x ∈ θz′(z ∩z′) ∩ θz′(z′ ∩z′′) = θz′(z ∩z′ ∩z′′) 10.1(4) = θz(z ∩z′ ∩z′′) 10.3(2) ⊆ θz(z ∩z′′), hence z ∼x z′′. we define the quotient set γx := ex/ ∼x. the equivalence class of z ∈ ex is denoted [z]x ∈ γx. next we define the set x̂ := { (x, [z]x) : x ∈ x and [z]x ∈ γx } and the projection map p: x̂ → x which assigns (x,z) to x. for each z ∈ e , we define the map γz : z → x̂ by γz(x) = (x, [z]x) for x ∈ z. we define a σ-algebra â and a σ-ideal n̂ on x̂ by â := p(x̂) ∩ { a : γ−1z (a) ∈ az ,∀z ∈ e } , n̂ := p(x̂) ∩ { n : γ−1z (n) ∈ nz ,∀z ∈ e } . actually, â and n̂ are the finest σ-algebra and σ-ideal such that the maps γz become [(az, nz), (â , n̂ )]-measurable. clearly, ( x̂, â , n̂ ) is saturated. let us check that the projection map p: x̂ → x is [(â , n̂ ), (a , n )]measurable. if a ∈ a then for any z ∈ e we have γ−1z ( p−1(a) ) = (p◦γz)−1(a) = z ∩a ∈ az, so by definition p−1(a) ∈ â . one proves similarly that p−1(n) ∈ n̂ for all n ∈ n . theorem 10.5. let (x, a , n ) be a saturated msn that has a compatible family of lower density 〈θz〉z∈e , where e ⊆ a is a family such that conditions (1) and (2) of definition 10.4 hold. then the germ space ( x̂, â , n̂ ) constructed in definition 10.4 together with p is the 4c version of (x, a , n ). it is also its lld version in case card e 6 c and (z, az) is countably separated for all z ∈ e . proof. the second conclusion is a consequence of the first and of theorem 7.6. step 1: we prove that (x̂, â , n̂ ) possesses a lower density θ, obtained by “patching together” the lower densities θz for z ∈ e . for every a ∈ â , we set θ(a) := x̂ ∩ { (x, [z]x) : x ∈ θz ( γ−1z (a) )} . radon-nikodýmification of arbitrary measure spaces 57 the condition x ∈ θz ( γ−1z (a) ) does not depend on the representative z of [z]x. indeed, if z ′ ∼x z for some z′ ∈ ex, then x ∈ θz ( γ−1z (a) ) ∩ θz(z ∩z′) = θz ( γ−1z (a) ∩z ′). note that the sets γ−1z (a)∩z ′ and γ−1z′ (a)∩z coincide n -almost everywhere, as( γ−1z (a) ∩z ′) (γ−1z′ (a) ∩z) ⊆ z ∩z′ ∩{y : [z]y 6= [z′]y} ⊆ z ∩z′ \ θz(z ∩z′) is negligible by 10.1(2). consequently, θz ( γ−1z (a) ∩z ′) = θz(γ−1z′ (a) ∩z) 10.1(2) = θz′ ( γ−1z′ (a) ∩z ) 10.3(2) ⊆ θz′ ( γ−1z′ (a) ) and in turn x ∈ θz′ ( γ−1z′ (a) ) , as expected. next we show that θ satisfies the four properties required to be a lower density: • let a,b ∈ â such that a b ∈ n̂ . then γ−1z (a) γ −1 z (b) ∈ n for all z ∈ e , which implies (x, [z]x) ∈ θ(a) ⇐⇒ x ∈ θz(γ−1z (a)) ⇐⇒ x ∈ θz(γ−1z (b)) 10.1(1) ⇐⇒ (x, [z]x) ∈ θ(b) and we conclude that θ(a) = θ(b). • let a ∈ â . by construction, γ−1z (θ(a)) = θz ( γ−1z (a) ) , for all z ∈ e . this gives that γ−1z (a θ(a)) = γ −1 z (a) γ −1 z (θ(a)) ∈ n . by definition of the σ-ideal n̂ , we infer that a θ(a) ∈ n̂ . • that θ(∅) = ∅ is straightforward. • let a,b ∈ â . we have (x, [z]x) ∈ θ(a∩b) ⇐⇒ x ∈ θz ( γ−1z (a∩b) ) ⇐⇒ x ∈ θz ( γ−1z (a) ∩γ −1 z (b) ) ⇐⇒ x ∈ θz ( γ−1z (a) ) ∩ θz ( γ−1z (b) ) ⇐⇒ (x, [z]x) ∈ θ(a) ∩ θ(b). 58 p. bouafia, t. de pauw step 2: we establish that p is a “local isomorphism”. set ẑ := p−1(z) for all z ∈ e . we also call pz and sz the respective restrictions of p and γz to ẑ → z and z → ẑ. first, we remark that pz ◦sz = idz. let us show that ẑ \sz(z) ∈ n̂ . for z′ ∈ e , we find that x ∈ γ−1z′ (ẑ \sz(z)) ⇐⇒ x ∈ z ′ and (x, [z′]x) ∈ ẑ \sz(z) ⇐⇒ x ∈ z ∩z′ and [z′]x 6= [z]x ⇐⇒ x ∈ z ∩z′ \ θz′(z ∩z′). so γ−1z′ ( ẑ \ sz(z) ) ∈ nz′. as this holds for all z′ ∈ e , we deduce that ẑ \sz(z) is n̂ -negligible. since ẑ ∩{ξ : (sz ◦pz)(ξ) 6= ξ} = ẑ \sz(z), this shows that sz ◦pz and id ẑ coincide n̂ -almost everywhere. as a consequence, the morphisms pz and sz induced by pz and sz are reciprocal isomorphisms of msn between (z, az, nz) and ( ẑ, â ẑ , n̂ ẑ ) . they are supremum preserving, according to proposition 3.7(a). step 3: ( x̂, â , n̂ ) is “locally determined” (in the sense of proposition 10.2(c)) by the family ê := { ẑ : z ∈ e } . let a a subset of x̂. by definition of â , we have a ∈ â ⇐⇒ ∀z ∈ e : γ−1z (a) ∈ az ⇐⇒ ∀z ∈ e : s−1z (a∩ ẑ) ∈ az ⇐⇒ ∀z ∈ e : a∩ ẑ ∈ â ẑ . the direct implication of the last equivalence is justified as follows: if s−1z ( a∩ ẑ ) is measurable, then so is p−1z ( s−1z (a∩ ẑ) ) , from which a∩ ẑ differs only by an n̂ negligible set. we prove analogously that a set n ⊆ x̂ is negligible if and only if n ∩ ẑ ∈ n̂ for all z ∈ e . step 4: p is supremum preserving. let f ⊆ a be a collection which has an n -essential supremum denoted s. clearly, p−1(s) is an n̂ -essential upper bound of p−1(f ) := { p−1(f) : f ∈ f } . let u be an arbitrary n̂ -essential upper bound of p−1(f ). we need to prove that p−1(s)\u ∈ n̂ , that is, γ−1z ( p−1(s) \u ) ∈ n̂ for all z ∈ e . but γ−1z ( p−1(s) \u ) = (p◦γz)−1(s) \γ−1z (u) = z ∩s \γ −1 z (u). by lemma 3.4 we recognize z∩s as an n -essential supremum of {z∩f : f ∈ f}. this last collection can be also written { γ−1z (p −1(f)) : f ∈ f } , of which γ−1z (u) is an n -essential upper bound, leading to z∩s\γ −1 z (u) ∈ n . radon-nikodýmification of arbitrary measure spaces 59 step 5: ( x̂, â , n̂ ) is 4c. this is an application of proposition 10.2 with the collection ê . we check that all the hypotheses are satisfied. for z ∈ e , the submsn ( ẑ, â ẑ , n̂ ẑ ) is ccc because of the isomorphism pz : ( ẑ, â ẑ , n̂ ẑ ) → (z, az, nz). since p is supremum preserving, x̂ is an n̂ -essential supremum of ê . the “local determination” property was established in step 3. step 6: the pair (( x̂, â , n̂ ) ,p ) satisfies the universal property of definition 4.11. we finish the proof in a way similar to the proof of theorem 7.1. let (y, b, m ) a 4c msn and q: (y, b, m ) → (x, a , n ) a supremum preserving morphism, represented by a map q ∈ q. for every z ∈ e , we define qz = γz ◦ ( q q−1(z) ) : q−1(z) → x̂. we claim that 〈qz〉z∈e is a compatible family subordinated to 〈 q−1(z) 〉 z∈e . indeed, for distinct z,z ′ ∈ e , q−1(z) ∩q−1(z′) ∩{qz 6= qz′} = q−1 ( z ∩z′ ∩{γz 6= γz′} ) = q−1 ( z ∩z′ \ θz(z ∩z′) ) is negligible, using that θz is a lower density and q is [(b, m ), (a , n )]measurable. then, by proposition 6.4, the family 〈qz〉z∈e has a gluing that we denote r: y → x̂. that r is [(b, m ), (â , n̂ )]-measurable and supremum preserving follows from lemma 6.2. indeed, each γz is supremum preserving. this follows from the same property of sz, proved in step 2, and the distributivity lemma 3.4. we need to show that {p ◦ r 6= q} is m -negligible. in fact, for any z ∈ e and y ∈ q−1(z), we note that p(qz(y)) = p(γz(q(y)) = q(y), so q−1(z)∩{p◦r 6= q}⊆ ( q−1(z) ∩{r 6= qz} ) is m -negligible. we then use that (y, b, m ) has locally determined negligible sets (see proposition 5.3(e) and the preceding paragraph 5.2) to conclude that p◦r = q almost everywhere. we have found a supremum preserving morphism r: (y, b, m ) → ( x̂, â , n̂ ) , namely the one induced from r, such that p◦r = q. we now prove that this factorization is unique. let r be a supremum preserving morphism such that p ◦ r = q and r ∈ r. for z ∈ e and almost every y ∈ q−1(z), we have p(r(y)) = q(y) ∈ z. therefore r(y) ∈ ẑ for almost all y ∈ q−1(z). for such a y, we have q(y) = p(r(y)) = pz(r(y)), which implies that qz(y) = γz(q(y)) = sz(q(y)) = sz(pz(r(y)). but sz◦pz and idẑ coincide almost everywhere on ẑ as we saw in step 2. this implies that r(y) = qz(y) for almost all y ∈ q−1(z). the map r must be a gluing of 〈qz〉z∈e , so it is unique up to equality almost everywhere according to proposition 6.4. 60 p. bouafia, t. de pauw 11. applications here, we apply theorem 10.5 to two different situations. for this result to apply to an msn (x, a , n ), the following conditions need to be met: (i) (x, a , n ) is saturated. (ii) an n -generating family e ⊆ a is given. (iii) for every z ∈ e , the msn (z, az, nz) is ccc. (iv) for every z ∈ e , the measurable space (z, az) is countably separated. (v) card e 6 c. (vi) for every z ∈ e , a lower density θz is given for (z, az, nz), so that θz(z) = z. (vii) for every z,z′ ∈ e and a ∈ a such that a ⊆ z ∩ z′, one has θz(a) = θz′(a). in that case, the corresponding germ space ( x̂, â , n̂ ) constructed in 10.4 is the 4c version and the lld version of (x, a , n ). 11.1. (purely unrectifiable negligibles) fix integers 1 6 k 6 m − 1. recall [4, 3.2.14] that a subset n ⊆ rm is called purely (h k,k)unrectifiable whenever h k(n ∩m) = 0 for every k-rectifiable set m ⊆ rm. this is equivalent to h k(n ∩m) = 0 for every k-dimensional embedded submanifold m ⊆ rm of class c1 with h k(m) < ∞, by [4, 3.1.15]. we denote by npu,k the collection of purely (h k,k)-unrectifiable subsets of rm. it is a σ-ideal of p(rm). we also introduce the borel σ-algebra b(rm) of rm and its completion b(rm) := { b n : b ∈ b(rm),n ∈ npu,k } . we shall show that the msn (rm, b(rm), npu,k) can be associated with a germ space, as in 10.4. we notice that, by definition, this msn is saturated. we let e be the collection of all k-dimensional (embedded) submanifolds m ⊆ rm of class c1, [4, 3.1.19], such that h k m is locally finite (that is h k(m ∩b) < ∞ for every bounded borel set b ⊆ rm). clearly, each member of e is borel. (ii) we now show that e is npu,k-generating. let u ∈ b(rm) be such that rm \u 6∈ npu,k. by definition of this σ-ideal, there exists m ∈ e such that h k((rm\u)∩m) > 0. in other words, m \u 6∈ npu,k, i.e. u is not an npu,k-essential upper bound of e . (iii) we next claim that (m, b(rm)m, (npu,k)m ) is ccc, for every m ∈ e . to this end, we notice that for every m ∈ e the following holds: for every s ⊆ m : s ∈ npu,k if and only if h k(s) = 0. (f) radon-nikodýmification of arbitrary measure spaces 61 in other words, ( m, b(rm)m, (npu,k)m ) is the saturation of the msn associated with the measure space ( m, b(m), h k m ) . since the latter is σ-finite, the claim follows from proposition 4.5. we also record the following useful consequence of (f), for m ∈ e : if s ∈b(rm)m then s = b n for some b ∈b(m) and n ⊆ m with h k(n) = 0. (�) indeed, s = b′ n ′, b′ ∈ b(rm), n ′ ∈ npu,k. thus s = m∩s = (m∩b′) (m∩n ′), which proves (�). in particular, s is h k-measurable, even though some s ∈ b(rm) may not be h k-measurable. (iv) consider m ∈ e . we observe that the canonical embedding( m, b(rm)m ) → ( rm, b(rm) ) is, indeed, injective and measurable. therefore, ( m, b(rm)m ) is countably separated, according to proposition 6.7. (v) since e ⊆ b(rm) we infer that card e 6 c, according to [17, 3.3.18]. (vi) in order to define lower densities, we recall [4, 2.10.19] the density numbers θk∗(φ,x) and θ ∗k(φ,x), defined by means of closed euclidean balls, associated with an outer measure φ on rm and x ∈ rm. given m ∈ e we abbreviate φm = h k m and we define θm (a) = m ∩ { x : θk∗(φm,x) = 1 } , whenever a ∈ b(rm)m . given x ∈ r m, the function r 7→ φm (b(x,r)) is right continuous, since φm is locally finite. it easily follows that x 7→ θk∗(φm,x) is borel measurable and, in turn, that θm (a) ∈ b(rm). in particular, θm maps b(rm)m to itself. the following is the main point of the construction: for every x ∈ m : θm∗ (φm,x) = 1. (♣) see for instance the proof of [3, 3.6.1]. for instance, it follows that θm (m) = m. we now turn to checking that θm is a lower density. if a,b ⊆ m are such that a b ∈ npu,k then h k(a b) = 0, recall (iii). consequently, φm (a∩ b(x,r)) = φm (b ∩ b(x,r)) for all x ∈ rm and r > 0. thus, θk∗(a,x) = θ k ∗(b,x). since x is arbitrary, θm (a) = θm (b). this proves condition of 10.1. condition (3) of 10.1 is trivial. in view of proving 10.1(3) we let a ∈ b(rm)m . according to condition (1) just proved and (�), there is no restriction to assume that a is borel. we ought to show that the equation θk∗(φm a,x) = 1a(x) holds for h k-almost every x ∈ m. letting ψ = φm a, we infer from the besicovitch covering theorem as in [11, 2.12] 62 p. bouafia, t. de pauw that limr→0+ ψ(b(x,r)) φm (b(x,r)) = 1a(x) for φm -almost every every x ∈ rn. in view of (♣), it ensues that the sought for equation holds h k-almost everywhere on m. to establish that θm is a lower density, it remains to proves 10.1(4). let a,b ∈ b(rm)m . we observe that θk∗ ( φm (a∩b),x ) > θk∗(φm,x) − θ ∗k(φm (m \a),x) − θ∗k ( φm (m \b),x ) for all x ∈ m. now, as a and b are φm -measurable, according to (�), if x ∈ θm (a) ∩ θm (b), then it follows from (♣) that θ∗k ( φm (m \a),x ) = θ∗k ( φm (m \b),x ) = 0 and, in turn, referring to (♣) again, that θk∗ ( φm (a ∩ b),x ) = 1. thus, x ∈ θm (a∩b). we have shown that θm (a) ∩ θm (b) ⊆ θm (a∩b). the other inclusion is trivial, so that θm is, indeed, a lower density. (vii) let m,m ′ ∈ e and a ∈ b(rm) be such that a ⊆ m ∩m ′. notice that a = a∩m ′ = a∩m and φm a∩m ′ = φm′ a∩m. therefore, if x ∈ θm (a), then 1 = θk∗ ( φm a,x ) = θk∗ ( φm a∩m,x ) = θk∗ ( φm′ a∩m,x ) = θk∗ ( φm′ a,x ) . since also x ∈ m ′, we conclude that x ∈ θm′(a). switching the rôles of m and m ′ we conclude that θm (a) = θm′(a). it is interesting to try to understand the corresponding germ space. each (x,m) ∈ r̂m consists of a pair where x ∈ rm belongs to the base space rm and m is an equivalence class of a k-dimensional submanifolds passing through x. if m 3 x ∈ m ′ are two such submanifolds, then m ∼x m ′ if and only if lim r→0+ h k ( m ∩m ′ ∩b(x,r) ) α(k)rk = 1. this relation is finer than the usual notion of a germ of a k-dimensional submanifold passing through x. of course if m and m ′ belong to the same, classically defined, germ, i.e. if there exists a neighborhood v of x in rm such that m∩v = m ′∩v , then m ∼x m ′. notwithstanding, the following example illustrates the difference. let x ∈ rm, let w ⊆ rm be a k-dimensional affine subspace containing x, and let c ⊆ w be closed with empty interior and such that θk∗(φw c,x) = 1. choose a k-dimensional submanifold m ⊆ rm of class c1 “that sticks to w exactly along c”, that is w ∩m = c. it follows that w ∼x m, yet m ∩v 6= w ∩v , for every neighborhood v of x. we note, radon-nikodýmification of arbitrary measure spaces 63 however, that if m ∼x m ′, then tan(m,x) = tan(m ′,x). the construction here could be repeated by replacing e by e ′, the collection of all borel measurable, countably (h k,k)-rectifiable subsets m of x such that h k m is locally finite, and θk∗(φm,x) = 1 for every x∈m. the latter does not hold in general for rectifiable sets, unlike the case of (embedded) submanifolds. it is critical when establishing that condition 10.1(4) holds. 11.2. (integral geometric measure) here, we show that the methods of 11.1 apply, in fact, to a special measure space. we keep the same notations as in 11.1 and we let i k∞ be the integral geometric outer measure on r m defined in [4, 2.10.5(1)] or [11, 5.14]. the measure space ( rm, b(rm), i k∞ ) is not semi-finite (for the case 1 = k = m− 1, see [4, 3.3.20]). thus, recalling 8.8, we introduce the following: ǐ k∞(a) = sup { i k∞(a∩b) : b ∈ b(r m),b ⊆ a and i k∞(b) < ∞ } , for a ∈ b(rm). the measure space ( rm, b(rm), ǐ k∞ ) is semi-finite, and ǐ k∞(a) = 0 whenever a ∈ b(rm) is purely i k∞-infinite, i.e. a itself and all its borel subsets of nonzero measure have infinite measure. we denote by ( rm, b̃(rm), ĩ k∞ ) its completion. our goal is to describe its 4c, lld, and strictly localizable version. the corresponding msn ( rm, b̃(rm), n ĩ k∞ ) is readily saturated. we will check conditions (ii) through (vii) at the beginning of this section. (ii) we claim that e is n ĩ k∞ -generating in the msn ( rm, b̃(rm), n ĩ k∞ ) , where e is as in 11.1. we know that the collection a := b̃(rm) ∩ { a : ĩ k∞(a) < ∞ } is n ĩ k∞ -generating, by 4.2. it is easy to check that it suffices to establish the following: for every a ∈ a there is a sequence 〈mn〉n∈n in e such that a\ ⋃ n∈n mn ∈ nĩ k∞. let a ∈ a . by definition of completion of a measure space, there are b ∈ b(rm), n ∈ nǐ k∞, and n ′ ⊆ n such that a = b n ′. since ĩ k∞(n ′) = 0, it suffices to prove the existence of a sequence 〈mn〉n∈n in e such that b \ ⋃ n∈n mn ∈ nĩ k∞. since ǐ k ∞(b) = ĩ k ∞(b) = ĩ k ∞(a) < ∞, there are borel sets f and n such that b = f ∪ n, i k∞(f) < ∞, and ǐ k∞(n) = 0, by 8.8(3). it follows from the besicovitch structure theorem [4, 3.3.14] that f is (i k∞,k)-rectifiable. in particular, there is a sequence 〈mn〉n∈n in e such that f \ ⋃ n∈n mn ∈ nh k ⊆ ni k∞. since b \f ∈ nǐ k∞ ⊆ nĩ k∞, the proof is complete. 64 p. bouafia, t. de pauw in order to establish (iii) through (vii), it suffices to observe that for each m ∈ e the msns ( m, b(rm)m, (npu,k)m ) and ( m, b̃(rm)m, (nĩ k∞)m ) are the same. we recall from 11.1(iii) that the former is the saturation of (m, b(rm)m, h k m). let us prove that the latter has the same property. let s ∈ b̃(rm)m . there are b ∈ b(r m) and n ∈ n ĩ k∞ such that s = b n. since s = s ∩ m = (b ∩ m) (m ∩ n), there is no restriction to assume that both b and n are contained in m. therefore, we ought to show that h k(n) = 0. there exists a borel set n ′⊆m containing n and such that ǐ k∞(n ′) = 0. we observe that ǐ k∞ m = i k ∞ m = h k m, where the second equality follows from [4, 3.2.26], and the first follows from 8.8(2) and the fact that m has σ-finite i k∞ measure. thus, h k(n ′) = 0 and we are done. it follows that the germ space ( r̂m, â , n̂ ) constructed in 11.1 is, in fact, also the 4c and lld version of the msn ( rm, b̃(rm), n ĩ k∞ ) . furthermore, if î k∞ denotes the pre-image measure of ĩ k ∞ along the projection map p: r̂m → rm, then ( r̂m, ̂̃ b(rm), î k∞ ) is the strictly localizable version of( rm, b̃(rm), ĩ k∞ ) . 11.3. (hausdorff measures) here, we briefly comment on why the lower densities set up so far in this section do not help to describe explicitly the 4c and lld version of the saturation of the msn ( rm, b(rm), nh k ) . the main reason is that we would need to enlarge the collection e for it to be generating, since there are (much) less h k-negligible sets than there are purely k-unrectifiable sets. in doing so we loose (♣), which was critical for implementing the techniques of the previous section. in fact, if m ⊆ rm is borel, φm = h k m is locally finite, and θk(φm,x) = 1 for h k-almost every x ∈ m, then m is countably (h k,k)-rectifiable, see e.g. [11, 17.6(1)]. since we ought to include non h k-negligible, purely k-unrectifiable sets in an nh k-generating family, our only choice is, if possible, to change the definition of the lower densities θm . so far, we do not know how to construct, in this case, a compatible family of lower densities. references [1] f. borceux, “ handbook of categorical algebra, 1. basic category theory ”, encyclopedia of mathematics and its applications, vol. 50, cambridge university press, cambridge, 1994. radon-nikodýmification of arbitrary measure spaces 65 [2] th. de pauw, undecidably semilocalizable metric measure spaces, to appear in commun. contemp. math. 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[14] d. pavlov, gelfand-type duality for commutative von neumann algebras, j. pure appl. algebra 226 (4) (2022), paper no. 106884, 53 pp. [15] i.e. segal, equivalences of measure spaces, amer. j. math. 73 (1951), 275 – 313. [16] s. shelah, the lifting problem with the full ideal, j. appl. anal. 4 (1) (1998), 1 – 17. [17] s.m. srivastava, “ a course on borel sets ”, graduate texts in mathematics, 180, springer-verlag, new york, 1998. [18] a. tarski, über additive und multiplikative mengenkörper und mengenfunktionen, soc. sci. lett. varsovie. c. r. cl. iii. sci. math. phys. 30 (1937), 151 – 181. [19] s. ulam, zur maßtheorie in der allgemeinen mengenlehre, fund. math. 16 (1) (1930), 140 – 150. [20] a.c. zaanen, the radon-nikodym theorem. i, ii, nederl. akad. wetensch. proc. ser. a 64 = indag. math. 23 (1961), 157 – 170, 171 – 187. foreword measurable spaces with negligibles supremum preserving morphisms localizable, 4c and strictly localizable msns localizable locally determined msns gluing measurable functions existence of 4c and lld versions strictly localizable version of a measure space a directional radon-nikodým theorem 4c version deduced from a compatible family of lower densities applications references e extracta mathematicae vol. 33, núm. 2, 191 – 208 (2018) characterization of some classes related to the class of browder linear relations attaif farah, maher mnif department of mathematics, faculty of science of sfax, university of sfax soukra road, km 3.5, po box 1171, 3000, sfax, tunisia farah-lotfi1@hotmail.fr , maher.mnif@gmail.com presented by manuel gonzález received march 3, 2018 abstract: in this paper, we introduce the sets of left and right invertible linear relations and we give some of their properties. furthermore, we study the connection between these sets and the classes of fredholm linear relations. the obtained results are used to give some characterizations of some classes related to the class of browder linear relations. key words: closed linear relation, leftand right-browder linear relation, left and right invertible linear relation. ams subject class. (2000): primary 47a06, 47a53, 47a05. 1. introduction first, let us notice that, throughout this paper, (x, || ||) and (y, || ||) will represent complex banach spaces. a linear relation t : x → y is a mapping from a subspace d(t) = {u ∈ x : t(u) ̸= ∅} ⊆ x, called the domain of t , which takes values in p(y )\{∅} (the collection of nonempty subsets of y ) and is such that t(αx1 + βx2) = αt(x1) + βt(x2) for all non-zero scalars α, β ∈ k and x1, x2 ∈ d(t). the class of all linear relations from x to y will be denoted by lr(x, y ). we write lr(x) = lr(x, x). if t maps the points of its domain to singletons, then t is said to be an operator, which is equivalent to t(0) = {0}. the class of linear bounded operators defined on all x is denoted by b(x, y ). a linear relation t is uniquely defined by its graph g(t) = {(u, v) ∈ x ×y : u ∈ d(t), v ∈ t(u)}. the inverse of t is the relation t −1 given by: g(t −1) = {(v, u) ∈ y × x : (u, v) ∈ g(t)}. if g(t) is closed, then t is said to be closed. the class of such relations is denoted by cr(x, y ). we denote by r(t) = t(d(t)) the range of t and 191 192 a. farah, m. mnif by n(t) := {x ∈ x : (x, 0) ∈ g(t)} the kernel of t. if r(t) = y , then t is called surjective, and if n(t) = {0}, then t is called injective. we may write n(t) = dim n(t) and d(t) = codim r(t) and the index of t , namely i(t), is defined by i(t) = n(t) − d(t), provided that n(t) and d(t) are not both infinite. for s, t ∈ lr(x, y ) and λ ∈ k, the linear relations s+t , s+̂t , s⊕t and λs are defined by g(s+t) := {(x, y+z) ∈ x×y : (x, y) ∈ g(s) and (x, z) ∈ g(t)}, g(s+̂t) := {(x+y, z+t) ∈ x×y : (x, z) ∈ g(s) and (y, t) ∈ g(t)}, this last sum is direct when g(s) ∩ g(t) = {(0, 0)}. in such case, we write s ⊕ t, g(λs) := {(x, λy) ∈ x × y : (x, y) ∈ g(s)}, and s ⊂ t means that g(s) ⊂ g(t). for t ∈ lr(x, y ) and s ∈ lr(y, z), the product st is given by g(st) := {(x, z) ∈ x ×z : (x, y) ∈ g(t), (y, z) ∈ g(s) for some y ∈ y }. let t ∈ lr(x). if α ∈ k, then α − t stands for αi − t, where i is the identity operator in x. the resolvent set of t is the set ρ(t) = {z ∈ c : (z − t)−1 ∈ b(x)}. if m is a subspace of x, then tm is the linear relation whose graph is g(t) ∩ (m × m). recall that the class of upper semi-fredholm linear relations is denoted by: ϕ+(x, y ) = {t ∈ cr(x, y ) : r(t) is closed and n(t) < ∞}. moreover, the class of lower semi-fredholm linear relations is denoted by: ϕ−(x, y ) = {t ∈ cr(x, y ) : r(t) is closed and d(t) < ∞}. t is called a fredholm relation, if t ∈ ϕ+(x, y ) ∩ ϕ−(x, y ). the class of all fredholm relations is denoted by ϕ(x, y ). recall that a closed subspace m in a normed space x is said to be complemented in x if there exists a closed subspace n of x such that x = m + n and {0} = m ∩ n (in short, x = m ⊕ n)). if a linear relation t ∈ lr(x, y ) is upper semi-fredholm and r(t) is complemented in y , then t is said to be left-fredholm linear relation. a linear relation t ∈ lr(x, y ) is right-fredholm relation if it is lower semi-fredholm and n(t) is complemented in d(t). the set of left-fredholm linear relations (right-fredholm linear relations) is denoted by ϕl(x, y ) (ϕr(x, y )). let t ∈ lr(x). we define t n ∈ lr(x), n ∈ n by t 0 = i, t 1 = t and t n = tt n−1. we define n∞(t) = ∪nn(t n) and r∞(t) = ∩nr(t n). the singular chain manifold of t ∈ lr(x), rc(t) is defined by rc(t) := ( ∞∪ n=1 n(t) )∩( ∞∪ n=1 t n(0) ) . class of browder linear relations 193 the ascent and the descent of t ∈ lr(x) are defined as follows: asc(t) := min{p ∈ n : n(t p) = n(t p+1)}, des(t) := min{p ∈ n : r(t p) = r(t p+1)}, respectively, whenever these minima exists. if no such numbers exist, the ascent and descent of t are defined to be ∞. a relation t ∈ cr(x) is upper semi-browder if it is upper semi-fredholm with finite ascent. if t ∈ cr(x) is lower semi-fredholm with finite descent, then t is lower semibrowder. let b+(x) (b−(x)) denotes the set of all upper (lower) semibrowder linear relations. the set of browder linear relations is defined by b(x) = b+(x) ∩ b−(x). t ∈ cr(x) is said to be left-browder relation if it is left-fredholm with finite ascent. if t ∈ cr(x) is right-fredholm with finite descent, then t is right-browder relation. let bl(x) (br(x)) denotes the set of all left-(right-) browder linear relations. a study of left and right browder linear relations has been carried by a number of authors in the recent past (see [5], [7], [9]). in a recent paper of (2016) [7], the authors prove that a left (right) browder linear relation t in a banach space can be expressed in the form t = a+b where a is an injective (onto) left (right) fredholm linear relation and b is a bounded finite rank operator with bt ⊂ tb. the purpose of the present paper is to consider the notion of left and right invertible linear relations and we give some characterizations of leftand right-browder closed linear relations. to make the paper easily accessible, some results from the theory of linear relations due to cross [8] are recalled in section 2. in section 3, we extend to the general case of closed linear relations in banach spaces, some results concerning upper and the lower semi-browder closed operators proved by snez̆ana c̆ in [11, theorem 3 and theorem 4]. in particular, we prove that the upper (lower) semi-browder linear relation t is a upper (lower) semi-fredholm and almost bounded below (onto) linear relation. finally, in section 4, the definition of left (right) invertible linear relation is given, and some properties of these relations are shown wich have been used to characterize the left (right) browder linear relations. in particular, we prove that the linear relation t is left (right) browder if and only if there exists a bounded operator projector p , such that tp −pt = t −t, dim r(p) < ∞, (tp)d = t(0) for some d ∈ n and t + p is left (right) invertible linear relation if and only if there exists a compact operator b satisfying tb − bt = t − t and t − b is a left (right) invertible linear relation. these results are generalizations of the results in the 194 a. farah, m. mnif case of linear operators shown by snez̆ana c̆. z̆ivković-zlatanović, dragan s. djordjević and robin e. harte [11, theorem 5 and theorem 6]. 2. auxiliary results in this section, we recall some auxiliary results from the theory of linear relations in banach spaces. let t be a linear relation in a banach space x. recall that t is said to be continuous if for each neighborhood v in r(t), the inverse image t −1(v ) is a neighborhood in d(t), bounded if it is continuous and its domain is whole x, open if its inverse is continuous. in order to give some characterizations of these classes of linear relations, one introduces the following notations. let qt denotes the quotient map from x onto x/t(0). we note that qt t is single-valued and so we can define ∥tx∥ := ∥qt tx∥, x ∈ d(t) and ∥t∥ := ∥qt t∥ called the norm of tx and t respectively, and the minimum modulus of t is the quantity γ(t) := sup{λ ≥ 0 : λ dist(x, n(t)) ≤ ∥tx∥, x ∈ d(t)}. in [8, ii.3.2 and ii.5.3] ronald cross proves that: (i) t is continuous if and only if ∥t∥ < ∞; (ii) t is open if and only if γ(t) > 0; (iii) t is closed if and only if qt t is a closed operator and t(0) is a closed subspace. recall that t is said to be regular linear relation if r(t) is closed and t verifies one of the equivalent conditions: (i) n(t) ⊆ r(t m), for all nonnegative integer m; (ii) n(t n) ⊆ r(t), for all nonnegative integer n; (iii) n(t n) ⊆ r(t m), for all nonnegative integers n and m. the kato decomposition of left and right-fredholm linear relations are collected in the following lemma. lemma 2.1. ([7, theorem 5.1 and theorem 6.1]) let t ∈ cr(x, y ). then: class of browder linear relations 195 (i) if t ∈ ϕl(x, y ), then there exist two closed subspaces m and n of x such that x = m ⊕ n with n ⊂ d(t) and dim n < ∞; t = tm ⊕ tn, such that tm is a regular left-fredholm linear relation in m and tn is a bounded nilpotent operator in n. (ii) if t ∈ ϕr(x, y ), be such that d(t) = x and ρ(t) ̸= ∅, then there exist two closed subspaces m and n of x such that x = m ⊕ n with n ⊂ d(t) and dim n < ∞; t = tm ⊕ tn, such that tm is a regular right-fredholm linear relation in m and tn is a bounded nilpotent operator in n. let m, l be two subspaces of a banach space x and let δ(m, l) = sup x∈m ∥x∥≤1 dis(x, l). the gap between m and l is defined by δ̂(m, l) = max{δ(m, l), δ(l, m)}. finally, we give the main result of this section. theorem 2.1. let t be a bounded regular linear relation with ρ(t) ̸= ∅. then (i) if t is almost bounded below, then t is bounded below. (ii) if t is almost onto, then t is onto. proof. (i) we have t is almost bounded below, then there exists δ > 0 such that t −λ is injective and open for all 0 < |λ| < δ. hence n(t −λ) = {0} and γ(t − λ) > 0. on the other hand, we have t is regular, then γ(t) > 0. by using [2, lemma 2.10 (ii)] and [1, theorem 23 (5)], we deduce that δ̂ ( n(t − λ), n(t) ) ≤ | λ | min { γ(t − λ), γ(t) } ≤ | λ | γ(t) − 3 | λ | . hence limλ→0 δ̂ ( n(t − λ), n(t) ) = 0. therefore there exists λ > 0 such that δ̂ ( n(t − λ), n(t) ) < 1. thus by [10, corollary 10] we have dim n(t) = dim n(t − λ). then n(t) = {0}. therefore t is bounded below. 196 a. farah, m. mnif (ii) suppose that t is almost onto, then there exists δ > 0 such that for all 0 < |λ| < δ we have r(t −λi) = x. hence n((t −λi)′) = r(t −λi)⊥ = {0}. by [12, proposition iii.1.5] we have n(t ′ −λi) = {0}. therefore t ′ is almost bounded below and regular relation. then, by (i), we have t ′ is bounded below. hence n(t ′) = {0}, then r(t) = x. thus, t is onto. 3. some properties of upper and lower semi-browder linear relations the goal of this section is to discuss some properties of upper and lower semi-browder linear relations that will be used in the last section. definition 3.1. let t ∈ lr(x) and s ∈ b(x). we say that s commutes with t if s(d(t)) = d(t) and for all x ∈ d(t), we have, stx = tsx. we shall write comm−1ϵ (t) = { s ∈ b(x) : s invertible, commutes with t and ∥s∥ < ϵ } . proposition 3.1. let x be a banach space and t ∈ cr(x) be such that d(t) = x and ρ(t) ̸= ∅. let s ∈ comm−1ϵ (t). then (i) s−1 commutes with t. (ii) s′ commutes with t ′. (iii) for all n ∈ n∗, sn commutes with t n. proof. (i) let x ∈ d(t). then there exists u ∈ d(t) such that su = x. we have tsu = stu, hence tx = sts−1x, and s−1tx = ts−1x. therefore s−1 commutes with t. (ii) first we claim that s′(d(t ′)) = d(t ′). indeed, for y′ ∈ d(t ′) we have ∥s′y′(tx)∥ = ∥(y′s)(tx)∥ = ∥(y′t)(sx)∥ ≤ ∥y′t∥∥s∥∥x∥ for every x ∈ d(t). hence, s′y′t is continuous. we have s′y′t(0) = s′(t ′y′(0)) = s′(0) = 0, then, by [8, proposition iii.1.2] we deduce that s′y′ ∈ d(t ′). therefore, s′(d(t ′)) ⊂ d(t ′). we have s ∈ comm−1ϵ (t) then, s′ is bijective and ∥s′∥ = ∥s∥. let y′ ∈ d(t ′). then there exists a unique functional z′ ∈ x′ such that y′ = s′z′ = z′s. it follows that z′ = y′s−1 and by (i) we get: ∥(z′t)x∥ = ∥y′(s−1(tx))∥ = ∥y′(t(s−1x))∥ = ∥(y′t)(s−1x)∥ ≤ ∥y′t∥∥s−1∥∥x∥. class of browder linear relations 197 therefore, z′t is continuous. we have z′t(0) = y′s−1t(0) = y′t(0) = 0, then, by [8, proposition iii.1.2] we deduce that z′ ∈ d(t ′). now, we show that t ′s′ = s′t ′. we have s and s′ are bijective, then r(s′) = x′, d(s′) = x′, d(t) ⊂ r(s) = x, and r(t) ⊂ d(s) = x. hence by [9, theorem iii.1.6], we have (st)′ = t ′s′ and (ts)′ = s′t ′. therefore t ′s′ = (st)′ = (ts)′ = s′t ′. then s′ commutes with t ′. (iii) for n = 2 we will show that s2(d(t 2)) = d(t 2) and for all x ∈ d(t 2), we have, s2t 2x = t 2s2x. indeed, let x ∈ d(t 2). then x ∈ d(t) and tx ∩ d(t) ̸= ∅. using that x ∈ d(t) and s(d(t)) = d(t), we get s2x ∈ d(t). on another hand, we have ts2x = s2tx, tx ∩ d(t) ̸= ∅ and s(d(t)) = d(t) then ts2x ∩ d(t) ̸= ∅. so, s2x ∈ d(t 2). therefore d(t 2) ⊂ s2(d(t 2)). let x ∈ d(t 2). then x ∈ d(t) and tx ∩ d(t) ̸= ∅. we have s(d(t)) = d(t), then there exists y ∈ d(t) such that x = s2y. it remains to prove that ty ∩ d(t) ̸= ∅. we have, tx ∩ d(t) = ts2y ∩ d(t) = s2ty ∩ d(t) ̸= ∅ . then ty ∩ d(t) ̸= ∅. hence s2(d(t 2)) ⊂ d(t 2). therefore s2(d(t 2)) = d(t 2). let x ∈ d(t 2). then s2t 2x = ssttx = ststx = tstsx = ttssx = t 2s2x. the case n > 2, is deduced by using an induction argument. now, we prove the following result useful for the proof of the first main result of this section. proposition 3.2. let t ∈ lr(x) and s ∈ comm−1ϵ (t). then n(t − s) ⊂ r∞(t). proof. first we show by induction that, if x ∈ n(t − s) then for all n ≥ 1, we have t nx = snx + t n(0). the case n = 1 is obvious. assume that t nx = snx + t n(0) and we shall prove that t n+1x = sn+1x + t n+1(0). indeed, t n+1x = tsnx + tt n(0) = sntx + t n+1(0) = sn(sx + t(0)) + t n+1(0) = sn+1x + t n+1(0). now, let x ∈ n(t −s). we have s−1 commutes with t , then for all n ≥ 1, t n(s−1)nx = (s−1)nt nx = x + t n(0). hence n(t − s) ⊂ r(t n). therefore n(t − s) ⊂ r∞(t). 198 a. farah, m. mnif now, we are ready to state the first main result of this section. theorem 3.1. let t ∈ b+(x) be such that rc(t) = {0}. then, the following statements holds: (i) t ∈ ϕ+(x), and there exists ϵ > 0 such that for all s ∈ comm−1ϵ (t), we have t − s is bounded below. (ii) t ∈ ϕ+(x), and almost bounded below. proof. (i) we have t ∈ ϕ+(x), then, by [5, lemma 2.5], t n ∈ ϕ+(x) for all n ∈ n. let x1 = r∞(t). then x1 is a closed subspace. let t1 : x1 → x1; the restriction of t to x1. then, by using [1, lemma 20] we deduce that β(t1) = 0 and α(t1) < ∞. then, t1 ∈ ϕ(x). clearly we have s(x1) ⊂ x1. writing s1 : x1 → x1, the restriction of s to x1. then, by using [12, proposition 2.4 and proposition 2.6], we deduce that α(t1 − s1) ≤ α(t1); β(t1 − s1) ≤ β(t1), and i(t1 − s1) = i(t1). from proposition 3.2 we deduce that α(t − s) = α(t1 − s1). therefore α(t − s) = i(t1 − s1) = i(t1) = α(t1). now, by using rc(t) = {0} and asc(t) ≤ p for some p ∈ n, we get n(t k) ∩ r(t p) = {0} for all k ∈ n. hence α(t1) = 0 and therefore α(t − s) = 0. furthermore by [4, proposition 3], t − s has a closed range, then t − s is bounded below. (ii) is obvious. now, we are in position to give the second main result of this section. theorem 3.2. let x be a banach space and t ∈ b−(x) be such that d(t) = x and ρ(t) ̸= ∅. then (i) t ∈ ϕ−(x) and there exists ϵ > 0 such that for all s ∈ comm−1ϵ (t), we have t − s is onto. (ii) t ∈ ϕ−(x) and almost onto. proof. (i) let t ∈ b−(x) be such that d(t) = x and ρ(t) ̸= ∅. then, by using [6, theorem 2.1], [5, lemma 2.3] and [8, v.1.1] we deduce that t ′ ∈ b+(x) and rc(t ′) = 0. hence by theorem 3.1, there exists ϵ > 0 such that for all a ∈ comm−1ϵ (t ′), t ′ − a is bounded below. let s ∈ comm−1ϵ (t). then by proposition 3.1, we have s′ ∈ comm−1ϵ (t ′). therefore β(t − s) = α((t − s)′) = α(t ′ − s′) = 0. then t − s is onto. (ii) is obvious. class of browder linear relations 199 4. characterization of leftand right-browder linear relations this section concerns the characterization of leftand right-browder linear relations in banach spaces. 4.1. characterization of left-browder linear relations. we begin by introducing the new concept of left invertible linear relation and give some of its properties. definition 4.1. let t ∈ lr(x). we say that t is left invertible, if there exists a bounded operator a such that for all x ∈ d(t), atx = x. in this case we say that a is a left inverse of t. the following lemmas give the relationship between the notion of bounded below linear relations and the notion of left invertible linear relations. lemma 4.1. let t be an everywhere defined closed bounded below linear relation. if r(t) is complemented in x, then t is left invertible. proof. since t is injective and open then t −1 is a continuous operator. we have r(t) is complemented, then there exists a closed subspace f of x such that x = r(t) ⊕ f and there exists a continuous projector p such that r(p) = f and n(p) = r(t). take a = t −1(i − p) + p . then a is a bounded operator and we have for all x ∈ d(t), atx = x. therefore t is left invertible. lemma 4.2. let t be a closed left invertible linear relation. then t is bounded below. proof. let a be a bounded operator such that for all x ∈ d(t), atx = x. let x ∈ n(t). then t(x) = t(0). hence x = at(x) = at(0) = 0. therefore n(t) = {0}. on the other hand, for all x ∈ d(t), t −1t(x) = x = at(x). then for all y ∈ r(t) we have t −1(y) = a(y). hence ∥t −1y∥ = ∥ay∥ ≤ ∥a∥∥y∥, for all y ∈ d(t −1). hence t −1 is a continuous relation. thus t is bounded below. proposition 4.1. let t be an everywhere defined closed left invertible linear relation on a banach space x such that t(0) is complemented. then t ∈ ϕl(x). 200 a. farah, m. mnif proof. using lemma 4.2 and [8, v.18] we deduce that t ∈ ϕ+(x). let a be a left inverse of t, we have (ta)2x = tatax = tax. then ta is a multivalued projector. let n = r(t) = r(ta) = n(i − ta) and m = n(ta) = r(i − ta). we claim that m ∩ n = t(0). indeed, let x ∈ m ∩ n. then there exists y ∈ x such that x ∈ tay and 0 ∈ tax. hence 0 ∈ tatay = tay. therefore tay = ta(0) = t(0) and, as a result, m ∩ n ⊂ t(0). conversely, let x ∈ t(0). then x ∈ ta(0) and so x ∈ r(ta) and tax = ta(0). hence x ∈ n(ta). thus m ∩ n = r(ta) ∩ n(ta) = t(0). we have m, n and m + n = x are closed, then by [5, lemma 3.1 (i)], p = ta is a continuous multivalued projector. on the other hand, we have m ∩ n = t(0) is complemented in x. then by using [5, lemma 3.1 (ii)], we deduce that r(t) is complemented in x. therefore t ∈ ϕl(x). lemma 4.3. if t is an injective everywhere defined linear relation and s be a bounded operator such that st ⊂ ts, then, for all n ∈ n, t −1(t + s)−n(0) = (t + s)−n(0) ⊆ t(x). proof. we have (t +s)−n(0) = (t +s)−nt −1(0) = (t(t +s)n)−1(0). by using [9, proposition 3 (iii)] we deduce that (t +s)−n(0) = ((t +s)nt)−1(0) = t −1(t + s)−n(0). then t(t + s)−n(0) = (t + s)−n(0) + t(0). hence (t + s)−n(0) ⊂ t(t + s)−n(0) ⊂ t(x). proposition 4.2. let t be a bounded closed bounded below linear relation and s be a compact operator such that st ⊂ ts. then asc(t +s) < ∞. proof. we have t is injective then by lemma 4.3, we deduce that t −1(t + s)−n(0) = (t + s)−n(0) ⊆ t(x). (4.1) if k > 0 be such that ∥x∥ ≤ k∥tx∥ for each x ∈ x, then dis(x, (t + s)−n(0)) ≤ k dis(qt+stx, qt+s(t + s)−n(0)). indeed; if x ∈ x and yn ∈ (t + s)−n(0) are arbitrary, then by (4.1), there exists zn ∈ (t + s)−n(0) for which yn ∈ tzn, (tzn = yn + t(0) = yn + t(0)); dis(x, (t + s)−n(0)) ≤ ∥x − zn∥ ≤ k∥t(x − zn)∥ ≤ k∥qt t(x − zn)∥ ≤ k∥qt+stx − qt+syn∥. class of browder linear relations 201 we deduce that dis(x, (t + s)−n(0)) ≤ k dis(qt+stx, qt+s(t + s)−n(0)). assume that asc(t + s) = ∞, then there exists (xn) ⊂ x such that ∥xn∥ = 1; xn ∈ (t + s)−n−1(0) and dis(xn, (t + s)−n(0)) ≥ 12. it follows that if n and m ≥ n + 1 are arbitrary, then k∥qt+ssxm − qt+ssxn∥ ≥ k∥qt+ssxn − qt+s(t + s)xm + qt+stxm∥. we have xn ∈ (t + s)−n−1(0), then 0 ∈ (t + s)n+1(xn). hence, using [9, proposition 3], we get 0 = s(0) ∈ s(t + s)n+1(xn) ⊂ (t + s)n+1(s(xn)). therefore s(xn) ∈ (t + s)−n−1(0). thus, qt+ss(xn) ∈ qt+s(t + s)−n−1(0) ⊂ qt+s(t + s)−m(0). now, we have xm ∈ (t + s)−m−1(0), then (t + s)(xm) ⊂ (t + s)(t + s)−m−1(0) = (t + s)−m(0) + (t + s)(0). hence qt+s(t + s)(xm) ∈ qt+s(t + s)−m(0). therefore k∥qt+ssxm − qt+ssxn∥ ≥ k dis(qt+stxm, qt+s(t + s)−m(0)) ≥ dis(xm, (t + s)−m(0)) ≥ 1 2 . which contradicts the compactness of the operator qt+ss. definition 4.2. we say that a relation t ∈ cr(x) is almost left invertible if there exists δ > 0 such that for all 0 < |λ| < δ we have t − λi is left invertible. the following theorem is our first main result of this section where we give several sufficient and necessary conditions for a closed bounded linear relation to be left-browder. theorem 4.1. let t be a bounded closed linear relation such that ρ(t) ̸= ∅ and t(0) is complemented in x. then the following properties are equivalent: (i) t ∈ bl(x). (ii) t ∈ ϕl(x) and there exists ϵ > 0 such that for all s ∈ comm−1ϵ (t), we have t − s is bounded below. (iii) t ∈ ϕl(x) and almost bounded below. 202 a. farah, m. mnif (iv) there exists a bounded operator projector p , such that tp − pt = t −t, dim r(p) < ∞, t is completely reduced by the pair (n(p), r(p)) with tn(p) is regular left invertible linear relation in n(p) and tr(p) is a bounded nilpotent operator in r(p). (v) there exists a bounded operator projector p , such that tp − pt = t − t, dim r(p) < ∞, (tp)d = t(0) for some d ∈ n and t + p is left invertible linear relation. (vi) there exists a compact operator b satisfying tb − bt = t − t and t − b is left invertible. proof. (i) ⇒ (ii) : an immediate consequence of theorem 3.1. (ii) ⇒ (iii) : obvious. (iii) ⇒ (iv) : from lemma 2.2 it follows that there exist two closed subspaces m and n of x such that x = m ⊕n with dim n < ∞; t = tm ⊕tn, such that tm is regular left-fredholm linear relation in m and tn is bounded nilpotent operator in n. let p be the projector such that r(p) = n and n(p) = m. we claim that tp −pt = t −t. indeed, let x ∈ x. then there exist x1 ∈ m and x2 ∈ n such that x = x1 + x2. hence (tp − pt)x = tpx − ptx = t(x2) − p(tm(x1) + tn(x2)) = tm(0) + tn(x2) − tn(x2) = tm(0) = t(0). therefore tp − pt = t − t . we have tr(p) = tn and tn(p) = tm. evidently tr(p) is a bounded nilpotent operator. we claim now that tn(p) is a regular left invertible linear relation. indeed, we have tm = tn(p) is regular and left fredholm linear relation in n(p). so r(tn(p)) is complemented in n(p). on another hand t(0) = tn(p)(0) is complemented in x. so there exists a closed subspace f of x such that t(0) ⊕ f = x. hence (t(0) ⊕ f) ∩ n(p) = n(p). since t(0) ⊂ n(p), then t(0) ⊕ f ∩ n(p) = n(p). therefore t(0) = tn(p)(0) is complemented in n(p). we have t is almost bounded below, then there exists δ > 0 such that for all 0 < |λ| < δ there exists kλ > 0 such that for all x ∈ x; ∥x∥ ≤ kλ∥(t − λi)x∥. so for all x ∈ n(p); ∥x∥ ≤ kλ∥(t − λi)x∥ = kλ∥tn(p)x − λx + tr(p)(0) − λ0∥ = ∥(tn(p) − λin(p))x∥. class of browder linear relations 203 therefore tn(p) is almost bounded below in n(p). by using theorem 2.1, we deduce that tn(p) is bounded below. using lemma 4.1, we get tn(p) is left invertible. (iv) ⇒ (v) : let p be the projector in (iv) and x = u + v such that u ∈ n(p) and v ∈ r(p). we have tr(p) is a nilpotent operator. then there exists d ∈ n such that t d r(p) = 0. hence: (tp)dx = (tp)(tp) . . . (tp)︸︷︷︸ d−1 t(v) = (tp)(tp) . . . (tp)︸︷︷︸ d−1 (tn(p)(0) + tr(p)(v)). and by iteration, we get (tp)dx = t dr(p)(v) + t(0) = t(0). now, if we show that t + p is bounded below and r(t + p) is complemented, then we can use lemma 4.1 to deduce that t + p is left invertible. for that, since t is closed and p is a bounded linear operator, then t + p is closed.on another hand we have tr(p) is a bounded nilpotent operator, so tr(p) +i is invertible, and hence n(t +p) = n(tn(p))⊕n(tr(p) +i) = {0} and r(t +p) = r(tn(p))⊕r(tr(p) +i) = r(tn(p))⊕r(p) which is closed. therefore, t + p is injective with closed range. then by the closed graph theorem and lemma 2.7 we get t + p is bounded below. now, by proposition 4.1, tn(p) is ϕl(n(p)). then r(tn(p)) is complemented in n(p). so there exists a closed subspace f1 such that r(tn(p))⊕f1 = n(p). therefore r(t + p) + f1 = x. let x ∈ r(t + p) ∩ f1. then x = xr(tn(p )) + xr(p) and x = xf1. so, xr(p) = xf1 − xr(tn(p )). by according to xr(p) ∈ r(p), xf1 − xr(tn(p )) ∈ n(p) and r(tn(p)) ∩ f1 = {0}, we may deduce that xr(p) = xf1 = xr(tn(p )) = 0. therefore x = 0. so r(t + p) ⊕ f1 = x. hence r(t + p) is complemented in x. (v) ⇒ (vi) : as, p is a bounded operator with finite rank, then p is compact. so, just take b = −p , we deduce the desired result. (vi) ⇒ (i) : let b be a compact operator satisfying tb −bt = t −t and t − b is left invertible. by proposition 4.1, we deduce that t − b ∈ ϕl(x). by using [2, theorem 11] we infer that t ∈ ϕl(x). we claim now that asc(t) < +∞. indeed, let y ∈ btx. we have tbx−btx = t(0), then there exist z ∈ tbx and α ∈ t(0) such that y−z = α. therefore y ∈ tbx+t(0) = 204 a. farah, m. mnif tbx. so g(bt) ⊂ g(tb). then b(t − b) ⊂ bt − bb ⊂ tb − bb = (t − b)b. by using lemma 4.2 and proposition 4.2 we deduce that asc(t) < ∞ and, as a result, t ∈ bl(x). 4.2. characterization of right-browder linear relation. we begin by introducing the new concept of right invertible linear relation and giving some of its properties. definition 4.3. let t ∈ lr(x). we say that t is right invertible, if there exists a bounded operator b such that tb = i+t(0) and r(b) ⊂ d(t). in this case we say that b is a right inverse of t. proposition 4.3. let t ∈ ϕr(x) be such that t is bounded and onto. then t is right invertible. proof. we have t is onto and closed, then t is open. hence t −1 is continuous. since t ∈ ϕr(x), then n(t) is complemented. hence there exists a continuous projector p such that r(p) = n(t). let p1 = i − p and a = p1t −1. then, a is a continuous selection of t −1. hence tax = x+t(0) for all x ∈ x and r(a) ⊂ r(t −1) = d(t). therefore t is right invertible. remark 4.1. let t ∈ cr(x) be everywhere defined. if t is right invertible and t(0) is complemented, then there exists a bounded operator b such that tb = i + t(0) and t(0) = n(b). proof. we have t is right invertible, then there exists a bounded operator a such that ta = i +t(0). using that t(0) is complemented we deduce that there exists a closed subspace g ⊂ x such that t(0)⊕g = x. let b = apg, where pg is the projector onto g with kernel t(0). hence for all x ∈ x, we have tbx = tapgx = pgx + t(0) = xg + t(0) = xg + xt(0) + t(0) = x + t(0). therefore tb = i + t(0) and n(b) ⊂ t(0). now, let x ∈ t(0). then bx = apgx = a(0) = 0. hence t(0) ⊂ n(b). therefore t(0) = n(b). class of browder linear relations 205 proposition 4.4. let t ∈ cr(x) be everywhere defined. if t is right invertible with t(0) is complemented, then t ∈ ϕr(x). proof. let a be a right inverse of t and x ∈ x. then tax = x + t(0). hence x ∈ ta(x) ⊂ r(ta) ⊂ r(t). therefore t is onto and, as a result, t ∈ ϕ−(x). now, by remark 4.1 we deduce that there exists a bounded operator b such that tb = i + t(0) and t(0) = n(b). let x ∈ x = d(t). then there exists y ∈ x such that x ∈ t −1(y). we have tby = y + t(0), then t −1y = by + t −1(0). hence x ∈ r(b) + n(t). let x ∈ r(b) ∩ n(t). then 0 ∈ tx and there exists y ∈ x such that x ∈ by. hence 0 ∈ tby = y + t(0). therefore y ∈ t(0). since n(b) = t(0) we deduce that x = 0 and so, x = r(b) ⊕ n(t). let s : (x/n(b)) ⊕ n(t) → x defined by s(x, y) = bx + y. we have r(s) = r(b) + n(t) = x, then s is onto. let x ∈ x/n(b) and y ∈ n(t) be such that s(x, y) = bx + y = 0. then bx = 0 and y = 0, hence x = 0 and y = 0. therefore s is injective. we have s is bijective and continuous, then s−1 is continuous. since (x/n(b))⊕{0} is closed and s((x/n(b))⊕{0}) = r(b), then we deduce that r(b) is closed. hence n(t) is complemented and t ∈ ϕr(x). definition 4.4. we say that a relation t ∈ lr(x) is almost right invertible if there exist δ > 0 such that for all 0 < |λ| < δ we have t − λi is right invertible. we finish this section by giving a characterization of right browder linear relations. theorem 4.2. let x be a banach space and t ∈ cr(x) be such that d(t) = x, t(0) is complemented and ρ(t) ̸= ∅. then the following properties are equivalent: (i) t ∈ br(x). (ii) t ∈ ϕr(x) and there exists ϵ > 0 such that for all s ∈ comm−1ϵ (t), we have t − s is onto. (iii) t ∈ ϕr(x), and almost onto. (iv) there exists a bounded projector operator p , such that tp −pt = t − t, dim r(p) < ∞, t is completely reduced by the pair (n(p), r(p)), with tn(p) is a regular right invertible linear relation in n(p) and tr(p) is a bounded nilpotent operator in r(p). 206 a. farah, m. mnif (v) there exists a bounded projector operator p , such that tp − pt = t −t, dim r(p) < ∞, (tp)d = t(0) for some d ∈ n, and t +p is right invertible. (vi) there exists a compact operator b satisfying tb − bt = t − t and t − b is right invertible. proof. (i) ⇒ (ii) : an immediate consequence of theorem 3.2. (ii) ⇒ (iii) : obvious. (iii) ⇒ (iv) : if t ∈ ϕr(x), then by lemma 2.2 and as in the proof of theorem 4.1 there exists a projector p = p 2, such that tp − pt = t − t, tr(p) is a bounded nilpotent operator and tn(p) is a regular right-fredholm linear relation. we claim now that tn(p) is a bounded regular right invertible linear relation. indeed, we have t is almost onto, then there exists δ > 0 such that for all 0 < |λ| < δ, t − λ is onto. then, n(p) ∩ r(t − λi) = n(p). let y ∈ n(p) ∩ r(t − λi) = n(p). then, there exists x ∈ x such that y ∈ tx − λx. therefore there exist xr(p) ∈ r(p) and xn(p) ∈ n(p) such that y ∈ tr(p)xr(p) − λxr(p) + tn(p)xn(p) − λxn(p). then, −tr(p)xr(p) + λxr(p) ∈ −y + tn(p)xn(p) − λxn(p). by using −tr(p)xr(p) + λxr(p) ∈ r(p) and −y + tn(p)xn(p) − λxn(p) ∈ n(p) we deduce that −tr(p)xr(p) + λxr(p) = 0. then, y ∈ r(tn(p) − λin(p)). therefore, tn(p) − λin(p) is onto. so tn(p) is almost onto. by using tn(p) is regular and by theorem 2.2, we deduce that tn(p) is onto. finally, according to tn(p) ∈ ϕr(n(p)) and proposition 4.3, we infer that tn(p) is right invertible. (iv) ⇒ (v) : suppose that there exists a projector p , such that tp −pt = t − t, tr(p) is a nilpotent operator of degree d and tn(p) is onto. as in the proof of theorem 4.1 we get that (tp)d = t(0). from r(t + p) = r(tn(p)) ⊕ r(tr(p) + i) = n(p) ⊕ r(p) = x, we see that t +p is onto. now, we claim that n(t +p) is complemented. indeed, first we note that n(t + p) = n(tn(p)). on another hand, n(tn(p)) is complemented in n(p), then there exists a closed subspace f such that n(tn(p)) ⊕ f = n(p). hence n(tn(p)) + f + r(p) = x. let x ∈ (n(tn(p)) + r(p)) ∩ f. then x = xn(tn(p )) + xr(p) and x = xf . hence class of browder linear relations 207 xn(tn(p )) + xr(p) = xf . so, xr(p) = xf − xn(tn(p )). by according to xr(p) ∈ r(p) and xf − xn(tn(p )) ∈ n(p), we may deduce that xr(p) = 0 and xf = xn(tn(p )). therefore xf = 0 and xn(tn(p )) = 0. thus x = 0. so n(t + p) is complemented in x. then t + p ∈ ϕr(x) such that t + p is onto. by using proposition 4.3 we deduce that t + p is right invertible. (v) ⇒ (vi) : as, p is a bounded operator with finite rank, then p is compact. so, just take k = −p , we deduce that there exists a compact operator k satisfying tk − kt = t − t and t − k is right invertible. (vi) ⇒ (i) : we have t − k is right invertible, then by proposition 4.4 we deduce the desired result. we have (t − k)(0) = t(0) is closed, k(t − k) ⊂ (t − k)k and t − k is onto, then by [9, proposition 14] we deduce that d(t) < ∞. therefore t ∈ br(x). references [1] t. alvarez, on regular linear relations, acta math. sin. (engl. ser.) 28 (1) (2012), 183 – 194. [2] t. alvarez, on the perturbation of semi-fredholm relations with complemented ranges and null spaces, acta math. sin. (engl. ser.) 26 (8) (2010), 1545 – 1554. published online: july 15, 2010. [3] t. alvarez, m. benharrat, relationship between the kato spectrum and the goldberg spectrum of a linear relation, mediterr. j. math. 13 (1) (2016), 365 – 378. [4] t. alvarez, d. wilcox, perturbation theory of multivalued atkinson operators in normed spaces, bull. austral. math. soc. 76 (2) (2007), 195 – 204. [5] t. alvarez, y. chamkha, m. mnif, leftand right-atkinson linear relation matrices, mediterr. j. math. 13 (4) (2016), 2039 – 2059. [6] t. alvarez, f. fakhfakh, m. mnif, coperturbation function and lower semi-browder multivalued linear operators, linear multilinear algebra 61 (4) (2013), 494 – û516. [7] t. alvarez, f. fakhfakh, m. mnif, left-right fredholm and left-right browder linear relations, filomat 31 (2) (2017), 255 – 271. [8] r.w. cross, “ multivalued linear operators ”, monographs and textbooks in pure and applied mathematics, 213. marcel dekker, inc., new york, 1998. [9] f. fakhfakh, m. mnif, perturbation theory of lower semi-browder multivalued linear operators, publ. math. debrecen 78 (3-4) (2011), 595 – 606. [10] v. müller, “ spectral theory of linear operators and spectral systems in banach algebras ”, second edition, operator theory: advances and applications, 139. birkhäuser verlag, basel, 2007. 208 a. farah, m. mnif [11] c. snezana zivkovic-zlatanovic, dragan s. djordjevic, robin e. harte, on left and right browder operators, j. korean math. soc. 48 (5) (2011), 1053 – 1063. [12] d. wilcox, essential spectra of linear relations, linear algebra appl. 462 (2014), 110 – 125. e extracta mathematicae vol. 33, núm. 2, 229 – 249 (2018) sobolev spaces and potential spaces associated to hermite polynomials expansions iris a. lópez p. universidad simón bolivar, departamento de matemáticas puras y aplicadas, aptd 89000, caracas 1080-a, venezuela iathamaica@usb.ve presented by carmen calvo received june 14, 2017 abstract: the aim of this paper is to study the relation existing between potential spaces and sobolev spaces, induced by the ornstein-uhlenbeck differential operator and associated to hermite polynomials expansions, where we consider the multidimensional gaussian measure. by means of analytical methods we prove that potential spaces and sobolev spaces are spaces with equivalent norms. key words: hermite polynomials, gaussian measure, sobolev spaces, potential spaces, meyer’s multiplier theorem. ams subject class. (2000): 42c10, 26a33. 1. introduction in the classical harmonic analysis theory the study of the differentiability and smoothness of functions, f : rd → r, can be described in the context of banach spaces of functions. thus, sobolev spaces w p k (λd) and potential spaces l p s(λd) are taking into consideration, where we denote λd as the lebesgue measure, s > 0, k ∈ n and 1 < p < ∞. on the one hand, sobolev spaces w p k (λd) allows us to consider functions, such that, f ∈ l p(λd) and its partial derivatives ∂αf ∈ lp(λd), with |α| ≤ k, where the derivatives are understood in a suitable weak sense to make the space complete. on the other hand, potential spaces l p s(λd) are defined by using fractional powers of the laplacean (−△)−s/2, 0 < s < d, and its variants (i − △)−s/2, s > 0. these fractional powers are known as riesz potentials and bessel potentials respectively and we have that the potential spaces l p s(λd) are subspaces of l p(λd), such that, f = (i − △)−s/2g, with g ∈ lp(λd). riesz potentials, bessel potentials and their properties have been deeply studied and by means of these operators it has been obtained that w p k (λd) = l p k(λd) with k ∈ n and 1 < p < ∞, 229 230 i. a. lópez p. (see [15, 6]). this identity is very meaningful because the condition defining to w p k (λd) spaces is not easy to check for any function given, nevertheless, the condition defining to l p k(λd) can be described in terms of bessel potentials which have integral representations. similar identity holds in the hermite setting and a probabilistic proof of this fact has been given by h. sugita in [16] and s. watanabe in [17], where the notion of sobolev spaces of wiener functionals has been introduced to develop malliavin’s calculus. this question has been studied by b. bongioanni and j. l torrea in [1, 2], where they considered sobolev spaces associated to hermite functions expansions and laguerre functions expansions. however, in [7] we introduced the fractional derivative for the gaussian measure γd and by means of analytical methods we obtained that w p k (γ1) = l p k(γ1), with k ∈ n and 1 < p < ∞, but only in the unidimensional case. therefore, the purpose of this paper is to extend this identity to the multidimensional case. if d ≥ 1, k ∈ n and 1 < p < ∞, we shall prove that w p k (γd) = l p k(γd) and once more, bessel potentials and adjoint gaussian-riesz transforms are key tools in the proof of this fact. in the laguerre polynomials and jacobi polynomials setting, some similar results have been obtained in [5] and [8], respectively. particularly, if 1 < p < ∞, 0 < s < 1 and k ≥ 1 we obtain that w p k (γd) ⊂ l p s(γd) and the inclusion is proper. the paper is organized as follows. section 2 contains some basic facts notation and we obtain the result of this paper in section 3. 2. preliminary definitions. let β = (β1, . . . ,βd) ∈ nd ∪ {⃗0} be a multi-index, so β! = ∏d i=1 βi! and |β| = ∑d i=1 βi. let us denote by ∂i = ∂ ∂xi , for each 1 ≤ i ≤ d, and ∂β = ∂ β1 1 · · ·∂ βd d . let p be the subspace of polynomials functions on r d. we denote the gaussian measure γd(dx) = e−|x| 2 πd/2 dx, with x ∈ rd and the ornstein-uhlenbeck differential operator is defined as l = 1 2 △x − ⟨x,∇x⟩ . sobolev spaces and potential spaces 231 let us consider the normalized hermite polynomials of order β, in d variables, defined as hβ(x) = 1( 2|β|β! )1/2 d∏ i=1 (−1)βiex 2 i ∂βi ∂x βi i ( e−x 2 i ) , with h0(x) = 1, (see [18, pages 105–106]) and it is well known that the hermite polynomials are eigenfunctions of l; this way, lhβ(x) = − |β|hβ(x). (2.1) also, ∂jhβ(x) = √ 2βjhβ−ej, for each 1 ≤ j ≤ d and if α is a multi-index, ∂αhβ(x) =   2|α|/2 ( d∏ j=1 βj(βj − 1) · · · (βj − αj + 1) )1/2 hβ−α(x), if αj ≤ βj, ∀j = 1, . . . ,d, 0, otherwise. now, given a function f ∈ l1(γd) its β-fourier-hermite coefficient is defined by c f β = ∫ rd f(x)hβ(x)γd(dx) and let cn be the closed subspace of l 2(γd) generated by the linear combinations of {hβ : |β| = n}. the orthogonality of the hermite polynomials with respect to γd, lets us see that {cn} is an orthogonal decomposition of l2(γd) l2(γd) = ∞⊕ n=0 cn which is called the wiener chaos decomposition, see [17]. now, we denote jn the orthogonal projection of l2(γd) onto cn. if f ∈ l2(γd), we have that jnf = ∑ |β|=n c f βhβ and its hermite expansion is given by f = ∑ n≥0 jnf. then, following [16] there exists a positive constant cp,n, such that, ∥jnf∥p,γd ≤ cp,n∥f∥p,γd for 1 < p < ∞. 232 i. a. lópez p. also, if f ∈ l2(γd) the operator lf = ∑ n≥0 −njnf, defined on the domain d2(l) = {f ∈ l2(γd) : ∑ n≥0 ∑ |β|=n |c f β| 2 < ∞} is a self-adjoint extension of l considered on dense subspace of l2(γd). more precisely, l has a clousure wich also will denote by l. in this context, the ornstein-uhlenbeck semigroup, {tt}t≥0, is defined as ttf(x) = 1 (1 − e−2t)d/2 ∫ rd e − e −2t(|x|2+|y|2)−2e−t⟨x,y⟩ 1−e−2t f(y)γd(dy), (2.2) where its kernel is given by mehler formula 1 (1 − e−2t)d/2 e − e −2t(|x|2+|y|2)−2e−t⟨x,y⟩ 1−e−2t = ∑ n≥0 ∑ |β|=n e−|β|thβ(x)hβ(y). it is well known that {tt}t≥0 is a symmetric diffusion semigroup, with infinitesimal generator l, see [14, 16] and moreover ∥tt(i − j0 − · · · − jn−1)f∥p,γd ≤ cp,ne −nt∥f∥p,γd, 1 < p < ∞. (2.3) by means of bochner subordination formula the poisson-hermite semigroup {pt}t≥0 is defined as ptf(x) = 1 √ π ∫ ∞ 0 e−u √ u tt2/4uf(x)du (2.4) and similarly, {pt}t≥0 is a strongly continuous semigroup on l p(γd) with infinitesimal generator (−l)1/2, (see [14]). from (2.1) we obtain that tthβ(x) = e −t|β|hβ(x) and pthβ(x) = e −t √ |β|hβ(x) (2.5) moreover, tt(jnf) = e −ntjnf and pt(jnf) = e − √ ntjnf. (2.6) now, if s > 0, similar to the classical case, the gaussian fractional integral of order s, i γ s , is defined by iγs := (−l) −s/2π0, where π0 = i − j0. sobolev spaces and potential spaces 233 this (formal) definition is correct for all hermite polynomials hβ, since by using (2.1) we have that iγs hβ(x) = 1 |β|s/2 hβ(x), ∀|β| > 0, (2.7) and define i γ s h0(x) = 0, see [3, 13]. in the case that f ∈ l1(γd), an integral representation of i γ s is obtained in [13], which is given by iγs f = 1 γ(s) ∫ ∞ 0 ts−1pt(i − j0)fdt. (2.8) by using (2.5), we can see that (2.8) coincides with (2.7), if f = hβ, ∀ |β| > 0 and consequently, (2.8) coincides with (2.7) if f is a nonconstant polynomial or f ∈ l2(γd). again, from (2.3) and (2.4) we obtain that ∥pt(i − j0)f∥p,γd ≤ cpe −t∥f∥p,γd 1 < p < ∞, (2.9) since e−t = 1 √ π ∫ ∞ 0 e−u √ u e−t 2/4udu. therefore, from (2.8) and (2.9) we can conclude ∥iγs f∥p,γd ≤ cp∥f∥p,γd 1 < p < ∞. (2.10) now, for α ∈ nd, we consider the riesz transform of order |α|, associated to l, defined as, (see [10]) rα|α| := ∂ αi γ |α| (2.11) and if f ∈ l1(γd) with c f 0 = 0, then rα|α|f(x) = cd,α ∫ rd ∫ 1 0 r|α|−1 ( −logr 1 − r2 ) |α|−2 2 hα ( y − rx √ 1 − r2 ) e − |y−rx| 2 1−r2 (1 − r2)d/2+1 drf(y)dy. lp(γd) estimates for 1 < p < ∞ of the gaussian-riesz transform, have been showed by several authors using probabilistic and analytic methods (see for 234 i. a. lópez p. example [4, 10, 11] among other authors). particularly, if f = hβ, for each i = 1, . . . ,d and βi ≥ αi, we have rα|α|hβ(x) = ( 2|α| |β||α| )1/2 [ d∏ i=1 βi(βi − 1) · · · (βi − αi + 1) ]1/2 hβ−α(x) and the j-th gaussian-riesz transform of first order, r j 1 = ∂ji γ 1 , j = 1, . . . ,d, with respect to hermite polynomials can be expressed as r j 1hβ = √ 2βj |β| hβ−ej. in [7], the j-th adjoint operator, (r j 1) ∗ of the gaussian-riesz transform r j 1, has been defined as ⟨ (r j 1) ∗f,g ⟩ γd = ⟨f,rj1g⟩γd and we can observe that i γ 1 = (−l) −1/2 is a self-adjoint operator, then integrating by parts with respect xj, we obtain ⟨f,rj1g⟩γd = ⟨f,∂ji γ 1 g⟩γd = ⟨δjf,i γ 1 g⟩γd = ⟨i γ 1 δjf,g⟩γd, where δj(·) = −∂j(·) + 2xj(·). this way, we can express, for each j = 1, . . . ,d, (r j 1) ∗ := i γ 1 δj = (−l) −1/2δj. by means of the identity√ 2(βj + 1)hβ+ej − 2xjhβ + √ 2βjhβ−ej = 0, where h−ej = 0, ∀j = 1, . . . ,d, (see [18, pages 105–106]), we have that δjhβ = √ 2(βj + 1)hβ+ej (2.12) and therefore, (r j 1) ∗hβ = √ 2(βj + 1) |β| + 1 hβ+ej. (2.13) also, in [7] we obtain the boundedness of j-th adjoint operator of the gaussian-riesz transform (r j 1) ∗ for 1 < p < ∞. this follows easily from hölder’s inequality and the lp(γd) continuity of the riesz transform. so,∥∥∥(rj1)∗f∥∥∥ p,γd = sup ∥g∥q≤1 ∣∣∣⟨f,rj1g⟩γd∣∣∣ ≤ cq∥f∥p,γd. sobolev spaces and potential spaces 235 similarly, the j-th adjoint gaussian-riesz operator (r j k) ∗ of higher order k, with k ≥ 1, is defined by (r j k) ∗ := i γ kδ k j = (−l) −k/2δkj , for each j = 1, . . . ,d, if α ∈ nd, we can define the higher order adjoint operator of the riesz transform by (rα|α|) ∗ := i γ |α|δ αd d ◦ · · · ◦ δ α1 1 , thus, we get (rα|α|) ∗hβ(x) = ( 2|α| |β + α||α| )1/2 [ d∏ i=1 (βi + 1) · · · (βi + αi) ]1/2 hβ+α(x) and if 1 < p < ∞ with f ∈ lp(γd), we have that ∥(rα|α|) ∗f∥p,γd ≤ cp∥f∥p,γd. now, in [7] the gaussian fractional derivative of order s > 0, d γ s , is defined formally as dγs := (−l) s/2 and for hermite polynomials, from (2.1), we have that dγshβ(x) = |β| s/2 hβ(x). (2.14) particularly, from (2.14), we have that d γ sh0(x) = 0 and, similar to the classical case, the gaussian fractional derivative of a constant function is equal to zero, see [14, 12]. in the case of 0 < s < 1, we can write dγsf = 1 cs ∫ ∞ 0 t−s−1(ptf − f)dt, (2.15) where cs = ∫ ∞ 0 u−s−1(e−u − 1)du, for f ∈ p and f ∈ l2(γd) (see [7]). this way, (2.15) is an integral representation of d γ sf. by using (2.5) we get that (2.15) coincides with (2.14), if f = hβ, ∀|β| > 0 and 0 < s < 1. similarly, by means of the property pt1 = 1 we conclude that (2.15) coincides with (2.14), if f = h0. moreover, if f ∈ p, by (2.7) and (2.14) we obtain that iγs (d γ sf) = d γ s (i γ s f) = π0f. (2.16) 236 i. a. lópez p. following s. watanabe [17] and h. sugita [16], we consider the gaussianbessel potentials defined by (i − l)−s/2f = ∞∑ n=0 (1 + n)−s/2jnf, for f ∈ p. by means of the gamma function we obtain that 1 γ(s/2) ∫ ∞ 0 t s 2 −1e−(1+n)tdt = (1 + n)−s/2, thus, (2.6) lets us write (i − l)−s/2f = 1 γ(s/2) ∫ ∞ 0 t s 2 −1e−tttfdt, for f ∈ p (2.17) and by use of the contraction property of the semigroup {tt}t≥0, we obtain that ∥(i − l)−s/2f∥p,γd ≤ ∥f∥p,γd. then, the gaussian-bessel potential spaces of order s ≥ 0, lps(γd), with 1 < p < ∞, can be defined as the completion of the polynomials with respect to the norm ∥f∥p,s := ∥∥∥(i − l)s/2f∥∥∥ p,γd ; in other words, l p s(γd) is a subspace of l p(γd) consisting of all f which can be written in the form f = (i − l)−s/2ψ, with ψ ∈ lp(γd), where ∥f∥p,s = ∥ψ∥p,γd. these potential spaces present the following inclusion properties (see [7, 17] for more details). i) if p ≤ q then lqs(γd) ⊆ l p s(γd), for each s > 0. ii) if 0 < s ≤ r then lpr(γd) ⊆ l p s(γd), for each 1 < p < ∞. then, in [7] the following theorem has been obtained theorem 2.1. let s ≥ 0 and 1 < p < ∞. then f ∈ lps(γd) if and only if d γ sf ∈ lp(γd). moreover, bp,s ∥f∥p,s ≤ ∥d γ sf∥p,γd ≤ ap,s ∥f∥p,s . sobolev spaces and potential spaces 237 particularly, we can observe that if {pn} ∈ p, such that, limn→∞ pn = f in l p s(γd), then limn d γ spn exists in l p s(γd) and does not depend on the choice of a sequence {pn}n. theorem 2.1 is a corollary of the following theorem, obtained by p.a. meyer, see [9, 16]. this theorem shall be an important tool to develop our result in section 3. theorem 2.2. (meyer’s multiplier theorem) let tϕ be given by tϕ = ∑ n≥0 ϕ(n)jn, where {ϕ(n)}n≥0 is a real sequence. assume that h(z) is a function, which is analytic on some neighborhood of the origin. if there are n0 ∈ n, and a positive constant s, such that h(n−s) = ϕ(n) ∀n ≥ n0, then tϕ can uniquely extend to a bounded linear operator on lp(γd), for each 1 < p < ∞. as an application of meyer’s multiplier theorem, in [7], the following result has been obtained. proposition 2.1. given 1 < p < ∞ and s ≥ 1. if f ∈ lps(γd), then f ∈ lps−1(γd) and for each j = 1, . . . ,d, ∂jf ∈ l p s−1(γd). moreover, ∥f∥p,s−1 + d∑ j=1 ∥∂jf∥p,s−1 ≤ ap,s,d ∥f∥p,s . finally, let us consider the gaussian sobolev space defined as w p k (γd) := {f : ∂ αf ∈ lp(γd), α ∈ nd, |α| ≤ k}, where ∂0f = f, equipped with the norm ∥f∥wp k := ∑ |α|≤k ∥∂αf∥p,γd (see [15, pages 121–122]; where w p k (λd) spaces are considered). then for each 1 < p < ∞ and k ≥ m, we can see that ∥f∥wpm ≤ ∥f∥wpk and therefore, w p k (γd) ⊆ w p m(γd). also, from the definition of w p k (γd) spaces we can see that f ∈ w p k (γd), if and only if, f and ∂jf ∈ w p k−1(γd) for each j = 1, . . . ,d. moreover, the two norms ∥f∥wp k and ∥f∥ w p k−1 + d∑ j=1 ∥∂jf∥ w p k−1 238 i. a. lópez p. are equivalent. remark. in [7], as has been mentioned previously, we proved that if k ≥ 1, 1 < p < ∞ and f ∈ lpk(γ1), then there exist a positive constant, bp,k, such that, ∥f∥p,k ≤ bp,k ∥∥∥∥ dkdxk f ∥∥∥∥ p,γ1 for the unidimensional case. nevertheless, in order to motivate the results to be developed in section 3, we recall how this inequality was proved. in fact, if f ∈ p we define the operator tk as tkf = ∑ n≥0 (n + k)k 2k(n + k) · · · (n + 1) cfnhn. then using meyer’s multipliers theorem with the function h(z) = (1 + kz)k 2k(1 + kz) · · · (1 + z) , we obtain that ∥tkf∥p,γd ≤ cp ∥f∥p,γd . now, we introduce the operator uk as ukhn = ( n + k 2 )k/2 ( (n + k) · · · (n + 1) )−1/2 hn+k and if f ∈ p we get that ukf = ∑ n≥0 c f nuk(hn). denoting (rk) ∗ = (−l)−k/2δk as the unidimensional adjoint gaussian riesz transform of order k, with k ≥ 1, we can see that (rk) ∗hn(x) = 2 k/2 [(n + 1) · · · (n + k)] 1/2 (n + k)k/2 hn+k. therefore, for each k ≥ 1 and n ≥ 1, we have uk(hn) = [(rk) ∗ ◦ tk](hn). for this reason, if f ∈ p, by means of the lp(γd)-continuity of the operators (rk)∗ and tk we obtain that ∥ukf∥p,γd ≤ cp,k ∥f∥p,γd . (2.18) sobolev spaces and potential spaces 239 but by definition (uk ◦ rk)(hn) = hn and if f ∈ p with c f 0 = 0, we get d γ kf = ( uk ◦ rk ◦ d γ k ) f = ( uk ◦ dk dxk ◦ iγk ◦ d γ k ) f = uk ( dk dxk f ) . therefore, by using the theorem 2.1 and (2.18) we have that ∥f∥p,k ≤ bp,k ∥∥∥∥ dkdxk f ∥∥∥∥ p,γ1 , and we can use the density of the polynomials in lp(γ1) in the general case. now, let us consider the multidimensional case with d > 1 and we would like to repeat a similar argument. in this case, we could define the operators tα(hβ) = |β + α||α| 2|α| d∏ i=1 (|β| + 1) · · · (|β| + αi) hβ and uα(hβ) = ( |β + α| 2 )|α|/2 [ d∏ i=1 (βi + 1) · · · (βi + αi) ]−1/2 hβ+α for α ∈ nd. thus, if f ∈ p we introduce tαf = ∑ n≥0 tα(jnf), and uαf = ∑ n≥0 uα(jnf) and by using meyer’s multipliers theorem with h(z) = 2−|α|(1 + |α|z) d∏ i=1 (1 + αiz) · · · (1 + z) and ϕ(|β|) = 2−|α| ( 1 + |α| |β| ) d∏ i=1 ( 1 + 1 |β| ) · · · ( 1 + αi |β| ), we obtain that tα has a l p(γd)-continuous extension. however, we can see that [(rα|α|) ∗ ◦ tα](hβ) = ( |β + α||α| 2|α| )1/2   d∏ i=1 (βi + 1) · · · (βi + αi) d∏ i=1 (|β| + 1) · · · (|β| + αi)   1/2 hβ+α 240 i. a. lópez p. and therefore uαf ̸= [(rα|α|) ∗ ◦ tα]f, if f ∈ p. in consequence, this reasoning fails in the multidimensional case. this way, to prove that ∥f∥p,k ≃ ∥f∥wp k , we need to developed a different argument. consequently, we are able to present the results of this paper in the following section. 3. the results by using (2.11) and (2.16) we can write ∂αf = rα|α|d γ |α|f, if f is a nonconstant polynomial. then, by means of the lp(γd)-continuity of the gaussian-riesz transform and theorem 2.1, we get ∥∂αf∥p,γd ≤ cp,|α| ∥f∥p,|α| (3.1) and therefore, the density of the polynomials in l p |α|(γd), with |α| ≤ k, the fact that ∥f∥p,|α| ≤ ∥f∥p,k and (3.1), allows us to conclude that ∥f∥wp k ≤ cp.k∥f∥p,k. (3.2) consequently, l p k(γd) ⊆ w p k (γd) for each k ∈ n. (3.3) now, we shall prove the converse inequality in (3.2). first, we establish the following lemma, where the relation between riesz potentials and bessel potentials are obtained in the gaussian context; see [3] for similar results and compare with [15, pages 133–134]. lemma 3.1. let s ≥ 0 and 1 < p < ∞. i) suppose f ∈ p, then there exists a constant cp > 0, such that,∥∥(dγs ◦ (i − l)−s/2)f∥∥p,γd ≤ cp∥f∥p,γd. ii) suppose f ∈ p, then we have∥∥((i − l)s/2 ◦ iγs )f∥∥p,γd ≤ ap∥f∥p,γd. sobolev spaces and potential spaces 241 iii) there exists a pair of operators t1 and t2, which are bounded operators on lp(γd), so that (i − l)s/2 = t1 + dγs ◦ t2. proof. items i) and ii) have been obtained in [3]. however, we present it with details for the sake of completeness. i) if f ∈ p then f = ∑ n≥0 jnf. particularly, we observe that( dγs ◦ (i − l) −s/2)j0f = cf0(dγs ◦ (i − l)−s/2)h0 = 0 and we can express( dγs ◦ (i − l) −s/2)f = ∑ n≥0 ns/2 (1 + n)s/2 jnf. therefore, the result follows from theorem 2.2 considering the function h(z) = (z + 1) −s/2 . ii) again, we consider f ∈ p, such that f ∈ (c0)⊥, then( (i − l)s/2 ◦ iγs ) f = ∑ n>0 (1 + n)s/2 ns/2 jnf and similarly, by means of meyer’s multiplier theorem with the function h(z) = (z + 1) s/2 , we get that ∥∥((i − l)s/2 ◦ iγs )f∥∥p,γd ≤ ap∥f∥p,γd. particularly, if f is a constant function, then f = c f 0h0 and since by definition i γ s h0 = 0, we have that( (i − l)s/2 ◦ iγs ) f = c f 0 ( (i − l)s/2 ◦ iγs ) h0 = 0. iii) we choose t1 = i and t2 = ( (i − l)s/2 ◦ iγs ) − iγs . then if f ∈ p, such that, f ∈ (c0)⊥, we obtain ∥t2f∥p,γd ≤ ∥∥((i − l)s/2 ◦ iγs )f∥∥p,γd + ∥iγs f∥p,γd and iii) follows from ii) and (2.10). particularly, if f is a constant function, we have that i γ s f = 0 and therefore, t2f = 0. 242 i. a. lópez p. remark. it should be emphasized, that polynomials are dense in lp(γd), for 1 < p < ∞. this way, the lemma 3.1 states that the operators defined by dγs ◦ (i − l) −s/2 and (i − l)s/2 ◦ iγs , are bounded operators on lp(γd), on every 1 < p < ∞. now, on the one hand, by using (2.12) and (2.13) we observe that (r j 1) ∗(∂jhβ) = (r j 1) ∗(√2βjhβ−ej) = 2βj|β|1/2 hβ, for each j = 1, ...,d and |β| ̸= 0. thus, d∑ j=1 (r j 1) ∗(∂jhβ) = 2|β|1/2hβ and if g ∈ p, we obtain that d∑ j=1 (r j 1) ∗(∂jjng) = 2n 1/2jng. in consequence, d γ 1g = 1 2 d∑ j=1 (r j 1) ∗(∂jg). (3.4) particularly, if g is a constant function we can see that d γ 1g = 0 and since ∂jg = 0, then (3.4) is also true. on other hand, if s ≥ 1, we claim that (i − l)−(s−1)/2(∂jg) = ( t0 ◦ ∂j(i − l)−(s−1)/2 ) g, (3.5) where we set t0 := (i − l)(s−1)/2 ◦ i γ s−1. in fact, suppose g ∈ p, such that, g ∈ (c0)⊥. then g = ∑ n>0 jng and ∂j(i − l)−(s−1)/2g = ∑ n>0 ( 1 n + 1 )(s−1)/2 ∑ |β|=n √ 2βjc g βhβ−ej. sobolev spaces and potential spaces 243 also, (i − l)−(s−1)/2∂jg = ∑ n>0 ∑ |β|=n √ 2βjc g β |β|(s−1)/2 hβ−ej = ∑ n>0 ( 1 + n n )(s−1)/2 ( 1 1 + n )(s−1)/2 ∑ |β|=n √ 2βjc g βhβ−ej = ((i − l)(s−1)/2 ◦ iγs−1 ◦ ∂j(i − l) −(s−1)/2)g, which proves (3.5). particularly, if g is a constant function we have that (i − l)−(s−1)/2(∂jg) = 0 and ∂j(i − l)−(s−1)/2g = c g 0∂jh0 = 0 and therefore, (3.5) is also true. then, we are able to prove the following proposition (see [15, pages 136– 138]). proposition 3.1. given 1 < p < ∞ and s ≥ 1. suppose that f ∈ l p s−1(γd) and ∂jf ∈ l p s−1(γd) for each j = 1, . . . ,d. then, f ∈ l p s(γd) and also, ∥f∥p,s ≤ bp,s ( ∥f∥p,s−1 + d∑ j=1 ∥∂jf∥p,s−1 ) . proof. let f ∈ lps−1(γd) ∩ p. then, there exists ψ ∈ p, such that f = (i − l)−(s−1)/2ψ and therefore, f = ∑ n≥0 ( 1 n + 1 )(s−1)/2 jnψ and ∂jf = ∑ n≥0 ( 1 n + 1 )(s−1)/2 ∑ |β|=n √ 2βjc ψ βhβ−ej. since ψ ∈ lp1(γd) ∩ p, we can write ψ = (i −l) −1/2h and according to the lemma 3.1, part iii), where we consider s = 1, t1 = i and t2 = ( (i − l)1/2 ◦ iγ1 ) − iγ1 , 244 i. a. lópez p. we obtain that h = (i − l)1/2ψ = t1ψ + (d γ 1 ◦ t2)ψ. therefore, ∥h∥p,γd ≤ ∥t1ψ∥p,γd + ∥∥(dγ1 ◦ t2)ψ∥∥p,γd. (3.6) now, we can see that ∥h∥p,γd = ∥∥(i − l)1/2ψ∥∥ p,γd = ∥∥(i − l)1/2(i − l)(s−1)/2f∥∥ p,γd = ∥f∥p,s and ∥t1ψ∥p,γd = ∥ψ∥p,γd = ∥∥(i − l)(s−1)/2f∥∥ p,γd = ∥f∥p,s−1. on the other hand, by means of (3.4) we have that (d γ 1 ◦ t2)ψ = (t2 ◦ d γ 1)ψ = 1 2 t2 [ d∑ j=1 (r j 1) ∗(∂jψ) ] and therefore, ∥(dγ1 ◦ t2)ψ∥p,γd = 1 2 ∥∥∥∥∥t2 [ d∑ j=1 (r j 1) ∗(∂jψ) ]∥∥∥∥∥ p,γd ≤ ap ∥∥∥∥∥ d∑ j=1 (r j 1) ∗(∂jψ) ∥∥∥∥∥ p,γd ≤ cp d∑ j=1 ∥∂jψ∥p,γd. since f ∈ lps−1(γd) ∩ p and ∂jf ∈ l p s−1(γd) ∩ p, for each j = 1, . . . ,d, according to (3.5) with g = ψ we have that t0(∂jf) = (i − l)−(s−1)/2∂jψ and consequently, ∥∂jψ∥p,γd = ∥∥(t0 ◦ (i − l)(s−1)/2)(∂jf)∥∥p,γd ≤ ap ∥∥(i − l)(s−1)/2(∂jf)∥∥p,γd = cp∥∂jf∥p,s−1, sobolev spaces and potential spaces 245 because t0 ◦ (i − l)(s−1)/2 = (i − l)(s−1)/2 ◦ t0. then, from (3.6) we obtain that ∥f∥p,s ≤ bp,s ( ∥f∥p,s−1 + d∑ j=1 ∥∂jf∥p,s−1 ) . in the general case, the density of the polynomials in l p s(γd) is used. remark. in the proof of the proposition 3.1, if f is a constant function, we can see that f = ψ = h. then, trivially we can write h = (i − l)1/2ψ = t1ψ + (d γ 1 ◦ t2)ψ because (i − l)1/2ψ = ψ, t1ψ = ψ and t2ψ = 0. this way, we obtain that ∥f∥p,s−1 = ∥f∥p,s . the proposition 2.1 and the proposition 3.1 let us obtain the following corollary. corollary 3.1. suppose 1 < p < ∞ and s ≥ 1. then, f ∈ lps(γd) if and only if f ∈ lps−1(γd) and for each j = 1, . . . ,d, ∂jf ∈ l p s−1(γd). moreover, the two norms, ∥f∥p,s and ∥f∥p,s−1 + ∑d j=1 ∥∂jf∥p,s−1, are equivalent. also, by means of the proposition 3.1 we get the following result. corollary 3.2. given 1 < p < ∞ and k ≥ 1. suppose f ∈ wpk (γd), then there exists a positive constant ap,k such that ∥f∥p,k ≤ ap.k ( ∥f∥p,γd + ∑ |α|≤k ∥∂αf∥p,γd ) . proof. by using the proposition 3.1, with s = 1, we have ∥f∥p,1 ≤ bp,1 ( ∥f∥p,γd + d∑ j=1 ∥∂jf∥p,γd ) . if s = 2 we can see that ∥f∥p,2 ≤ bp,2 ( ∥f∥p,1 + d∑ i=1 ∥∂if∥p,1 ) 246 i. a. lópez p. and for each i = 1, ...,d, ∥∂if∥p,1 ≤ ap,1 ( ∥∂if∥p,γd + d∑ j=1 ∥∥∂2ijf∥∥p,γd ) . then by means of the estimates above, we get ∥f∥p,2 ≤ cp,2 ( ∥f∥p,γd + d∑ j=1 ∥∂jf∥p,γd + d∑ i=1 d∑ j=1 ∥∥∂2ijf∥∥p,γd ) , and the result of this corollary is obtained by induction. therefore, we can conclude that w p k (γd) ⊆ l p k(γd) for each k ∈ n (3.7) and in consequence, by using (3.3) and (3.7) we obtain the following theorem which it is the principal result of this paper. theorem 3.1. suppose k is a positive integer and 1 < p < ∞. then l p k(γd) = w p k (γd). particularly, as a final comment we observe that remark. if 0 < s < 1, 1 < p < ∞ and k ≥ 1, then wpk (γd) ⊂ l p s(γd) and the inclusion is proper. in fact, if 0 < s < 1 and k ≥ 1, the inclusion properties of the potential spaces allows us to write w p k (γd) = l p k(γd) ⊂ l p s(γd). now, let us consider d = 1, k = 1, p = 2, 0 < s < 1 and the function f defined as f(x) = {√ x if x ≥ 0, 0 if x < 0. first, one checks that f ∈ l2(γ1) but f ′ /∈ l2(γ1) and therefore f /∈ w21 (γ1). now, for each s ∈ (0,1) we define the family of functions gs(x) = {√ 2sx if x ≥ 0, 0 if x < 0, sobolev spaces and potential spaces 247 and from (2.2) we get that the semigroup, ttgs, can be written as ttgs(x) = √ 2s√ π(1 − e−2t) ∫ ∞ 0 e − |y−e −tx|2 1−e−2t √ ydy. by using the change of variable u = y−e−tx√ 1−e−2t in the above identity, we have that ttgs(x) = √ 2s π ∫ ∞ − xe −t (1−e−2t)1/2 (√ 1 − e−2tu + e−tx )1/2 e−u 2 du ≥ e−t/2 √ x √ 2s π ∫ ∞ 0 e−u 2 du, so, ttgs(x) ≥ 2s/2e−t/2 √ x, for all t > 0 and x ≥ 0. therefore, (i − l)−s/2gs(x) = 1 γ(s/2) ∫ ∞ 0 t s 2 −1e−tttgs(x)dt ≥ 2s/2 √ x ( 1 γ(s/2) ∫ ∞ 0 t s 2 −1e−2tdt ) and then, by use of the change of variable t = u/2, we obtain (i − l)−s/2gs(x) ≥ f(x) for each x ≥ 0. (3.8) from (2.17) if h1 ≥ h2, we have that (i − l)−s/2h1 ≥ (i − l)−s/2h2, (3.9) because tt is linear and tth ≥ 0, if h ≥ 0. also, if h1 ≥ h2, we can see that (i − l)s/2h1 ≥ (i − l)s/2h2. in fact, suppose that h1 ≥ h2 but (i − l)s/2h1 < (i − l)s/2h2, then by using (3.9) we obtain a contradiction. thus, by applying the operator (i − l)s/2 in (3.8), we get gs(x) ≥ (i − l)s/2f(x) and consequently, ∥f∥2,s = ∥∥(i − l)s/2f∥∥ 2,γ1 ≤ ∥gs∥2,γ1 = 2 s/2∥f∥2,γ1 ≤ √ 2∥f∥2,γ1, since s ∈ (0,1). therefore, we can conclude that f ∈ l2s(γ1). this way, we have obtained a function f, such that, f /∈ w21 (γ1) but f ∈ l2s(γ1), if 0 < s < 1. 248 i. a. lópez p. references [1] b. bongioanni, j. l. torrea, sobolev spaces associated to the harmonic oscillator, proc. indian acad. sci. math. sci. 116 (3) (2006), 337-360. [2] b. bongioanni, j. l. torrea, what is a sobolev space for laguerre function systems?, studia math. 192 (2) (2009), 147-172. 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[9] p. a. meyer, wuelques résultats analytiques sur le semi-groupe d’ornsteinuhlenbeck en dimension infinie, in “theory and application of random fields”, lect. notes control inf. sci. 49, springer, berlin, 1983, 201-214. [10] p. a. meyer, transformations de riesz pour les lois gaussiennes, in “seminar of probability xviii”, lecture notes in math. 1059, springer, berlin, 1984, 179-193. [11] g. pisier, riesz transforms: a simpler analytic proof of p. a. meyer’s inequality, in “seminar of probability xxii”, lecture notes in math. 1321, springer, berlin, 1988, 485-501. [12] b. ross, the development of fractional calculus 1695-1900, historia math. 4 (1977), 75-89. [13] p. sjögren, operators associated with the hermite semigroup a survey, j. fourier anal. appl. 3 (1997), 813-823. [14] e. m. stein, “topics in harmonic analysis related to the littlewood-paley theory”, annals of mathematics studies 63, princeton university press, princeton, n. j., 1970. [15] e. stein, “singular integrals and differentiability properties of functions”, princeton mathematical series 30, princeton university press, princeton, n. j., 1970. [16] h. sugita, sobolev spaces of wiener functionals and malliavin’s calculus, j. math. kyoto univ. 25 (1) (1985), 31-48. sobolev spaces and potential spaces 249 [17] s. watanabe, “lectures on stochastic differential equations and malliavin calculus”, notes by m. gopalan nair and b. rajeev, tata institute of fundamental research lectures on mathematics and physics 73, springer-verlag, berlin, 1984. [18] g. szegö, “orthogonal polynomials”, american mathematical society colloquium publications 23, amer. math. soc., providence, r.i., 1959. e extracta mathematicae vol. 33, núm. 1, 109 – 126 (2018) on some inequalities for strongly reciprocally convex functions m. bracamonte 1,2, j. medina 2, m. vivas 1,2 1 facultad de ciencias naturales y matemática, departamento de matemática escuela superior politécnica del litoral (espol), campus “ gustavo galindo” km. 30.5 vı́a perimetral, guayaquil, ecuador 2 departamento de matemática, universidad centroccidental lisandro alvarado barquisimeto, venezuela mireyabracamonte@ucla.edu.ve , mrbracam@espol.edu.ec , jesus.medina@ucla.edu.ve mvivas@ucla.edu.ve , mjvivas@espol.edu.ec , mjvivas@puce.edu.ec presented by david yost received may 9, 2016 abstract: we establish some hermite-hadamard and fejér type inequalities for the class of strongly reciprocally convex functions. key words: strongly reciprocally convex functions, hermite-hadamrd, fejér. ams subject class. (2010): 26d15, 52a40, 26a51. 1. introduction due to its important role in mathematical economics, engineering, management science, and optimization theory, convexity of functions and sets has been studied intensively; see [1, 5, 7, 8, 9, 11, 13, 15, 16] and the references therein. let r be the set of real numbers and i ⊆ r be a interval. a function f : i → r is said to be convex in the classical sense if it satisfies the following inequality f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y) for all x, y ∈ i and t ∈ [0, 1]. we say that f is concave if −f is convex. in recent years several extensions and generalizations have been considered for classical convexity, and the theory of inequalities has made essential contributions in many areas of mathematics. a significant subclass of convex functions is that of strongly convex functions introduced by b.t. polyak [20]. strongly convex functions are widely used in applied economics, as well as in nonlinear optimization and other branches of pure and applied mathematics. in this paper we present a new class of strongly convex functions, mainly the class of strongly harmonically convex functions. our investigation is devoted 109 110 m. bracamonte, j. medina, m. vivas to the classical results related to convex functions due to charles hermite, jaques hadamard [10] and lipót fejér [8]. the hermite-hadamard inequalities and fejér inequalities have been the subject of intensive research, and many applications, generalizations and improvements of them can be found in the literature (see, for instance, [1, 7, 15, 18, 19, 21, 24] and the references therein). many inequalities have been established for convex functions but the most famous is the hermite-hadamard inequality, this asserts that the mean value of a continuous convex functions f : [a, b] ⊆ r → r lies between the value of f at the midpoint of the interval [a, b] and the arithmetic mean of the values of f at the endpoints of this interval, that is, f ( a + b 2 ) ≤ 1 b − a ∫ b a f(x) dx ≤ f(a) + f(b) 2 . (1.1) moreover, each side of this double inequality characterizes convexity in the sense that a real-valued continuous function f defined on an interval i is convex if its restriction to each compact subinterval [a, b] ⊆ i verifies the left hand side of (1.1) (equivalently, the right hand side on (1.1)). see [17]. in [8], lipót fejér established the following inequality which is the weighted generalization of hermite-hadamard inequality (1.1): if f : [a, b] → r is a convex function, then the inequality f ( a + b 2 )∫ b a p(x) dx ≤ 1 b − a ∫ b a f(x)p(x) dx ≤ f(a) + f(b) 2 ∫ b a p(x) dx (1.2) holds, where p : [a, b] → r is nonnegative, integrable and symmetric about x = (a + b)/2. various generalizations have been pointed out in many directions, for recent developments of inequalities (1.1) and (1.2) and its generalizations, see [5, 6, 7, 4, 9, 13]. in [13], imdat iscan gave the definition of harmonically convex functions: definition 1.1. [13] let i be an interval in r\{0}. a function f : i → r is said to be harmonically convex on i if the inequality f ( xy tx + (1 − t)y ) ≤ tf(y) + (1 − t)f(x) (1.3) holds, for all x, y ∈ i and t ∈ [0, 1] . strongly reciprocally convex functions 111 if the inequality in (1.3) is reversed, then f is said to be harmonically concave. the following result of the hermite-hadamard type for harmonically convex functions holds. theorem 1.2. let f : i ⊆ r\{0} → r be a harmonically convex function and a, b ∈ i with a < b. if f ∈ l[a, b], then the following inequalities hold f ( 2ab a + b ) ≤ ab b − a ∫ b a f(x) x2 dx ≤ f(a) + f(b) 2 . (1.4) in [4], f. chen and s. wu proved the following fejér inequality for harmonically convex functions. theorem 1.3. ([4]) let f : i ⊆ r \ {0} → r be a harmonically convex function and a, b ∈ i with a < b. if f ∈ l(a, b), then one has f ( 2ab a + b )∫ b a p(x) x2 dx ≤ ∫ b a f(x) x2 p(x) dx ≤ f(a) + f(b) 2 ∫ b a p(x) x2 dx , (1.5) where p : [a, b] → r is nonnegative and integrable and satisfies p ( ab x ) = p ( ab a + b − x ) . 2. strongly reciprocally convex functions in 1966 polyak [20] introduced the notions of strongly convex and strongly quasi-convex functions. in 1976 rockafellar [23] studied the strongly convex functions in connection with the proximal point algorithm. they play an important role in optimization theory and mathematical economics. nikodem et al. have obtained some interesting properties of strongly convex functions (see [7, 12, 14]). definition 2.1. (see [12, 16, 22]) let d be a convex subset of r and let c > 0. a function f : d → r is called strongly convex with modulus c if f(tx + (1 − t)y) ≤ tf(x) + (1 − t)f(y) − ct(1 − t)(x − y)2 (2.1) for all x, y ∈ d and t ∈ [0, 1]. 112 m. bracamonte, j. medina, m. vivas the usual notion of convex function correspond to the case c = 0. for instance, if f is strongly convex, then it is bounded from below, its level sets {x ∈ i : f(x) ≤ λ} are bounded for each λ and f has a unique minimum on every closed subinterval of i [18, p. 268]. any strongly convex function defined on a real interval admits a quadratic support at every interior point of its domain. the proofs of the next two theorems can be found in [22]. theorem 2.2. let d be a convex subset of r and let c be a positive constant. a function f : d → r is strongly convex with modulus c if and only if the function g(x) = f(x) − cx2 is convex. theorem 2.3. the following are equivalent: (i) f(tx+(1−t)y) ≤ tf(x)+(1−t)f(y)−t(1−t)c(x−y)2, for all x, y ∈ (a, b) and t ∈ [0, 1]. (ii) for each x0 ∈ (a, b), there is a linear function t such that f(x) ≥ f(x0) + t(x − x0) + c(x − x0)2 for all x, y ∈ (a, b). (iii) for differentiable f, for each x0 ∈ (a, b): f(x) ≥ f(x0)+f ′(x0)(x−x0)+ c(x − x0)2, for all x, y ∈ (a, b). (iv) for twice differentiable f, f ′′(x) ≥ 2c, for all x, y ∈ (a, b). in [3] we proved the following sandwich theorem for harmonically convex functions: theorem 2.4. let f, g be real functions defined on the interval (0, +∞). the following conditions are equivalent: (i) there exists a harmonically convex function h : (0, +∞) → r such that f (x) ≤ h (x) ≤ g (x) for all x ∈ (0, +∞). (ii) the following inequality holds f ( xy tx + (1 − t)y ) ≤ tg(y) + (1 − t)g(x) (2.2) for all x, y ∈ (0, +∞) and t ∈ [0, 1]. on the other hand, in [2] we introduced the notion of harmonically strongly convex function as follows: strongly reciprocally convex functions 113 definition 2.5. let i be an interval in r\{0} and let c ∈ r+. a function f : i → r is said to be harmonically strongly convex with modulus c on i, if the inequality f ( xy tx + (1 − t)y ) ≤ tf(y) + (1 − t)f(x) − ct(1 − t)(x − y)2, (2.3) holds, for all x, y ∈ i and t ∈ [0, 1]. the symbol shc(i,c) will denote the class of functions that satisfy the inequality (2.3). we also establish some hermite-hadamard and fejér type inequalities for the class of harmonically strongly convex functions. next we will explore a generalization of the concept of harmonically convex functions which we will call reciprocally strongly convex functions, it is a concept parallel to the definition presented in the definition 2.5. definition 2.6. let i be an interval in r \ {0} and let c ∈ (0, ∞). a function f : i → r is said to be reciprocally strongly convex with modulus c on i, if the inequality f ( xy tx + (1 − t)y ) ≤ tf(y) + (1 − t)f(x) − ct(1 − t) ( 1 x − 1 y )2 , (2.4) holds, for all x, y ∈ i and t ∈ [0, 1]. the symbol src(i,c) will denote the class of functions that satisfy the inequality (2.4). theorem 2.7. let i ⊂ r \ {0} be a real interval and c ∈ (0, ∞). if f ∈src(i,c), then f es harmonically convex. proof. since ct(1 − t) ( 1 x − 1 y )2 ≥ 0, it is a immediate consequence of the definition. for the rest of this paper we will use i ⊂ r \ {0} to denote a real interval and c ∈ (0, ∞). theorem 2.8. let f : i → r be a function. f ∈src(i,c) if and only if the function g : i → r, defined by g(x) := f(x) − c x2 es harmonically convex. 114 m. bracamonte, j. medina, m. vivas proof. assume that f ∈src(i,c), then g ( xy tx + (1 − t)y ) = f ( xy tx + (1 − t)y ) − c ( tx + (1 − t)y xy )2 ≤ tf(y) + (1 − t)f(x) − ct(1 − t) ( 1 y − 1 x )2 − c ( t 1 y + (1 − t) 1 x )2 = tf(y) + (1 − t)f(x) − ct(1 − t) ( 1 y2 − 2 xy + 1 x2 ) c ( t2 y2 + 2t(1 − t) xy + (1 − t)2 x2 ) = tf(y) + (1 − t)f(x) − c ( t y2 − 2t xy + t x2 − t2 y2 + 2t2 xy − t2 x2 + t2 y2 + 2t xy − 2t2 xy + 1 x2 − 2t x2 + t2 x2 ) = tf(y) + (1 − t)f(x) − c ( t y2 + 1 x2 − t x2 ) = tf(y) + (1 − t)f(x) − c ( t y2 + (1 − t) 1 x2 ) = t ( f(y) − c y2 ) + (1 − t) ( f(x) − c x2 ) = tg(y) + (1 − t)g(x), for all x, y ∈ i and t ∈ [0, 1]. which proves that g is harmonically convex. conversely, if g is harmonically convex, then f ( xy tx + (1 − t)y ) = g ( xy tx + (1 − t)y ) + c ( tx + (1 − t)y xy )2 ≤ tg(y) + (1 − t)g(x) + c ( t 1 y + (1 − t) 1 x )2 = tg(y) + (1 − t)g(x) + c ( t2 y2 + 2t(1 − t) xy + (1 − t)2 x2 ) strongly reciprocally convex functions 115 = tg(y) + (1 − t)g(x) + c ( t(1 − 1 + t) y2 + 2t(1 − t) xy + (1 − t)(1 − t) x2 ) = tg(y) + (1 − t)g(x) + c ( t(1 − 1 + t) y2 + 2t(1 − t) xy + (1 − t)(1 − t) x2 ) = tg(y) + (1 − t)g(x) + c ( t y2 − t(1 − t) y2 + 2t(1 − t) xy + 1 − t x2 − t(1 − t) x2 ) = t ( g(y) + c 1 y2 ) + (1 − t) ( g(x) + c 1 x2 ) − ct(1 − t) ( 1 y2 − 2 xy + 1 x2 ) = tf(y) + (1 − t)f(x) − ct(1 − t) ( 1 y − 1 x )2 , for all x, y ∈ i and t ∈ [0, 1], showing that f ∈src(i,c). example 2.9. (a) the constant function is harmonically convex but not reciprocally strongly convex. (b) the function f : (0, +∞) → r defined by f(x) = −x2, is not a harmonically convex function, since f is a not convex and nonincreasing function. based on theorem 2.7, we obtain f /∈ src(i,c). (c) since g(x) = log(x) is a harmonically convex function, the function f(x) := log(x) + c x2 is a reciprocally strongly convex function. lemma 2.10. if f is a reciprocally strongly convex function, then the function φ = f + ϵ is also a reciprocally strongly convex function, for any constants ϵ. in fact, φ ( xy tx + (1 − t)y ) = f ( xy tx + (1 − t)y ) + ϵ ≤ tf(y) + (1 − t)f(x) + ct(1 − t) ( 1 x − 1 y )2 + ϵ = tf(y) + tϵ + (1 − t)f(x) + (1 − t)ϵ + ct(1 − t) ( 1 x − 1 y )2 = t(f(y) + ϵ) + (1 − t)(f(x) + ϵ) + ct(1 − t) ( 1 x − 1 y )2 = tφ(y) + (1 − t)φ(x) + ct(1 − t) ( 1 x − 1 y )2 . 116 m. bracamonte, j. medina, m. vivas theorem 2.11. if f : [a, b] ⊂ r\{0} → r and if we consider the function g : [ 1 b , 1 a ] → r, defined by g(t) = f ( 1 t ) , then f ∈src([a,b],c) if and only if g is strongly convex in [ 1 b , 1 a ] . proof. if for all x, y ∈ [a, b] and t ∈ [0, 1], we have f ( 1 t1 y + (1 − t) 1 x ) ≤ tf(y) + (1 − t)f(x) − ct(1 − t) ( 1 x − 1 y )2 ; this last inequality may be changed by another equivalent one: g (tw + (1 − t)u) ≤ tg (w) + (1 − t)g (u) − ct(1 − t) (u − w)2 , where u, w ∈ [ 1 b , 1 a ] and t ∈ [0, 1]. to complete the proof. it is easy to see that the result is also valid for intervals (a, b) ⊂ r \ {0}. theorem 2.12. the following are equivalent: (i) f ∈src((a,b),c). (ii) for each x0 ∈ (a, b), there is a linear function t such that f ( 1 x ) ≥ c(x−x0)2+t(x−x0)+f ( 1 x0 ) , for all x ∈ ( 1 b , 1 a ) . (2.5) (iii) for differentiable f and x0 ∈ (a, b), f ( 1 x ) ≥ f ( 1 x0 ) − f ( 1 x0 ) x − x0 x2 + c(x − x0)2, (2.6) for all x, y ∈ (a, b). (iv) for twice differentiable f, 1 x4 [ f ′′ ( 1 x ) + 2xf ′ ( 1 x )] ≥ 2c , for all x ∈ ( 1 b , 1 a ) . proof. (i) ⇒ (ii) : assume that f ∈ src((a,b),c). since all the assumptions of theorem 2.11 are satisfied, then the function g(x) := f ( 1 x ) is strongly strongly reciprocally convex functions 117 convex in ( 1 b , 1 a ) . then by theorem 2.3, for each x0 ∈ ( 1 b , 1 a ) , there is a linear function t such that g(x) ≥ g(x0) + t(x − x0) + c(x − x0)2, for all x, y ∈ ( 1 b , 1 a ) . this is equivalent to the inequality (2.5). (i) ⇒ (iii) : assume that f ∈src((a,b),c). by theorem 2.11, the function g(x) := f ( 1 x ) is strongly convex in ( 1 b , 1 a ) , then by theorem 2.3, for each x0 ∈ ( 1 b , 1 a ) , g(x) ≥ g(x0) + g′(x0)(x − x0) + c(x − x0)2, for all x, y ∈ (a, b). this is equivalent to the inequality (2.6). (ii) ⇒ (i) , (iii) ⇒ (i) are shown using the reciprocals of the theorem and lemma that we have used in the above part. (i) ⇐⇒ (iv) : suppose f is twice differentiable over (a, b). f ∈ src((a,b),c) if only if the function g(x) := f ( 1 x ) is strongly convex in ( 1 b , 1 a ) (by the theorem 2.11). it follows from theorem 2.3 that g is a strongly convex function with modulus c if only if g′′(x) ≥ 2c. hence it is equivalent to 1 x4 [ f ′′ ( 1 x ) + 2xf ′ ( 1 x )] ≥ 2c , for all x ∈ ( 1 b , 1 a ) . 3. main results in this section, we derive our main results. 3.1. hermite-hadamard type inequalities the following result is a counterpart of the hermite-hadamard inequality for strongly reciprocally convex functions. theorem 3.1. let i ⊂ r \ {0} be a real interval. if f : i → r is a strongly reciprocally convex function with modulus c, a, b ∈ i with a < b and f ∈ l[a, b] then f ( 2ab a + b ) + c 12 ( b − a ab )2 ≤ ab b − a ∫ b a f(x) x2 dx ≤ f(a) + f(b) 2 − c 6 ( b − a ab )2 . (3.1) proof. by theorem 2.11 the function g : i → r, defined by g(x) := f(x) − c x2 is harmonically convex, since f ∈src(i,c). 118 m. bracamonte, j. medina, m. vivas consequently, by the hermite-hadamard type inequality for harmonically convex functions (see [13, theorem 1]), we have g ( 2ab a + b ) ≤ ab b − a ∫ b a g(x) x2 dx ≤ g(a) + g(b) 2 , f ( 2ab a + b ) − c ( a + b 2ab )2 ≤ ab b − a ∫ b a f(x) − c x2 x2 dx ≤ f(a) − c a2 + f(b) − c b2 2 . this last inequality can be simplified to f ( 2ab a + b ) − c ( a + b 2ab )2 ≤ ab b − a ∫ b a f(x) x2 dx − abc 3(b − a) [ b3 − a3 a3b3 ] ≤ f(a) + f(b) 2 − c 2 ( a2 + b2 a2b2 ) , which in turn is equivalent to the inequality f ( 2ab a + b ) + c 12 ( b − a ab )2 ≤ ab b − a ∫ b a f(x) x2 dx ≤ f(a) + f(b) 2 − c 6 ( b − a ab )2 . remark 3.2. letting c → 0+, in the inequalities (3.1), we obtain (1.4), which is the hermite-hadamard type inequalities for harmonically convex functions. we establish some new inequalities of hermite-hadamard type for functions whose derivatives are strongly reciprocally convex. we need the following lemma, which can be found in [13]. lemma 3.3. ([13]) let f : i ⊂ r \ {0} → r is a differentiable function on i◦ and a, b ∈ i with a < b. if f ′ ∈ l[a, b], then f(a) + f(b) 2 − ab b − a ∫ b a f(x) x2 dx = ab(b − a) 2 ∫ 1 0 1 − 2t (tb + (1 − t)a)2 f ′ ( ab tb + (1 − t)a ) dt . strongly reciprocally convex functions 119 theorem 3.4. let f : i ⊂ (0, +∞) → r be a differentiable function on i◦, a, b ∈ i with a < b, and f ′ ∈ l[a, b]. if |f ′|q is strongly reciprocally convex with modulus c on [a, b] for q ≥ 1, then ∣∣∣∣f(a) + f(b)2 − abb − a ∫ b a f(x) x2 dx ∣∣∣∣ ≤ ab(b − a) 2 λ 1− 1 q 1 [ λ2|f ′(a)|q + λ3|f ′(b)|q − c ( 1 b − 1 a )2 λ4 ]1 q , (3.2) where λ1 = 1 ab − 2 (b − a)2 ln ( (a + b)2 4ab ) , λ2 = − 1 b(b − a) + 3a + b (b − a)3 ln ( (a + b)2 4ab ) , λ3 = 1 a(b − a) − 3b + a (b − a)3 ln ( (a + b)2 4ab ) , λ4 = − 1 b(b − a) + 1 (b − a)4 [ [a(a + 2b) + b(b + 2a)] ln ( (a + b)2 4ab ) − (a + b)2(2a − b) 2b + b2 − 3a2 ] . proof. from lemma 3.3, and letting p := q q − 1 , we get ∣∣∣∣f(a) + f(b)2 − abb − a ∫ b a f(x) x2 dx ∣∣∣∣ = ∣∣∣∣ab(b − a)2 ∫ 1 0 1 − 2t (tb + (1 − t)a)2 f′ ( ab tb + (1 − t)a ) dt ∣∣∣∣ ≤ ab(b − a) 2 ∫ 1 0 ∣∣∣∣ 1 − 2t(tb + (1 − t)a)2 ∣∣∣∣ ∣∣∣∣f′ ( ab tb + (1 − t)a )∣∣∣∣dt (3.3) = ab(b − a) 2 ∫ 1 0 ∣∣∣∣ 1 − 2t(tb + (1 − t)a)2 ∣∣∣∣ 1 p (∣∣∣∣ 1 − 2t(tb + (1 − t)a)2 ∣∣∣∣ 1 q ∣∣∣∣f′ ( ab tb + (1 − t)a )∣∣∣∣ ) dt . 120 m. bracamonte, j. medina, m. vivas we apply hölder’s inequality to the right-hand side of (3.3) and using the hypothesis that |f ′|q ∈ src([a,b,c), we get ≤ ab(b − a) 2 [∫ 1 0 (∣∣∣∣ 1 − 2t[tb + (1 − t)a]2 ∣∣∣∣ 1 p )p dt ]1 p · [∫ 1 0 (∣∣∣∣ 1 − 2t[tb + (1 − t)a]2 ∣∣∣∣ 1 q ∣∣∣∣f′ ( ab tb + (1 − t)a )∣∣∣∣ )q dt ]1 q = ab(b − a) 2 [∫ 1 0 ∣∣∣∣ 1 − 2t[tb + (1 − t)a]2 ∣∣∣∣dt ]1− 1 q · [∫ 1 0 ∣∣∣∣ 1 − 2t[tb + (1 − t)a]2 ∣∣∣∣ ∣∣∣∣f′ ( ab tb + (1 − t)a )∣∣∣∣q dt ]1 q ≤ ab(b − a) 2 [∫ 1 0 ∣∣∣∣ 1 − 2t[tb + (1 − t)a]2 ∣∣∣∣dt ]1− 1 q (3.4) · [∫ 1 0 |1 − 2t| [tb + (1 − t)a]2 ( t|f′(a)|q + (1 − t)|f′(b)|q − ct(1 − t) ( 1 b − 1 a )2) dt ]1 q . it can be shown that λ1 := ∫ 1 0 |1 − 2t| [tb + (1 − t)a]2 dt = 1 ab − 2 (b − a)2 ln ( (a + b)2 4ab ) , λ2 := ∫ 1 0 |1 − 2t|t [tb + (1 − t)a]2 dt = ∫ 1 2 0 (1 − 2t)t [tb + (1 − t)a]2 dt − ∫ 1 1 2 (1 − 2t)t [tb + (1 − t)a]2 dt = − 1 b(b − a) + b + 3a (b − a)3 ln ( (a + b)2 4ab ) , λ3 := ∫ 1 0 |1 − 2t|(1 − t) [tb + (1 − t)a]2 dt = λ1 + λ2 , λ4 := ∫ 1 0 t(1 − t)|1 − 2t| [tb + (1 − t)a]2 dt , = − 1 b(b − a) + 1 (b − a)4 [ [a(a + 2b) + b(b + 2a)] ln ( (a + b)2 4ab ) − (a + b)2(2a − b) 2b + b2 − 3a2 ] . strongly reciprocally convex functions 121 now if we replace this values in (3.4), we get (3.2). 3.2. fejér type inequalities the following result is a counterpart of the fejér inequality for strongly reciprocally convex functions. theorem 3.5. let i ⊂ r \ {0} be a real interval. if f : i → r is a strongly reciprocally convex function with modulus c, a, b ∈ i with a < b and f ∈ l[a, b] then f ( 2ab a + b )∫ b a p(x) x2 dx + c ∫ b a [ 1 x2 − ( a + b 2ab )2] p(x) x2 dx ≤ ∫ b a f(x) x2 p(x) dx (3.5) ≤ f(a) + f(b) 2 ∫ b a p(x) x2 dx − c ∫ b a [ 1 2 ( a2 + b2 a2b2 ) − 1 x2 ] p(x) x2 dx , where p : [a, b] → [0, ∞) is an integrable function and satisfies p ( ab x ) = p ( ab a + b − x ) . (3.6) proof. by theorem 2.11 the function g : i → r, defined by g(x) := f(x) − c x2 is harmonically convex, then in virtue of theorem 1.3, we have that g ( 2ab a + b )∫ b a p(x) x2 dx ≤ ∫ b a g(x) x2 p(x) dx ≤ g(a) + g(b) 2 ∫ b a p(x) x2 dx . the above inequality is equivalent to [ f ( 2ab a + b ) − c ( a + b 2ab )2]∫ b a p(x) x2 dx ≤ ∫ b a f(x) − c x2 x2 p(x) dx ≤ f(a) − c a2 + f(b) − c b2 2 ∫ b a p(x) x2 dx . 122 m. bracamonte, j. medina, m. vivas this last inequality can be simplified to f ( 2ab a + b )∫ b a p(x) x2 dx − c ( a + b 2ab )2 ∫ b a p(x) x2 dx + c ∫ b a p(x) x4 dx ≤ ∫ b a f(x) x2 dx ≤ f(a) + f(b) 2 ∫ b a p(x) x2 dx − c 2 ( 1 a2 + 1 b2 )∫ b a p(x) x2 dx + c ∫ b a p(x) x4 dx , which in turn is equivalent to the inequality f ( 2ab a + b )∫ b a p(x) x2 dx + c ∫ b a [ 1 x2 − ( a + b 2ab )2] p(x) x2 dx ≤ ∫ b a f(x) x2 p(x) dx ≤ f(a) + f(b) 2 ∫ b a p(x) x2 dx − c ∫ b a [ 1 2 ( a2 + b2 a2b2 ) − 1 x2 ] p(x) x2 dx . remarks 3.6. (a) letting c → 0+, in inequality (3.5), we obtain (1.5) which is the fejér type inequality for harmonically convex functions. (b) putting p(x) ≡ 1 into theorem 3.5, we obtain the inequality (3.1). now, we establish a new fejér-type inequality for strongly reciprocally convex functions. theorem 3.7. suppose f : i ⊂ r \ {0} → r is a strongly reciprocally convex function with modulus c on i. if a, b ∈ i, a < b, and f ∈ l[a, b], then f ( 2ab a + b )∫ b a p(x) x2 dx + c 2ab ∫ b a p (x) x4 [2ab − (a + b)x] dx ≤ ∫ b a f(x) x2 p(x) dx (3.7) ≤ a [f(a) + f(b)] b − a ∫ b a (b − x) p(x) x3 dx − c ab ∫ b a (b − x)(x − a) p(x) x4 dx , where p : [a, b] → r is a nonnegative integrable function that satisfies (3.6). strongly reciprocally convex functions 123 proof. according to (3.6), for x = tb + (1 − t)a, we have p ( ab tb + (1 − t)a ) = p ( ab ta + (1 − t)b ) . (3.8) since f ∈ src([a,b],c), from the definition 2.6, we obtain f ( 2xy x + y ) ≤ f(y) + f(x) 2 − c 4 ( 1 x − 1 y )2 , x, y ∈ [a, b] . (3.9) let x = ab tb + (1 − t)a and y = ab ta + (1 − t)b in (3.9), then f ( 2ab a + b ) ≤ f ( ab ta+(1−t)b ) + f ( ab tb+(1−t)a ) 2 − c 4 ( tb + (1 − t)a ab − ta + (1 − t)b ab )2 . thus, f ( 2ab a + b ) p ( ab tb + (1 − t)a ) ≤ 1 2 [ f ( ab ta + (1 − t)b ) p ( ab ta + (1 − t)b ) + f ( ab tb + (1 − t)a ) p ( ab tb + (1 − t)a )] − c 4 ( tb + (1 − t)a ab − ta + (1 − t)b ab )2 p ( ab tb + (1 − t)a ) . integrating both sides of the above inequalities with respect to t over [0, 1], we obtain f ( 2ab a + b )∫ 1 0 p ( ab tb + (1 − t)a ) dt ≤ 1 2 ∫ 1 0 f ( ab ta + (1 − t)b ) p ( ab ta + (1 − t)b ) dt + 1 2 ∫ 1 0 f ( ab tb + (1 − t)a ) p ( ab tb + (1 − t)a ) dt − c 4 ∫ 1 0 ( tb + (1 − t)a ab − ta + (1 − t)b ab )2 p ( ab tb + (1 − t)a ) dt . 124 m. bracamonte, j. medina, m. vivas by simple computation, f ( 2ab a + b ) ab b − a ∫ b a p(x) x2 dx ≤ 1 2 ab b − a ∫ b a f (x) x2 p (x) dx + ab b − a ∫ b a f (x) x2 p (x) dx − c 4 2 b − a ∫ b a p (x) x4 [2ab − (a + b)x] dx . on the other hand, f ( ab ta + (1 − t)b ) p ( ab ta + (1 − t)b ) ≤ [ tf(b) + (1 − t)f(a) − ct(1 − t) ( 1 a − 1 b )2] p ( ab ta + (1 − t)b ) . again, integrating both sides of the above inequalities with respect to t over [0, 1], we obtain∫ 1 0 f ( ab ta + (1 − t)b ) p ( ab ta + (1 − t)b ) dt ≤ ∫ 1 0 [ tf(b) + (1 − t)f(a) − ct(1 − t) ( 1 a − 1 b )2] p ( ab ta + (1 − t)b ) dt . by simple computation,∫ b a f (x) x2 p (x) dx ≤ a [f(a) + f(b)] b − a ∫ b a (b − x) p (x) x3 dx − c ab ∫ b a (b − x) (x − a) p (x) x4 dx . this concludes the proof. remarks 3.8. 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[24] c. zălinescu, “ convex analysis in general vector spaces ”, world scientific publishing co., inc., river edge, nj, 2002. e extracta mathematicae vol. 33, núm. 1, 51 – 66 (2018) the geometry of l(3l2∞) and optimal constants in the bohnenblust-hille inequality for multilinear forms and polynomials sung guen kim ∗ department of mathematics, kyungpook national university, daegu 702 − 701, south korea, sgk317@knu.ac.kr presented by ricardo garćıa received december 23, 2016 abstract: we classify the extreme and exposed 3-linear forms of the unit ball of l(3l2∞). we introduce optimal constants in the bohnenblust-hille inequality for symmetric multilinear forms and polynomials and investigate about their relations. key words: extreme points, exposed points, the optimal constants in the bohnenblust-hille inequality for symmetric multilinear forms and polynomials. ams subject class. (2010): 46a22. 1. introduction we write be for the closed unit ball of a real banach space e and the dual space of e is denoted by e∗. x ∈ be is called an extreme point of be if y, z ∈ be with x = 12(y + z) implies x = y = z. x ∈ be is called an exposed point of be if there is a f ∈ e∗ so that f(x) = 1 = ∥f∥ and f(y) < 1 for every y ∈ be \ {x}. it is easy to see that every exposed point of be is an extreme point. we denote by extbe and expbe the sets of extreme and exposed points of be, respectively. let n ∈ n, n ≥ 2. a mapping p : e → r is a continuous n-homogeneous polynomial if there exists a continuous n-linear form l on the product e × · · · × e such that p(x) = l(x, . . . , x) for every x ∈ e. we denote by l(ne) the banach space of all continuous n-linear forms on e endowed with the norm ∥l∥ = sup∥xj∥=1, 1≤j≤n |l(x1, . . . , xn)|. ls( ne) denotes the closed subspace of l(ne) consisting all continuous symmetric n-linear forms on e. p(ne) denotes the banach space of all continuous nhomogeneous polynomials from e into r endowed with the norm ∥p∥ = ∗this research was supported by the basic science research program through the national research foundation of korea(nrf) funded by the ministry of education, science and technology (2013r1a1a2057788). 51 52 s. g. kim sup∥x∥=1 |p(x)|. note that the spaces l(ne), ls(ne), p(ne) are very different from a geometric point of view. in particular, for integral multilinear forms and integral polynomials one has ([2], [9], [32]) extbli(ne) = {ϕ1ϕ2 · · · ϕn : ϕi ∈ extbe∗} and extbpi(ne) = {±ϕ n : ϕ ∈ e∗, ∥ϕ∥ = 1}, where li(ne) and pi(ne) are the spaces of integral n-linear forms and integral n-homogeneous polynomials on e, respectively. for more details about the theory of multilinear mappings and polynomials on a banach space, we refer to [10]. in 1998, choi et al. ([4], [5]) characterized the extreme points of the unit ball of p(2l21) and p( 2l22). kim [15] classified the exposed 2-homogeneous polynomials on p(2l2p) (1 ≤ p ≤ ∞). kim ([17], [19], [23]) classified the extreme, exposed, smooth points of the unit ball of p(2d∗(1, w)2), where d∗(1, w)2 = r2 with the octagonal norm of weight w. in 2009, kim [16] initiated extremal problems for bilinear forms on a classical finite dimensional real banach space and classified the extreme, exposed, smooth points of the unit ball of ls(2l2∞). kim ([18], [20]–[22]) classified the extreme, exposed, smooth points of the unit balls of ls(2d∗(1, w)2) and l(2d∗(1, w)2). we refer to ([1]–[9], [11]–[32] and references therein) for some recent work about extremal properties of multilinear mappings and homogeneous polynomials on some classical banach spaces. let k = r or c. the bohnenblust-hille inequality for n-linear forms ([3] and references therein) tells us that there exists a sequence of positive scalars (c(n : k))∞n=1 in [1, ∞] such that( ∞∑ j1,...,jn=1 ∣∣t(ej1, . . . , ejn)∣∣ 2nn+1 )n+1 2n ≤ c(n : k)∥t∥ for all continuous n-linear forms t : c0 × · · · × c0 → k. the optimal constant in the bohnenblust-hille inequality for n-linear forms c(n : k) is defined by c(n : k) := sup {( ∞∑ j1,...,jn=1 |t(ej1, . . . , ejn)| 2n n+1 )n+1 2n : t ∈ l(nc0 : k), ∥t∥ = 1 } . the geometry of l(3l2∞) 53 we introduce the optimal constant in the bohnenblust-hille inequality for symmetric n-linear forms cs(n : k) is defined by cs(n : k) := sup {( ∞∑ j1,...,jn=1 |t(ej1, . . . , ejn)| 2n n+1 )n+1 2n : t ∈ ls(nc0 : k), ∥t∥ = 1 } . it is obvious that cs(n : k) ≤ c(n : k). we also introduce the optimal constant in the bohnenblust-hille inequality for n-homogeneous polynomials cp(n : k) is defined by cp(n : k) := sup {( ∞∑ j=1 |p(ej)| 2n n+1 )n+1 2n : p ∈ p(nc0 : k), ∥p∥ = 1 } . recently, diniz et al. [12] showed that c(2 : r) = √ 2. in this paper, we classify the extreme and exposed 3-linear forms of the unit ball of l(3l2∞). we introduce optimal constants in the bohnenblust-hille inequality for symmetric multilinear forms and polynomials and investigate about their relations. 2. the extreme points of the unit ball of l(3l2∞) let t ∈ l(3l2∞) be given by t ( (x1, x2), (y1, y2), (z1, z2) ) = ax1y1z1 + bx2y2z2 + c1x2y1z1 + c2x1y2z1 + c3x1y1z2 + d1x1y2z2 + d2x2y1z2 + d3x2y2z1 for some a, b, cj, dj ∈ r and for j = 1, 2, 3. for simplicity, we will denote t = (a, b, c1, c2, c3, d1, d2, d3). theorem 2.1. let t = (a, b, c1, c2, c3, d1, d2, d3) ∈ l(3l2∞). then ∥t∥ = max { |a + c1 + c2 + d3| + |b + c3 + d1 + d2|, |a − c2 − c3 + d1| + |b + c1 − d2 − d3|, |a − b + c3 − d3| + |c1 − c2 − d1 + d2|, |a + b − c1 − d1| + |c2 − c3 + d2 − d3| } . 54 s. g. kim proof. note that extbl2∞ = {(1, 1), (1, −1), (−1, 1), (−1, −1)}. by the kreinmilman theorem, bl2∞ = co(extbl2∞). by the continuity and trilinearity of t , ∥t∥ = max { |t((1, 1), (1, 1), (1, 1))|, |t((1, −1), (1, 1), (1, 1))| |t((1, 1), (1, −1), (1, 1))|, |t((1, 1), (1, 1), (1, −1))|, |t((1, −1), (1, −1), (1, 1))|, |t((1, −1), (1, 1), (1, −1))|, |t((1, 1), (1, −1), (1, −1))|, |t((1, −1), (1, −1), (1, −1))| } = max { |a + c1 + c2 + d3| + |b + c3 + d1 + d2|, |a − c2 − c3 + d1| + |b + c1 − d2 − d3|, |a − b + c3 − d3| + |c1 − c2 − d1 + d2|, |a + b − c1 − d1| + |c2 − c3 + d2 − d3| } . note that if ∥t∥ = 1, then |a| ≤ 1, |b| ≤ 1, |cj| ≤ 1, |dj| ≤ 1, for j = 1, 2, 3. theorem 2.2. extbl(3l2∞) = { (a, b, c1, c2, c3, d1, d2, d3) : a = 1 8 (ϵ1 + ϵ2 + ϵ3 + ϵ4 + ϵ5 + ϵ6 + ϵ7 + ϵ8), b = 1 8 (ϵ1 − ϵ2 − ϵ3 − ϵ4 + ϵ5 + ϵ6 + ϵ7 − ϵ8), c1 = 1 8 (ϵ1 − ϵ2 + ϵ3 + ϵ4 − ϵ5 − ϵ6 + ϵ7 − ϵ8), c2 = 1 8 (ϵ1 + ϵ2 − ϵ3 + ϵ4 − ϵ5 + ϵ6 − ϵ7 − ϵ8), c3 = 1 8 (ϵ1 + ϵ2 + ϵ3 − ϵ4 + ϵ5 − ϵ6 − ϵ7 − ϵ8), d1 = 1 8 (ϵ1 + ϵ2 − ϵ3 − ϵ4 − ϵ5 − ϵ6 + ϵ7 + ϵ8), d2 = 1 8 (ϵ1 − ϵ2 + ϵ3 − ϵ4 − ϵ5 + ϵ6 − ϵ7 + ϵ8), d3 = 1 8 (ϵ1 − ϵ2 − ϵ3 + ϵ4 + ϵ5 − ϵ6 − ϵ7 + ϵ8), ϵj = ±1, for j = 1, 2, . . . , 8 } . the geometry of l(3l2∞) 55 proof. let t = (a, b, c1, c2, c3, d1, d2, d3) ∈ l(3l2∞) with ∥t∥ = 1. note that t ( (1, 1), (1, 1), (1, 1) ) = a + b + c1 + c2 + c3 + d1 + d2 + d3, t ( (1, −1), (1, 1), (1, 1) ) = a − b − c1 + c2 + c3 + d1 − d2 − d3, t ( (1, 1), (1, −1), (1, 1) ) = a − b + c1 − c2 + c3 − d1 + d2 − d3, t ( (1, 1), (1, 1), (1, −1) ) = a − b + c1 + c2 − c3 − d1 − d2 + d3, t ( (1, −1), (1, −1), (1, 1) ) = a + b − c1 − c2 + c3 − d1 − d2 + d3, t ( (1, −1), (1, 1), (1, −1) ) = a + b − c1 + c2 − c3 − d1 + d2 − d3, t ( (1, 1), (1, −1), (1, −1) ) = a + b + c1 − c2 − c3 + d1 − d2 − d3, t ( (1, −1), (1, −1), (1, −1) ) = a − b − c1 − c2 − c3 + d1 + d2 + d3. let a = (aij)1≤i,j≤8 be the 8 × 8 matrix such that ai1 = 1 (i = 1, . . . , 8), ai2 = 1 (i = 1, 5, 6, 7), ak2 = −1 (k = 2, 3, 4, 8), ai3 = 1 (i = 1, 3, 4, 7), ak3 = −1 (k = 2, 5, 6, 8), ai4 = 1 (i = 1, 2, 4, 6), ak4 = −1 (k = 3, 5, 7, 8), ai5 = 1 (i = 1, 2, 3, 5), ak5 = −1 (k = 4, 6, 7, 8), ai6 = 1 (i = 1, 2, 7, 8), ak6 = −1 (k = 3, 4, 5, 6), ai7 = 1 (i = 1, 3, 6, 8), ak7 = −1 (k = 2, 4, 5, 7), ai8 = 1 (i = 1, 4, 5, 8), ak8 = −1 (k = 2, 3, 6, 7). by calculation, det(a) = −212, so a is invertible. note that at = ( t ( (1, 1), (1, 1), (1, 1) ) , t ( (1, −1), (1, 1), (1, 1) ) , t ( (1, 1), (1, −1), (1, 1) ) , t ( (1, 1), (1, 1), (1, −1) ) , t ( (1, −1), (1, −1), (1, 1) ) , t ( (1, −1), (1, 1), (1, −1) ) , t ( (1, 1), (1, −1), (1, −1) ) , t ( (1, −1), (1, −1), (1, −1) ))t and ∥at∥∞ = ∥t∥. note also that t = a−1 ( t ( (1, 1), (1, 1), (1, 1) ) , t ( (1, −1), (1, 1), (1, 1) ) , t ( (1, 1), (1, −1), (1, 1) ) , t ( (1, 1), (1, 1), (1, −1) ) , t ( (1, −1), (1, −1), (1, 1) ) , t ( (1, −1), (1, 1), (1, −1) ) , t ( (1, 1), (1, −1), (1, −1) ) , t ( (1, −1), (1, −1), (1, −1) ))t . 56 s. g. kim we claim that t ∈ extbl(3l2∞) if and only if 1 = ∣∣t((1, 1), (1, 1), (1, 1))∣∣ = ∣∣t((1, −1), (1, 1), (1, 1))∣∣ = ∣∣t((1, 1), (1, −1), (1, 1))∣∣ = ∣∣t((1, 1), (1, 1), (1, −1))∣∣ = ∣∣t((1, −1), (1, −1), (1, 1))∣∣ = ∣∣t((1, −1), (1, 1), (1, −1))∣∣ = ∣∣t((1, 1), (1, −1), (1, −1))∣∣ = ∣∣t((1, −1), (1, −1), (1, −1))∣∣. (⇒): otherwise. then we have 8 cases as follows: case 1 : ∣∣t((1, 1), (1, 1), (1, 1))∣∣ < 1 or case 2 : ∣∣t((1, −1), (1, 1), (1, 1))∣∣ < 1 or case 3 : ∣∣t((1, 1), (1, −1), (1, 1))∣∣ < 1 or case 4 : ∣∣t((1, 1), (1, 1), (1, −1))∣∣ < 1 or case 5 : ∣∣t((1, −1), (1, −1), (1, 1))∣∣ < 1 or case 6 : ∣∣t((1, −1), (1, 1), (1, −1))∣∣ < 1 or case 7 : ∣∣t((1, 1), (1, −1), (1, −1))∣∣ < 1 or case 8 : ∣∣t((1, −1), (1, −1), (1, −1))∣∣ < 1. case 1: ∣∣t((1, 1), (1, 1), (1, 1))∣∣ < 1. let ϵ1 := t ( (1, 1), (1, 1), (1, 1) ) , ϵ2 := t ( (1, −1), (1, 1), (1, 1) ) , ϵ3 := t ( (1, 1), (1, −1), (1, 1) ) , ϵ4 := t ( (1, 1), (1, 1), (1, −1) ) , ϵ5 := t ( (1, −1), (1, −1), (1, 1) ) , ϵ6 := t ( (1, −1), (1, 1), (1, −1) ) , ϵ7 := t ( (1, 1), (1, −1), (1, −1) ) , ϵ8 := t ( (1, −1), (1, −1), (1, −1) ) . then, at = (ϵ1, ϵ2, ϵ3, ϵ4, ϵ5, ϵ6, ϵ7, ϵ8) t. let n0 ∈ n such that |ϵ1| + 1n0 < 1. let t1, t2 ∈ l( 3l2∞) be the solutions of at1 = ( ϵ1+ 1 n0 , ϵ2, ϵ3, ϵ4, ϵ5, ϵ6, ϵ7, ϵ8 )t , at2 = ( ϵ1− 1 n0 , ϵ2, ϵ3, ϵ4, ϵ5, ϵ6, ϵ7, ϵ8 )t . the geometry of l(3l2∞) 57 note that tj ̸= t, ∥tj∥ = ∥atj∥∞ = 1 for j = 1, 2. it follows that a (1 2 (t1 + t2) ) = (ϵ1, ϵ2, ϵ3, ϵ4, ϵ5, ϵ6, ϵ7, ϵ8) t = at, which shows that 1 2 (t1 + t2) = a −1(ϵ1, ϵ2, ϵ3, ϵ4, ϵ5, ϵ6, ϵ7, ϵ8) t = t, so t is not extreme. by the similar argument in the case 1, if any of cases 2–8 holds, then we may reach to a contradiction. (⇐): let ϵj ∈ r be given for j = 1, 2, . . . , 8. consider the following system of 8 simultaneous linear equations: at = (ϵ1, ϵ2, ϵ3, ϵ4, ϵ5, ϵ6, ϵ7, ϵ8) t, ie, a + b + c1 + c2 + c3 + d1 + d2 + d3 = ϵ1, a − b − c1 + c2 + c3 + d1 − d2 − d3 = ϵ2, a − b + c1 − c2 + c3 − d1 + d2 − d3 = ϵ3, a − b + c1 + c2 − c3 − d1 − d2 + d3 = ϵ4, a + b − c1 − c2 + c3 − d1 − d2 + d3 = ϵ5, a + b − c1 + c2 − c3 − d1 + d2 − d3 = ϵ6, a + b + c1 − c2 − c3 + d1 − d2 − d3 = ϵ7, a − b − c1 − c2 − c3 + d1 + d2 + d3 = ϵ8. (∗) we get the unique solution of (∗) as follows: t = a−1(ϵ1, ϵ2, ϵ3, ϵ4, ϵ5, ϵ6, ϵ7, ϵ8)t, ie, a = 1 8 (ϵ1 + ϵ2 + ϵ3 + ϵ4 + ϵ5 + ϵ6 + ϵ7 + ϵ8), b = 1 8 (ϵ1 − ϵ2 − ϵ3 − ϵ4 + ϵ5 + ϵ6 + ϵ7 − ϵ8), c1 = 1 8 (ϵ1 − ϵ2 + ϵ3 + ϵ4 − ϵ5 − ϵ6 + ϵ7 − ϵ8), c2 = 1 8 (ϵ1 + ϵ2 − ϵ3 + ϵ4 − ϵ5 + ϵ6 − ϵ7 − ϵ8), c3 = 1 8 (ϵ1 + ϵ2 + ϵ3 − ϵ4 + ϵ5 − ϵ6 − ϵ7 − ϵ8), d1 = 1 8 (ϵ1 + ϵ2 − ϵ3 − ϵ4 − ϵ5 − ϵ6 + ϵ7 + ϵ8), d2 = 1 8 (ϵ1 − ϵ2 + ϵ3 − ϵ4 − ϵ5 + ϵ6 − ϵ7 + ϵ8), d3 = 1 8 (ϵ1 − ϵ2 − ϵ3 + ϵ4 + ϵ5 − ϵ6 − ϵ7 + ϵ8). (∗∗) 58 s. g. kim let t1 = (a+ϵ, b+δ, c1 +γ1, c2 +γ2, c3 +γ3, d1 +ρ1, d2 +ρ2, d3 +ρ3) ∈ l(3l2∞) and t2 = (a−ϵ, b−δ, c1 −γ1, c2 −γ2, c3 −γ3, d1 −ρ1, d2 −ρ2, d3 −ρ3) ∈ l(3l2∞) be such that 1 = ∥t1∥ = ∥t2∥ for some ϵ, δ, γj, ρj for j = 1, 2, 3. then, for k = 1, 2, 1 ≥ ∣∣tk((1, 1), (1, 1), (1, 1))∣∣ = 1 + |ϵ + δ + γ1 + γ2 + γ3 + ρ1 + ρ2 + ρ3|, 1 ≥ ∣∣tk((1, −1), (1, 1), (1, 1))∣∣ = 1 + |ϵ − δ − γ1 + γ2 + γ3 + ρ1 − ρ2 − ρ3|, 1 ≥ ∣∣tk((1, 1), (1, −1), (1, 1))∣∣ = 1 + |ϵ − δ + γ1 − γ2 + γ3 − ρ1 + ρ2 − ρ3|, 1 ≥ ∣∣tk((1, 1), (1, 1), (1, −1))∣∣ = 1 + |ϵ − δ + γ1 + γ2 − γ3 − ρ1 − ρ2 + ρ3|, 1 ≥ ∣∣tk((1, −1), (1, −1), (1, 1))∣∣ = 1 + |ϵ + δ − γ1 − γ2 + γ3 − ρ1 − ρ2 + ρ3|, 1 ≥ ∣∣tk((1, −1), (1, 1), (1, −1))∣∣ = 1 + |ϵ + δ − γ1 + γ2 − γ3 − ρ1 + ρ2 − ρ3|, 1 ≥ ∣∣tk((1, 1), (1, −1), (1, −1))∣∣ = 1 + |ϵ + δ + γ1 − γ2 − γ3 + ρ1 − ρ2 − ρ3|, 1 ≥ ∣∣tk((1, −1), (1, −1), (1, −1))∣∣ = 1 + |ϵ − δ − γ1 − γ2 − γ3 + ρ1 + ρ2 + ρ3|. therefore, we have 0 = ϵ + δ + γ1 + γ2 + γ3 + ρ1 + ρ2 + ρ3, 0 = ϵ − δ − γ1 + γ2 + γ3 + ρ1 − ρ2 − ρ3, 0 = ϵ − δ + γ1 − γ2 + γ3 − ρ1 + ρ2 − ρ3, 0 = ϵ − δ + γ1 + γ2 − γ3 − ρ1 − ρ2 + ρ3, 0 = ϵ + δ − γ1 − γ2 + γ3 − ρ1 − ρ2 + ρ3, 0 = ϵ + δ − γ1 + γ2 − γ3 − ρ1 + ρ2 − ρ3, 0 = ϵ + δ + γ1 − γ2 − γ3 + ρ1 − ρ2 − ρ3, 0 = ϵ − δ − γ1 − γ2 − γ3 + ρ1 + ρ2 + ρ3. hence, a(ϵ, δ, γ1, γ2, γ3, ρ1, ρ2, ρ3) t = 0. by (∗∗), 0 = ϵ = δ = γ1 = γ2 = γ3 = ρ1 = ρ2 = ρ3. hence, t is extreme. therefore, we complete the proof. corollary 2.3. if t = (a, b, c1, c2, c3, d1, d2, d3) ∈ extbl(3l2∞), then |a|, |b|, |cj|, |dj| ∈ {0, 14, 1 2 , 3 4 , 1} for j = 1, 2, 3. the geometry of l(3l2∞) 59 theorem 2.4. ([26]) extbls(3l2∞) = { ± (1, 0, 0, 0, 0, 0, 0, 0), ±(0, 1, 0, 0, 0, 0, 0, 0, ), ± (1 2 , 0, 0, 0, 0, − 1 2 , − 1 2 , − 1 2 ) , ± ( 0, 1 2 , − 1 2 , − 1 2 , − 1 2 , 0, 0, 0 ) , ± (1 4 , − 3 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 ) , ± ( − 3 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 ) , ± (3 4 , 1 4 , 1 4 , 1 4 , 1 4 , − 1 4 , − 1 4 , − 1 4 ) , ± (1 4 , 3 4 , − 1 4 , − 1 4 , − 1 4 , 1 4 , 1 4 , 1 4 )} . theorem 2.5. extbls(3l2∞) = extbl(3l2∞) ∩ ls( 3l2∞). proof. it follows from theorems 2.2 and 2.4. remarks. (1) 24 = |extbls(3l2∞)| < |extbl(3l2∞)| = 2 8. (2) let t = (a, b, c1, c2, c3, d1, d2, d3) ∈ l(3l2∞). then, by scaling, we may assume that dj ≥ 0 for j = 1, 2, 3. proof. let t1((x1, x2), (y1, y2), (z1, z2)) := t((ϵ1x1, x2), (ϵ2y1, y2)), (ϵ3z1, z2), where ϵk = ±1 be given satisfying ϵjdj ≥ 0 for j = 1, 2, 3. question. is it true that extbls(nl2∞) = extbl(nl2∞) ∩ ls( nl2∞) for n ≥ 4? 3. the exposed points of the unit ball of l(3l2∞) theorem 3.1. let f ∈ l(3l2∞)∗ with α = f(x1y1z1), β = f(x2y2z2), γ1 = f(x2y1z1), γ2 = f(x1y2z1), γ3 = f(x1y1z2), δ1 = f(x1y2z2), δ2 = f(x2y1z2), δ3 = f(x2y2z1). 60 s. g. kim then, ∥f∥ = max {∣∣∣∣aα + bβ + ∑ j=1,2,3 cjγj + ∑ j=1,2,3 djδj ∣∣∣∣ : a = 1 8 (ϵ1 + ϵ2 + ϵ3 + ϵ4 + ϵ5 + ϵ6 + ϵ7 + ϵ8), b = 1 8 (ϵ1 − ϵ2 − ϵ3 − ϵ4 + ϵ5 + ϵ6 + ϵ7 − ϵ8), c1 = 1 8 (ϵ1 − ϵ2 + ϵ3 + ϵ4 − ϵ5 − ϵ6 + ϵ7 − ϵ8), c2 = 1 8 (ϵ1 + ϵ2 − ϵ3 + ϵ4 − ϵ5 + ϵ6 − ϵ7 − ϵ8), c3 = 1 8 (ϵ1 + ϵ2 + ϵ3 − ϵ4 + ϵ5 − ϵ6 − ϵ7 − ϵ8), d1 = 1 8 (ϵ1 + ϵ2 − ϵ3 − ϵ4 − ϵ5 − ϵ6 + ϵ7 + ϵ8), d2 = 1 8 (ϵ1 − ϵ2 + ϵ3 − ϵ4 − ϵ5 + ϵ6 − ϵ7 + ϵ8), d3 = 1 8 (ϵ1 − ϵ2 − ϵ3 + ϵ4 + ϵ5 − ϵ6 − ϵ7 + ϵ8), ϵj = ±1, for j = 1, 2, . . . , 8 } proof. proof. it follows from theorem 2.2 and the krein-milman theorem. theorem 3.2. expbl(3l2∞) = extbl(3l2∞). proof. we will show that extbl(3l2∞) ⊂ expbl(3l2∞). by theorem 2.2, corollary 2.3 and remarks(2), it suffices to show that if t =(1, 0, 0, 0, 0, 0, 0, 0), ( − 1 2 , 1 2 , 1 2 , 1 2 , 0, 0, 0, 0 ) ( − 3 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 ) , (3 4 , − 1 4 , − 1 4 , − 1 4 , 1 4 , 1 4 , 1 4 , 1 4 ) , then t ∈ expbl(3l2∞). claim: t = (1, 0, 0, 0, 0, 0, 0, 0) ∈ expbl(3l2∞). let f ∈ l(3l2∞)∗ with α = 1, 0 = β = γj = δj for j = 1, 2, 3. note that, by corollary 2.3 and theorems 2.2 and 3.1, ∥f∥ = 1 = f(t) and |f(s)| < 1 for all s ∈ extbl(3l2∞)\{±t}. the claim follows from theorem 2.3 of [21]. the geometry of l(3l2∞) 61 claim: t = ( − 1 2 , 1 2 , 1 2 , 1 2 , 0, 0, 0, 0 ) ∈ expbl(3l2∞). let f ∈ l(3l2∞)∗ with −α = 1 2 = β = γ1 = γ2, 0 = γ3 = δj for j = 1, 2, 3. note that, by corollary 2.3 and theorems 2.2 and 3.1, ∥f∥ = 1 = f(t) and |f(s)| < 1 for all s ∈ extbl(3l2∞)\{±t}. the claim follows from theorem 2.3 of [21]. claim: t = ( − 3 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 , 1 4 ) ∈ expbl(3l2∞). let f ∈ l(3l2∞)∗ with α = − 1 2 , 5 14 = β = γj = δj for j = 1, 2, 3. note that, by corollary 2.3 and theorems 2.2 and 3.1, ∥f∥ = 1 = f(t) and |f(s)| < 1 for all s ∈ extbl(3l2∞)\{±t}. the claim follows from theorem 2.3 of [21]. claim: t = ( 3 4 , −1 4 , −1 4 , −1 4 , 1 4 , 1 4 , 1 4 , 1 4 ) ∈ expbl(3l2∞). let f ∈ l(3l2∞)∗ with α = 1 2 , − 5 14 = β = γ1 = γ2 = −γ3 = −δj for j = 1, 2, 3. note that, by corollary 2.3 and theorems 2.2 and 3.1, ∥f∥ = 1 = f(t) and |f(s)| < 1 for all s ∈ extbl(3l2∞)\{±t}. the claim follows from theorem 2.3 of [21]. we complete the proof. theorem 3.3. ([26]) expbls(3l2∞) = extbls(3l2∞). theorem 3.4. expbls(3l2∞) = expbl(3l2∞) ∩ ls( 3l2∞). proof. it follows from theorems 3.2 and 3.3. question. is it true that expbls(nl2∞) = expbl(nl2∞) ∩ls( nl2∞) for n ≥ 4? 4. optimal constants in the bohnenblust-hille inequality for symmetric multilinear forms and polynomials theorem 4.1. 1 ≤ cp(n : k) ≤ n n n! cs(n : k) ≤ n n n! c(n : k) for all n ≥ 2. proof. it is enough to show that cp(n : k) ≤ n n n! cs(n : k). let p ∈ p(nc0 : k), ∥p∥ = 1. by the polarization formula, ∥p̌∥ ≤ n n n! ∥p∥ = n n n! , where p̌ is the corresponding symmetric n-linear form to p . hence,( ∞∑ j=1 | n! nn p(ej)| 2n n+1 )n+1 2n ≤ ( ∞∑ j1,··· ,jn=1 ∣∣∣∣ n!nn p̌(ej1, · · · , ejn) ∣∣∣∣ 2n n+1 )n+1 2n ≤ cs(n : k). 62 s. g. kim theorem 4.2. cs(2 : r) = c(2 : r) = √ 2. proof. it is enough to show that cs(2 : r) = √ 2. let t ( (x1, x2), (y1, y2) ) = 1 2 x1y1 − 1 2 x2y2 + 1 2 x1y2 + 1 2 x2y1. then t ∈ ls(2l2∞), ∥t∥ = 1. by a result of [12], √ 2 ≤ ( ∑2 i,j=1 |t(ei, ej)| 4 3 ) 3 4 ≤ cs(2 : r) ≤ c(2 : r) = √ 2. theorem 4.3. ([16]) extbls(2l2∞) = { ±(1, 0, 0, 0), ±(0, 1, 0, 0), ± 1 2 (1, −1, 1, 1), ± 1 2 (1, −1, −1, −1) } . theorem 4.4. sup {( 2∑ i,j=1 |t(ei, ej)| 4 3 )3 4 : t ∈ ls(2l2∞), ∥t∥ = 1, t /∈ extbls(2l2∞) } = cs(2 : r). proof. let l := sup {( 2∑ i,j=1 |t(ei, ej)| 4 3 )3 4 : t ∈ ls(2l2∞), ∥t∥ = 1, t /∈ extbls(2l2∞) } . for |c| < 1 2 , let tc ( (x1, x2), (y1, y2) ) = 1 2 x1y1 − 1 2 x2y2 + cx1y2 + cx2y1. then tc ∈ ls(2l2∞), ∥tc∥ = 1. by theorem 4.3, tc /∈ extbls(2l2∞). it follows that cs(2 : r) ≥ l ≥ sup {( 2∑ i,j=1 |tc(ei, ej)| 4 3 )3 4 : |c| < 1 2 } = sup |c|< 1 2 ( 2 (1 2 )4 3 + 2|c| 4 3 )3 4 = √ 2 = cs(2 : r). the geometry of l(3l2∞) 63 theorem 4.5. let n ≥ 2. then, 2 n+1 2n ≤ cp(n : r). proof. let w := sup {( ∞∑ j=1 |p(ej)| 2n n+1 )n+1 2n : p ∈ p(nc0), ∥p∥ = 1, p ( (xm) ∞ m=1 ) = ∞∑ j=1 ajx n j for some aj ∈ r } . claim: w = 2 n+1 2n . let p ∈ p(nc0), ∥p∥ = 1, p((xm)∞m=1) = ∑∞ j=1 ajx n j for some aj ∈ r. let a := {j ∈ n : aj ≥ 0} and b := n\a. note that, for every k ∈ n, 1 ≥ ∣∣∣∣∣p ( ∑ j∈a, j≤k ej )∣∣∣∣∣ = ∑ j∈a, j≤k |aj| and 1 ≥ ∣∣∣∣∣p ( ∑ j∈b, j≤k ej )∣∣∣∣∣ = ∑ j∈b, j≤k |aj|. hence, ∑ j∈a |aj| ≤ 1, ∑ j∈b |aj| ≤ 1. it follows that( ∞∑ j=1 |p(ej)| 2n n+1 )n+1 2n = (∑ j∈a |aj| 2n n+1 + ∑ j∈b |aj| 2n n+1 )n+1 2n ≤ (∑ j∈a |aj| + ∑ j∈b |aj| )n+1 2n ≤ 2 n+1 2n , which shows that w ≤ 2 n+1 2n . let p0 ( (xm) ∞ m=1 ) = xn1 − x n 2 ∈ p( nc0) for (xm) ∞ m=1 ∈ c0. then ∥p0∥ = 1. hence, 2 n+1 2n = ( ∞∑ j=1 |p0(ej)| 2n n+1 )n+1 2n ≤ w ≤ 2 n+1 2n . therefore, 2 n+1 2n ≤ cp(n : r). we complete the proof. 64 s. g. kim corollary 4.6. c(2 : r) < 2 3 4 ≤ cp(2 : r) ≤ 2 √ 2. theorem 4.7. let t : l2∞(r) × l2∞(r) → r be given by t(x, y) =∑2 i,j=1 aijxiyj, with aij ∈ r, a12 = a21. then the symmetric bilinear forms satisfying ( 2∑ i,j=1 |t(ei, ej)| 4 3 )3 4 = √ 2∥t∥ are given by t(x, y) = a(x1y1 − x2y2 + x1y2 + x2y1) or t(x, y) = a(−x1y1 + x2y2 + x1y2 + x2y1) for all a ∈ r\{0}. proof. it follows from theorem 4.1 of [3]. theorem 4.8. let t ∈ ls(2l2∞), ∥t∥ = 1, t(ej, ej) ̸= 0 for j = 1, 2. then, ( 2∑ i,j=1 |t(ei, ej)| 4 3 )3 4 = cs(2 : r) if and only if t ∈ extbls(2l2∞). proof. it follows from theorems 4.2, 4.3 and 4.7. theorem 4.9. sup{( 2∑ i,j=1 |t(ei, ej)| 6 4 ) 4 6 : t ∈ extbl(3l2∞)} = (7 + 3 3 2 ) 2 3 4 < c(3 : r). proof. diniz et al. 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[32] w.m. ruess, c.p. stegall, extreme points in duals of operator spaces, math. ann. 261 (4) (1982), 535 – 546. e extracta mathematicae vol. 33, núm. 1, 33 – 50 (2018) representing matrices, m-ideals and tensor products of l1-predual spaces s. dutta, d. khurana, a. sensarma department of mathematics and statistics indian institute of technology kanpur, india sudipta@iitk.ac.in , divyakh@iitk.ac.in , aryasen@iitk.ac.in presented by jesús m.f. castillo received november 28, 2016 abstract: motivated by bratteli diagrams of approximately finite dimensional (af) c∗algebras, we consider diagrammatic representations of separable l1-predual spaces and show that, in analogy to a result in af c∗-algebra theory, in such spaces, every m-ideal corresponds to directed sub diagram. this allows one, given a representing matrix of a l1-predual space, to recover a representing matrix of an m-ideal in x. we give examples where the converse is true in the sense that given an m-ideal in a l1-predual space x, there exists a diagrammatic representation of x such that the m-ideal is given by a directed sub diagram and an algorithmic way to recover a representing matrix of m-ideals in these spaces. given representing matrices of two l1-predual spaces we construct a representing matrix of their injective tensor product. key words: representing matrix, generalized diagram, directed sub diagram, m-ideals, tensor products. ams subject class. (2010): 46b25, 46b20. 1. introduction in 1971 lazar and lindenstrauss (see [3]) introduced notion of representing matrices for separable l1-predual spaces. the idea to construct representing matrix of a l1-predual space depends on following result in [3, theorem 3.2], which essentially says that any separable l1-predual space is built up by putting together increasing union of `n∞, n = 1, 2, . . . ∞’s. theorem 1.1. let x be a separable infinite dimensional banach space such that x∗ is isometric to l1(µ) for some positive measure µ. let f be a finite dimensional space whose unit ball is a polytope. then there exists a sequence {en}∞n=1 of finite dimensional subspaces of x such that e1 ⊃ f, en+1 ⊃ en and en = `mn∞ for every n and x = ∪∞n=1en. we now describe the notion of representing matrices. by theorem 1.1 any separable l1-predual space is ∪∞n=1`n∞ and different such spaces are con33 34 s. dutta, d. khurana, a. sensarma structed depending on how one embeds `n∞ → `n+1∞ . let {ei}ni=1 denote the standard unit vector basis of ` n ∞. by admissible basis of `n∞ we mean a basis of the form {θieπ(i)}ni=1 where θi = ±1 and π is a permutation of {1, . . . ,n}. it is easy to see that if {ui} is an admissible basis of `n∞ then for any m > n a linear operator t : `n∞ → `m∞ is an isometry if and only if there exists an admissible basis {vi}mi=1 of ` m ∞ such that tui = vi + σ m j=n+1a i jvj with σni=1|a i j| ≤ 1 for every n + 1 ≤ j ≤ m. now for any separable l1-predual space with the representation x = ∪n∈nen where en ⊆ en+1 and each en is isometric to `n∞ we may choose admissible basis {ein}ni=1 of en such that, after relabelling, tne i n = e i n+1 + a i ne n+1 n+1 with σni=1|a i n| ≤ 1. a triangular matrix a = ( ain )1≤i≤n n≥1 associated with x in this manner is called a representing matrix of x. the construction of the representing matrix is best understood in the context of c(k), k is totally disconnected. for use in the later part of this paper, we illustrate this with an example by constructing of representing matrix for such a space. let k be a totally disconnected compact metric space. then there exists a sequence { ∏ n} ∞ n=1 of partitions of k into disjoint closed sets so that for every n, { ∏ n} has n elements, { ∏ n+1} is a refinement of { ∏ n} and %n = max a∈ ∏ n d(a) → 0 where d(a) denotes diameter of a. let en be the linear span of the characteristic functions of the sets in ∏ n. then it follows trivially that each en is isometric to ` n ∞, en ⊆ en+1 and c(k) = ⋃∞ n=1 en. let us denote ∏ n = {k 1 n,k 2 n, . . . ,k n n} for all n ∈ n. we may write 1k11 = 1k12 + 1k22 . now ∏ 3 = {k 1 3,k 2 3,k 3 3}, 1k12 = 1k13 + 1k33 and 1k22 = 1k23 . we continue this procedure to get a representing matrix of c(k) which is 0, 1-valued [3, theorem 5.1]. a l1-predual space x has a rich collection of structural subspaces of x, namely m-ideals. m-ideals in a l1-predual space are themselves l1preduals and in some sense deterministic for the isometric properties of the representing matrices, m-ideals, tensor products 35 space, meaning, any isometric property of a l1-predual space can be read off from some isometric properties of its m-ideals. on the other hand, representing matrices ‘encode’ every possible information of the structure of a l1-predual space. a separable predual x of l1 may be thought of as an isometric version (commutative, where *-isomorphism is replaced by linear isometry) of approximately finite dimensional (af) real c∗-algebras. two sided norm closed ideals in an af c∗-algebra are completely determined by hereditary directed sub diagrams of its bratteli diagram (see [1]). the analogous notion of closed two sided ideals in a c∗-algebra in banach space category is m-ideals. here we present a representing diagram of a separable l1-predual space, the diagram itself arise out of representing matrix of such a space. we show that every directed sub diagram of a representing diagram represents an m-ideal in the corresponding space. since by definition of representing diagram, it is always hereditary, this is an exact analogy to the corresponding result for af c∗-algebras. we believe the converse is also true and we establish it in some cases. we now briefly describe the plan of this paper. in section 2 we present our main idea of diagrammatic representation of a separable l1-predual space x and directed sub diagram. we show any directed sub diagram corresponds to an m-ideal in x and the residual diagram corresponds to x/m. if m is an m-summand then we show the diagram for x splits into two directed sub diagram. this recovers the result in [7]. we believe that the converse, that any m-ideal in a l1-predual space x is represented by a directed sub diagram of some diagram is true. however there is a problem here. there are m-ideals which have empty sub diagram. nevertheless we present converse for c(k) spaces (with extra assumption for general k). we also observe that for a(k) -the space of affine continuous function on k, where k is a separable poulsen simplex (note that a(k) is isometric to the gurariy space in this case) given any m-ideal, there exists a diagrammatic representation of corresponding space such that the given m-ideal is represented by a directed sub diagram. in section 3 we describe a ‘fill in the gaps’ algorithm for construction of representing matrix from information that x = ∪∞n=1` mn ∞ . this in one hand provides way to construct representing matrix for an m-ideal given by directed sub diagram and on the other, allows one to write down representing matrix of x⊗̌y , x, y l1-preduals, knowing the representing matrix of x and y . we also show that for c[0, 1], given an m-ideal, there exists a diagram36 s. dutta, d. khurana, a. sensarma matic representation of c[0, 1] such that the given m-ideal is represented by a directed sub diagram. through out this work we only consider separable l1-predual spaces. recall that a subspace m of banach space is called an m-ideal if there exists a projection (called l-projection) p : x∗ → x∗ such that ker p = m⊥ and x∗ = range p ⊕1 ker p , where ⊕1 denote the `1-sum. in this case range p is isometric to m∗. m is said to be an m-summand in x if x = m ⊕∞ n. trivially any m-summand is an m-ideal. acknowledgements. a sensarma wishes to acknowledge the support received from csir, india, senior research fellowship (award letter no. 09/092(0872)/2013-emr-i) 2. directed diagrams and m-ideals for a l1-predual space x with a representing matrix a = ( ain )1≤i≤n n≥1 we will consider the following diagrammatic representation of x. a diagram d of a l1-predual space x = ∪∞n=1`n∞, and representing matrix a = ( ain )1≤i≤n n≥1 consists of nodes and weighted arrows. the nodes at the n-th level of the diagram are {ein : 1 ≤ i ≤ n} where span{ein : 1 ≤ i ≤ n} is isometric to `n∞, n ∈ n. for a node ein, there can be at most two arrows from ein one reaching to e i n+1 and another to e n+1 n+1. any arrow from e i n to e i n+1 has weight 1 and there is an arrow from ein to e n+1 n+1, then it has a weight a i n. for example if all ain 6= 0 then we have the following diagram: e11 e12 e22 e13 e23 e33 e14 e24 e34 e44 a1 1 a 1 2 a2 2 a 13 a 2 3 a3 3 representing matrices, m-ideals, tensor products 37 in case some ain’s are zero we do not put arrows from e i n to e n+1 n+1. for example diagram for a space with a11,a 2 2,a 3 3 = 0, will look like the following: e11 e12 e22 e13 e23 e33 e14 e24 e34 e44 a 1 2 a 2 3 a 13 in the following we describe the diagram for the space c with representing matrix a such that a1n = 1, n ≥ 1 and a j n = 0,j 6= 1 (see [3]): e11 e12 e22 e13 e23 e33 e14 e24 e34 e44 1 1 1 note that every representing matrix of a l1-predual space corresponds to a unique diagram d and vice-versa. for a given diagram d we will denote the corresponding space by xd. now we will introduce the notion of generalized diagram for a l1-predual space x, where x = ∪xn and xn is isometric to `mn∞ for an increasing 38 s. dutta, d. khurana, a. sensarma sequence (mn). let {e1mn, . . . ,e mn mn } be the admissible basis of xn. any isometry tmn : ` mn ∞ → ` mn+1 ∞ is uniquely specified by scalars (a i mn+j ), 1 ≤ j ≤ mn+1 −mn, 1 ≤ i ≤ mn such that tmne i mn = eimn+1 + a i mn+1 emn+1mn+1 + · · · + a i mn+1 emn+1mn+1 , i = 1, 2, . . . ,mn. for a node eimn , there will be one arrow from e i mn to eimn+1 . if a i mn+j 6= 0, then there will be a weighted arrow from eimn to e mn+j mn+1 , 1 ≤ j ≤ mn+1 −mn with weight aimn+j. definition 2.1. a sub diagram s of d will be called a directed sub diagram if whenever ein ∈s for some n,i ∈ n, i ≤ n then (a) ein+1 ∈s, (b) if ain 6= 0, e n+1 n+1 ∈s. a sub diagram s ⊆ d is directed if whenever ein ∈ s for some n,i ∈ n, i ≤ n and there is an arrow from ein to e j n+1 then e j n+1 ∈s. we define directed sub diagram of a generalized diagram similarly. if we take s ⊆d, and, s is directed then the original isometric embedding of xn into xn+1 is preserved (see introduction). hence xs will be an isometric subspace of xd. moreover there exists a norm one projection p : x ∗ → x∗ with ker p = x⊥s . to see this observe that xs = ∪ ∞ n=1` mn ∞ , hence xs is itself a l1-predual space which is an isometric subspace of xd. we prove that for any directed sub diagram s the space xs is an m-ideal in xd and the diagram d\s represents the space xd/xs. theorem 2.2. let x be a l1-predual space with a given diagram d. then for any directed sub diagram s of d the subspace xs is an m-ideal in x. proof. let x = ∪xn, where xn ⊂ xn+1, xn is isometric to `n∞ for each n. let p : x∗ → x∗ be a norm one projection with ker p = xs⊥, that is, x∗ = xs ⊥⊕f where f = range p . we need to prove that x∗ = xs⊥⊕1 f. let mn = span{ein : ein ∈s, 1 ≤ i ≤ n} and fn = span{ein : ein 6∈ s, 1 ≤ i ≤ n} for each n ∈ n. then xn = mn⊕∞fn and x∗n = m⊥n ⊕1f⊥n . for any x∗ ∈ x∗ we can write x∗ = x∗1 + x ∗ 2 where x ∗ 1 ∈ xs ⊥ and x∗2 ∈ f. then x ∗|xn = x∗1|xn + x ∗ 2|xn and ||x∗|xn|| = ||x∗1|xn|| + ||x ∗ 2|xn||. for given � > 0, we can choose some m ∈ n representing matrices, m-ideals, tensor products 39 such that ||x∗|xn|| ≥ ||x∗||− �, ||x∗1|xn|| ≥ ||x ∗ 1||− � and ||x ∗ 2|xn|| ≥ ||x ∗ 2||− � for all n ≥ m. now ||x∗1|| + ||x ∗ 2|| ≥ ||x ∗|| ≥ ||x∗|xn|| = ||x ∗ 1|xn|| + ||x ∗ 2|xn|| ≥ ||x ∗ 1|| + ||x ∗ 2||− 2�. thus it follows that ||x∗|| = ||x∗1|| + ||x ∗ 2|| for all x ∗ ∈ x∗. from this we can conclude that x∗ = xs ⊥ ⊕1 f. remark 2.3. let x be a l1-predual space with a given generalized diagram d. same proof as in theorem 2.2 shows that directed sub diagram s of d represents the subspace xs which is an m-ideal in x. next theorem is analogous to [1, theorem iii.4.4]) in the case of l1predual spaces. theorem 2.4. let x be a l1-predual space with a given diagram d and s a directed sub diagram of d. then the diagram d\s represents the space x/xs. proof. let x = ∪xn where xn is isometric to `n∞. as before, let mn = span{ein : ein ∈ s, 1 ≤ i ≤ n} and fn = span{ein : ein 6∈ s, 1 ≤ i ≤ n} for each n ∈ n. then xs = ∪mn, mn = `m∞ for some m ≤ n, is the m-ideal corresponding to the directed diagram s and xn = mn ⊕∞ fn. consider the norm one projection pn : xn → fn where pn ( n∑ i=1 aie i n ) = ∑ ein /∈s aie i n. let in : fn → fn+1 be the isometry determined by arrows of the diagram d\s, that is, for ein ∈d\s, in(e i n) = e i n+1 + a i ne n+1 n+1 if e i n+1, e n+1 n+1 ∈d\s, in(e i n) = e i n+1 if e i n+1 ∈d\s, e n+1 n+1 6∈ d\s, in(e i n) = a i ne n+1 n+1 if e n+1 n+1 ∈d\s, e i n+1 6∈ d\s, in(e i n) = 0 if e i n+1, e n+1 n+1 6∈ d\s. it is straightforward to verify that pn+1|xn = in ◦pn. we now define p : ∪xn →∪fn by px = pnx if x ∈ xn. it follows that p is well defined and extends as a quotient map from x to the space determined by ∪fn which is the space determined by the diagram d\s. this completes the proof. 40 s. dutta, d. khurana, a. sensarma we now investigate the converse of theorem 2.2. explicitly stated the problem is the following. problem 2.5. let x be a l1-predual space and m an m-ideal in x. then there exists a diagram d representing x and a directed sub diagram s of d such that m = xs. we believe the answer to problem 2.5 is affirmative. we will present evidences towards this for m-summands in general and m-ideals in some class of l1-predual spaces. the following proposition shows that any m-summand in a l1-predual space is represented by a directed sub diagram. proposition 2.6. let x be a l1-predual space and m be an m-summand in x. then there exists a diagram d representing x such that m corresponds to some directed sub diagram s of d. proof. let n be the complement of m in x, that is, x = m⊕∞n. then by [7, proposition 2.4] it follows that x has a representing matrix of the form a =   0 a12 0 a 1 4 0 a 1 6 . . . 0 a23 0 a 2 5 0 . . . 0 a34 0 a 3 6 . . . 0 a45 0 . . . 0 a56 . . . 0 . . . . . . . . . . . . . . . . . . . . . . . .   where ba = (b i n) with b i n = a 2i−1 2n , ca = (c i n) with c i n = a 2i 2n, n ∈ n, 1 ≤ i ≤ n, are the matrices for m and n respectively. let s1 and s2 be the diagrams corresponding to matrices ba and ca respectively. now it follows that s1 and s2 are directed sub diagrams of the diagram of x corresponding to the representing matrix a. remark 2.7. directed sub diagrams s1 and s2 considered in proposition 2.6 are disjoint in the sense that no arrows of s1 enters into s2 and vice-versa. we now consider m-ideals in c(k)-spaces. we need to recall few notation and a result from [7]. representing matrices, m-ideals, tensor products 41 let x be a l1-predual space with x = ∪∞n=1`n∞ and {e i n : 1 ≤ i ≤ n} are admissible bases of `n∞, n ∈ n. define φj ∈ x∗, j ∈ n, by φj(e i n) = { 0 if i 6= j, 1 if i = j; i = 1, . . . ,n; j ≤ n; n ∈ n. by ext bx∗ we will denote the extreme point of bx∗ . lemma 2.8. [7, lemma 1.2] let x and {φj} be as above. then (a) φj ∈ ext bx∗ for all j ∈ n, and (b) {±φi : i ∈ n} = ext bx∗ , where closure is taken in weak∗-topology of bx∗ . remark 2.9. for each i, ker φi represents the space xsi for some directed sub diagram si of a given diagram d of x where the line passing through eii is a part of the diagram d\si. the idea of the proof for the following result is to use the flexibility provided by lemma 2.8 for the choice of φi in a totally disconnected compact metric space k. recall that for any c(k) space where k is a compact metric space, an m-ideal is given by jd = {f ∈ c(k) : f|d = 0}, where d is some closed subset of k. proposition 2.10. let k be a totally disconnected compact metric space and d a closed subset of k. then there exists a diagram d representing c(k) and a directed sub diagram d ⊆s such that jd = xs. proof. since k be a totally disconnected we can get a sequence { ∏ n} ∞ n=1,∏ n = {k 1 n,k 2 n, . . . ,k n n} of partitions of k into disjoint closed sets, ∏ n+1 is a refinement of ∏ n and %n = maxa∈ ∏ n diam (a) → 0 (see introduction). let d0 = {dn : n ∈ n} be a countable dense set in d. choose φ1 = δd1 . for n ≥ 2 by renaming the elements in ∏ n we assume that d1 ∈ k 1 n. for n = 2 if (a) d0 ∩k22 6= ∅, we find the least n0 such that dn0 ∈ d0 ∩k 2 2 and choose φ2 = δdn0 . we will assume for all n ≥ 3, dn0 ∈ k 2 n, again by possibly renaming the members of ∏ n, (b) otherwise choose and fix any k ∈ k22 and take φ2 = δk. we will assume for all n ≥ 3, k ∈ k2n. 42 s. dutta, d. khurana, a. sensarma we will follow the same procedure for n ≥ 3. we need to ensure that each dn will be chosen. let n be the least number among all k’s such that dk ∈ kin for some i,n. let m 6= n and dm ∈ kin as well. since diam (kin) → 0 we can choose some suitable large m ∈ n such that dm ∈ kmm and m is the least among all k’s such that dk ∈ k m m . so following the algorithm above we define φm = δdm . let d be the diagram representing c(k) given by the partition { ∏ n} after renaming the elements of { ∏ n} as considered above. since d0 is dense in d, we have jd = ⋂ d∈d ker δd = ⋂ d∈d0 ker δd. thus jd = xs, where s is the intersection of directed diagrams corresponding to kernel of φi = δdi , di ∈ d0. next result shows affirmative answer to problem 2.5 for general c(k) space with additional assumption on an m-ideal. by int d we mean interior of a set d. proposition 2.11. let k be any compact metric space and d a closed subset of k such that d = int d. then the m-ideal jd corresponds to the space xs for some directed sub diagram s of given diagram d of c(k), provided, s is not an empty diagram. proof. let φj = δkj , kj ∈ k. since {φj} are weak*-dense in extreme points of the dual unit ball of c(k) and d = int d, we have a sub collection φji ⊆ int d such that φji = δkji and kji is dense in d. it follows jd = ⋂ ker φji and hence jd is represented by the directed sub diagram s of d which is generated by intersection of directed sub diagram representing ker φji , where φji = δkji . remarks 2.12. (1) if we assume k to be a ‘nice’ compact metric space, then given d a closed subset in k, we can construct a diagrammatic representation of c(k) such that jd corresponds to a directed sub diagram. we will do it in next section as we need algorithm to construct representing matrix of a l1-predual space x when it is given in the form x = ∪n≥1`mn∞ . (2) let k be (the) separable poulsen simplex. then the space a(k) the space of real valued affine continuous functions on k is the separable representing matrices, m-ideals, tensor products 43 gurariy space. it was proved in ([8]) that any infinite dimensional mideal in separable gurariy space is isometric to itself. thus any representing diagram of the gurariy space represents m-ideals in it and problem 2.5 has affirmative solution for the gurariy space. we note that an empty diagram is always a directed sub diagram of any given diagram d. it may be the case that an m-ideal in a l1-predual space corresponds to an empty diagram. we give an easy example towards this. example 2.13. consider the matrix a such that a1n = 1 for all n and ain = 0 for all i > 1. it is proved in [3] that a represents c. consider the m-ideal j = {(xn) ∈ c : xn = 0, n ≥ 2}. then j = ∩ker φn, φn = δn, n ≥ 2. in second figure on page 5, except the line segment starting from the node enn and the line segment that starts from the node e 1 1 and ends at e 1 n−1, all the diagram represents the space ker φn. it is straightforward to verify that ∩n≥2 ker φn is empty. another difficulty in solving problem 2.5 affirmatively in general is empty diagram may represent a space which is not an m-ideal. we give an example of this in a typical non g-space. note that an empty diagram is always directed. example 2.14. let x = { f ∈ c[1,ω0] : f(ω0) = f(1)+f(2) 2 } . then x is a l1 -predual space which is not a g-space (see [5]). we will consider the following admissible basis for x (see [2]): e11 = ( 1, 1, 1, 1, . . . ) , e12 = ( 1, 0, 1 2 , 1 2 , 1 2 , . . . ) , e22 = ( 0, 1, 1 2 , 1 2 , 1 2 , . . . ) , e13 = ( 1, 0, 0, 1 2 , 1 2 , 1 2 , . . . ) , e23 = ( 0, 1, 0, 1 2 , 1 2 , 1 2 , . . . ) , e33 = ( 0, 0, 0, 1, 0, . . . ) , . . . . for n ∈ [1,ω0], we denote by jn the m-ideal {f ∈ c[1,ω0] : f(n) = 0}. each jn is of codimension 1 in c[1,ω0]. we consider the m-ideal in c[1,ω0], j[3,ω0] = {f ∈ c[1,ω0] : f(n) = 0, n ≥ 3}. now consider the subspace j[3,ω0]∩x = ∩n≥3jn∩x of x. as in example 2.13 it is easy to check that the intersection of corresponding directed sub diagrams of jn ∩x for n ≥ 3 is empty diagram. however, j[3,ω0] ∩ x is not an m-ideal in x. to see this we observe that j[3,ω0] ∩ x is the range of norm one projection p : x → x given by p(f) = (f(1),−f(1), 0, 0, . . . ). thus if j[3,ω0] ∩ x is an m-ideal then it is an m-summand as well. so for any f ∈ x, ‖f‖ = max{‖pf‖,‖(i − p)f‖}. 44 s. dutta, d. khurana, a. sensarma however if we consider the element f ∈ x where f(1) = 1, f(2) = 0 and f(n) = 1/2 for all n ≥ 3, i.e, f = (1, 0, 1/2, 1/2, 1/2, . . . ) then ( 1, 0, 1 2 , 1 2 , 1 2 , . . . ) = ( 1 2 ,−1 2 , 0, 0, 0, . . . ) + ( 1 2 , 1 2 , 1 2 , 1 2 , 1 2 , . . . ) and the norm of both side will not match. thus j[3,ω0] ∩ x is not an m-summand in x. 3. fill in the gaps in this section we provide an algorithm to construct representing matrix of a l1-predual space x where x is given by x = ∪xn and xn is isometric to `mn∞ for an increasing sequence (mn). this construction is implicit in the description of representing matrix given in [3]. however we fix an algorithm (there may be several as seen below) and use it for finding representing matrix of x⊗̌y the injective tensor product of two separable l1-predual spaces, knowing the representing matrices of x and y . first we need to provide following justification to our construction. fact: let x be a l1-predual space such that x = ∪xn, where xn ⊆ xn+1 and xn is isometric to ` mn ∞ for some increasing sequence (mn). if z is a l1-predual space with z = ∪zn, where zn ⊆ zn+1, zn is isometric to `n∞, zmn = xn and the isometry tn : xn → xn+1 is same as composition of isometries of zmn to zmn+1, zmn+1 to zmn+2, . . . ,zmn+1−1 to zmn+1 given by the representing matrix of z, then z is isometric to x. we now describe the proposed algorithm. let x = ∪xn where for each n, xn = `mn∞ with admissible basis {eimn} mn i=1. any isometry tmn from ` mn ∞ to ` mn+1 ∞ in terms of admissible basis is given by tmne i mn = eimn+1 + a i mn+1 emn+1mn+1 + . . . + a i mn+1 emn+1mn+1 , i = 1, 2, . . . ,mn . (1) hence given x as above and isometric embeddings `mn∞ → ` mn+1 ∞ we know exactly (mn+1 −mn)mn numbers of( aij ) , i = 1, . . . ,mn , j = mn + 1, . . . ,mn+1 −mn . let us assume c = ( cin )1≤i≤n n≥1 is a representing matrix for x. we will write representing matrices, m-ideals, tensor products 45 {eimn} in terms of {e i mn+1 } according to isometries given by c: eimn = e i mn+1 + cimne mn+1 mn+1 = eimn+2 + c i mn+1 emn+2mn+2 + c i mn (emn+1mn+2 + c mn+1 mn+1 emn+2mn+2) = eimn+2 + c i mn emn+1mn+2 + (c i mn+1 + cimnc mn+1 mn+1 )emn+2mn+2 = . . . . this way we will have (mn+1−mn) 2 (mn+1 + mn−1) numbers of (c j n) unknowns. we will put cmn+imn+j = 0 , i = 1, . . . ,mn+1 −mn , 1 ≤ j ≤ i . it is a straight forward verification that this way we will have (mn+1−mn) 2 (mn+1 −mn) of cin’s zero. thus remaining (mn+1 −mn)mn of cin’s equal the number of known variables ain’s and can be expressed in terms of linear equations. we emphasize that the above way of choosing (cin) is not unique and different ways will give us different representing matrices. note that here we can not recover first m1 − 1 columns of the representing matrix by the above algorithm so it can be chosen arbitrarily (see [6, theorem 4.7]). remark 3.1. let the admissible basis of xn is {eimn : 1 ≤ i ≤ mn}. if we follow the above algorithm of ‘fill in the gaps’ from xn to xn+1 where xn is isometric to `mn∞ and xn+1 is isometric to ` mn+1 ∞ then the basis elements emn+imn+1, e mn+i mn+2 , . . . , emn+imn+1−1 are same as e mn+i mn+1 for all i ≥ 1. we illustrate this procedure by considering two special cases. first one is simple trial case with mn = 2n and our second example provides us with representing matrix of c[0, 1] with entries 0 and 1 2 . example 3.2. let c = ( cin )1≤i≤n n≥1 be a representing matrix of x and xn = span { e12n, . . . ,e 2n 2n } and tn : xn → xn+1 is an isometric embedding with ei2n = e i 2(n+1) + a i 2n+1e 2n+1 2(n+1) + ai2(n+1)e 2(n+1) 2(n+1) , 1 ≤ i ≤ 2n, n ≥ 1 . if we write the expression for ei2n according to the matrix c then we get ei2n = e i 2(n+1) + c i 2ne 2n+1 2(n+1) + ( ci2n+1 + c i 2nc 2n+1 2n+1 ) e 2(n+1) 2(n+1) . from above two expressions for ei2n we have a i 2n+1 = c i 2n and a i 2(n+1) = ci2n+1 + ci2nc 2n+1 2n+1. 46 s. dutta, d. khurana, a. sensarma now if we proceed by above algorithm and put c2n+12n+1 = 0, n ∈ n we get ai2n+1 = c i 2n, a i 2(n+1) = ci2n+1, 1 ≤ i ≤ 2n, n ≥ 1, and, we have the following representing matrix for x, c =   − a13 a 1 4 a 1 5 a 1 6 . . . ... a23 a 2 4 a 2 5 a 2 6 . . . ... ... 0 a35 a 3 6 . . . ... ... ... a45 a 4 6 . . . ... ... ... ... 0 . . . ... ... ... ... ... ...   . the fact stated above indeed justifies that the resulting matrix is a representing matrix of x. example 3.3. consider the function φ : r → r, φ(t) = 1 + t for t ∈ [−1, 0], φ(t) = 1 − t for t ∈ [0, 1], and φ(t) = 0 for t /∈ [−1, 1]. define gk,2n = φ(2 nt−k), t ∈ [0, 1]. we can write c[0, 1] = ∪xn, xn = span{gk,2n : k = 0, 1, . . . , 2n} where {gk,2n : k = 0, 1, . . . , 2n} is an admissible basis of xn. then for all n = 0, 1, . . . and k = 1, 2, . . . , 2n − 1 we have (see [4]) gk,2n = 1 2 g2k−1,2n+1 + g2k,2n+1 + 1 2 g2k+1,2n+1 , g0,2n = g0,2n+1 + 1 2 g1,2n+1 , g2n,2n = 1 2 g2n+1−1,2n+1 + g2n+1,2n+1 . let c = ( cin )1≤i≤n n≥1 be a representing matrix of c[0, 1]. first we have to write the expression for gk,2n according to c. now comparing the equations with the above and put c 2n+j 2n+i = 0, 1 ≤ i ≤ 2 n+1 − 2n − 1, 1 ≤ j ≤ i, we will get a representing matrix of c[0, 1] with entries 0 and 1 2 only. we now answer problem 2.5 in affirmative for c[0, 1]. theorem 3.4. let d be a closed subset of [0, 1]. then there exists a diagram d representing c[0, 1] such that the m-ideal jd corresponds to the space xs for some directed sub diagram s of d, provided s is not an empty diagram. representing matrices, m-ideals, tensor products 47 proof. let d0 = {dn : n ∈ n} be a countable dense set in d. we can extend d0 to a set m = {ki : i ∈ n} such that m = [0, 1]. consider e12 = 1−t, t ∈ [0, 1] and e22 = t, t ∈ [0, 1]. without loss of generality choose an element k1 ∈ [0, 1] and consider e13 = 1 − 1 k1 t if t ∈ [0,k1] , e13 = 0 if t ∈ [k1, 1] ; e23 = 0 if t ∈ [0,k1] , e 2 3 = t−k1 1 −k1 if t ∈ [k1, 1] and e33 = 1 k1 t if t ∈ [0,k1] , e33 = 1 − t 1 −k1 if t ∈ [k1, 1] . here e12, e 2 2, e 1 3, e 2 3, e 3 3 satisfy the following equations; e 1 2 = e 1 3 + (1 − k1)e 3 3, e22 = e 2 3 + k1e 3 3. now with out loss of generality choose k2 ∈ [0,k1] and k3 ∈ [k1, 1]. consider e15 = 1 − 1 k2 t if t ∈ [0,k2] , e15 = 0 if t ∈ [k2, 1] ; e25 = 0 if t ∈ [0,k3] , e 2 5 = t−k3 1 −k3 if t ∈ [k3, 1] ; e35 = 0 if t ∈ [0,k2] , e 3 5 = t−k2 k1 −k2 if t ∈ [k2,k1] , e35 = k3 − t k3 −k1 if t ∈ [k1,k3] , e35 = 0 if t ∈ [k3, 1] ; e45 = 1 k2 t if t ∈ [0,k2] , e45 = k1 − t k1 −k2 if t ∈ [k2,k1] , e45 = 0 if t ∈ [k1, 1] and e 5 5 = 0 if t ∈ [0,k1] , e55 = t−k1 k3 −k1 if t ∈ [k1,k3] , e55 = 1 − t 1 −k3 if t ∈ [k3, 1] . by the construction e13, e 2 3, e 3 3, e 1 5, e 2 5, e 3 5, e 4 5, e 5 5 satisfy the following equations: e13 = e 1 5 + k1 −k2 k2 e45 , e 2 3 = e 2 5 + k3 −k1 1 −k1 e55 and e 3 3 = e 3 5 + k2 k1 e45 + 1 −k3 1 −k1 e55 . similarly we can construct ei2n+1, 1 ≤ i ≤ 2 n + 1. take an element f ∈ c[0, 1]. define a sequence (pn) ∞ n=0 in the following way. let p0 = f(0)e 1 2, p1 = p0 + (f(1) −p0(1))e22 , p2 = p1 + (f(k1) −p1(k1))e 3 3 , p3 = p2 + (f(k2) −p2(k2))e45 , p4 = p3 + (f(k3) −p3(k3))e 5 5 , and so on. here p0 and f takes the same value at 0 while p1 and f takes the same value at 0 and 1 and interpolates linearly in between, p2 and f takes 48 s. dutta, d. khurana, a. sensarma same value at 0, 1 and k1 and interpolates linearly in between, and so on. it is straightforward to check that limn ||pn −f||∞ = 0. therefore we can write c[0, 1] = ∪en, where en = span { ei2n+1 : 1 ≤ i ≤ 2 n + 1 } and en is isometric to ` 2n+1 ∞ . we know that the support of e i 2n+1 is going to zero as n approaches to infinity and it consists a single element of {ki : i ∈ n}. each i ∈ n, ki will be in some tji = ∩ ∞ n=1 supp{e j 2n+1} and any two tji ’s are disjoint. here we consider the generalized diagram of c[0, 1] with respect to above basis and from n-th to (n + 1)-th step we choose 2n−1 of ki’s and these ki’s lie in the support of exactly one of the basis elements of e 2n+1 2n+1 , . . . , e2 n+1 2n+1 . now if we follow the algorithm for ‘fill in the gaps’ from n-th to (n + 1)-th step and consider ki ∈ supp(e 2n+j 2n+1 ) chosen above, then ki ∈ supp(e 2n+j m ) and ki /∈ supp(elm), l 6= 2n + j for 2n + 1 ≤ m ≤ 2n+1 − 1, j ≥ 1 (see remark 3.1). so by following the same procedure of choosing φi as in proposition 2.10 we will get for any ki there exists a φm such that φm = δki and the set {ki}∞i=1 is dense in [0, 1] (see [2, lemma 2]). given that d0 is dense in d so jd = ∩d∈d ker δd = ∩d∈d0 ker δd. thus jd = xs, where s is the intersection of directed diagrams corresponding to kernel of φi = δdi , di ∈ d0. this completes the proof. representing matrix for x⊗̌y : if x and y are separable l1-predual spaces, then it is known that x⊗̌y is also a separable l1-predual space. we adopt the above algorithm to find a representing matrix for x⊗̌y . let x and y has representing matrices ( ain )1≤i≤n n≥1 and ( bin )1≤i≤n n≥1 respectively corresponding to the admissible basis {ein : 1 ≤ i ≤ n} and {fin : 1 ≤ i ≤ n}. then x⊗̌y = ∪∞n=1en2 , where en2 is isometric to ` n2 ∞ with admissible bases {ein ⊗ f j n, i = 1, . . . ,n; j = 1, . . . ,n}. we will denote this collection as{ ei n2 : 1 ≤ i ≤ n2 } with the following convention: (a) first n2 terms of the admissible basis of e(n+1)2 is same as the admissible basis of en2 . for example if e i n2 = ek (n−1)2 ⊗ e l (n−1)2 then e i (n+1)2 = ek n2 ⊗el n2 . (b) we will choose { en 2+i (n+1)2 : 1 ≤ i ≤ (n + 1)2 −n2 } by the following way. take en 2+1 (n+1)2 = e1n+1 ⊗ f n+1 n+1 , e n2+2 (n+1)2 = en+1n+1 ⊗ f 1 n+1. for i = 2k + 1, k ∈ n, en 2+2k+1 (n+1)2 = ekn+1 ⊗f n+1 n+1 and for i = 2k + 2, k ∈ n, e n2+2k+2 (n+1)2 = en+1n+1 ⊗f k n+1. representing matrices, m-ideals, tensor products 49 we will now follow algorithm for ‘fill in the gaps’ described above. let us illustrate this with first few steps. let c = ( cin )1≤i≤n n≥1 be the representing matrix of x⊗̌y . according to the above convention e11 = e 1 1⊗f 1 1 and e 1 4 = e 1 2⊗f 1 2 , e 2 4 = e 1 2⊗f 2 2 , e 3 4 = e 2 2⊗f 1 2 , e44 = e 2 2 ⊗ f 2 2 . by expanding e 1 1 in terms of {e i 4} 4 i=1 according to the given representing matrix of x and y we get e11 = e 1 4 + b 1 1e 2 4 + a 1 1e 3 4 + a 1 1b 1 1e 4 4 . similarly expansion of e11 in terms of {e i 4} 4 i=1 according to the representing matrix c of x⊗̌y , e11 = e 1 4 + c 1 1e 2 4 + (c 1 2 + c 1 1c 2 2)e 3 4 + (c 1 2c 3 3 + c 1 1c 2 3 + c 1 3 + c 1 1c 2 2c 3 3)e 4 4. by following the algorithm we will get c11 = b 1 1, c 1 2 = a 1 1, c 2 2 = 0, c 1 3 = a 1 1b 1 1, c23 = 0, c 3 3 = 0. by expanding {e i 4} 4 i=1 in terms of {e i 9} 9 i=1 according to given representing matrices for x, y and matrix c we will get c14 = b 1 2 , c 2 4 = b 2 2 , c 3 4 = 0 , c 4 4 = 0 , c15 = a 1 2 , c 2 5 = 0 , c 3 5 = a 2 2 , c 4 5 = 0 , c 5 5 = 0 , c16 = 0 , c 2 6 = 0 , c 3 6 = b 1 2 , c 4 6 = b 2 2 , c 5 6 = 0 , c 6 6 = 0 , c17 = 0 , c 2 7 = a 1 2 , c 3 7 = 0 , c 4 7 = a 2 2 , c 5 7 = 0 , c 6 7 = 0 , c 7 7 = 0 , c18 = a 1 2b 1 2 , c 2 8 = a 1 2b 2 2 , c 3 8 = a 2 2b 1 2 , c 4 8 = a 2 2b 2 2 , c 5 8 = 0 , c68 = 0 , c 7 8 = 0 , c 8 8 = 0 . proceeding as above we will get representing matrix of x⊗̌y as c =   b11 a 1 1 a 1 1b 1 1 b 1 2 a 1 2 0 0 a 1 2b 1 2 . . . ... 0 0 b22 0 0 a 1 2 a 1 2b 2 2 . . . ... ... 0 0 a22 b 1 2 0 a 2 2b 1 2 . . . ... ... . . . 0 0 b22 a 2 2 a 2 2b 2 2 . . . ... ... . . . . . . 0 0 0 0 . . . ... ... . . . . . . . . . 0 0 0 . . . ... ... ... ... . . . 0 0 . . . ... ... ... ... ... ... ... 0 . . . ... ... ... ... ... ... ... ... . . .   . 50 s. dutta, d. khurana, a. sensarma remark 3.5. from above description of representing matrix for x⊗̌y we can actually read off representing matrices ( ain )1≤i≤n n≥1 and ( bin )1≤i≤n n≥1 for x and y respectively. for example representing matrix of y is given by b =   c11 c 1 4 c 1 9 c 1 42 . . . c1 n2 . . . ... c24 c 2 9 c 2 42 . . . c2 n2 . . . ... ... c59 c 5 42 . . . . . . c2 2+1 n2 . . . ... ... . . . c10 42 . . . . . . c3 2+1 n2 . . . ... ... . . . . . . . . . . . . . . . . . . . . . ... ... . . . . . . . . . . . . . . . . . . . . . ... ... ... ... . . . . . . . . . . . . . . . ... ... ... ... ... . . . . . . . . . . . . ... ... ... ... ... ... c (n−1)2+1 n2 . . . ... ... ... ... ... ... ... . . .   . thus if a l1-predual space has a representing matrix like c, it is actually tensor product of two l1-predual spaces with representing matrices a and b. references [1] k.r. davidson, “ c∗-algebras by example ”, fields institute monographs, 6, american mathematical society, providence, ri, 1996. [2] a.b. hansen, y. sternfeld, on the characterization of the dimension of a compact metric space k by the representing matrices of c(k), israel j. math. 22 (2) (1975), 148 – 167. [3] a.j. lazar, j. lindenstraus, banach spaces whose duals are l1 spaces and their representing matrices, acta math. 126 (1971), 165 – 193. [4] a. lima, v. lima, e. oja, absolutely summing operators on c[0,1] as a tree space and the bounded approximation property, j. funct. anal. 259 (11) (2010), 2886 – 2901. [5] j. lindenstraus, “ extension of compact operators ”, mem. amer. math. soc. no. 48, american mathematical society, providence, ri, 1964. [6] j. lindenstrauss, g. olsen, y. sternfeld, the poulsen simplex, ann. inst. fourier (grenoble) 28 (1) (1978), vi, 91 – 114. [7] w. lusky, on separable lindenstrauss spaces, j. functional analysis 26 (2) (1977), 103 – 120. [8] t.s.s.r.k. rao, on almost isometric ideals in banach spaces, monatsh. math. 181 (1) (2016), 169 – 176. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 36, num. 1 (2021), 99 – 145 doi:10.17398/2605-5686.36.1.99 available online january 28, 2021 ancient solutions of the homogeneous ricci flow on flag manifolds s. anastassiou 1, i. chrysikos 2 1 center for research and applications of nonlinear systems (crans) department of mathematics, university of patras, rion 26500, greece 2 faculty of science, university of hradec králové rokitanskeho 62, hradec králové 50003, czech republic sanastassiou@gmail.com , ioannis.chrysikos@uhk.cz received september 13, 2020 presented by carolyn s. gordon accepted january 6, 2021 abstract: for any flag manifold m = g/k of a compact simple lie group g we describe noncollapsing ancient invariant solutions of the homogeneous unnormalized ricci flow. such solutions emerge from an invariant einstein metric on m, and by [13] they must develop a type i singularity in their extinction finite time, and also to the past. to illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized ricci flow on any flag manifold m = g/k with second betti number b2(m) = 1, for a generic initial invariant metric. we describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose αlimit set consists of fixed points at infinity of m g. based on the poincaré compactification method, we show that these fixed points correspond to invariant einstein metrics and we study their stability properties, illuminating thus the structure of the system’s phase space. key words: ricci flow, homogeneous spaces, flag manifolds, ancient solutions, scalar curvature. msc (2020): primary: 53c44, 53c25, 53c30, secondary: 37c10. introduction given a riemannian manifold (mn,g), recall that the unnormalized ricci flow is the geometric flow defined by ∂g(t) ∂t = −2 ricg(t) , g(0) = g , (0.1) where ricg(t) denotes the ricci tensor of the one-parameter family g(t). the above system consists of non-linear second order partial differential equations on the open convex cone m of riemannian metrics on m. a smooth family{ g(t) : t ∈ [0,t) ⊂ r } ∈ m defined for some 0 < t ≤ ∞, is said to be a solution of the ricci flow with initial metric g, if it satisfies the system (0.1) issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.36.1.99 mailto:sanastassiou@gmail.com mailto:ioannis.chrysikos@uhk.cz https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 100 s. anastassiou, i. chrysikos for any x ∈ m and t ∈ [0,t). the ricci flow was introduced in the celebrated work of hamilton [33] and nowadays is the essential tool in the proof of the famous poincaré conjecture and thurston’s geometrization conjecture, due to the seminal works [48, 49] of g. perelman. in general, and since a system of partial differential equations is involved, it is hard to produce explicit examples of ricci flow solutions. however, the ricci flow for an initial invariant metric reduces to a system of odes. more precisely, homogeneity implies bounded curvature (see [21]), and thus the isometries of the initial metric will be in fact the isometries of any involved metric. hence, when g is an invariant metric, any solution g(t) of (0.1) is also invariant. as a result, in some cases it is possible to solve the system explicitly and proceed to a study of their asymptotic properties, or even specify analytical properties related to different type of singularities and deduce curvature estimates, see [16, 28, 7, 29, 18, 38, 11, 13, 1, 30], and the articles quoted therein. especially for the non-compact case, note that during the last decade the ricci flow for homogeneous, or cohomogeneity-one metrics, together with the so-called bracket flow play a key role in the study of the alekseevsky conjecture, see [47, 41, 42, 43, 39, 40, 12, 14]. in this work we examine the ricci flow, on compact homogeneous spaces with simple spectrum of isotropy representation, in terms of graev [31, 32], or of monotypic isotropy representation, in terms of buzano [18], or pulemotov and rubinstein [52]. nowadays, such spaces are of special interest due to their rich applications in the theory of homogeneous einstein metrics, prescribed ricci curvature, ricci iteration, ricci flow and other (see [4, 53, 10, 15, 31, 8, 9, 18, 32, 24, 11, 50, 52, 30]). here, we focus on flag manifolds m = g/k of a compact simple lie group g and examine the dynamical system induced by the vector field corresponding to the homogeneous ricci flow equation. on such cosets (even for g semisimple), the homogeneous ricci flow cannot possess fixed (stationary) points, since by the theorem of alekseevsky and kimel’fel’d [3] invariant ricci flat metrics must be flat and so they cannot exist. however, as follows from the maximum principle, the homogeneous ricci flow on any flag manifold m = g/k must admit ancient invariant solutions, which in fact by the work of böhm [11] must have finite extinction time (note that since any flag manifold is compact and simply connected and carries invariant einstein metrics, e.g., invariant kähler-einstein metrics always exist, the first conclusion above occurs also by a result of lafuente [38, corollary 4.3]). to be more specific, recall that a solution g(t) of the ricci flow is called ancient if it has as interval of definition the open set (−∞,t), for some t < ∞. ancient solutions of the homogeneous ricci flow 101 such solutions are important, because they arise as limits of blow ups of singular solutions to the ricci flow near finite time singularities, see [25]. for our case it follows that for any flag manifold m = g/k one must be always able to specify a g-invariant metric g, such that any (maximal) ricci flow solution g(t) with initial condition g(0) = g has an interval of definition of the form (ta,t), with −∞ ≤ ta < 0 and t < ∞. indeed, for any flag space below we will provide explicit solutions of this type, which are ancient. they arise by using an invariant kähler-einstein metric, which always exists, or any other possible existent invariant einstein metric g0, and they are defined on open intervals of the form (−∞,t), where t = 1 2λ = n 2 scal(g0) −1, with λ = scal(g0) n > 0 being the corresponding einstein constant (see proposition 2.4). all such solutions become extinct when t → t, in the sense that g(t) → 0, i.e., they tend to 0. as ancient solutions, they have positive scalar curvature scal(g(t)) ([20]) and the asymptotic behaviour of scal(g(t)), at least for the trivial one, can be easily treated, see proposition 2.4 which forms a specification of [38, theorem 1.1] on flag manifolds (see also example 3.3 and example 3.4). moreover, for any invariant einstein metric on m = g/k with b2(m) = 1, we show the existence of an unstable manifold and compute its dimension. for the invariant einstein metrics which are not kähler, we obtain a 2or 3-dimensional unstable manifold, depending on the specific case, which implies the existence of non–trivial ancient solutions emerging from the corresponding einstein metric. actually, by [13], it also follows that these ancient solutions develop a type i singularity. this means (see [25, 18, 11, 44]) lim t→−∞ ( |t| · supx∈m‖rm(g(t))‖g(t)(x,t) ) < ∞, where rm(g(t)) denotes the curvature tensor of (m = g/k,g(t)), or equivalently that there is a constant 0 < cg0 < ∞ such that (t − t) · supm‖rm(g(t))‖g(t) ≤ cg0 for any t ∈ (−∞,t = 1 2λ ). finally, by [13] we also deduce that the predicted non-trivial ancient solutions are non-collapsed (and the same the trivial one, see corollary 2.8). in this point we should mention that not any compact homogeneous space m = g/k of a compact (semi)simple lie group g admits (unstable) einstein metrics (see for example [55, 46] for non-existence 102 s. anastassiou, i. chrysikos results). so, even assuming that the universal covering of mn = g/k is not diffeomorphic to rn (which for the compact case is equivalent to say that mn is not a n-torus), the predicted solutions of lafuente can be in general hard to be specified. note now that for any flag manifold m = g/k of a compact simple lie group g, the symmetric space of invariant metrics m g is the phase space of the homogeneous ricci flow and it is flat, i.e., m g ∼= rr+ for some r ≥ 1. therefore, the dynamical system of the homogeneous ricci flow can be converted to a qualitative equivalent dynamical system of homogeneous polynomial equations and the well-known poincaré compactification ([51]) strongly applies. the main idea back of this method is to identify rn with the northern and southern hemispheres through central projections, and then extend x to a vector field p(x) on sn (see subsection 2.2). here, for any (non-symmetric) flag manifold m = g/k of a compact simple lie group g with b2(m) = 1, we present the global study of the dynamical system induced by the vector field corresponding to the unnormalized ricci flow for an initial invariant metric, which is generic. in particular, the main contribution of this work is the description via the poincaré compactification method, of the fixed points of the homogeneous ricci flow at the so-called infinity of m g (see definition 2.11 and remark 2.13). based on this method we can study the stability properties of such fixed points, which we prove that are in bijective correspondence with the existent invariant einstein metrics on m = g/k, and moreover that coincide with the α-limit set of an invariant line, i.e., a solution of the homogeneous ricci flow which has as trace a line of m g. it turns out that such solutions are ancient and non-collapsed, and develop type i singularities. note that through the compactification procedure of poincaré, we are able to distinguish the unique invariant kähler-einstein metric from the other invariant einstein metrics in terms of (un)stable manifolds, in particular for every invariant einstein metric we compute the dimension of the corresponding (un)stable manifold. moreover, for the case r = 2 we discuss the ω-limit of any solution of the homogeneous ricci flow, for details see theorem 3.1. since we are interested in the unnormalized ricci flow on flag manifolds with b2(m) = 1, we should finally mention that this dynamical system has been very recently examined by [30], for flag spaces with three isotropy summands (however via a different method of the poincaré compactification), while a related study of certain examples of flag manifolds with two isotropy is given in [29]. note finally that for flag manifolds with r = 2, this work is complementary to [18], in the sense that there were studied homogeneous anancient solutions of the homogeneous ricci flow 103 cient solutions on compact homogeneous spaces with two isotropy summands. however, the specific class of flag manifolds with r = 2 was excluded (see at the end of the article [18]), and the filling of this small gap was a motivation of the present work. the structure of the paper is given as follows: in section 1 we refresh basics from the theory of homogeneous spaces, introduce the homogeneous ricci flow and recall some details from the structure and geometry of flag manifolds. next, in section 2 we shortly present some basic results for ancient solutions emerging from an invariant einstein metric, and also the poincaré compactification adapted to our scopes. finally, section 3 is about the global study of the homogeneous ricci flow for all non-symmetric flag manifolds m = g/k of a compact simple lie group g with b2(m) = 1, where a proof of our main theorem, i.e., theorem 3.1, is presented. 1. preliminaries we begin by recalling preliminaries of the homogeneous ricci flow. after that we will refresh useful notions of the structure and geometry of generalized flag manifolds. 1.1. homogeneous ricci flow. recall that a homogeneous riemannian manifold is a homogeneous space m = g/k (see [36, 19, 5] for details on homogeneous spaces) endowed with a g-invariant metric g, that is τ∗ag = g for any a ∈ g, where τ : g×g/k → g/k denotes the transitive g-action. equivalently, is a riemannian manifold (m,g) endowed with a transitive action of its isometry group iso(m,g). if m is connected, then each closed subgroup g ⊆ iso(m,g) which is transitive on m induces a presentation of (m,g) as a homogeneous space, i.e., m = g/k, where k ⊂ g is the stabilizer of some point o ∈ m. in this case, the transitive lie group g can be also assumed to be connected (since the connected component of the identity of g is also transitive on m). usually, to emphasize on the transitive group g, we say that (m,g) is a g-homogeneous riemannian manifold. however, note that may exist many closed subgroups of iso(m,g) acting transitively on (m,g). next we shall work with connected homogeneous manifolds. as it is well-known, the geometric properties of a homogeneous space can be examined by restricting our attention to a point. set o = ek for the identity coset of (mn = g/k,g) and let tog/k be the corresponding tangent space. since we assume the existence of a g-invariant metric g, k ⊂ g can be identified with a closed subgroup of o(n) ≡ o(tog/k) (or of so(n) if g/k is 104 s. anastassiou, i. chrysikos oriented), so k is compact and hence any homogeneous riemannian manifold (mn = g/k,g) is a reductive homogeneous space. this means that there is a complement m of the lie algebra k = lie(k) of the stabilizer k inside the lie algebra g = lie(g) of g, which is adg(k)-invariant, i.e., g = k ⊕ m and adg(k)m ⊂ m, where adg ≡ ad : g → aut(g) denotes the adjoint representation of g. note that in general the reductive complement m may not be unique, and for a general homogeneous space g/k a sufficient condition for its existence is the compactness of k. on the other hand, once such a decomposition has been fixed, there is always a natural identification of m with the tangent space tog/k = g/k, given by x ∈ m ←→ x∗o = d dt ∣∣ t=0 τexp tx(o) ∈ tog/k , where exp tx is the one-parameter subgroup of g generated by x. under the linear isomorphism m = tog/k, the isotropy representation χ : k → aut(m), defined by χ(k) := (dτk)o for any k ∈ k, is equivalent with the representation adg |k : k × m → m. hence, χ(k)x = adg(k)x for any k ∈ k and x ∈ m. in terms of lie algebras we have χ∗(y )x = [y,x]m for any y ∈ k and x ∈ m, or in other words χ∗(y ) = ad(y )|m. the homogeneous spaces m = g/k that we will examine below (with k compact), are assumed to be almost effective, which means that the kernel ker(τ) (which is a normal subgroup both of g and k), is finite. thus, the isotropy representation χ is assumed to have a finite kernel, and then we may identify k with the lie algebra χ∗(k) = lie(χ(k)) of the linear isotropy group χ(k) ⊂ aut(m). when only the identity element e ∈ g acts as the identity transformation on m = g/k, then the g-action is called effective and the isotropy representation χ is injective. if we assume for example that g ⊆ iso(m,g) is a closed subgroup, then the action of g to g/k is effective. note that an almost effective action of g on g/k gives rise to an effective action of the group g′ = g/ker(τ) (of the same dimension with g), so we will not worry much for the effectiveness of m = g/k. recall that the space of g-invariant symmetric covariant 2-tensors on a (almost) effective homogeneous space m = g/k with a reductive decomposition g = k⊕m, is naturally isomorphic with the space of symmetric bilinear forms on m, which are invariant under the isotropy action of k on m. as a consequence, the space m g of g-invariant riemannian metrics on m = g/k coincides with the space of inner products 〈 , 〉 on m satisfying 〈x,y 〉 = 〈χ(k)x,χ(k)y 〉 = 〈adg(k)x, adg(k)y 〉 , ancient solutions of the homogeneous ricci flow 105 for any k ∈ k and x,y ∈ m. the correspondence is given by 〈x,y 〉 = g(x,y )o. moreover, when k is compact and m = h ⊥ with respect to b = −bg, where bg is the killing form of g, then one can extend the above correspondence between elements g = 〈 , 〉 ∈ m g and adg(k)-invariant bselfadjoint positive-definite endomorphisms l : m → m of m, i.e., 〈x,y 〉 = b(lx,y ), for any x,y ∈ m. to simplify the text, whenever is possible next we shall relax the notation adg(k) to ad(k) and scalar products on m as above, will be just referred to as ad(k)-invariant scalar products. note that the ad(k)-invariance of 〈 , 〉 implies its ad(k)-invariance, which means that the endomorphism ad(z)|m : m → m is skew-symmetric with respect to 〈 , 〉, for any z ∈ k. when k is connected, the inclusions ad(k)m ⊂ m and [k,m] ⊂ m are equivalent and hence one can pass from the ad(k)-invariance to ad(k)-invariance and conversely. from now on will denote by p(m)ad(k) the space of all ad(k)-invariant inner products on m. given a riemannian manifold (mn,g), a solution of the ricci flow is a family of riemannian metrics {gt}∈ m satisfying the system (0.1). if the initial metric g = g(0) ∈ m is a g-invariant metric with respect to some closed subgroup g ⊆ iso(m,g), i.e., (m = g/k,g) is a homogeneous riemannian manifold and so g ∈ m g, then the solution {g(t)} is called homogeneous, i.e., {g(t)}∈ m g. indeed, the isometries of g are isometries for any other evolved metric, and by [37] it is known that the isometry group is preserved under the ricci flow. thus, after considering a reductive decomposition g = k ⊕ m of (m = g/k,g), the homogeneity of g allows us to reduce the ricci flow to a system of odes for a curve of ad(k)-invariant inner products on p(m)ad(k), where m ∼= tog/k is a reductive complement. in particular, due to the identification m g ∼= p ad(k)(m) we may write g(t) = 〈 , 〉t and then (0.1) takes the form d dt 〈 , 〉t = −2 ric〈 , 〉t , 〈 , 〉0 ≡〈 , 〉 = g , where ric〈 , 〉t denotes the ad(k)-invariant bilinear form on m, corresponding to the ricci tensor of g(t). note that since g0 = g(0) is an invariant metric, the solution g(t) of (0.1) must be unique among complete, bounded curvature metrics (see [21]). remark 1.1. when one is interested in more general homogeneous spaces m = g/k and a reductive decomposition may not exist, the above setting can be appropriately transferred to g/k ∼= tog/k. however, the “reductive setting” serves well the goals of this paper and it is sufficient for our subsequent computations and description. 106 s. anastassiou, i. chrysikos 1.2. flag manifolds. let g be a compact semisimple lie group with lie algebra lie(g) = g. a flag manifold1 is an adjoint orbit of g, i.e., m = ad(g)w = {ad(g)w : g ∈ g}⊂ g for some left-invariant vector field w ∈ g. let k = {g ∈ g : ad(g)w = w}⊂ g be the isotropy subgroup of w and let k = lie(k) be the corresponding lie algebra. since g acts on m transitively, m is diffeomorphic to the (compact) homogeneous space g/k, that is ad(g)w = g/k. in particular, k = {x ∈ g : [x,w] = 0} = ker ad(w), where ad : g → end(g) is the adjoint representation of g. moreover, the set sw = {exp(tw) : t ∈ r} is a torus in g and the isotropy subgroup k is identified with the centralizer in g of sw, i.e., k = c(sw). hence rankg = rankk and k is connected. thus, equivalently a flag manifold is a homogeneous space of the form g/k, where k = c(s) = {g ∈ g : ghg−1 = h for all h ∈ s} is the centralizer of a torus s in g. when k = c(t) = t is the centralizer of a maximal torus t in g, the g/t is called a full flag manifold. flag manifolds admit a finite number of invariant complex structures, in particular flag spaces g/k of a compact, simply connected, simple lie group g exhaust all compact, simply connected, de rham irreducible homogeneous kähler manifolds (see for example [4, 8, 23, 2] for further details). so, for any flag manifold m = g/k we may work with g simply connected (if for instance a flag manifold of so(n) is given, one can always pass to its universal covering by using the double covering spin(n)). for our scopes, it is also sufficient to focus on the de rham irreducible case, which is equivalent to say that g is simple, see [36]. hence, in the following we can always assume that m = g/k satisfies these conditions, and as before we shall denote by bg the killing form of the lie algebra g. the ad(g)-invariant inner product b := −bg induces a bi-invariant metric on g by left translations, and we may fix, once and for all, a b-orthogonal ad(k)-invariant decomposition g = k⊕m. g-invariant riemannian metrics on g/k will be identified with ad(k)invariant inner products 〈 , 〉 on the reductive complement m = tog/k. note that the restriction b ∣∣ m induces the so-called killing metric gb ∈ m g, which is the unique invariant metric for which the natural projection π : (g,b) −→ (g/k,gb) is a riemannian submersion. the second betti number of any flag manifold m = g/k is encoded in the corresponding painted dynkin diagram. to recall the procedure, let gc = hc ⊕ ∑ α∈r g c α be the usual root space decomposition of the complexification 1also called complex flag manifold, or generalized flag manifold. ancient solutions of the homogeneous ricci flow 107 gc of g, with respect to a cartan subalgebra hc of gc, where r ⊂ ( hc )∗ is the root system of gc. via the killing form of gc we identify ( hc )∗ with hc. let π = {α1, . . . ,α`} (dim hc = `) be a fundamental system of r and choose a subset πk of π. we denote by rk = { β ∈ r : β = ∑ αi∈πk kiαi } the closed subsystem spanned by πk. then, the lie subalgebra k c = hc ⊕ ∑ β∈rk g c β is a reductive subalgebra of gc, i.e., it admits a decomposition of the form kc = z ( kc ) ⊕ kcss, where z(kc) is its center and kcss = [ kc,kc ] the semisimple part of kc. in particular, rk is the root system of k c ss, and thus πk can be considered as the associated fundamental system. let k be the connected lie subgroup of g generated by k = kc ∩ g. then the homogeneous manifold m = g/k is a flag manifold, and any flag manifold is defined in this way, i.e., by the choise of a triple ( gc, π, πk ) , see also [4, 8, 23, 2]. set πm = π\πk and rm = r\rk, such that π = πk t πm , and r = rk t rm , respectively. roots in rm are called complementary roots. let γ = γ(π) be the dynkin diagram of the fundamental system π. definition 1.2. let m = g/k be a flag manifold. by painting black the nodes of γ corresponding to πm , we obtain the painted dynkin diagram of g/k (pdd in short). in this diagram the subsystem πk is determined as the subdiagram of white roots. remark 1.3. conversely, given a pdd, one may determine the associated flag manifold m = g/k as follows: the group g is defined as the unique simply connected lie group generated by the unique real form g of the complex simple lie algebra gc (up to inner automorphisms of gc), which is reconstructed by the underlying dynkin diagram. moreover, the connected lie subgroup k ⊂ g is defined by using the encoded by the pdd splitting π = πk tπm . the semisimple part of k is obtained from the (not necessarily connected) subdiagram of white simple roots, while each black root, i.e., each root in πm , gives rise to a u(1)-summand. thus, the pdd determines the isotropy group k and the space m = g/k completely. by using certain rules to determine whether different pdds define isomorphic flag manifolds (see [4]), one can obtain all flag manifolds g/k of a compact simple lie group g (see for example the tables in [2]). proposition 1.4. ([17, 4]) the second betti number of a flag manifold m = g/k equals to the totality of black nodes in the corresponding pdd, i.e., the cardinality of the set πm . 108 s. anastassiou, i. chrysikos note that for any flag manifold m = g/k, the b-orthogonal reductive complement m decomposes into a direct sum of ad(k)-inequivalent and irreducible submodules, which we call isotropy summands, see [4, 8, 2] and the references therein. this means that when m ∼= tom is viewed as a k-module, then there is always a b-orthogonal ad(k)-invariant decomposition m = m1 ⊕···⊕mr , (1.1) for some r ≥ 1, such that: • k acts (via the isotropy representation) irreducibly on any m1, . . . ,mr; • mi � mj are inequivalent as ad(k)-representations for any i 6= j. in fact, a decomposition as in (1.1) satisfying the given conditions must be unique, up to a permutation of the isotropy summands, see for example [52]. when r = 1, m = g/k is an isotropy irreducible compact hermitian symmetric space (hss in short), and all these cosets can be viewed as flag manifolds with b2(m) = 1. note that from the class of full flag manifolds only cp1 ∼= s2 = su(2)/u(1) is an irreducible hss, and hence a flag manifold with b2(m) = 1. in this text we are mainly interested in non-symmetric flag manifolds m = g/k with b2(m) = 1, and in this case r is bounded by the inequalities 2 ≤ r ≤ 6 (see below). lemma 1.5. ([4, 8]) let m = g/k be a flag manifold of a compact simple lie group. then, the isotropy representation of m is monotypic and decomposes as in (1.1), for some r ≥ 1. moreover, any g-invariant metric on m = g/k is given by g = 〈 , 〉 = r∑ i=1 xi ·b|mi , (1.2) where xi ∈ r+ are positive real numbers for any i = 1, . . . ,r. thus, m g coincides with the open convex cone rr+ = {(x1, . . . ,xr) ∈ r r : xi > 0 for any i = 1 . . .r}. invariant metrics as in (1.2) are called diagonal (see [55] for details). by schur’s lemma, the ricci tensor ricg of such a diagonal invariant metric g, needs to preserve the splitting (1.1) and consequently, ricg is also diagonal, i.e., ricg(mi,mj) = 0, whenever i 6= j. as before, ricg is determined by a symmetric ad(k)-invariant bilinear form on m, although not necessarily ancient solutions of the homogeneous ricci flow 109 positive definite, and hence it has the expression ricg ≡ ric〈 , 〉 = r∑ i=1 yi ·b ∣∣ mi = r∑ i=1 (xi · rici) ·b ∣∣ mi , for some yi = xi · rici ∈ r, where 〈 , 〉 ∈ p ad(k)(m) is the ad(k)-invariant inner product corresponding to g ∈ m g and rici are the so-called ricci components. there is a simple description of rici, and hence of ricg (and also of the scalar curvature scalg = tr ricg), in terms of the metric parameters xi, the dimensions di = dimr mi and the so-called structure constants of g/k with respect to the decomposition (1.1), ckij ≡ [ k i j ] = ∑ α,β,γ b ( [xα,yβ],zγ )2 , i,j,k ∈{1, . . . ,r}, where {xα}, {yβ}, {zγ} are b-orthonormal bases of mi, mj, mk, respectively. these non-negative quantities were introduced in [55] and they have a long tradition in the theory of (compact) homogeneous einstein spaces, see for example [46, 53, 10, 15, 24]. following [32], next we shall refer to rational polynomials depending on some real variables x1,x −1 1 , . . . ,xm,x −1 m for some positive integer m, by the term laurent polynomials. if such a rational polynomial is homogeneous, then it will be called a homogeneous laurent polynomial. now, as a conclusion of most general results presented by [55, 46], one obtains the following proposition 1.6. ([55, 46]) let m = g/k be a flag manifold of a compact simple lie group g. then, (i) the components rick of the ricci tensor ricg corresponding to g = ∑r i=1 xi · b|mi ∈ m g are homogeneous laurent polynomials in x1,x −1 1 , . . . ,xr,x −1 r of degree −1, given by rick = 1 2xk + 1 4dk r∑ i,j=1 xk xixj [ k i j ] − 1 2dk r∑ i,j=1 xj xkxi [ j k i ] , (k = 1, . . . ,r) . (ii) the scalar curvature scalg = ∑r i=1 di·rici corresponding to g = ∑r i=1 xi· b|mi ∈ m g is a homogeneous laurent polynomial in x1,x −1 1 , . . . ,xr,x −1 r of degree −1, given by scalg = 1 2 r∑ i=1 di xi − 1 4 r∑ i,j,m=1 [ m i j ] xm xixj . note that scalg : m → r is a constant function. 110 s. anastassiou, i. chrysikos 2. the homogeneous ricci flow on flag manifolds in this section we fix a flag manifold m = g/k of a compact, simply connected, simple lie group g, whose isotropy representation m decomposes as in (1.1). since m g endowed with the l2-metric coincides with the open convex cone rr+, for the study of the ricci flow, as an initial invariant metric we fix the general invariant metric g = ∑r i=1 xi·b|mi; this often will be simply denoted by g = (x1, . . . ,xr) ∈ rr+ ∼= m g. then, the ricci flow equation (0.1) with initial condition g(0) = g descents to the following system of odes (in component form):{ ẋk = −2xk · rick(x1, . . . ,xr) , : 1 ≤ k ≤ r } . (2.1) since g ∈ m g, and any invariant metric evolves under the ricci flow again to a g-invariant metric, every solution of the homogeneous ricci flow needs to be of the form g(t) = r∑ i=1 xi(t) ·b|mi, g(0) = g = 〈 , 〉 , where the smooth functions xi(t) are positive on the same maximal interval t ∈ (ta, tb) for which g(t) is defined. usually, solutions have a maximal interval of definition (ta, tb), where 0 ∈ (ta, tb), −∞ ≤ ta < 0 and tb ≥ +∞. next we are mainly interested in ancient solutions. 2.1. invariant ancient solutions. recall that (see for example [18, 38]). definition 2.1. a solution g(t) of (2.1) which is defined on an interval of the form (−∞, tb) with tb < +∞, is called ancient. note that ancient solutions typically arise as singularity models of the ricci flow and it is well-known that all ancient solutions have non-negative scalar curvature, see for instance [20, corollary 2.5]. we first proceed with the following lemma 2.2. the homogeneous ricci flow (2.1) on a flag manifold m = g/k, does not possess fixed points in m g ∼= rr+. in other words, considering the associated flow to (2.1), i.e., the map ψt : m g −→ m g , g 7−→ g(t) , ancient solutions of the homogeneous ricci flow 111 where t ∈ (−�,�) for some � ∈ (0,∞), there is no g ∈ m g ∼= rr+ such that ψt(g) = g for all t. proof. obviously, fixed points of the homogeneous ricci flow need to correspond to invariant ricci-flat metrics, and conversely. since m = g/k is compact, according to alekseevsky and kimel’fel’d [3] such a metric must be necessarily flat. but then m must be a torus, a contradiction. however, the system (2.1) can admit more general solutions, different than stationary points. indeed, any flag manifold mn = g/k is compact and simply connected, and hence its universal covering is not diffeomorphic with an euclidean space. hence, by [38, corollary 4.3] it follows that proposition 2.3. ([38]) for any flag manifold m = g/k of a compact, simply connected, simple lie group g, there exists a g-invariant metric g such that any ricci flow solution g(t) with initial condition g(0) = g, has an interval of definition of the form (ta, tb), with −∞≤ ta < 0 and tb < ∞. in the following section, for all flag manifolds m = g/k with b2(m) = 1 we will construct solutions of (2.1), specify the maximal interval of their definition (ta, tb) and study their asymptotic behaviour at the infinity of the corresponding phase space m g (in terms of the poincaré compactification, see definition 2.11). these solutions, both trivial and non-trivial, emerge from invariant einstein metrics, in particular the trivial are shrinking type solutions of (2.1), in terms of [22, p. 98] for instance. moreover, they are ancient and hence the scalar curvature along such solutions is positive, while they extinct in finite time. in fact, for any flag manifold m = g/k one can state the following basic result. proposition 2.4. let m = g/k be a flag manifold of a compact, connected, simply connected, simple lie group g, with m g ∼= rr+, for some r ≥ 1. let g0 be any g-invariant einstein metric on m = g/k with einstein constant λ and consider the 1-parameter family g(t) = (1 − 2λt)g0. then, (i) g(t) is an ancient solution of (2.1) defined on the open interval ( −∞, 1 2λ ) with g(t) → 0 as t → 1 2λ (from below). hence, its scalar curvature scal(g(t)) is a monotonically increasing function with the same interval of definition, and satisfies lim t→−∞ scal(g(t)) = 0 , lim t→ 1 2λ scal(g(t)) = +∞ . in particular, scal(g(t)) > 0 for any t ∈ ( −∞, 1 2λ ) . 112 s. anastassiou, i. chrysikos (ii) similarly, the ricci components ricti ≡ rici(g(t)) corresponding to the solution g(t) = (1 − 2λt)g0, satisfy the following asymptotic properties: lim t→−∞ ricti = 0 , lim t→ 1 2λ ricti = +∞ . in particular, ricti > 0, for any t ∈ ( −∞, 1 2λ ) . proof. (i) let g0 be a g-invariant einstein metric on m = g/k (e.g., one can fix as g0 an invariant kähler-einstein metric, which always exists). set c(t) = (1 − 2λt) and note that c(t) = (1 − 2λt) > 0 for any t ∈ ( −∞, 1 2λ ) . then, since g(t) = c(t)g0 is a 1-parameter family of invariant metrics, we have g(t) = r∑ i=1 (1 − 2λt)x0i ·b|mi = r∑ i=1 c(t)x0i ·b|mi = r∑ i=1 xi(t) ·b|mi , where xi(t) := c(t)x 0 i , for any 1 ≤ i ≤ r with xi(0) = x 0 i , where without loss of generality we assume that g0 = (x 0 1, . . . ,x 0 r) for some x 0 i ∈ r r + and some n 3 r ≥ 1. as it is well-known from the general theory of ricci flow, and it is trivial to see, g(t) is a solution of (2.1) which has as interval of definition the open set ( −∞, 1 2λ ) . hence it is an ancient solution, since 0 < 1 2λ < +∞ (recall that λ > 0 is the einstein constant of an invariant einstein metric on a compact homogenous space), and moreover limt→−∞g(t) = +∞ and limt→ 1 2λ g(t) = 0. on the other hand, by proposition 1.6, the scalar curvature scalg0 ≡ scal(g0) is a homogeneous laurent polynomial of degree −1. hence, scal(c(t)g0) = c(t) −1 scal(g0) = 1 (1 − 2λt) scal(g0) . consequently scal(g(t)) = 1 (1−2λt) scal(g0) > 0, for any t ∈ ( −∞, 1 2λ ) , since scal(g0) > 0. obviously, scal′(g(t)) = 2λscal(g0) (1 − 2λt)2 > 0 , for any ( −∞, 1 2λ ) and when t tends to 1 2λ from below, we see that scal(g(t)) → +∞. since scal(g(t)), as a smooth function of t, is defined only for t ∈( −∞, 1 2λ ) , we conclude. moreover, for t →−∞ we get scal(g(t)) → 0. ancient solutions of the homogeneous ricci flow 113 (ii) similarly, by proposition 1.6, the components rici ≡ rici(g0) = ric0i (1 ≤ i ≤ r) of the ricci tensor of g0 are homogeneous laurent polynomials of degree −1. hence, rici(c(t)g0) = 1 c(t) rici(g0) = 1 c(t) ric0i , i.e., rici(g(t)) = c(t) −1 ric0i . the conclusion now easily follows, since g0 is einstein and so ric0i = λ, which is independent of t, for any 1 ≤ i ≤ r. remark 2.5. (i) the conclusions for the asymptotic behaviour of scalar curvature for the solutions g(t) verify a more general statement for the limit behaviour of the scalar curvature of homogeneous ancient solutions, obtained by lafuente [38, theorem 1.1 (i)] in terms of the so-called bracket flow 2. later, for the convenience of the reader, in section 3 we will illustrate proposition 2.4 by certain examples (see example 3.3 and example 3.4). (ii) recall by [22, p. 545] that rict ≡ ric(g(t)) = ric(g(0)) = λg0 = λ (1 − 2λt) g(t) . thus, the trivial solutions g(t) given in proposition 2.4, being homothetic to invariant einstein metrics, they also satisfy the einstein equation (and therefore the ricci flow equation), and hence lim t→−∞ rict = λg0 = lim t→( 1 2λ )− rict . the ad(k)-invariant g(t)-self-adjoint operator rg(t) ≡ rt : m → m (ricci endomorphism) corresponding to the ricci tensor rict is defined by rict(x,y ) = gt(rt(x),y ) for any x,y ∈ m. it satisfies the relation rt = λc(t)at = λa0 = r0, for any t ∈ (−∞, 1 2λ ), where a0 = ∑r i=1 x 0 i ·idmi is the positive definite ad(k)invariant g0-self-adjoint operator corresponding to the diagonal metric g0. the endomorphism of the 1-parameter family g(t) = c(t)g0, given by at = c(t)a0, is also positive definite for any t ∈ (−∞, 1 2λ ), and the same satisfies rt. (iii) obviously, proposition 2.4 and the above conclusions can be extended to any homogeneous space g/k of a compact semisimple lie group g modulo a compact subgroup k ⊂ g, with a monotypic isotropy representation admitting an invariant einstein metric g0. a simple example is given below. on the 2 the solutions g(t) described above have as maximal interval of definition the open set (−∞, 1 2λi ), so the second part of [38, thmeorem 1.1] does not apply in our situation. 114 s. anastassiou, i. chrysikos other side, there are examples of effective compact homogeneous spaces, with non-monotypic isotropy representation, i.e., mi ∼= mj for some 1 ≤ i 6= j ≤ r, for which the invariant (einstein) metrics are still diagonal. to take a taste, consider the stiefel manifold v2(r`+1) = g/k = so(` + 1)/so(` − 1) and assume for simplicity that ` 6= 3. this is a compact homogeneous space, admitting a u(1)-fibration over the grassmannian gr+2 (r `+1) = so(` + 1)/so(`− 1) ×so(2) . let so(`+1) = so(`−1)⊕m be a b-orthogonal reductive decomposition. then, it is not hard to see that m = m0 ⊕ m1 ⊕ m2, where m0 is 1-dimensional and m1 ∼= m2 are two irreducible submodules of dimension `−1, both isomorphic to the standard representation of so(`− 1). hence, the isotropy representation of v2(r`+1) is not monotypic. however, the invariant metrics on v2(r`+1) can be shown that are still diagonal. this is based on the action of the generalized weyl group (gauge group) ng(k)/k on the space p(m) ad(k) of ad(k)-invariant inner products on m (see for example [45, 6]). for the specific case of v2(r`+1), the group ng(k)/k is isomorphic to a circle and this action was used in [34, p. 121] to eliminate the off-diagonal components of the invariant metrics (note that for ` 6= 3, v2(r`+1) admits a unique so(2` + 1)invariant einstein metric). hence, the results discussed above can also be extended in that more general case. example 2.6. let m = g/k be an isotropy irreducible homogeneous space of a compact simple lie group g. consider a b-orthogonal reductive decomposition g = k ⊕ m. then, m g = r+ and the killing form gb = b|m is the unique invariant einstein metric (up to a scalar). hence, g(t) = (1 − 2λbt)gb is a homogeneous ancient solution of the corresponding homogeneous ricci flow, defined on the open interval (−∞, 1 2λb ), where λb > 0 is the einstein constant of gb. this applies in particular to any symmetric flag manifold m = g/k of a compact simple lie group g (i.e., a compact isotropy irreducible hss where r = 1). definition 2.7. a homogeneous ancient solution of (2.1) is called noncollapsed, if the corresponding curvature normalized metrics have a uniform lower injectivity radius bound. non-collapsed homogeneous ancient solutions of the ricci flow on compact homogeneous space have been recently studied in [13]. in this work, among other results, the authors proved that: ancient solutions of the homogeneous ricci flow 115 • (α) if g,k are connected and does not exist some intermediate group k ⊂ l ⊂ g such that l/k is a torus, i.e., if g/k is not a homogeneous torus bundle g/k → g/l over g/l, then any ancient solution on g/k is non-collapsed ([13, remark 5.3]). • (β) non-trivial homogeneous ancient solutions of ricci flow must develop a type i singularity close to their extinction time and also to the past (see [13, corollary 2, page 2, and pages 24 – 25]). hence we deduce that corollary 2.8. let m= g/k be a flag manifold as in proposition 2.4. then any non-trivial ancient solution of the homogeneous ricci flow, if it exists, emanates from an invariant einstein metric, is non-collapsed and develops a type-i singularity close to the extinsion time (and also to the past). in particular, the trivial ancient solutions of the form g(t) = (1−2λt)g0, where g0 ∈mg is an invariant einstein metric on m with einstein constant λ, are non-collapsed and as t → 1 2λ the volume of m= g/k with respect to g(t) tends to 0, v olg(t)(g/k) → 0, i.e., m= g/k shrinks to a point in finite time. proof. for the record, let us assume that there exists some intermediate closed subgroup k ⊂ l ⊂ g such that l/k is torus ts for some s ≥ 1. then obviously, it must be rankl > rankk. but rankk = rankg and then inclusion l ⊂ g gives a contradiction. hence, m = g/k cannot be homogeneous torus bundle. thus, according to (α) any possible homogeneous ancient solution of (2.1) on m = g/k must be non-collapsed. the other claims and the assertions for type i behaviour, follow now by (β). remark 2.9. recall that on a compact homogeneous space m = g/k the total scalar curvature functional s(g) = ∫ m scal(g)dvg , restricted on the set m g1 of g-invariant metrics of volume 1, coincides with the smooth function m g1 3 g 7−→ scal(g) ≡ scalg, since the scalar curvature scalg of g is a constant function on m. in this case, g-invariant einstein metrics of volume 1 on g/k are precisely the critical points of the restriction s|m g1 . a g-invariant einstein metric is called unstable if is not a local maximum of s|m g1 . by [13, lemmma 5.4] it also known that for any unstable homogeneous einstein metric on a compact homogeneous space g/k, there exists a 116 s. anastassiou, i. chrysikos non-collapsed invariant ancient solution emanating from it. for general flag manifolds, up to our knowledge, is an open question if all possible existent invariant einstein metrics are unstable or not. however, by [9, thmeorem 1.2] it is known that on a flag manifold with r = 2, i.e., with two isotropy summands, there exist two non-isometric invariant einstein metrics which are both local minima of s|m g1 , and hence unstable in the above sense. thus, for r = 2 the existence of non-collapsed ancient solutions of system (2.1) can be also obtained by a direct combination of the results in [9, 13]. 2.2. the poincaré compactification procedure. to study the asymptotic behaviour of homogeneous ricci flow solutions, even of more general abstract solutions than these given in proposition 2.4, one can successfully use the compactification method of poincaré which we refresh below, adapted to our setting (see also [28, 7, 29]). indeed, the poincaré compactification procedure is used to study the behaviour of a polynomial system of ordinary differential equations in a neighbourhood of infinity, see [27, 54] for an explicit description. we describe it here, in a form suitable for our purposes. let (m = g/k,g) be a flag manifold with m g ∼= rr+ for some r ≥ 1. by using the expressions of proposition 1.6 and after multiplying the right-hand side of the equations in (2.1), with a suitable positive factor, we obtain a qualitative equivalent dynamical system consisting of homogeneous polynomial equations of positive degree. let us denote this system obtained from the homogeneous ricci flow via this procedure, by{ ẋk = rfk(x1, . . . ,xr) : k = 1, . . . ,r } . (2.2) so, for any 1 ≤ k ≤ r, rfk(x1, . . . ,xr) are homogeneous polynomials, whose maximum degree is defined to be the degree d of the system (2.2). let us proceed with the following definition. definition 2.10. the vector field associated to the system (2.2) will be denoted by x(x1, . . . ,xr) := ( rf1(x1, . . . ,xr), . . . ,rfr(x1, . . . ,xr) ) and referred to as the homogeneous vector field associated to the homogeneous ricci flow on (m = g/k,g). consider the subset of the unit sphere sr ⊂ rr+1, containing all points having non-negative coordinates, i.e., sr≥ = { y ∈ rr+1 : ‖y‖ = 1, yi ≥ 0, i = 1, . . . ,r + 1 } ⊂ sr . ancient solutions of the homogeneous ricci flow 117 it is also convenient to identify rr≥ with the subset of r r+1 defined by t := { (y1, . . . ,yr,yr+1) ∈ rr+1 : yr+1 = 1, yi ≥ 0, ∀ i = 1, . . . ,r } ∼= rr≥ . consider now the central projection f : rr≥ ∼= t → s r ≥, assigning to every p ∈ t the point t(p) ∈ sr≥, defined as follows: t(p) is the intersection of the straight line joining the initial point p with the origin of rr+1. the explicit form of f is given by f(y1, . . . ,yr, 1) = 1 ‖(y1, . . . ,yr, 1)‖ (y1, . . . ,yr, 1) . through this projection, rr≥ can be identified with the subset of s r ≥ with yi+1 > 0. moreover, the equator of s r ≥, that is sr−1≥ = { y ∈ sr≥ : yr+1 = 0 } , is identified as the infinity of rr≥. to be more explicit, let us state this as a definition. definition 2.11. a point at infinity of m g ∼= rr≥ is understood to be point of the equator sr−1≥ . note that sr≥ is diffeomorphic with the standard r–dimensional simplex ∆r = { (y1, . . . ,yr+1) ∈ rr+1 : ∑ iyi = 1, ∀ yi ≥ 0, i = 1, . . . ,r + 1 } and thus with the so-called non-negative part of the real projective space rpr≥ = ( rr+1≥ \{0} ) /r+ (see [26, chapter 4]). via push-forward, the central projection carries the vector field x onto sr≥. the vector field obtained by this procedure, i.e., the vector field p(x)(y) := yd−1r+1f∗x(y) is called the poincaré compactification of x, and this is an analytic vector field defined on all sr≥. actually, in order to perform computations with p(x) one needs its expressions in a chart. thus, consider the chart u = { y ∈ sr : y1 > 0 } of sr, and project it to the plane { (y1, . . . ,yr+1) ∈ rr+1 : y1 = 1) } ∼= rr≥. then, via the corresponding central projection, assign to every point of u the 118 s. anastassiou, i. chrysikos point of intersection of the straight line joining the origin with the original point. this second central projection denoted by f : u → rr≥, has obviously the form f(y1,y2, . . . ,yr+1) = ( 1, y2 y1 , . . . , yr+1 y1 ) . we can now compute the local expression of p(x) in the chart u; as above, let us denote by xi the coordinates on rr≥. then, we obtain proposition 2.12. the local expression of the poincaré compactification p(x) of the homogeneous vector field x associated to the ricci flow on (m = g/k,g), reads as ẋi = x d r ( −xirf1 + rfi+1 ) for any i = 1, . . . ,r − 1 , ẋr = x d r ( −xrrf1 ) , (2.3) where 3 rfi(x1, . . . ,xr) := rfi ( 1 xr , x2 xr , . . . , xr−1 xr ) . remark 2.13. in the expressions given by (2.3), the factor xdr is canceled by the dominators of the rational polynomials rfi(x1, . . . ,xr). after doing so and by locating the fixed points of the resulting dynamical system, under the condition xr = 0, we obtain the so-called fixed points at the infinity of m g ∼= rr+ of the homogeneous ricci flow on (m = g/k,g). example 2.14. for r = 2 the expression of p(x) in the local chart u is given by ẋ1 = x d 2 ( −x1rf1 + rf2 ) , ẋ2 = x d 2 ( −x2rf1 ) , where rfi(x1,x2) = rfi ( 1 x2 , x1 x2 ) for any i = 1, 2. we may omit the term 1 ‖(x1,x2,1)‖ , by using a time reparametrization. fixed points of the homogeneous ricci flow on (m = g/k,g) at infinity of m g = r2+, can be studied by setting x2 = 0. for r = 3 the expression of p(x) in the local chart u reads by ẋ1 = x d 3 ( −x1rf1 + rf2 ) , ẋ2 = x d 3 ( −x2rf1 + rf3 ) , ẋ3 = x d 3 ( −x3rf1 ) , where rfi = rfi ( 1 x3 , x1 x3 , x2 x3 ) , for any i = 1, 2, 3. in an analogous way, fixed points at infinity can be studied by setting x3 = 0, while similarly are treated cases with r > 3. 3 here, we have omitted the term 1‖(x1,...,xr,1)‖ , since it does not affect the qualitative behaviour of the system. ancient solutions of the homogeneous ricci flow 119 3. global study of hrf on flag spaces m = g/k with b2(m) = 1 we turn now our attention to the global study of the classical ricci flow equation for an initial invariant metric on (non symmetric) flag manifolds m = g/k with b2(m) = 1. again we can work with g simple. let us begin first with a few details about this specific class of flag spaces. 3.1. flag manifolds m = g/k with b2(m) = 1. according to proposition 1.4, flag manifolds m = g/k of a compact simple lie group g with b2(m) = 1 are defined by painting black in the dynkin diagram of g only a simple root, i.e., πm = {aio} for some aio ∈ π. the number of the isotropy summands of a flag space m = g/k with b2(m) = 1 can be read from the pdd, at least when we encode with it the so-called dynkin marks of the simple roots; these are the positive integers coefficients appearing in the expression of the highest root of g as a linear combination of simple roots. for flag manifolds with b2(m) = 1, i.e., πm = π/πk = {aio : 1 ≤ io ≤ ` = rankg} we have the relation r = dynk(αio) (see [23, 24]), where r is the integer appearing in (1.1). in other words, a flag manifold m = g/k with b2(m) = 1 and r isotropy summands is obtained by painting black a simple root αio with dynkin mark r, and conversely. since for a compact simple lie group g the maximal dynkin mark equals to 6 (and occurs for g = e8 only), we result with the bound 1 ≤ r ≤ 6. so, from now on assume that m = g/k is a flag manifold as above with b2(m) = 1 and 2 ≤ r ≤ 6. the classification of such flag manifolds can be found in [24]; there, in combination with earlier results of kimura, arvanitoyeorgos and the second author [35, 8, 9] is proved that any such space admits a finite number of non-isometric (non-kähler) invariant einstein metrics, a result which supports the still open finiteness conjecture of böhmwang-ziller ([15]). note that m = g/k admits a unique invariant complex structure, and thus a unique invariant kähler-einstein metric, with explicit form (see [17, 24]) gke = r∑ i i ·b|mi = b|m1 + 2b|m2 + . . . + rb|mr . (3.1) the isotropy summands satisfy the relations [k,mi] ⊂ mi , [mi,mi] ⊂ k + m2i , [mi,mj] ⊂ mi+j + m|i−j| (i 6= j). 120 s. anastassiou, i. chrysikos hence, the non-zero structure constants are listed as follows: r non-zero structure constants ckij 2 c211 ([9]) 3 c211, c 3 12 ([35, 7]) 4 c211, c 3 12, c 4 13, c 4 22 ([8]) 5 c211, c 3 12, c 4 13, c 5 14, c 4 22, c 5 23 ([24]) 6 c211, c 3 12, c 4 13, c 5 14, c 6 15, c 4 22, c 5 23, c 6 24, c 6 33 ([24]) for r = 2, 3, the use of the kähler-einstein metric is sufficient for an explicit computation of ckij, and this yields a general expression of them in terms of di = dim mi (see [35, 9, 7]). for 4 ≤ r ≤ 6 more advanced techniques are necessary for the computation of ckij, which depend on the specific coset (see [8, 24]). case r = 2 : all flag manifolds m = g/k with g simple and r = 2 have b2(m) = 1, see [9] for details. we recall that: c211 = d1d2 d1 + 4d2 , ric1 = 1 2x1 − c211 x2 2d1x 2 1 , ric2 = 1 2x2 + c211 4d2 (x2 x21 − 2 x2 ) , scalg = 2∑ i=1 di · rici = 1 2 (d1 x1 + d2 x2 ) − c211 4 (x2 x21 + 2 x2 ) . (3.2) case r = 3 : let m = g/k be a flag manifold with b2(m) = 1 and r = 3. such flag spaces have been classified by [35], see also [7, 30]. they all correspond to exceptional compact simple lie groups, but the correspondence is not a bijection. one should mention that not all flag manifolds with r = 3 are exhausted by this type of homogeneous spaces; this means that still one can construct flag spaces with r = 3 and b2(m) = 2. we recall that c211 = d1d2 + 2d1d3 −d2d3 d1 + 4d2 + 9d3 , c312 = d3(d1 + d2) d1 + 4d2 + 9d3 , ric1 = 1 2x1 − c211x2 2d1x 2 1 + c312 2d1 ( x1 x2x3 − x2 x1x3 − x3 x1x2 ) , ancient solutions of the homogeneous ricci flow 121 ric2 = 1 2x2 + c211 4d2 (x2 x21 − 2 x2 ) + c312 2d2 ( x2 x1x3 − x1 x2x3 − x3 x1x2 ) , ric3 = 1 2x3 + c312 2d3 ( x3 x1x2 − x1 x2x3 − x2 x1x3 ) , scalg = 1 2 (d1 x1 + d2 x2 + d3 x3 ) − c211 4 (x2 x21 + 2 x2 ) (3.3) − c312 2 ( x1 x2x3 + x2 x1x3 + x3 x1x2 ) . case r = 4 : let m = g/k be a flag manifold with g simple, r = 4 and b2(m) = 1. there exist four such homogeneous spaces and all of the them correspond to an exceptional lie group. to give the reader a small taste of pdds, we present these flag spaces below, together with the corresponding pdd and dynkin marks. as for the case r = 3, one should be aware that still exist flag manifold with r = 4, but b2(m) = 2. m = g/k with b2(m) = 1 and r = 4 pdd f4 /su(3) ×su(2) ×u(1) cα1 2 cα2 3 >sα3 4 cα4 2 e7 /su(4) ×su(3) ×su(2) ×u(1) cα1 1 cα2 2 cα3 3 sα4 4 cα72 cα5 3 cα6 2 e8 /so(10) ×su(3) ×u(1) cα1 2 cα2 3 sα3 4 cα4 5 cα83 cα5 6 cα6 4 cα7 2 e8 /su(7) ×su(2) ×u(1) cα1 2 cα2 3 cα3 4 cα4 5 cα83 cα5 6 sα6 4 cα7 2 we also recall that ric1 = 1 2x1 − c211 2d1 x2 x21 + c312 2d1 ( x1 x2x3 − x2 x1x3 − x3 x1x2 ) + c413 2d1 ( x1 x3x4 − x3 x1x4 − x4 x1x3 ) , ric2 = 1 2x2 − c422 2d2 x4 x22 + c211 4d2 (x2 x21 − 2 x2 ) + c312 2d2 ( x2 x1x3 − x1 x2x3 − x3 x1x2 ) , 122 s. anastassiou, i. chrysikos ric3 = 1 2x3 + c312 2d3 ( x3 x1x2 − x2 x1x3 − x1 x2x3 ) + c413 2d3 ( x3 x1x4 − x1 x3x4 − x4 x1x3 ) , ric4 = 1 2x4 + c422 4d4 (x4 x22 − 2 x4 ) + c413 2d4 ( x4 x1x3 − x1 x3x4 − x3 x1x4 ) , scalg = 1 2 4∑ i=1 di xi − c312 2 ( x1 x2x3 + x2 x1x3 + x3 x1x2 ) − c413 2 ( x1 x3x4 + x3 x1x4 + x4 x1x3 ) − c211 4 (x2 x21 + 2 x2 ) − c422 4 (x4 x22 + 2 x4 ) . let us finally present the values of ckij and the corresponding dimensions: m = g/k c422 c 2 11 c 3 12 c 4 13 f4 /su(3) ×su(2) ×u(1) 2 2 1 2/3 e7 /su(4) ×su(3) ×su(2) ×u(1) 2 8 4 4/3 e8 /so(10) ×su(3) ×u(1) 2 16 8 8/5 e8 /su(7) ×su(2) ×u(1) 14/3 14 7 14/5 d1 d2 d3 d4 f4 /su(3) ×su(2) ×u(1) 12 18 4 6 e7 /su(4) ×su(3) ×su(2) ×u(1) 48 36 16 6 e8 /so(10) ×su(3) ×u(1) 96 60 32 6 e8 /su(7) ×su(2) ×u(1) 84 70 28 14 case r = 5 : according to [24], there is only one flag manifold m = g/k with g simple, b2(m) = 1 and r = 5; this is the coset space m = g/k = e8 /u(1) ×su(4) ×su(5) and is determined by painting black the simple root α4 of e8, i.e., πm = {α4}, with dynk(α4) = 5 (and hence r = 5). the ricci components rici are given by ric1 = 1 2x1 − c211 2 d1 x2 x12 + c312 2 d1 ( x1 x2x3 − x2 x1x3 − x3 x1x2 ) + c413 2 d1 ( x1 x3x4 − x3 x1x4 − x4 x1x3 ) + c514 2 d1 ( x1 x4x5 − x4 x1x5 − x5 x1x4 ) , ancient solutions of the homogeneous ricci flow 123 ric2 = 1 2x2 + c211 4 d2 ( x2 x12 − 2 x2 ) − c422 2 d2 x4 x22 + c312 2 d2 ( x2 x1x3 − x1 x2x3 − x3 x2x1 ) + c523 2 d2 ( x2 x3x5 − x3 x2x5 − x5 x2x3 ) , ric3 = 1 2x3 + c312 2 d3 ( x3 x1x2 − x2 x3x1 − x1 x3x2 ) + c413 2 d3 ( x3 x1x4 − x1 x3x4 − x4 x1x3 ) + c523 2 d3 ( x3 x2x5 − x2 x3x5 − x5 x3x2 ) , ric4 = 1 2x4 + c422 4 d4 ( x4 x22 − 2 x4 ) + c413 2 d4 ( x4 x1x3 − x1 x3x4 − x3 x4x1 ) + c514 2 d4 ( x4 x1x5 − x1 x4x5 − x5 x1x4 ) , ric5 = 1 2x5 + c523 2 d5 ( x5 x2x3 − x2 x3x5 − x3 x2x5 ) + c514 2 d5 ( x5 x1x4 − x1 x4x5 − x4 x1x5 ) . the non-zero structure constants have been computed in [24, proposition 6] and it is useful to recall them: c211 = 12 , c 3 12 = 8 , c 4 13 = 4 , c 5 14 = 4/3 , c 4 22 = 4 , c 5 23 = 2 . moreover, d1 = 80, d2 = 60, d3 = 40, d4 = 20 and d5 = 8. case r = 6 : by [24] it is known that there is also only one flag manifold m = g/k with g simple, b2(m) = 1 and r = 6. this is isometric to the homogeneous space m = g/k = e8 /u(1) ×su(2) ×su(3) ×su(5), which is determined by painting black the simple root α5 of e8, i.e., πm = {α5}, with dynk(α5) = 6. we know that d1 = 60, d2 = 60, d3 = 40, d4 = 30, d5 = 12 and d6 = 10. also, the values of the non-zero structure constants have the form (see [24, proposition 12]) c211 = 8 , c 3 12 = 6 , c 4 13 = 4 , c 5 14 = 2 , c 6 15 = 1 , c422 = 6 , c 5 23 = 2 , c 6 24 = 2 , c 6 33 = 2 , and the components rici of the ricci tensor ricg corresponding to g are given 124 s. anastassiou, i. chrysikos by ric1 = 1 2x1 − c211 2 d1 x2 x12 + c312 2 d1 ( x1 x2x3 − x2 x1x3 − x3 x1x2 ) + c413 2 d1 ( x1 x3x4 − x3 x1x4 − x4 x1x3 ) c514 2 d1 ( x1 x4x5 − x4 x1x5 − x5 x1x4 ) + c615 2 d1 ( x1 x5x6 − x5 x1x6 − x6 x1x5 ) , ric2 = 1 2x2 + c211 4 d2 ( x2 x12 − 2 x2 ) − c422 2 d2 x4 x22 + c312 2 d2 ( x2 x1x3 − x1 x2x3 − x3 x2x1 ) + c523 2 d2 ( x2 x3x5 − x3 x2x5 − x5 x2x3 ) + c624 2 d2 ( x2 x4x6 − x4 x2x6 − x6 x2x4 ) , ric3 = 1 2x3 − c633 2 d3 x6 x32 + c312 2 d3 ( x3 x1x2 − x2 x3x1 − x1 x3x2 ) + c413 2 d3 ( x3 x1x4 − x1 x3x4 − x4 x1x3 ) + c523 2 d3 ( x3 x2x5 − x2 x3x5 − x5 x3x2 ) , ric4 = 1 2x4 + c422 4 d4 ( x4 x22 − 2 x4 ) + c413 2 d4 ( x4 x1x3 − x1 x3x4 − x3 x4x1 ) + c514 2 d4 ( x4 x1x5 − x1 x4x5 − x5 x1x4 ) + c624 2 d4 ( x4 x2x6 − x2 x4x6 − x6 x2x4 ) , ric5 = 1 2x5 + c514 2 d5 ( x5 x1x4 − x1 x4x5 − x4 x1x5 ) + c523 2 d5 ( x5 x2x3 − x2 x3x5 − x3 x2x5 ) + c615 2 d5 ( x5 x1x6 − x1 x5x6 − x6 x1x5 ) , ric6 = 1 2x6 + c633 4 d6 ( x6 x32 − 2 x6 ) + c615 2 d6 ( x6 x1x5 − x1 x5x6 − x5 x1x6 ) + c624 2 d6 ( x6 x2x4 − x2 x4x6 − x4 x2x6 ) . 3.2. the main theorem. let m = g/k be a flag manifold with b2(m) = 1 and 2 ≤ r ≤ 6. the system of the homogeneous ricci flow is given by { ẋi = −2xi · rici : i = 1, . . . ,r } . (3.4) ancient solutions of the homogeneous ricci flow 125 for any case separately, a direct computation shows that system (3.4) does not possess fixed points in m g ∼= rr+, i.e., points (x1, . . . ,xr) ∈ r r r satisfying the system {ẋi = ẋ2 = · · · = ẋr = 0}, which verifies lemma 2.2. we now agree on the following notation: we denote by ej a fixed point of the homogeneous ricci flow (hrf) at infinity of m g, as defined before. we shall write n for the number of all such fixed points. we will also denote by dunstbj (respectively d stb j ) the dimension of the unstable manifold (respectively stable manifold) in m g (respectively in the infinity of m g), corresponding to ej. in this terms we obtain the following theorem 3.1. let m = g/k be a non-symmetric flag manifold with b2(m) = 1, and let r (2 ≤ r ≤ 6) be the number of the corresponding isotropy summands. then, the following hold: (1) the hrf admits exactly n fixed points ej at the infinity of m g, where for any coset g/k the number n is specified in table 1. these fixed points are in bijective correspondence with non-isometric invariant einstein metrics on m = g/k, and are specified explicitly in the proof. (2) the dimensions of the stable/unstable manifolds corresponding to ej are given in table 1, where e1 represents the fixed point corresponding to the unique invariant kähler-einstein metric on g/k. we see that (i) for any m = g/k, the fixed point e1 has always an 1-dimensional unstable manifold in m g, and a (r− 1)-dimensional stable manifold in the infinity of m g. (ii) any other fixed point ek with 2 ≤ k ≤ n, has always a 2-dimensional unstable manifold, while its stable manifold is (r − 2)-dimensional and is contained in the infinity of m g, with the following three exceptions: • the fixed point e5 for the space m∗ := e8 /u(1) × su(3) × so(10) in the case with r = 4; • the fixed point e5 in the case of m = e8 /u(1) × su(4) × su(5) with r = 5; • the fixed point e4 in the case of m = e8 /u(1) ×su(2) ×su(3) × su(5) with r = 6. these three exceptions have a 3-dimensional unstable manifold and a (r − 3)-dimensional stable manifold. similarly, the stable manifold for these cases is contained entirely in the infinity of m g. 126 s. anastassiou, i. chrysikos (3) each unstable manifold of any ej, contains a non-collapsed ancient solution, given by gj : ( −∞, 1 2λj ) −→ m g , t 7−→ gj(t) = (1 − 2λjt) · ej , j = 1, . . . ,n , where λj is the einstein constant of the corresponding einstein metric gj(0), j = 1, . . .n, on m = g/k (these are also specified below). all such solutions gj(t) tend to 0 when t → t = 12λj > 0, and m = g/k shrinks to a point in finite time. (4) when r = 2, any other possible solution of the ricci flow with initial condition in m g has {0} as its ω–limit set. conditions dunstbj d stb j r for m = g/k n dunstb1 d stb 1 (2 ≤ j ≤ n) (2 ≤ j ≤ n) 2 2 1 1 2 0 3 3 1 2 2 1 4 m � m∗ 3 1 3 2 2 4 m ∼= m∗ 5 1 3 (for j 6= 5) 2 (for j 6= 5) 2 (for j = 5) 3 (for j = 5) 1 5 6 1 4 (for j 6= 5) 2 (for j 6= 5) 3 (for j = 5) 3 (for j = 5) 2 6 5 1 5 (for j 6= 4) 2 (for j 6= 4) 4 (for j = 4) 3 (for j = 4) 3 table 1: the exact number n of the fixed points ek of hrf at infinity of m g for any non-symmetric flag space m = g/k with b2(m) = 1, and the dimensions dstbk , d unstb k , for any 1 ≤ k ≤ n. proof. we split the proof in cases, depending on the possible values of r. case r = 2. in this case the system (3.4) reduces to:{ ẋ1 = − (d1 + 4d2)x1 −d2x2 (d1 + 4d2)x1 , ẋ2 = − 8d2x 2 1 + d1x 2 2 2(d1 + 4d2)x 2 1 } . (3.5) to search for fixed points of the hrf at infinity of m g, we first multiply the right-hand side of these equations with the positive factor 2(d1 + 4d2)x 2 1. this ancient solutions of the homogeneous ricci flow 127 multiplication does not qualitatively affect system’s behaviour, and we result with the following equivalent system:{ ẋ1 = rf1(x1,x2) = −2(d1 + 4d2)x21 + 2d2x1x2 , ẋ2 = rf2(x1,x2) = − ( 8d2x 2 1 + d1x 2 2 )} , where the right-hand side consists of two homogeneous polynomials of degree 2. this is the maximal degree d of the system, as discussed above. we can therefore apply the poincaré compactification procedure to study its behaviour at infinity. for the formulas given in example 2.14, i.e.,{ ẋ1 = x 2 2 ( −x1rf1(x1,x2) + rf2(x1,x2) ) , ẋ2 = x 2 2 ( −x2rf1(x1,x2) )} , we compute rf1(x1,x2) = rf1 ( 1 x2 , x1 x2 ) = − 2(d1 + 4d2 −d2x1) x22 , rf2(x1,x2) = rf2 ( 1 x2 , x1 x2 ) = − (8d2 + d1x 2 1) x22 . hence finally we result with the system{ ẋ1 = −(x1 − 2)(−4d2 + d1x1 + 2d2x1) , ẋ2 = 2(d1 + 4d2 −d2x1)x2 } , which is the desired expression in the u chart. to study the behaviour of this system at infinity of r2+, we set x2 = 0. then, the second equation becomes ẋ2 = 0, confirming that infinity remains invariant under the flow. to locate fixed points, we solve the equation ẋ1 = h(x1) = −(x1 − 2)(−4d2 + d1x1 + 2d2x1) = 0 . we obtain two exactly solutions, namely xa1 = 2 and x b 1 = 4d2/(d1 + 2d2), and since h′(xa1) = −2d1, h ′(xb1) = 2d1, both fixed points are hyperbolic. in particular, h′(xa1) < 0 and so x a 1 is always an attracting node with eigenvalue equal to −2d1. on the other hand, h′(xb1) > 0 and hence x b 1 is a repelling node, with eigenvalue equal to 2d1. recall now that the central projection f maps the sphere to the y1 = 1 plane. therefore, the fixed points xa1, x b 1 represent the points e1 = (1,x a 1) and e2 = (1,x b 1) in r 2 +, which correspond to the invariant einstein metrics gke = 1 ·b|m1 + 2 ·b|m2 , ge = 1 ·b|m1 + 4d2/(d1 + 2d2) ·b|m2 , 128 s. anastassiou, i. chrysikos respectively (see also [9]). to locate the ancient solutions, let ĝ(t) = t ( x1(0),x2(0) ) be a straight line in m g. at the point (a,b), belonging in the trace of ĝ(t), the vector normal to the straight line is the vector (−b,a). the straight line ĝ(t) must be tangent to the vector field x(x1,x2) defined by the homogeneous ricci flow, hence the following equation should hold: (−b,a) ·x(a,b) = 0 . this gives us two solutions, namely (a1,b1) = (1, 2) and (a2,b2) = (1, 4d2/(d1+ 2d2)), which define the lines γi(t) = t(ai,bi), i = 1, 2. the solutions g1(t), g2(t) of system (3.5) corresponding to the lines γ1(t) and γ2(t), are determined by the following equations g1(t) = (1 − 2λ1t) · (1, 2) = (1 − 2λ1t) · e1 , g2(t) = (1 − 2λ2t) · ( 1, 4d2 d1 + 2d2 ) = (1 − 2λ2t) · e2 , where λ1,λ2 are the einstein constants of gke = g1(0) and ge = g2(0), respectively, given by λ1 = d1 + 2d2 2(d1 + 4d2) , λ2 = d21 + 6d1d2 + 4d 2 2 2(d1 + 2d2)(d1 + 4d2) . obviously, these are both ancient solutions since are defined on the open set (−∞, 1 2λi ) (see also proposition 2.4), in particular gi(t) → 0 when t → 12λi , i = 1, 2. the assertion that gi(t) are non-collapsed follows by corollary 2.8. based on the definitions of the central projections f and f given before, we verify that f(f(γ1(t))) = (1, 2) and f(f(γ2(t))) = ( 1, 4d2 d1 + 2d2 ) . thus, we have that lim t→ 1 2λi γi(t) = 0 and lim t→−∞ γi(t) = ei , ∀ i = 1, 2, which proves the claim that these ancient solitons belong to the unstable manifolds of the fixed points located at infinity. to verify the claim in (4), we use the function v2(x1,x2) = x 2 1 + x 2 2 as a lyapunov function for the system (3.5). we compute that dv dt (x1(t),x2(t)) = − 2d2x 2 1(4x1 + 3x2) + d1 ( 2x31 + x 3 2 ) 2(d1 + 4d2)x 2 1 , ancient solutions of the homogeneous ricci flow 129 which for x1,x2 > 0 is always negative. hence, if we denote as (ta, tb), ta, tb ∈ r∪{±∞} the domain of definition of the solution curve g(t) = (x1(t),x2(t)), we conclude that g(t) tends to the origin, as t → tb. this completes the proof for r = 2. case r = 3. in this case, to reduce system (3.4) in a polynomial dynamical system, we must multiply with the positive factor 2d1d2(d1+4d2+9d3)x 2yz. this gives ẋ1 = rf1(x1,x2,x3) = −2d2x1 ( d1d3x 3 1 + d2d3x 3 1 −d1d3x1x 2 2 −d2d3x1x 2 2 + d21x1x2x3 + 4d1d2x1x2x3 + 9d1d3x1x2x3 −d1d2x 2 2x3 − 2d1d3x 2 2x3 + d2d3x 2 2x3 −d1d3x1x 2 3 −d2d3x1x 2 3 ) , ẋ2 = rf2(x1,x2,x3) = −d1x2 ( − 2d1d3x31 − 2d2d3x 3 1 + 2d1d3x1x 2 2 (3.6) + 2d2d3x1x 2 2 + 8d 2 2x 2 1x3 − 4d1d3x 2 1x3 + 20d2d3x 2 1x3 + d1d2x 2 2x3 + 2d1d3x 2 2x3 −d2d3x 2 2x3 − 2d1d3x1x 2 3 − 2d2d3x1x 2 3 ) ẋ3 = rf3(x1,x2,x3) = 2d1d2x1x3 ( d1x 2 1 + d2x 2 1 −d1x1x2 − 4d2x1x2 − 9d3x1x2 + d1x22 + d2x 2 2 −d1x 2 3 −d2x 2 3 ) . the right-hand side of the system above consists of homogeneous polynomials of degree 4. let us apply the poincaré compactification procedure and set x3 = 0, to obtain the equations governing the behaviour of the system at infinity of m g. by example 2.14 we deduce that in the u-chart, these must be given as follows: ẋ1 = x1 ( 2d21d3 + 4d1d2d3 + 2d 2 2d3 − 2d 2 1d3x 2 1 − 4d1d2d3x 2 1 − 2d 2 2d3x 2 1 − 8d1d22x2 + 4d 2 1d3x2 − 20d1d2d3x2 + 2d 2 1d2x1x2 + 8d1d 2 2x1x2 + 18d1d2d3x1x2 −d21d2x 2 1x2 − 2d1d 2 2x 2 1x2 − 2d 2 1d3x 2 1x2 − 3d1d2d3x21x2 + 2d 2 2d3x 2 1x2 + 2d 2 1d3x 2 2 − 2d 2 2d3x 2 2 ) , ẋ2 = −2d2x2 ( −d21 −d1d2 −d1d3 −d2d3 + d 2 1x1 + 4d1d2x1 + 9d1d3x1 −d21x 2 1 −d1d2x 2 1 + d1d3x 2 1 + d2d3x 2 1 −d 2 1x1x2 − 4d1d2x1x2 − 9d1d3x1x2 + d1d2x21x2 + 2d1d3x 2 1x2 −d2d3x 2 1x2 + d 2 1x 2 2 + d1d2x 2 2 + d1d3x 2 2 + d2d3x 2 2 ) , ẋ3 = 0 . 130 s. anastassiou, i. chrysikos as before, the last equation confirms that infinity remains invariant under the flow of the system. to locate fixed points, we have to solve the system of equations {ẋ1 = ẋ2 = 0}. for this, it is sufficient to study each case separately and replace the dimensions di. for any case we get exactly three fixed points, which we list as follows: • e8 /e6 ×su(2) × u(1) : in this case, we have d1 = 108, d2 = 54, d3 = 4 and the fixed points at infinity are located at: (2, 3) , (0.914286, 1.54198) , (1.0049, 0.129681) . • e8 /su(8)×u(1) : in this case, we have d1 = 112, d2 = 56, d3 = 16 and the fixed points at infinity are located at: (2, 3) , (0.717586, 1.25432) , (1.06853, 0.473177) . • e7 /su(5)×su(3)×u(1) : in this case, we have d1 = 60, d2 = 30, d3 = 8 and the fixed points at infinity are located at: (2, 3) , (0.733552, 1.27681) , (1.06029, 0.443559) . • e7 /su(6)×su(2)×u(1) : in this case, we have d1 = 60, d2 = 30, d3 = 4 and the fixed points at infinity are located at: (2, 3) , (0.85368, 1.45259) , (1.01573, 0.229231) . • e6 /su(3)×su(3)×su(2)×u(1) : in this case, we have d1 = 36, d2 = 18, d3 = 4 and the fixed points at infinity are located at: (2, 3) , (0.771752, 1.33186) , (1.04268, 0.373467) . • f4 /su(3)×su(2)×u(1) : in this case, we have d1 = 24, d2 = 12, d3 = 4 and the fixed points at infinity are located at: (2, 3) , (0.678535, 1.20122) , (1.09057, 0.546045) . • g2 /u(2) : in this case, we have d1 = 4, d2 = 2, d3 = 4 and the fixed points at infinity are located at: (2, 3) , (1.67467, 2.05238) , (0.186894, 0.981478) . ancient solutions of the homogeneous ricci flow 131 the projections of these solutions via f : u → r3≥, give us the points ei, i = 1, 2, 3. these are the points obtained from the coordinates of the fixed points given above, with one extra coordinate equal to 1, in the first entry. according to (3.1), the indicated fixed point e1 = (1, 2, 3) corresponds to the unique invariant kähler-einstein metric, and simple eigenvalue calculations show that it possesses a 2-dimensional stable manifold at the infinity of m g, and a 1-dimensional unstable manifold, which is contained in m g. all the other fixed points, correspond to non-kähler non-isometric invariant einstein metrics (see [35, 7]), and have a 2-dimensional unstable manifold and a 1-dimensional stable manifold, contained at the infinity of m g. let us now consider an invariant line of system (3.6), of the form ĝ(t) = t ( x1(0),x2(0),x3(0) ) . at a point (a,b,c) belonging to this line, the normal vectors are: (−b,a, 0), (0,−c,b), thus ĝ(t) is tangent to the vector field x(x1,x2,x3) defined by the homogeneous ricci flow if (−b,a, 0) ·x(a,b,c) = 0 , (0,−c,b) ·x(a,b,c) = 0 . these equations possess three solutions, with respect to (a,b,c). one of them is always (1, 2, 3), while the other two can be obtained after numerically solving equations above for every value of d1,d2,d3. these solutions give us the three non-collapsed ancient solutions gi(t) = (1 − 2λit) · ei with t ∈ (−∞, 12λi ), for any i = 1, 2, 3, where the einstein constants λi for the cases i = 2, 3 can be computed easily by replacing the corresponding einstein metrics g2(0), g3(0) in the ricci components, while for g1(t), λ1 is specified as follows: λ1 = d1 + 2d2 + 3d3 2d1 + 8d2 + 18d3 . moreover, it is easy to show, taking limits, that the α-limit set of each of these solutions is one of the located fixed points at infinity of m g, while the ω-limit set, for all of them, is {0}. finally, for any i we have gi(t) → 0 as t → 12λi , where t = 1 2λi depends on the dimensions di, i = 1, 2, 3, and so on the flag manifold m = g/k, while again the assertion that gi(t) are non-collapsed, for any i = 1, 2, 3, is a consequence of corollary 2.8. case r = 4. the proof follows the lines of the previous cases r = 2, 3. thus, we avoid to present similar arguments. consider for example the case of the flag manifold m = g/k with b2(m) = 1 and r = 4, corresponding 132 s. anastassiou, i. chrysikos to m∗ := e8 /u(1) × su(3) × so(10). after multiplication with the positive term 60x21x 2 2x3x4, the system (3.4) of hrf equation turns into the following polynomial system: ẋ1 = x1x2 ( − 5x4x31 + x 2 4x1x2 −x 3 1x2 + 5x4x1x 2 2 − 60x4x1x2x3 + 10x4x 2 2x3 + 5x4x1x 2 3 + x1x2x 2 3 ) , ẋ2 = 2x2x4 ( 4x31x2 − 4x1x 3 2 + x4x 2 1x3 − 22x 2 1x2x3 − 4x 3 2x3 + 4x1x2x 2 3 ) , ẋ3 = 3x1x2x3 ( 5x4x 2 1 + x 2 4x2 − 20x4x1x2 + x 2 1x2 + 5x4x 2 2 − 5x4x23 −x2x 2 3 ) , ẋ4 = −2x1x4 ( 8x24x 2 2 − 8x 2 1x 2 2 + 5x 2 4x1x3 + 20x1x 2 2x3 − 8x 2 2x 2 3 ) . note that the right-hand side of the equations above are all homogeneous polynomials of degree 6. hence we can apply proposition 2.12 to study the behaviour of this system at infinity of m g. in the u chart and by setting x4 = 0, we obtain ẋ1 = −x1 ( −x21 + x 2 1x 2 2 − 13x1x3 + 13x 3 1x3 + 44x1x2x3 − 60x 2 1x2x3 + 18x31x2x3 − 3x1x 2 2x3 + x 2 1x 2 3 − 2x2x 2 3 ) , ẋ2 = −2x1x2 ( − 2x1 + 2x1x22 − 10x3 + 30x1x3 − 5x 2 1x3 − 30x1x2x3 + 5x21x2x3 + 10x 2 2x3 −x1x 2 3 ) , ẋ3 = −x3 ( − 17x21 + 40x 2 1x2 − 15x 2 1x 2 2 − 5x1x3 + 5x 3 1x3 − 60x 2 1x2x3 + 10x31x2x3 + 5x1x 2 2x3 + 17x 2 1x 2 3 + 10x2x 2 3 ) , ẋ4 = 0 . again, last equation confirms that infinity remains invariant under the flow of the system. solving the system { ẋ1 = ẋ2 = ẋ3 = 0 } , we get exactly three fixed points, given by (2, 3, 4) , (1.09705, 0.770347, 1.29696) , (1.15607, 1.01783, 0.214618) (0.649612, 1.10943, 1.06103) , (0.763357, 1.00902, 0.191009) . these solutions give us, through the central projection f of the u chart on r4≥, the five fixed points e1,e2,e3,e4,e5. the fixed point e1 = (1, 2, 3, 4) corresponds to the unique invariant kähler-einstein metric. consider the jacobian matrix ancient solutions of the homogeneous ricci flow 133 corresponding to the system in u chart, that is jac :=   ∂f1 ∂x1 ∂f1 ∂x2 ∂f1 ∂x3 ∂f2 ∂x1 ∂f2 ∂x2 ∂f2 ∂x3 ∂f3 ∂x1 ∂f3 ∂x2 ∂f3 ∂x3   , where f1 := −x1 ( −x21 + x 2 1x 2 2 − 13x1x3 + 13x 3 1x3 + 44x1x2x3 − 60x 2 1x2x3 + 18x31x2x3 − 3x1x 2 2x3 + x 2 1x 2 3 − 2x2x 2 3 ) , f2 := −2x1x2 ( − 2x1 + 2x1x22 − 10x3 + 30x1x3 − 5x 2 1x3 − 30x1x2x3 + 5x21x2x3 + 10x 2 2x3 −x1x 2 3 ) , f3 := −x3 ( − 17x21 + 40x 2 1x2 − 15x 2 1x 2 2 − 5x1x3 + 5x 3 1x3 − 60x 2 1x2x3 + 10x31x2x3 + 5x1x 2 2x3 + 17x 2 1x 2 3 + 10x2x 2 3 ) . calculating the jacobian matrix at the corresponding fixed point (2,3,4), we find it to be equal to: jac(2,3,4) =   −1600 368 32960 −1248 192 320 1760 −1696   . this has 3 negative eigenvalues, thus we conclude that the fixed point e1 possesses a 3-dimensional stable manifold, contained in the infinity of m g, while the straight line joining e1 with the origin of m g corresponds to its 1-dimensional unstable manifold. computing the jacobian matrix at the other fixed points and calculating their eigenvalues, we conclude that all the other fixed points, corresponding to non-kähler, non-isometric invariant einstein metrics (see [8]), have a 2-dimensional stable manifold, located in the infinity of m g, and a 2dimensional unstable manifold, one direction of which is the straight line in m g tending towards the origin, with the exception of fixed point e5. the jacobian matrix there becomes jac(0.763357,1.00902,0.191009) =   1.33763 −1.13506 0.2514940.0597364 −4.80309 0.447799 −0.13328 −0.320307 3.98402   , 134 s. anastassiou, i. chrysikos which has one negative eigenvalue and two positive ones. thus, e5 possesses a 1-dimensional stable manifold, contained in the infinity of m g, and a 3dimensional unstable manifold, one direction of which is the straight line emanating from e5 and tending to the origin of m g. invariant lines, corresponding to non-collapsed ancient solutions gi(t) = (1 − λ2it) · ei, t ∈ (−∞, 12λi ), can be found as in the previous cases, confirming once again that their ω-limit sets are equal to {0}, while the α-limit set is the corresponding fixed point at infinity of m g. this proves our claims. the ricci flow equations on the rest three homogeneous spaces of that type can be treated similarly. case r = 5. after multiplication with the positive term 60x21x 2 2x3x4x5, system (3.4) turns into the following polynomial system: ẋ1 = −x1x2 ( x31x2x3 −x1x2x3x 2 4 + 3x 3 1x2x5 − 3x1x2x 2 3x5 + 6x31x4x5 − 6x1x 2 2x4x5 + +60x1x2x3x4x5 − 9x2 2x3x4x5 − 6x1x23x4x5 − 3x1x2x 2 4x5 −x1x2x3x 2 5 ) , ẋ2 = 2x2x4 ( −x21x 3 2 + x 2 1x2x 2 3 + 4x 3 1x2x5 − 4x1x 3 2x5 − 24x 2 1x2x3x5 − 3x32x3x5 + 4x1x2x 2 3x5 + 2x 2 1x3x4x5 + x 2 1x2x 2 5 ) , ẋ3 = 3x1x2x3 ( x1x 2 2x4 −x1x 2 3x4 + 2x 2 1x2x5 − 2x2x 2 3x5 + 4x 2 1x4x5 − 20x1x2x4x5 + 4x22x4x5 − 4x 2 3x4x5 + 2x2x 2 4x5 + x1x4x 2 5 ) , ẋ4 = −2x1x4 ( − 2x21x 2 2x3 + 2x 2 2x3x 2 4 − 6x 2 1x 2 2x5 + 24x1x 2 2x3x5 − 6x22x 2 3x5 + 6x 2 2x 2 4x5 + 3x1x3x 2 4x5 − 2x 2 2x3x 2 5 ) , ẋ5 = 5x1x2x5 ( 2x21x2x3 + 3x1x 2 2x4 − 12x1x2x3x4 + 3x1x 2 3x4 + 2x2x3x 2 4 − 2x2x3x 2 5 − 3x1x4x 2 5 ) . note that the right-hand side consists of homogeneous polynomials of degree 7 and we can use proposition 2.12 to study the behaviour of this system at infinity of m g. using the expressions given above, the system at infinity, written in the u chart and setting x5 = 0, reads as follows: ẋ1 = −x1 ( −x21x2 + 2x 3 1x3 − 2x1x 2 2x3 + x 2 1x2x 2 3 − 3x 2 1x4 + 3x 2 1x 2 2x4 − 14x1x3x4 + 14x31x3x4 + 48x1x2x3x4 − 60x 2 1x2x3x4 + 15x 3 1x2x3x4 − 2x1x22x3x4 + 3x 2 1x 2 3x4 − 4x2x 2 3x4 + x 2 1x2x 2 4 − 2x1x3x 2 4 ) , ancient solutions of the homogeneous ricci flow 135 ẋ2 = −x1x2 ( −x1x2 − 3x21x3 + 3x 2 2x3 + x1x2x 2 3 − 9x1x4 + 9x1x 2 2x4 − 18x3x4 + 60x1x3x4 − 6x21x3x4 − 60x1x2x3x4 + 9x 2 1x2x3x4 + 18x22x3x4 − 3x1x 2 3x4 + x1x2x 2 4 − 3x3x 2 4 ) , ẋ3 = −x3 ( − 5x21x2 + 5x 2 1x2x 2 3 − 15x 2 1x4 + 48x 2 1x2x4 − 9x 2 1x 2 2x4 − 6x1x3x4 + 6x31x3x4 − 60x 2 1x2x3x4 + 9x 3 1x2x3x4 + 6x1x 2 2x3x4 + 15x 2 1x 2 3x4 + 6x2x 2 3x4 − 3x 2 1x2x 2 4 ) , ẋ4 = −x1x4 ( − 11x1x2 − 15x21x3 + 60x1x2x3 − 15x 2 2x3 − 9x1x2x 2 3 − 3x1x4 + 3x1x22x4 − 6x3x4 + 6x 2 1x3x4 − 60x1x2x3x4 + 9x 2 1x2x3x4 + 6x22x3x4 + 3x1x 2 3x4 + 11x1x2x 2 4 + 15x3x 2 4 ) , ẋ5 = 0 . similarly with before, last equation confirms that infinity remains invariant under the flow of the system. now, by solving the system {ẋ1 = ẋ2 = ẋ3 = ẋ4 = 0}, we get exactly six fixed points at the infinity of m g, given by: (2, 3, 4, 5) , (0.599785, 1.08371, 0.901823, 1.22291) , (1.02137, 0.546007, 1.05352, 1.10879) , (1.08294, 1.04088, 0.532615, 1.10351) , (0.720713, 1.02546, 0.475234, 1.07095) , (1.03732, 1.04718, 1.03082, 0.29862) . as before, these fixed points induce via the central projection f the explicit presentations ei, i = 1, . . . , 6. the fixed point represented by e1 = (1, 2, 3, 4, 5) corresponds to the unique invariant kähler-einstein metric on m = e8 /u(1) × su(4) × su(5) and eigenvalues calculations show that it possesses a 4-dimensional stable manifold, contained in the infinity of m g, and a 1-dimensional unstable manifold which coincide with a straight tending to the origin. the other three fixed points, corresponding to non-kähler, nonisometric invariant einstein metrics (see [24]), have a 3-dimensional stable manifold, located in the infinity of m g, and a 2-dimensional unstable manifold, one direction of which is the straight line in m g tending to the origin, with the exception of the fixed point e5, which possesses a 2-dimensional stable manifold, contained in the infinity of m g, and a 3-dimensional unstable manifold. the ancient solutions gi(t) = (1 − 2λit) · ei, i = 1, . . . , 6 are defined on the open interval (−∞, 1/2λi), where the corresponding einstein constant 136 s. anastassiou, i. chrysikos λi is given by λ1 = 11/60 for e1 = (1, 2, 3, 4, 5) , λ2 = 0.37877 for e2 = (1, 0.599785, 1.08371, 0.901823, 1.22291) , λ3 = 0.365507 for e3 = (1, 1.02137, 0.546007, 1.05352, 1.10879) , λ4 = 0.339394 for e4 = (1, 1.08294, 1.04088, 0.532615, 1.10351) , λ5 = 0.386982 for e5 = (1, 0.720713, 1.02546, 0.475234, 1.07095) , λ6 = 0.337271 for e6 = (1, 1.03732, 1.04718, 1.03082, 0.29862) . using these constants and by taking limits, as above, we obtain the rest claims for r = 5. case r = 6. to reduce the system (3.4) to a polynomial dynamical system, a short computation shows that we must multiply with the positive term 60x21x 2 2x 2 3x4x5x6. this gives the following: ẋ1 = −x1x2x3 ( x31x2x3x4 −x1x2x3x4x 2 5 + 2x 3 1x2x3x6 − 2x1x2x3x 2 4x6 + 4x31x2x5x6 − 4x1x2x 2 3x5x6 + 6x 3 1x4x5x6 − 6x1x 2 2x4x5x6 + 60x1x2x3x4x5x6 − 8x22x3x4x5x6 − 6x1x 2 3x4x5x6 − 4x1x2x 2 4x5x6 − 2x1x2x3x25x6 −x1x2x3x4x 2 6 ) , ẋ2 = 2x2x3 ( −x21x 3 2x3x5 + x 2 1x2x3x 2 4x5 −x 2 1x 3 2x4x6 + x 2 1x2x 2 3x4x6 + 3x31x2x4x5x6 − 3x1x 3 2x4x5x6 − 26x 2 1x2x3x4x5x6 − 2x 3 2x3x4x5x6 + 3x1x2x 2 3x4x5x6 + 3x 2 1x3x 2 4x5x6 + x 2 1x2x4x 2 5x6 + x 2 1x2x3x5x 2 6 ) , ẋ3 = 3x1x2x3x6 ( x1x 2 2x3x4 −x1x 3 3x4 + 2x 2 1x2x3x5 − 2x2x 3 3x5 + 3x21x3x4x5 − 20x1x2x3x4x5 + 3x 2 2x3x4x5 − 3x 3 3x4x5 + 2x2x3x 2 4x5 + x1x3x4x 2 5 + x1x2x4x5x6 ) , ẋ4 = 2x1x3x4 ( 2x1x 3 2x3x5 − 2x1x2x3x 2 4x5 + 2x 2 1x 2 2x3x6 − 2x 2 2x3x 2 4x6 + 4x21x 2 2x5x6 − 24x1x 2 2x3x5x6 + 4x 2 2x 2 3x5x6 − 4x 2 2x 2 4x5x6 − 3x1x3x24x5x6 + 2x 2 2x3x 2 5x6 + 2x1x2x3x5x 2 6 ) , ancient solutions of the homogeneous ricci flow 137 ẋ5 = 5x1x2x3x5 ( x21x2x3x4 −x2x3x4x 2 5 + 2x 2 1x2x3x6 + 2x1x 2 2x4x6 − 12x1x2x3x4x6 + 2x1x23x4x6 + 2x2x3x 2 4x6 − 2x2x3x 2 5x6 − 2x1x4x25x6 + x2x3x4x 2 6 ) , ẋ6 = 6x1x2x6 ( x21x2x 2 3x4 + 2x1x 2 2x 2 3x5 − 8x1x2x 2 3x4x5 + 2x1x 2 3x 2 4x5 + x2x 2 3x4x 2 5 −x2x 2 3x4x 2 6 − 2x1x 2 3x5x 2 6 −x1x2x4x5x 2 6 ) . note that the right-hand side consists of homogeneous polynomials of degree 9. hence again we can apply the poincaré compactification procedure to study the behaviour of this system at infinity of m g. using the expressions given above, the system at infinity, written in the u chart and setting x6 = 0, reads as follows: ẋ1 = −x1x2 ( −x21x2x3 + 2x 3 1x2x4 − 2x1x2x 2 3x4 + x 2 1x2x3x 2 4 − 2x 2 1x2x5 + 2x31x3x5 − 2x1x 2 2x3x5 + 2x 2 1x2x 2 3x5 − 4x 2 1x4x5 + 4x 2 1x 2 2x4x5 − 12x1x3x4x5 + 12x31x3x4x5 + 52x1x2x3x4x5 − 60x 2 1x2x3x4x5 + 12x31x2x3x4x5 + 4x 2 1x 2 3x4x5 − 6x2x 2 3x4x5 + 2x 2 1x2x 2 4x5 − 2x1x3x24x5 + x 2 1x2x3x 2 5 − 2x1x2x4x 2 5 ) , ẋ2 = −x1x2 ( −x1x22x3 + x1x 2 2x3x 2 4 − 2x1x 2 2x5 − 3x 2 1x2x3x5 + 3x 3 2x3x5 + 2x1x 2 2x 2 3x5 − 10x1x2x4x5 + 10x1x 3 2x4x5 −15x2x3x4x5 + 60x1x2x3x4x5 − 3x21x2x3x4x5 − 60x1x 2 2x3x4x5 + 8x 2 1x 2 2x3x4x5 + 15x 3 2x3x4x5 − 2x1x2x23x4x5 + 2x1x 2 2x 2 4x5 − 3x2x3x 2 4x5 + x1x 2 2x3x 2 5 − 3x1x3x4x 2 5 ) , ẋ3 = −x2x3 ( −x21x2x3 − 4x 3 1x2x4 + 4x1x2x 2 3x4 + x 2 1x2x3x 2 4 − 6x 2 1x2x5 + 6x21x2x 2 3x5 − 12x 2 1x4x5 + 48x 2 1x2x4x5 − 4x 2 1x 2 2x4x5 − 6x1x3x4x5 + 6x31x3x4x5 − 60x 2 1x2x3x4x5 + 8x 3 1x2x3x4x5 + 6x1x 2 2x3x4x5 + 12x21x 2 3x4x5 + 6x2x 2 3x4x5 − 2x 2 1x2x 2 4x5 + x 2 1x2x3x 2 5 − 4x1x2x4x 2 5 ) , ẋ4 = −2x1x2x4 ( − 3x1x2x3 + 3x1x2x3x24 − 6x1x2x5 − 5x 2 1x3x5 + 30x1x2x3x5 − 5x22x3x5 − 4x1x2x 2 3x5 − 2x1x4x5 + 2x1x 2 2x4x5 − 3x3x4x5 + 3x21x3x4x5 − 30x1x2x3x4x5 + 4x 2 1x2x3x4x5 + 3x 2 2x3x4x5 + 2x1x 2 3x4x5 + 6x1x2x 2 4x5 + 5x3x 2 4x5 − 2x1x2x3x 2 5 ) , 138 s. anastassiou, i. chrysikos ẋ5 = −x1x5 ( − 7x1x22x3 − 12x 2 1x 2 2x4 + 48x1x 2 2x3x4 − 12x 2 2x 2 3x4 − 5x1x22x3x 2 4 − 2x1x 2 2x5 + 2x1x 2 2x 2 3x5 − 4x1x2x4x5 + 4x1x 3 2x4x5 − 6x2x3x4x5 + 6x21x2x3x4x5 − 60x1x 2 2x3x4x5 + 8x 2 1x 2 2x3x4x5 + 6x32x3x4x5 + 4x1x2x 2 3x4x5 + 2x1x 2 2x 2 4x5 + 7x1x 2 2x3x 2 5 + 12x22x4x 2 5 + 6x1x3x4x 2 5 ) , ẋ6 = 0 . by the last equation one deduces that the infinity of m g remains invariant under the flow of the system. solving now the system {ẋ1 = ẋ2 = ẋ3 = ẋ4 = ẋ5 = 0}, we get exactly five fixed points given by: (2, 3, 4, 5, 6) , (0.823084, 1.1467, 1.17377, 1.42664, 1.46519) , (0.986536, 0.636844, 1.06853, 1.13323, 0.921127) , (0.90422, 0.778283, 0.927483, 1.03408, 0.359949) , (0.954875, 0.965321, 1.00534, 0.290091, 1.01965) . these solutions, projected through the central projection f, induce the fixed points ei at infinity of m g, namely e1 = (1, 2, 3, 4, 5, 6) , e2 = (1, 0.823084, 1.1467, 1.17377, 1.42664, 1.46519) , e3 = (1, 0.986536, 0.636844, 1.06853, 1.13323, 0.921127) , e4 = (1, 0.90422, 0.778283, 0.927483, 1.03408, 0.359949) , e5 = (1, 0.954875, 0.965321, 1.00534, 0.290091, 1.01965) . these points are in bijective correspondence with non-isometric invariant einstein metrics on m = g/k = e8 /u(1) × su(2) × su(3) × su(5) (see [24]), and determine the non-collapsed ancient solutions gi(t) = (1−2λit)·ei, where the corresponding einstein constant λi has the form λ1 = 3/20 , λ2 = 0.313933 , λ3 = 0.348603 , λ4 = 0.367518 , λ5 = 0.349296 . all the other conclusions are obtained in an analogous way as before. this completes the proof. remark 3.2. for r ≥ 3 a statement of ω-limits of general solutions of (3.4), as in the assertion (4) of theorem 3.1, is not presented, since for these cases a lyapunov function is hard to be computed. ancient solutions of the homogeneous ricci flow 139 let us now take a view of the limit behaviour of the scalar curvature for the solutions gi(t) given in theorem 3.1, and verify the statements of proposition 2.4. we do this by treating examples with r = 2, 3. example 3.3. let m = g/k be a flag manifold with r = 2. then, an application of (3.2) shows that both scal(gi(t)) are positive hyperbolas, given by scal(g1(t)) = (d1 + d2)(d1 + 2d2) 2(d1 + 4d2) − 2t(d1 + 2d2) , scal(g2(t)) = (d1 + d2)(d 2 1 + 6d1d2 + 4d 2 2) 2 ( (d21 + 6d1d2 + 8d 2 2) − t(d 2 1 + 6d1d2 + 4d 2 2) ) , respectively. therefore, they both are increasing in ( −∞, 1 2λi ) , which is the open interval which are defined, i.e., scal′(g1(t)) and scal ′(g2(t)) are strictly positive. the limit of scal(gi(t)) as t → 12λi must be considered only from below, and it is direct to check that lim t→ 1 2λi scal(gi(t)) = +∞ , lim t→−∞ scal(gi(t)) = 0 , ∀ i = 1, 2 . hence, scal(gi(t)) > 0, for any t ∈ ( −∞, 1 2λi ) and for any i = 1, 2, as it should be for ancient solutions in combination with the non-existence of ricci flat metrics ([37, 38]). let us present the graphs of scal(g1(t)) and scal(g2(t)) for the flag spaces g2 /u(2) and f4 /sp(3) × u(1), both with r = 2. we list all the related details below, together with the graphs of scal(gi(t)) for the corresponding intervals of definition ( −∞, 1 2λi ) (see figure 1 and 2, for i = 1, 2, respectively). m = g/k d1 d2 1/2λ1 1/2λ2 scal(g1(t)) scal(g2(t)) g2 /u(2) 8 2 4/3 12/11 120 32 − 24t 1760 384 − 362t f4 /sp(3) ×u(1) 28 2 9/8 72/71 960 72 − 64t 34080 2304 − 2272t the graphs now of the ricci components of the solutions gj(t), which we denote by ricij(t) := rici(gj(t)) , 1 ≤ i ≤ r , 1 ≤ j ≤ n , 140 s. anastassiou, i. chrysikos where again n represents the number of fixed points of hrf at infinity of m g, are very similar. for example, for any flag manifold m = g/k with r = 2 and for the first ancient solution g1(t) passing through the invariant kähler-einstein metric g1(0) = gke, we compute ric11(t) = ric1(g1(t)) = ric21(t) = ric2(g1(t)) = d1 + 2d2 2(d1 + 4d2) − 2(d1 + 2d2)t . for the solution g2(t) passing from the non-kähler invariant einstein metric g2(0) = ge we obtain ric12(t) = ric1(g2(t)) = ric22(t) = ric2(g2(t)) = d21 + 6d1d2 + 4d 2 2 2(d21 + 6d1d2 + 8d 2 2) − 2(4d 2 2 + d 2 1 + 6d1d2)t . note that the equalities ric11(t) = ric21(t) and ric12(t) = ric22(t) occur since gi(t) (i = 1, 2) are both 1-parameter families of invariant einstein metrics on m = g/k (as we mentioned in section 2). for instance, for g2 /u2 with r = 2, the above formulas reduce to ric11(t) = ric21(t) = 12 32 − 24t , ric12(t) = ric22(t) = 88 192 − 176t , with ric11(0) = ric21(0) = λ1 = 3/8 and ric12(0) = ric22(0) = λ2 = 11/24, respectively. the corresponding graphs are given in figure 3. t= 4 3 10 8 6 4 2 t 1 2 3 4 5 scal(g1(t)) t= 9 8 10 8 6 4 2 t 5 10 15 scal(g1(t)) figure 1: the graph of scal(g1(t)) for g2 /u(2) (left) and f4 /sp(3) × u(1) (right). ancient solutions of the homogeneous ricci flow 141 t= 12 11 10 8 6 4 2 t 1 2 3 4 5 scal(g2(t)) t= 72 71 10 8 6 4 2 t 5 10 15 scal(g2(t)) figure 2: the graph of scal(g2(t)) for g2 /u(2) (left) and f4 /sp(3) × u(1) (right). t= 4 3 ric11(0)= �1 10 8 6 4 2 t 0.1 0.2 0.3 0.4 0.5 ric11(t) t= 12 11 ric22(0)= �2 10 8 6 4 2 t 0.1 0.2 0.3 0.4 0.5 ric22(t) figure 3: the graphs of ric11(t) = ric21(t) and ric12(t) = ric22(t) for g2 /u(2) with m = m1 ⊕m2. example 3.4. let m = g/k be a flag manifold with b2(m) = 1 and r = 3. let us describe the asymptotic properties of the scalar curvature scal(g1(t)), related to the solution g1(t) = (1 − 2λ1t) · e1 only, where e1 = (1, 2, 3) is the fixed point corresponding to the invariant kähler-einstein metric g1(0) ≡ gke. by (3.3) we see that scal(g1(t)) is the positive hyperbola given by scal(g1(t)) = (d1 + d2 + d3)(d1 + 2d2 + 3d3) 2(d1 + 4d2 + 9d3) + 2t(d1 + 4d2 + 6d3) . thus, scal(g1(t)) increases on the open interval (−∞, 12λ1 ), where g1(t) is dedined. note that the value scal(g(0)) = (d1 + d2 + d3)(d1 + 2d2 + 3d3) 2(d1 + 4d2 + 9d3) equals to the scalar curvature of the kähler-einstein metric g1(0). the limit of scal(g1(t)) as t → 12λ1 must be considered only from below, and it follows 142 s. anastassiou, i. chrysikos that lim t→ 1 2λ1 scal(g1(t)) = +∞ , lim t→−∞ scal(g1(t)) = 0 . for example for g2 /u2 and r = 3, we compute λ1 = 5/24, so scal(g1(t)) = 25 12 − 5t , t ∈ ( −∞, 12 5 ) , scal(g1(0)) = 25 12 and the graph of scal(g1(t)) is given by scal(g1(0)) t= 12 5 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[55] m. wang, w. ziller, existence and nonexcistence of homogeneous einstein metrics, invent. math. 84 (1986), 177 – 194. arxiv:math/0211159 arxiv:math/0303109v1 preliminaries homogeneous ricci flow. flag manifolds. the homogeneous ricci flow on flag manifolds invariant ancient solutions. the poincaré compactification procedure. global study of hrf on flag spaces m=g/k with b2(m)=1 flag manifolds m=g/k with b2(m)=1. the main theorem. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 36, num. 2 (2021), 241 – 278 doi:10.17398/2605-5686.36.2.241 available online october 21, 2021 support and separation properties of convex sets in finite dimension v. soltan department of mathematical sciences, george mason university 4400 university drive, fairfax, va 22030, usa vsoltan@gmu.edu received august 21, 2021 presented by h. martini accepted september 20, 2021 abstract: this is a survey on support and separation properties of convex sets in the n-dimensional euclidean space. it contains a detailed account of existing results, given either chronologically or in related groups, and exhibits them in a uniform way, including terminology and notation. we first discuss classical minkowski’s theorems on support and separation of convex bodies, and next describe various generalizations of these results to the case of arbitrary convex sets, which concern bounding and asymptotic hyperplanes, and various types of separation by hyperplanes, slabs, and complementary convex sets. key words: convex, cone, bound, hyperplane, support, separation. msc (2020): 52a20. contents 1 introduction 242 2 preliminaries 243 3 minkowski’s theorems 247 3.1 support hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . 247 3.2 bounding and separating hyperplanes . . . . . . . . . . . . . . 250 3.3 sufficient conditions for convexity of solid sets . . . . . . . . . . 252 4 supports and bounds of convex sets 254 4.1 support hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . 254 4.2 bounding hyperplanes and halfspaces . . . . . . . . . . . . . . . 256 5 separation of convex sets 260 5.1 classification of separating hyperplanes . . . . . . . . . . . . . 260 issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.36.2.241 mailto:vsoltan@gmu.edu https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 242 v. soltan 5.2 geometric conditions on hyperplane separation . . . . . . . . . 264 5.3 sharp separation of convex cones . . . . . . . . . . . . . . . . . 267 5.4 penumbras and separation . . . . . . . . . . . . . . . . . . . . . 269 5.5 hemispaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 1. introduction support and separation properties of convex sets are among the core topics of convexity theory. introduced and studied by the prominent mathematician hermann minkowski (see [63] and [64]), they became useful tools of convex geometry. further development of minkowski’s ideas occurred at the beginning of 20th century and was summarized in the monograph of bonnesen and fenchel [8]. a rapid growth of linear analysis in the first half of 20th century led to numerous generalizations of minkowski’s contribution to the case of infinitedimensional vector spaces and made this topic an organic part of the discipline. for instance, the survey of klee [51] from 1969 describes existing results on support and separation properties of convex sets in vector spaces of any dimension, with sporadic divisions into finiteand infinite-dimensional cases. however, besides obvious similarities between these cases, they display nowadays different goals: while finite-dimensional convexity deals, predominantly, with the properties of convex sets, a big part of similar results in linear analysis became a tool for various classifications of topological and normed spaces. a new wave of interest towards finite-dimensional convexity occurred in the second half of the 20th century due to advancements in linear programming, with polyhedral sets considered as geometric interpretation of solutions sets of systems of linear inequalities (see, e.g., dantzig [19] and černikov [18]), and later in convex analysis, optimization theory, and polyhedral geometry. various books on convex analysis (see, e.g., güler [37], panik [69], and rockafellar [72]) contain separate chapters on support and separation properties of convex sets, illustrating a continuous development of these topics. these books and numerous articles in the field also underline a shift of interest from the study of convex bodies toward arbitrary convex sets, possibly unbounded or nonclosed. despite a steady progress in research, no comprehensive survey on support and separation properties of convex sets in finite dimension was published during the last five decades. the present paper aims to fill in (at least partly) this support and separation properties 243 gap and overview new trends and results. it contains an account of existing facts, given either chronologically or in related groups, exhibiting them in a uniform way, including terminology and notation. we do not consider here algorithmic and computational aspects of the theory on support and separation of convex sets, which deserve their own surveys. following the necessary preliminaries, given in section 2, the main text is divided into three parts. section 3 describes classical minkowski’s theorems on support and separation properties of convex bodies and summarizes immediate contributions of his colleagues. contemporary approach to the study of these topics is given in sections 4 and 5. these sections cover existing results on support, bounding and asymptotic planes of arbitrary convex sets, and various types of separation of convex sets by hyperplanes, slabs, and complementary convex halfspaces. it is interesting to compare methods of research on support and separation of convex sets in classical and contemporary periods. since the main results of the classical period are related to full-dimensional compact sets, the methods of their proofs are predominantly based on compactness arguments and basic topology of open sets in finite dimension. contemporary results in this field deal with arbitrary convex sets, possibly unbounded and having intermediate dimension. consequently, their proofs employ the concept of relative interior and extensively use various types of cones associated with convex sets. 2. preliminaries this section describes necessary notation, terminology, and results on convex sets in the n-dimensional euclidean space rn (see, e.g., [76] for details). the elements of rn are called vectors (or points), and o stands for the zero vector of rn. an r-dimensional plane l in rn, where 0 ≤ r ≤ n, is a translate, l = a + s, a ∈ rn, of a suitable r-dimensional subspace s of rn, called the direction space of l. a hyperplane is a plane of dimension n − 1; it can be described as h = {x ∈ rn : x·e = γ}, e 6= o, γ ∈ r, (2.1) where x·e means the dot product of vectors x and e. nonzero multiples λe of the vector e in (2.1) are called normal vectors of h. the direction space of the hyperplane (2.1) is the (n− 1)-dimensional subspace given by s = {x ∈ rn : x·e = 0}, e 6= o. (2.2) 244 v. soltan every hyperplane of the form (2.1) determines the opposite closed halfspaces v1 = {x ∈ rn : x·e ≤ γ} and v2 = {x ∈ rn : x·e ≥ γ} and the pair of opposite open halfspaces w1 = {x ∈ rn : x·e < γ} and w2 = {x ∈ rn : x·e > γ}. the closed and open (line) segments with distinct endpoints u and v in rn and the halfline through v with endpoint u are denoted by [u,v], (u,v), and [u,v〉, respectively. the norm (or the length) of a vector x ∈ rn is denoted ‖x‖. given a point a ∈ rn and a scalar ρ > 0, the sphere and balls (closed and open) of radius ρ and center a are denoted sρ(a), bρ(a), and uρ(a), respectively. the topological interior, closure, and boundary of a nonempty set x ⊂ rn are given by int x, cl x, and bd x, respectively. the open ρ-neighborhood of x, denoted uρ(x), is the union of all open balls uρ(x) of radius ρ > 0 centered at x ∈ x. nonempty sets x1 and x2 in rn are called strongly disjoint provided uρ(x1)∩uρ(x2) = ∅ for a suitable ρ > 0; the latter occurs if and only if the inf -distance δ(x1,x2), defined by δ(x1,x2) = inf{‖x1 −x2‖ : x1 ∈ x1,x2 ∈ x2}, is positive. for a nonempty set x ⊂ rn, the notations span x and x⊥, stand, respectively, for the span and orthogonal complement of x. the set x is called proper if ∅ 6= x 6= rn. the affine span of x, denoted aff x, is the intersection of all planes containing x, and dim x is defined as the dimension of the plane aff x. also, the direction space of x is defined by dir x = aff x − aff x, and the orthogonal space of x by ort x = (dir x)⊥ (generally, ort x 6= x⊥). given nonempty sets x,y ⊂ rn and a scalar λ, we let x + y = {x + y : x ∈ x,y ∈ y}, λx = {λx : x ∈ x}. nonempty sets x and y in rn are called (directly) homothetic provided x = z + λy for suitable z ∈ rn and λ > 0. in what follows, k means a convex set in rn. to avoid trivial cases, we will be assuming that all convex sets involved are nonempty. a point x of a convex set k ⊂ rn is said to be relatively interior to k provided there is a scalar ρ = ρ(x) > 0 such that aff k ∩ uρ(x) ⊂ k. the set of all relatively support and separation properties 245 interior points of k is called the relative interior of k and is denoted rint k. the set rint k is nonempty and convex; furthermore, cl k = cl (rint k). the difference between topological and relative interiors can be illustrated by the following example: if k is a unit circular disk of the coordinate xy-plane of r3, then int k = ∅, while rint k is the interior of this disk. the relative boundary of a convex set k ⊂ rn, denoted rbd k, is defined by rbd k = cl k \ rint k. it is known that rbd k 6= ∅ if and only if k is not a plane. a convex body in rn is a compact convex set with nonempty interior, a (convex) polyhedron is the intersection of finitely many closed halfspaces, and a polytope (bounded polyhedron) is the convex hull of finitely many points. a contemporary approach to the study of support and separation properties of convex sets deals with a family of various associated cones. we recall that a nonempty set c in rn is called a cone with apex a ∈ rn if a + λ(x − a) ∈ c whenever λ ≥ 0 and x ∈ c. (obviously, this definition implies that a ∈ c, although a stronger condition λ > 0 can be beneficial; see, e.g., [57].) the cone c is called convex if it is a convex set. the apex set of a convex cone c ⊂ rn, denoted ap c, is the set of all apices of c. if a is an apex of a convex cone c, then ap c = c ∩ (2a−c). natural generalizations of cones are given by the sets of the form k = b + c, where b is a compact convex set and c is a convex cone; these sets are called m-decomposable if c is closed, and m-predecomposable otherwise (see goberna et al. [35] and iusemi et al. [41], respectively). for a convex set k ⊂ rn and a point a ∈ rn, the generated cone ca(k) with apex a is defined by ca(k) = {a + λ(x−a) : x ∈ k, λ ≥ 0}. both sets ca(k) and cl ca(k) are convex cones with apex a (we observe that the cone ca(k) may be nonclosed even if k is closed). furthermore, ca(k) is a plane if and only if a ∈ rint k. in particular, ca(k) = rn if and only if a ∈ int k. the recession cone of a convex set k ⊂ rn is defined by rec k = {e ∈ rn : x + λe ∈ k whenever x ∈ k and λ ≥ 0}. if k is closed, then rec k is a closed convex cone with apex o, and rec k 6= {o} if and only if k is unbounded. the lineality space of k is the subspace defined by lin k = rec k ∩ (−rec k). in a standard way, the (negative) polar cone of a convex set k ⊂ rn is defined by k◦ = {e ∈ rn : x·e ≤ 0 for all x ∈ k}. 246 v. soltan the set k◦ is a closed convex cone with apex o. furthermore, 1. k◦ = {o} if and only if o ∈ int k, and k◦ is a subspace if and only if o ∈ rint k. 2. k◦ = (co(k)) ◦, (k◦)◦ = cl co(k), lin k ◦ = k⊥. 3. if c ⊂ rn is a closed convex cone with apex o, then a nonzero vector e ∈ rn belongs to rint c◦ if and only if x·e < 0 for all x ∈ c \ lin c. o k◦ k co(k) figure 1: the polar cone of a convex set k. given a point z in the closure of a convex set k ⊂ rn, the polar cone (k −z)◦ is often called the normal cone of k at z and is denoted nz(k); it consists of all vectors e ∈ rn such that z is the nearest to e + z point in cl k. o k −z nz(k) figure 2: the normal cone nz(k). the union of all normal cones nz(k), where z ∈ cl k, is denoted nor k and called the normal cone of k. the set nor k is a cone with apex o, which is not necessarily closed or convex. one more cone associated with k is its barrier cone, defined as bar k = {e ∈ rn : ∃γ = γ(e) ∈ r such that x·e ≤ γ for all x ∈ k}. support and separation properties 247 it is known that bar k is a convex (not necessarily closed) cone with apex o. generally, rint (rec (cl k))◦ ⊂ nor k ⊂ bar k ⊂ (rec (cl k))◦, which implies the equalities rec (cl k) = (nor k)◦ = (bar k)◦. for instance, if k = {(x,y) : y ≥ x2}, then k◦ = {(0,y) : y ≤ 0} and rec k = {(0,y) : y ≥ 0}, (rec k)◦ = {(x,y) : y ≤ 0}, nor k = bar k = {o}∪{(x,y) : y < 0}. a point z of a convex set k ⊂ rn is called an extreme point of k provided the set k \{z} is convex (equivalently, the equality z = (1 −λ)u + λv, where u,v ∈ k and 0 < λ < 1, is possible only if u = v = z). similarly, z is an exposed point of k provided there is a hyperplane h ⊂ rn satisfying the condition h ∩ k = {z}. the sets ext k and exp k of extreme and exposed points of a closed convex set k ⊂ rn have the following properties. 4. exp k 6= ∅ ⇔ ext k 6= ∅ ⇔ k is line-free (that is, k contains no line). 5. exp k ⊂ ext k ⊂ cl (exp k). 3. minkowski’s theorems 3.1. support hyperplanes the concept of support hyperplane is attributed to minkowski [63, § 8]: given a nonempty set x ⊂ rn, a hyperplane h ⊂ rn supports x if it contains at least one point of x and does not cut x (that is no two points of x belong, respectively, to the opposite open halfspaces determined by h). clearly, the above condition “h does not cut x” can be equivalently reformulated as “x entirely lies in a closed halfspace determined by h.” the following well-known result is due to minkowski (see [63, § 16] for all n ≥ 3 and [64, pp. 139 – 141] for n = 3). theorem 3.1. ([63, § 16]) every boundary point of a convex body k ⊂ rn belongs to a hyperplane supporting k. since general theory of convex sets, with credible geometric arguments, was still in rudimentary stage, minkowski’s proof of theorem 3.1 used an analytic description of convex bodies in terms of radial distances (see [63, § 1]). the radial distance (which later became known as the minkowski gauge function) from a point a to a point b in rn is a real-valued function s(a,b) satisfying the following conditions: 248 v. soltan 1. s(a,b) > 0 if a 6= b, and s(a,a) = 0; 2. if c = a + t(b−a), where t ≥ 0, then s(a,c) = ts(a,b); 3. s(a,c) ≤ s(a,b) + s(b,c) whenever a,b,c ∈ rn. the standard surface f and the standard body k of the radial distance function s(o,x) are defined, respectively, by f = {x ∈ rn : s(o,x) = 1} and k = {x ∈ rn : s(o,x) ≤ 1}. (3.1) minkowski observed (see [63, § 8]) that the standard body k from (3.1) is a compact convex set containing the origin o in its interior, and, conversely, that a certain translate of a convex body can be viewed as the standard body of a suitable radial distance function. using the properties of s(a,b), and not the convexity of k, minkowski showed that any point of f belongs to a hyperplane supporting f (and thus supporting k). his proof consists of two steps. step 1. given a point z ∈ f and a scalar t > 1, there is a hyperplane h(t) contained in rn \k and meeting the open interval (z,tz) (see figure 3). f k o z tz h(t) figure 3: a hyperplane h(t) in rn \k. step 2. given a sequence of scalars t1, t2, . . . ( > 1) tending to 1 and a respective sequence of hyperplanes h(t1),h(t2), . . . obtained in the above step 1 and described as β0(ti) + β1(ti)x1 + · · · + βn(ti)xn = 0, i ≥ 1, one can choose n+1 infinite subsequences βj(ti1 ),βj(ti2 ), . . . , 0 ≤ j ≤ n, which converge, respectively, to scalars β0,β1, . . . ,βn such that the limit hyperplane β0 + β1x1 + · · · + βnxn = 0 supports f at z. the analytic nature of minkowski’s proof prompted various mathematicians to consider more geometric approaches. for instance, carathéodory [16] proved theorem 3.1 based on the following auxiliary result, afterward widely used in convex geometry (see figure 4). support and separation properties 249 theorem 3.2. ([16]) let k ⊂ rn be a convex body, and u be a point outside k. if z is the nearest to u point in k, then the hyperplane h through z orthogonal to the segment [u,z] supports k such that u and k lie in the opposite closed halfspaces determined by h. z u k h figure 4: illustration to theorem 3.2. straszewicz [86, pp. 19 – 21] gave one more method of the proof of theorem 3.1 which uses an induction argument. steinitz (see [83, § 11] and [84, § 26]) showed that theorem 3.1 and theorem 3.2 hold for the case of any proper n-dimensional convex set k ⊂ rn, but his contribution went unnoticed. various proofs of theorem 3.1 for the case of dimensions 2 and 3 were provided at that time. for instance, brunn [10] uses a similar to straszewicz [86] method (and later another method in [11]), while blaschke [7, pp. 53 – 54] finds a suitable orthogonal projection of a convex body k ⊂ r3 on a plane and uses the support property of this projection in the plane. we observe that in all three sources [16, 63, 86], the resulting support hyperplane of a given convex body is obtained as the limit of a suitable converging sequence of hyperplanes. proofs of theorem 3.1 which do not employ limit procedures appeared much later. these can be found in the papers of favard [28, chapter 2] and botts [9], as given below. theorem 3.3. ([9]) if z is a boundary point of a compact convex set k ⊂ rn, and v is a point of the unit sphere s1(z) at a largest distance from k, then the hyperplane through z orthogonal to the segment [v,z] supports k. support properties of convex bodies are used for various classifications of their boundary points. for instance, a point z of a convex body k ⊂ rn is called regular if it belongs to a unique support hyperplane of k. it is wellknown that the set of regular points of a convex body k ⊂ rn is everywhere 250 v. soltan dense in bd k. historical references here are due to jensen [44] and bernstein [4, 5], (for the planar case), kakeya [45], fujiwara [31], and reidemeister [71] (for the 3-dimensional case), and mazur [62] (for the case of linear normed spaces of any dimension). z v k figure 5: illustration to theorem 3.3. botts [9] gave the following characteristic property of regular points: if k ⊂ rn is a convex body, z0 ∈ bd k, and h is a hyperplane supporting k at z0, then z0 is regular if and only if for every sequence of points z1,z2, . . . ∈ bd k \{z0} converging to z0, the sequence of numbers δ(z1,h) ‖z1 −z0‖ , δ(z2,h) ‖z2 −z0‖ , . . . tends to 0, where δ(zi,h) denotes the distance from zi to h. one more classification of boundary points derives from observing contact sets of a convex body k ⊂ rn and its support hyperplanes. for example, the equality k = cl (conv (exp k)), proved by straszewicz [87], implies that k has at least n + 1 exposed points, and that the set exp k is finite if and only if k is a polytope. 3.2. bounding and separating hyperplanes an important class of hyperplanes with respect to a given convex body was considered by carathéodory [17] and used by him to describe the closed convex hull of a compact set in rn. namely, a hyperplane h ⊂ rn is said to bound a convex body k ⊂ rn provided h ∩k = ∅. equivalently, h bounds k if k lies in one of the open halfspaces determined by h (see figure 6). theorem 3.4. ([17]) if z ∈ rn is an exterior point of a convex body k ⊂ rn, and u is the nearest to z point in k, then the hyperplane through z orthogonal to the segment [u,z] bounds k. support and separation properties 251 the following separation theorem of minkowski [64] originated a variety of related results. theorem 3.5. ([64, p. 141]) if k1 and k2 are convex bodies in r3 with disjoint interiors, then there is a plane h such that the sets int k1 and int k2 belong to the opposite open halfspaces determined by h. u z k h figure 6: bounding hyperplane of k through a given point z. h z1 z2 k1 k2 figure 7: separating hyperplane of convex bodies k1 and k2. minkowski’s proof of this theorem is divided into two cases. case 1: k1 ∩ k2 = ∅. if points z1 ∈ k1 and z2 ∈ k2 are at a minimum possible distance, and if h is a plane orthogonal to the segment [z1,z2] and passing through an interior point of this segment, then k1 and k2 are contained in the opposite open halfspaces determined by h (see figure 7). case 2: k1 ∩k2 6= ∅. assuming that o ∈ int k1, any smaller homothetic copy tk1 of k1, where 0 < t < 1, is disjoint from k2. by the above case 1, there is a plane h(t) such that tk1 and k2 belong to the opposite open halfspaces determined by h(t). given a sequence of scalars t1, t2, . . . in the interval (0, 1) which tend to 1, one can chose a sequence of planes h(ti) separating, respectively, tik1 and k2 and converging to a plane h such that k1 and k2 belong to the opposite closed halfspaces determined by h. 252 v. soltan interestingly, minkowski did not use the term “separating plane”. this term (“zwischenebene”) was used later by brunn [12]. the assertion of theorem 3.5 was formulated by bonnesen and fenchel [8, p. 5] for the n-dimensional case, with the term “separating hyperplane”, without any reference on minkowski [64]. theorem 3.5 was generalized in 1936 by eidelheit [25] for the case of convex bodies in linear normed spaces of any dimension (see the survey [51] for further bibliography). o k1 k2 figure 8: double cone of normal vectors to separating hyperplanes. the following result, obtained by brunn [12], describes the normal vectors of all planes which separate a given pair of convex bodies (compare with theorem 5.3 below). theorem 3.6. ([12]) let k1 and k2 be disjoint convex bodies in r3. the set of normal vectors (drawn at the origin of r3) to all planes separating k1 and k2 is a convex double cone, i.e., is the union of two opposite convex cones with the improper common apex o (see figure 8). 3.3. sufficient conditions for convexity of solid sets the following theorem of minkowski [63] characterizes bounded convex surfaces in terms of their support property. theorem 3.7. ([63, § 17]) a set f ⊂ rn is the boundary of a convex body in rn provided there is a point z ∈ rn such that (a) every open halfline originated at z meets f, (b) every point of f belongs to a hyperplane supporting f . using similar arguments, brunn [10, p. 293] proved the theorem below. support and separation properties 253 theorem 3.8. ([10]) if a compact set x ⊂ rn has nonempty interior and any boundary point of x belongs to a hyperplane supporting x, then x is a convex body. straszewicz [86, pp. 22 – 24] independently proved theorem 3.8 under the additional assumption that x is connected and coincides with the closure of its interior. a similar result was obtained by haalmeijer [38]: if x ⊂ rn is the closure of an open connected set and y is a dense subset of bd x, then x is convex provided every point x ∈ y belongs to a hyperplane supporting x. tietze (see [89] and [90]) gave two local versions of theorem 3.8. following [89], an open half-ball qρ(z) ⊂ rn with center z and radius ρ > 0 means the intersection of the open ball uρ(z) and an open halfspace whose boundary hyperplane contains z. theorem 3.9. ([89, 90]) a compact set x ⊂ rn is convex if it satisfies any of the following two conditions: (a) x is the closure of an open connected set, and for every boundary point z of x there is an open half-ball qρ(z), ρ = ρ(z) > 0, disjoint from x; (b) x is connected, int x 6= ∅, and there is a scalar ρ > 0 such that for every boundary point z of x a suitable open half-ball qρ(z) is disjoint from x. x figure 9: illustration to theorem 3.9. the nonconvex sets x ⊂ r2 in figure 9 illustrate that the assumptions in both conditions (a) and (b) of theorem 3.9 are essential. indeed, at any boundary point z, the set x has a support open half-ball qρ(z), where ρ = ρ(z), while int x is not connected and no constant scalar ρ > 0 satisfies condition (b). a simplified proof of part (b) of theorem 3.9 was given by reinhardt [70]. also, this part was sharpened later in the following ways. 254 v. soltan 1. gericke [34] (see also nöbeling [67]) showed that the centers of disjoint with x half-balls qρ(z) can be chosen in bd x\l, where l is a suitable (n− 3)-dimensional plane. 2. süß [88] proved that condition (b) is satisfied if x is connected, has interior points, and there is a scalar ρ > 0 such that for every boundary point z of x there is a cylinder c(z) of variable height based on an (n− 1)-dimensional ball of radius ρ, with x ∩ int c(z) = ∅. 3. schmidt (see bieberbach [6, p. 20]) observed that it is sufficient to require the existence of a hyperplane h through z which does not meet the set x ∩uρ(z) (without the assumption that h supports x ∩uρ(z)). burago and zalgaller [13] (see pp. 395 and 415) observed that the word “compact” can be replaced by “closed” in both theorem 3.8 and theorem 3.9. also, they changed the language of theorem 3.9, replacing in (b) the requirement x ∩qρ(z) = ∅ with the following one: x ∩uρ(z) is supported at z by a suitable hyperplane. one more variation of theorem 3.8 comes from geometric measure theory. namely, a closed set x ⊂ rn with non-empty interior is convex if and only if it has locally finite perimeter and possesses a support hyperplane at each point of its reduced boundary (see caraballo [14, 15] for definitions and technical details). 4. supports and bounds of convex sets 4.1. support hyperplanes an extension of minkowski’s definition of support hyperplane says that a hyperplane h ⊂ rn supports a nonempty set x ⊂ rn provided h meets its closure, cl x, and does not cut x. analysis of the proof of theorem 3.3 shows that its assertion holds for the case of any proper convex set k ⊂ rn. consequently, theorem 3.1 can be generalized as follows: theorem 4.1. any boundary point of a convex set k ⊂ rn belongs to a hyperplane supporting k. if the dimension of a convex set k ⊂ rn is less than n, then a hyperplane h ⊂ rn supporting k may contain k entirely. nevertheless, in many instances it is important to know whether h properly supports k, that is whether k 6⊂ h. equivalent terms used in the literature are essential support (see steinitz [84, § 28]) and nontrivial support (see rockafellar [72, p. 100]). support and separation properties 255 theorem 4.2. ([72, theorem 11.6]) let k ⊂ rn be a convex set which is not a plane, and let f be a nonempty convex subset of cl k (for instance, f is a singleton). there is a hyperplane containing f and properly supporting k if and only if f ⊂ rbd k. it is easy to see that a hyperplane h properly supports a convex set k if and only if it meets cl k such that h ∩ rint k = ∅. the following assertion is a variation of theorem 4.2. theorem 4.3. ([76, corollary 9.11]) if a plane l ⊂ rn meets the closure of a convex set k ⊂ rn such that l∩ rint k = ∅, then there is a hyperplane containing l and properly supporting k. the next result shows that the existence of a support hyperplane is a local property. theorem 4.4. ([76, problem 9.3]) let k ⊂ rn be a convex set, z be a point in cl k, and bρ(z) ⊂ rn be a closed ball of radius ρ > 0 centered at z. a hyperplane h ⊂ rn through z supports (properly supports) k if and only if h supports (respectively, properly supports) the set k ∩bρ(z). z k h figure 10: local support of k at z. if a support hyperplane h ⊂ rn of a convex set k ⊂ rn is described by the equation (2.1), then its normal vector e and the level scalar γ can be characterized as follows. 1. h supports (properly supports) k at a point z ∈ cl k if and only if e ∈ nz(k) (respectively, e ∈ nz(k) \ ort k) and γ = z·e. 2. h supports (properly supports) k if and only if e ∈ nor k (respectively, e ∈ nor k \ ort k) and γ = sup{x·e : x ∈ k}. 256 v. soltan theorem 4.5. ([76, theorem 12.22]) if k ⊂ rn is a line-free convex set, then the set of nonzero vectors e ∈ rn for which the hyperplane of the form h = {x ∈ rn : x·e = γ}, where γ = sup{u·e : u ∈ k}, supports cl k at a single point is dense in nor k. one more assertion, independently proved by durier [24] for the case of convex bodies and by klee [52] for the case of line-free closed convex sets, complements theorem 4.5 and the inclusion ext k ⊂ cl (exp k). theorem 4.6. ([24, 52]) let k ⊂ rn be a line-free closed convex set, h be a hyperplane supporting k, and z be an extreme point of k that belongs to h. then there is a sequence of points zi ∈ exp k and a respective sequence of hyperplanes hi, i ≥ 1, satisfying the conditions: (a) hi ∩k = {zi}, i ≥ 1, (b) z = lim i→∞ zi and h = lim i→∞ hi. there are a few results on support properties of special convex sets. for instance: 3. if a hyperplane h ⊂ rn supports a convex cone c ⊂ rn, then every apex of c belongs to h. furthermore, the intersection of all hyperplanes supporting c is precisely the apex set of cl c (see, e.g., [76], theorem 9.43 and theorem 9.46). 4. if a hyperplane h ⊂ rn supports an m-predecomposable set k ⊂ rn, expressed as the sum of a compact convex set b and a convex cone c with apex o, then h supports b (see [79]). 4.2. bounding hyperplanes and halfspaces modifying carathéodory’s definition of bounding hyperplane, we say that a hyperplane h ⊂ rn bounds a convex set k ⊂ rn provided k is contained in a closed halfspace determined by h. furthermore: h nontrivially bounds k if k 6⊂ h; strictly bounds k if h ∩k = ∅; and strongly bounds k if h bounds a suitable open ρ-neighborhood uρ(k) of k. analysis of the proof of theorem 3.4 shows that it can be generalized in the following way. support and separation properties 257 theorem 4.7. let k be a proper convex set rn and z ∈ rn \ cl k. if u is the nearest to z point in cl k, then the hyperplane h ⊂ rn through z orthogonal to the segment [u,z] strongly bounds k and δ(h,k) = δ(z,k) = ‖z −u‖. the results below describe all hyperplanes bounding a convex set k ⊂ rn and having a given direction of normal vectors or containing a given point (or even a plane) of rn. the following auxiliary statements on a hyperplane h ⊂ rn immediately follow from definitions. 1. h bounds k if and only if h can be expressed in the form (2.1), where e ∈ bar k and sup{x·e : x ∈ k}≤ γ. 2. h properly bounds k if and only if h can be expressed in the form (2.1), where e ∈ bar k and γ satisfies both inequalities inf{x·e : x ∈ k} < γ and sup{x·e : x ∈ k}≤ γ. 3. h strictly bounds k if and only if h can be expressed in the form (2.1), where e ∈ bar k and x·e < γ for all x ∈ k. 4. h strongly bounds k if and only if h can be expressed in the form (2.1), where e ∈ bar k and sup{x·e : x ∈ k} < γ. theorem 4.8. ([82]) for a convex set k ⊂ rn and a point z ∈ rn, the assertions below hold. (a) there is a hyperplane through z bounding k if and only if o /∈ int (k−z), or, equivalently, (k −z)◦ 6= {o}. (b) a hyperplane through z bounds k is and only if it can be expressed as h = {x ∈ rn : x·e = z·e}, (4.1) where e ∈ (k −z)◦ \{o}. theorem 4.9. ([82]) for a convex set k ⊂ rn and a point z ∈ rn, the assertions below hold. (a) there is a hyperplane through z properly bounding k if and only if o /∈ rint (k −z), or, equivalently, (k −z)◦ \ lin (k −z)◦ 6= ∅. (b) a hyperplane through z properly bounds k if and only if it can be expressed in the form (4.1), where e ∈ (k −z)◦ \ lin (k −z)◦. 258 v. soltan theorem 4.10. ([82]) for a closed convex set k ⊂ rn and a point z ∈ rn, the assertions below hold. (a) there is a hyperplane through z strongly bounding k if and only if o /∈ cl (k −z). (b) if o /∈ cl (k − z) and e ∈ rint (k − z)◦, then the hyperplane (4.1) strongly bounds k. (c) if a hyperplane through z strongly bounds k, then it can be expressed in the form (4.1), where e ∈ (k −z)◦ \ lin (k −z)◦. if, additionally, k is compact, then e can be chosen in rint (k −z)◦. there are various extensions of theorem 3.4 to the case of hyperplanes bounding a convex set k ⊂ rn and containing a given plane l ⊂ rn. 5. if l does not meet the relative interior of k, then there is a hyperplane through l properly bounding k (rockafellar [72, theorem 11.2]). 6. if the boundary of k does not contain a halfline and l is disjoint from cl k, then there is a hyperplane through l strictly bounding cl k (klee [47]). 7. if δ(k,l) > 0, then there is a hyperplane h through l strongly bounding k such that δ(k,h) = δ(k,l) ([76, theorem 9.6]). the families of hyperplanes through a plane l ⊂ rn bounding (properly, strictly, or strongly) a given convex set k ⊂ rn can be described similarly to theorems 4.8 – 4.10. for instance, 8. there is a hyperplane which bounds k and contains l if and only if o /∈ int (k −l), or, equivalently, (k −l)◦ 6= {o}. 9. a hyperplane h ⊂ rn bounds k and contains l if and only if h can be expressed in the form (4.1), where e ∈ (k −l)◦ \{o} and z is any point in l. we recall (see, e.g., gale and klee [33]) that a plane l ⊂ rn is an asymptote of a set x ⊂ rn provided cl x ∩ l = ∅ and δ(x,l) = 0. existence of plane asymptotes is closely related to the properties of various algebraic operations on sets (see, e.g., auslender and teboulle [1] and [75], [76] for further references). for instance, the following assertions hold. 10. for a closed set x ⊂ rn and a plane l ⊂ rn, the sum x + l is closed if and only if there is a translate of l which is an asymptote of x. support and separation properties 259 x figure 11: a plane asymptote of a set x. 11. for a linear transformation f : rn → rm and a closed set x ⊂ rn, the set f(x) is closed if and only if there is a translate of null f which is an asymptote of x. the family of plane asymptotes of a given convex set is not hereditary. for example, if k is the convex set in r3 be given by k = { (x,y,z) : x ≥ 0, xy ≥ 1, z ≥ (x + y)2 } , then the xyand xz-coordinate planes are the only plane asymptotes of k. asymptotic properties of convex sets without boundary halflines are studied by klee [49], and of cones and m-predecomposable sets in [77] and [79]. as proved in [81], every plane asymptote l of a convex set k ⊂ rn contains a line-free closed convex cone which is an asymptote of k. the following assertions immediately follow the above definitions. 12. any hyperplane h ⊂ rn disjoint from a closed set x ⊂ rn either strongly bounds x or is an asymptote of x. 13. a hyperplane h ⊂ rn is an asymptote of a closed convex set k ⊂ rn if and only if it can be expressed in the form (2.1), where e ∈ bar k\nor k and γ = sup{x·e : x ∈ k}. we will say that a closed halfspace v ⊂ rn bounds a convex set k ⊂ rn provided k ⊂ v . furthermore, if h denotes the boundary hyperplane of v , then v supports (properly supports ) k is h supports (properly supports) k. also, v strictly (strongly ) bounds k is h disjoint (strongly disjoint) from k. various results on the existence or on description of different types of bounding halfspaces can be routinely derived from the above assertions on bounding hyperplanes. the results below deal with various representations of the closure of a given proper convex set k ⊂ rn as intersections of bounding halfspaces (see [76, chapter 9]). 260 v. soltan 14. cl k is the intersection of all closed halfspaces bounding (supporting, strictly, or strongly bounding) k. 15. if k is not a plane, x is a dense subset of rbd k, and v(x) is the family of all closed halfspaces v each properly supporting k at a point from x, then cl k = ∩(v : v ∈v(x)) and rint k = ∩(int v : v ∈v(x)). 16. if e is a dense subset of the normal cone nor k, then cl k is the intersection of a countable family f of closed halfspaces of the form ve(γ) = {x ∈ rn : x·e ≤ γ}, where e ∈ e, γ ≥ sup{u·e : u ∈ k}. 17. if k is line-free, then cl k = ∩(v : v ∈g), where g denotes the family of all closed halfspaces supporting cl k at its exposed points. 5. separation of convex sets 5.1. classification of separating hyperplanes various results on hyperplane separation of convex sets are usually formulated in the following terms. if k1 and k2 are convex sets in rn and h ⊂ rn is a hyperplane, then we say that 1. h separates k1 and k2 if k1 and k2 lie in the opposite closed halfspaces determined by h (possibly, k1 ∪k2 ⊂ h). 2. h properly separates k1 and k2 if h separates k1 and k2 such that k1 ∪k2 6⊂ h. 3. h definitely separates k1 and k2 if h separates k1 and k2 such that k1 6⊂ h and k2 6⊂ h. 4. h strictly separates k1 and k2 if h separates k1 and k2 such that both sets are disjoint from h (equivalently, k1 and k2 lie in the opposite open halfspaces determined by h). 5. h strongly separates k1 and k2 if h separates suitable open neighborhoods uρ(k1) and uρ(k2). support and separation properties 261 (b) proper separation k1 k2 h (c) definite separation k1 k2 h (d) strict separation k1 k2 h (e) strong separation k1 k2 h figure 12: types of hyperplane separation of convex sets. the above terminology gradually evolved in time: the term proper separation is due to rockafellar [72, p. 95], definite separation is called real by bair and jongmans [3], strict separation and strong separation are due to klee [51]. the following obvious assertions provide analytical equivalences of the above definitions on separation of convex sets k1 and k2 in rn by a hyperplane h ⊂ rn of the form (2.1): 1.′ h separates k1 and k2 if and only if e and γ can be chosen such that sup{x·e : x ∈ k1}≤ γ ≤ inf{x·e : x ∈ k2}. (5.1) 2.′ h properly separates k1 and k2 if and only if e and γ can be chosen to satisfy both inequalities (5.1) and inf{x·e : x ∈ k1} < sup{x·e : x ∈ k2}. (5.2) 3.′ h definitely separates k1 and k2 if and only if e and γ can be chosen to satisfy both inequalities (5.1) and inf{x·e : x ∈ k1} < γ < sup{x·e : x ∈ k2}. (5.3) 4.′ h strictly separates k1 and k2 if and only if e and γ can be chosen such that both conditions below are satisfied: u·e < inf{x·e : x ∈ k2} for every u ∈ k1, sup{x·e : x ∈ k1} < u·e for every u ∈ k2. (5.4) 262 v. soltan 5.′ h strongly separates k1 and k2 if and only if e and γ can be chosen such that sup{x·e : x ∈ k1} < γ < inf{x·e : x ∈ k2}. (5.5) the above types of separation can be refined by using asymmetric conditions (see klee [51] and [78] for further details): if convex sets k1 and k2 in rn are separated by a hyperplane h ⊂ rn, then we say that 6. h properly separates k1 from k2 provided k1 6⊂ h. 7. h strictly separates k1 from k2 provided k1 is disjoint from h. the relation between symmetric and asymmetric types of separation is described in the following theorem. theorem 5.1. ([76, theorem 10.6]) if k1 and k2 are convex sets and h1 and h2 are hyperplanes such that hi properly (strictly) separates ki from k3−i, i = 1, 2, then there is a hyperplane containing h1 ∩ h2 and properly (strictly) separating k1 and k2. the results below describe the hyperplanes which separate a pair of convex sets and have a given direction of normals or contain a given point. an initial step towards description of separating hyperplanes with a given direction of normals consists in reduction to the case of a single convex set. theorem 5.2. ([76, theorem 10.7]) let k1 and k2 be convex sets in rn, and h be a hyperplane of the form (2.1). then the assertions below hold. (a) a translate of h separates (properly separates) k1 and k2 if and only if the subspace (2.2) bounds (properly bounds) k1 −k2. (b) a translate of h strictly separates at least one of the sets k1 and k2 from the other if and only if the subspace (2.2) strictly bounds k1−k2. (c) a translate of h strongly separates k1 and k2 if and only if the subspace (2.2) strongly bounds k1 −k2. a combination of theorems 4.8 – 4.10 and theorem 5.2 implies the following assertions. theorem 5.3. ([78, 82]) given convex sets k1 and k2 in rn, the following assertions hold. support and separation properties 263 (a) there is a hyperplane separating k1 and k2 if and only if any of the following equivalent conditions holds: int k1 ∩ int k2 = ∅, o /∈ int (k1 −k2), (k1 −k2)◦ 6= {o}. furthermore, a translate of a hyperplane h ⊂ rn separates k1 and k2 if and only if h can be expressed in the form (2.1), where e ∈ (k1 −k2)◦ \{o}. (b) there is a hyperplane properly separating k1 and k2 if and only if any of the following equivalent conditions holds: rint k1 ∩ rint k2 = ∅, o /∈ rint (k1 −k2), (k1 −k2)◦ \ lin (k1 −k2)◦ 6= ∅. furthermore, a translate of a hyperplane h ⊂ rn properly separates k1 and k2 if and only if h can be expressed in the form (2.1), where e ∈ (k1 −k2)◦ \ lin (k1 −k2)◦. (c) there is a hyperplane definitely separating k1 and k2 if and only if either o /∈ cl (k1 −k2) or o ∈ cl (k1 −k2) and (k1 −k2)◦ \ (ort k1 ∪ ort k2) 6= ∅. furthermore, a translate of a hyperplane h ⊂ rn definitely separates k1 and k2 if and only if h can be expressed in the form (2.1) such that one of the following conditions is satisfied: (i) o /∈ cl (k1 −k2) and e ∈ rint (k1 −k2)◦ ∪ (rbd (k1 −k2)◦ \ (ort k1 ∪ ort k2)), (ii) o ∈ cl (k1 −k2) and e ∈ (k1 −k2)◦ \ (ort k1 ∪ ort k2). (d) there exists some hyperplane strongly separating k1 and k2 if and only if o /∈ cl (k1 − k2). furthermore, if o /∈ cl (k1 − k2) and e ∈ rint (k1 − k2)◦, then a suitable translate of a hyperplane of the form (2.1) strongly separates k1 and k2. a variation of theorem 5.3 allows us to describe all hyperplanes which separate a pair of convex sets and contain a given point of rn. this description uses the following auxiliary lemma. 264 v. soltan lemma 5.4. ([82]) convex sets k1 and k2 in rn are separated (properly separated) by a hyperplane h ⊂ rn through a given point z ∈ rn if and only if the generated cones cz(k1) and cz(k2) are separated (properly separated) by h. theorem 5.5. ([82]) given convex sets k1 and k2 in rn and a point z ∈ rn, let d1 = cz(k1) −z, d2 = cz(k2) −z. the assertions below hold. (a) there is a hyperplane through z separating k1 and k2 if and only if the cones d1 and d2 satisfy any of the following equivalent conditions: o /∈ int (d1 −d2), (d1 −d2)◦ 6= {o}, d◦1 ∩ (−d ◦ 2) 6= {o}. (b) there is a hyperplane through z properly separating k1 and k2 if and only if the cones d1 and d2 satisfy any of the following equivalent conditions: o /∈ rint (d1 −d2), (d1 −d2)◦ is not a subspace, d◦1 ∩ (−d ◦ 2) is not a subspace. 5.2. geometric conditions on hyperplane separation theorem 5.3 provides a unified description of all hyperplanes separating convex sets k1 and k2 in rn, which is formulated in terms of the polar cone (k1 − k2)◦. some other types of geometric conditions that guaranty the existence of a desired type of separation are given below. proper separation. the condition rint k1 ∩ rint k2 = ∅ was already mentioned in theorem 5.3. it was obtained in various forms of generality by fenchel [30, p. 48], klee [46], and rockafellar [72, theorem 11.3]. a related result on asymmetric type of proper separation is due to rockafellar [72, theorem 20.2]: given convex sets k1 and k2 in rn such that k2 is polyhedral, there is a hyperplane properly separating k1 from k2 if and only if rint k1 ∩k2 = ∅. we observe that the latter assertion does not hold if the set k2 is not polyhedral. for instance, if k1 and k2 are planar circular disks in r3 given by k1 = {(x,y, 0) : x2 + (y − 1)2 ≤ 1}, k2 = {(0,y,z) : y2 + (z − 1)2 ≤ 1}, then rint k1 ∩k2 = ∅, while k1 is not properly separated from k2. support and separation properties 265 strict separation. the only known geometric result on strict separation of convex sets is attributed to klee [47]: if none of the disjoint closed convex sets k1 and k2 in rn is a plane, and none of the sets rbd k1 and rbd k2 contains a halfline, then k1 and k2 are strictly separated by a hyperplane. strong separation. an obvious continuity argument shows that if convex sets k1 and k2 are strongly separated by a hyperplane h, then there is a slab of positive width which separates k1 and k2 and whose boundary hyperplanes are parallel to h. a natural question here is to determine the maximum possible width of such a slab. the answer to this question was given by dax [20] (see also [76, theorem 10.20]). h k1 k2 figure 13: strict separation of k1 and k2. δ(k1,k2) k2 k1 h1 h2 figure 14: separation of k1 and k2 by a slab of maximum width. theorem 5.6. ([20]) if convex sets k1 and k2 in rn are strongly disjoint (that is, δ(k1,k2) > 0), then there is a unique pair of parallel hyperplanes h1 and h2 in rn, both separating k1 and k2 and satisfying the condition δ(h1,h2) = δ(k1,k2). the above equality δ(h1,h2) = δ(k1,k2), without specifying the uniqueness of the pair {h1,h2}, was obtained later by gabidullina [32] for the case when at least one of the sets k1 and k2 is compact. a similar question on strong separation of convex sets k1 and k2 concerns the existence of a pair of nearest points z1 ∈ cl k1 and z2 ∈ cl k2. a simple 266 v. soltan geometric argument shows that in this case the hyperplanes through z1 and z2 orthogonal to [z1,z2] form a slab of maximum width separating k1 and k2. a sufficient condition for the existence of such a pair {z1,z2} can be found in [76]. z1 z2 k1 k2 figure 15: a nearest pair of points in k1 and k2. theorem 5.7. ([76, theorem 10.24]) if convex sets k1 and k2 in rn satisfy the condition rec (cl k1) ∩ rec (cl k2) = {o}, then δ(k1,k2) = ‖z1 − z2‖ for suitable points z1 ∈ cl k1 and z2 ∈ cl k2. in particular, a nearest pair {z1,z2} exists provided at least one of the sets k1 and k2 is bounded. a useful result on strong separation of convex sets was obtained by de wilde [21]. theorem 5.8. ([21]) if k1 and k2 are disjoint closed convex sets in rn, then the following conditions are equivalent: (a) both k1 and k2 are line-free and rec k1 ∩ rec k2 = {o}. (b) there are parallel disjoint hyperplanes h1 and h2 both separating k1 and k2 such that hi ∩ki is an exposed point of ki, i = 1, 2. analysis of the proof of theorem 5.8 shows that the exposed points h1∩k1 and h2∩k2 are not necessarily the nearest. nevertheless, h1 and h2 may be chosen to satisfy the condition δ(h1,h2) > δ(k1,k2)−ε for any given scalar ε > 0. maximal separation. klee [50] obtained various results regarding strict and strong separation of convex sets by hyperplanes. given a pair {f,g} of nonempty families of closed convex sets in rn, we say that f is maximal with respect to a certain type s of separation provided it satisfies the following conditions: 1. the sets f and g are s-separated whenever f ∈ f and g ∈ g, with f ∩g = ∅. support and separation properties 267 2. for every f ∈f, there is g ∈g such that f ∩g = ∅. 3. for every closed convex set f /∈f, there is g ∈g such that f ∩g = ∅ but f and g are not s-separated. following gale and klee [33], we say that a closed convex set k ⊂ rn is continuous provided k admits no boundary halfline and no line asymptote. given disjoint closed convex sets f and g in rn, the assertions below hold (see [50]). 4. each of the following conditions implies that f and g are strictly separated and represents a maximal theorem for strict separation: (a) f is continuous, (b) neither f nor g admits a line asymptote, (c) neither f nor g has a boundary halfline. 5. each of the following conditions implies that f and g are strongly separated and represents a maximal theorem for strong separation: (d) f is continuous, (e) neither f nor g admits a line asymptote. an extensive development and generalization of klee’s results on maximal separation is given in the book of fajardo, goberna, rodŕıguez, and vicentepérez [27]. theorem 1.2 and theorem 1.3 from this book give a comprehensive list of various maximal separation assertions for the case of evenly convex sets. (according to fenchel [29], a convex set in rn is called evenly convex if it is the intersection of a family of open halfspaces. it is easy to see that every proper closed convex sets is evenly convex.) 5.3. sharp separation of convex cones if convex cones c1 and c2 with a common apex in rn are separated by a hyperplane h ⊂ rn, then h supports both cones cl c1 and cl c2. consequently, ap (cl c1)∪ap (cl c2) ⊂ h. in this regard, we will say that h sharply separates c1 from c2 provided h ∩ cl c1 = ap (cl c1). similarly, h sharply separates c1 and c2 if h ∩ (cl c1) = ap (cl c1) and h ∩ cl c2 = ap (cl c2). the next two theorems give criteria for sharp separation of cones in terms of their polar cones. 268 v. soltan theorem 5.9. ([78]) if c1 and c2 are convex cones in rn with a common apex a ∈ rn, then the following conditions are equivalent. (a) c1 is sharply separated from c2. (b) the set e = rint (c1 −a)◦ ∩ (a−c2)◦ has positive dimension. theorem 5.10. ([74, 78]) if c1 and c2 are convex cones in rn with a common apex a ∈ rn, then the following conditions are equivalent. (a) c1 and c2 are sharply separated. (b) each of the cones c1 and c2 is sharply separated from the other. (c) the set d = rint (c1 −a)◦ ∩ rint (a−c2)◦ has positive dimension. analysis of the proof of theorem 5.10 reveals a simple corollary: if c1 is not a plane and is sharply separated from c2, then c1 is properly separated from c2. the converse assertion is not true. for instance, in r2, the cone c1 = {(x, 0) : x ∈ r} is separated sharply but not properly from the cone c2 = {(x,y) : 0 ≤ x, 0 ≤ y ≤ x}, while c2 is separated properly but not sharply from c1 (see figure 16). c2c1 figure 16: proper but not sharp separation of cones c1 and c2. a geometric criterion for sharp separation of convex cones is given in the following theorem. theorem 5.11. ([74]) let c1 and c2 be convex cones in rn with a common apex a ∈ rn. the conditions below are equivalent. (a) c1 and c2 are sharply separated by a hyperplane. (b) cl c1 ∩ cl c2 = ap (cl c1) ∩ ap (cl c2) and at least one of the cases below holds: (i) dim (c1 ∪c2) 6 n− 1, (ii) at least one of the cones c1 and c2 is not a plane. support and separation properties 269 in terms of continuous linear functionals on a linear topological space, theorem 5.11, formulated for the case of closed convex cones with a common apex o, was proved earlier by klee [46] under the assumption ap c1 ∩ ap c2 = {o}, and by bair and gwinner [2] under the condition that ap c1 ∩ ap c2 is a subspace. 5.4. penumbras and separation following rockafellar [72, p. 22], we recall that the penumbra of a convex set k1 with respect to another convex set k2, denoted below p(k1,k2), is defined by p(k1,k2) = ∪(µk1 + (1 −µ)k2 : µ ≥ 1) = {µx1 + (1 −µ)x2 : µ ≥ 1,x1 ∈ k1,x2 ∈ k2}. geometrically, p(k1,k2) is the union of all closed halflines initiated at the points of k1 in the directions of vectors from k1 − k2 (see fig. 17). it is possible to show (see [80]) that both sets p(k1,k2) and p(k2,k1) are convex and contain k1 and k2, respectively. the following theorem illustrates the role of penumbras in separation of convex sets. p(k2,k1) k2 k1 p(k1,k2) h figure 17: illustration to theorem 5.12. theorem 5.12. ([80]) let k1 and k2 be convex sets in rn. a hyperplane h ⊂ rn separates (respectively, properly, strictly, or strongly) k1 and k2 if and only if it separates (respectively, nontrivially, strictly, or strongly) the sets p(k1,k2) and p(k2,k1). given convex sets k1 and k2 in rn, denote by h1(k1,k2) (respectively, by h2(k1,k2) and h3(k1,k2)) the family of all hyperplanes properly (respectively, strictly and strongly) separating k1 and k2. also, let ei(k1,k2) = ∪(h : h ∈hi), i = 1, 2, 3. 270 v. soltan theorem 5.13. ([80]) if convex sets k1 and k2 in rn satisfy the condition rint k1 ∩ rint k2 = ∅, then e1(k1,k2) = rn \ (rint p(k1,k2) ∪ rint p(k2,k1)). furthermore, a hyperplane h ⊂ rn properly separates k1 and k2 if and only if h ⊂ e1(k1,k2) and h ∩ aff (k1 ∪k2) 6= ∅. corollary 5.14. ([80]) if convex sets k1 and k2 in rn satisfy the condition cl k1 ∩ cl k2 = ∅, then e2(k1,k2) ⊂ f2(k1,k2), where f2(k1,k2) = rn \ (p(cl k1, cl k2) ∪p(cl k2, cl k1))). furthermore, a hyperplane h ⊂ rn strictly separates cl k1 and cl k2 if and only if h ⊂ f2(k1,k2) and h ∩ aff (k1 ∪k2) 6= ∅. the inclusion e2(k1,k2) ⊂ f2(k1,k2) in corollary 5.14 may be proper. indeed, consider the closed convex sets k1 = {(x, 1) : 0 ≤ x ≤ 1} and k2 = {(x, 0) : x ∈ r}. then e2(k1,k2) = {(x,y) : 0 < y < 1}, while f2(k1,k2) = e2(k1,k2) ∪{(x, 1) : x < 0}∪{(x, 1) : x > 1}. theorem 5.15. ([80]) if convex sets k1 and k2 in rn are strongly disjoint, then e3(k1,k2) = rn \ (cl p(k1,k2) ∪ cl p(k2,k1)). the following assertions from [80] relate various properties of penumbras to some known classes of convex sets in rn. 1. if k1 is compact, then p(k1,k2) is an m-predecomposable set. 2. if both k1 and k2 are compact and k1 ∩k2 = ∅, then p(k1,k2) is an m-decomposable set. 3. if both k1 and k2 are polyhedra, then cl p(k1,k2) is a polyhedron. 4. if both k1 and k2 are polytopes, then p(k1,k2) is a polyhedron. support and separation properties 271 5.5. hemispaces the following concept was introduced by motzkin [66, lecture iii] in three dimensions, and, independently, by hammer [39] in vector spaces of any dimension: given a point v ∈ rn, any maximal (under inclusion) convex subset of rn\{v}, denoted sv, is called a semispace of rn at v (in [43] and [53] these sets are called hypercones ). the next properties of semispaces can be easily obtained (see [39, 43, 66]). 1. for a semispace sv ⊂ rn, both sets sv and rn \ sv are convex cones with apex v. 2. for a convex set k⊂rn\{v}, there is a semispace sv⊂rn containing k. 3. if c ⊂ rn is a convex cone with improper apex v ∈ rn and b ⊂ rn is a convex set missing v and disjoint from c, then there is a semispace sv ⊂ rn containing c and disjoint from b (jamison [43] for the case v = o). additional properties of semispaces in vector spaces of any dimension can be found in the papers [22, 43, 48, 56, 65]. theorem 5.16. ([39]) the family of all semispaces of rn is the smallest among all families f of convex sets in rn satisfying the following condition: every proper convex set k ⊂ rn is the intersection of some elements from f. the structure of semispaces can be described in different ways. the first one, briefly mentioned by hammer [40] (see the books [58, satz 1.10], and [76, theorem 10.32] for complete proofs), uses a nested family of planes {v} = l0 ⊂ l1 ⊂ ···⊂ ln−1 ⊂ ln = rn, dim li = i, 0 ≤ i ≤ n, (5.6) and their halfplanes e1, . . . ,en, where ei an open halfplane of li determined by li−1, 1 ≤ i ≤ n. theorem 5.17. ([40]) if sv ⊂ rn is a semispace at v ∈ rn, then there is a nested sequence of planes of the form (5.6) and a respective sequence of open halfplanes e1, . . . ,en such that sv = e1 ∪·· ·∪en. conversely, any set of the form e1 ∪·· ·∪en is a semispace at v. another way (given by hammer [39, 40] without proof) is based on the choice of a suitable basis for rn. theorem 5.18. ([39, 40]) if sv ⊂ rn is a semispace at v ∈ rn, then there is a basis e1, . . . ,en for rn such that sv consists of all vectors of the form 272 v. soltan v + α1e1 + · · ·+ αnen, where α21 + · · ·+ α 2 n > 0 and the first nonzero scalar in the sequence α1, . . . ,αn is positive. conversely, given any basis e1, . . . ,en for rn, the set of described above vectors is a semispace at v. the equivalence of description of semispaces in theorem 5.17 and theorem 5.18 follows from the simple geometric arguments: 4. if e1, . . . ,en is a basis for rn, then the open halfplanes ei from theorem 5.17 can be chosen as ei = v +{αn−i+1en−i+1 +· · ·+αnen : αn−i+1 > 0}, 1 ≤ i ≤ n. (5.7) 5. for any choice of planes (5.6) and of respective halfplanes e1, . . . ,en, nonzero vectors ei ∈ (en−i+1 −v) \ (ln−i −v), 1 ≤ i ≤ n, form a basis for rn such that the equalities (5.7) hold. independently, mart́ınez-legaz [59] described a similar separation result, based on lexicographic ordering � of rn. we recall that for distinct vectors x = (x1, . . . ,xn) and y = (y1, . . . ,yn), one can write x ≺ y if xi < yi, with i being the first index in {1, . . . ,n} for which xi 6= yi; also, x � y if x ≺ y or x = y. in a standard way, a invertible n × n matrix a is orthogonal if a−1 = at . theorem 5.19. ([59]) for a proper convex set k ⊂ rn and a point x0 /∈ k, there is an invertible (even orthogonal) n×n matrix a and a vector v ∈ rn such that ax ≺ v � ax0 whenever x ∈ k. although mart́ınez-legaz [59] made an observation that the sets from theorem 5.19 are similar in their properties to semispaces, it was singer [73] who proved the following assertion: a set m ⊂ rn is a semispace at v if and only if there is an invertible matrix n×n matrix a such that m = {x ∈ rn : ax ≺ v}. the result below is proved by tukey [91] (the condition that the vector space e should be normed is superfluous) and, independently, by stone [85] (see theorem 7 from chapter 3). theorem 5.20. ([85, 91]) any pair of disjoint convex sets k1 and k2 in a vector space e can be separated by complementary convex sets q1 and q2: k1 ⊂ q1, k2 ⊂ q2, q1 ∪q2 = e, q1 ∩q2 = ∅. support and separation properties 273 we observe that theorem 5.20 cannot be extended to the case of more than two convex sets. for instance, the convex cones c1,c2, and c3 in the plane, depicted in figure 18, cannot be enlarged into pairwise disjoint (even pairwise non-overlapping) convex sets whose union is the entire plane. c1 c2 c3 figure 18: no convex extensions of cones c1,c2 and c3 cover the whole plane. the following results are similar to those from theorem 5.20. 6. if c ⊂ rn is a convex cone with apex o such that c∩(−c) = {o}, then there is a convex cone c′ with apex o satisfying the conditions c ⊂ c′, c′ ∩ (−c′) = {o}, and c′ ∪ (−c′) = rn (ellis [26]). 7. let f ba a commuting family of affine transformations in rn, and let k1 and k2 be disjoint convex sets both invariant with respect to transformations from f. then there are complementary f-invariant convex sets q1 and q2 such that k1 ⊂ q1 and k2 ⊂ q2 (páles [68]). following jamison [42], we say that a proper convex subset q of rn is a hemispace provided its complement rn \q is a convex set. in [54] and [76], hemispaces are also called convex halfspaces. various properties of hemispaces in vectors spaces of any dimension can be found in the papers [23, 42, 55] (see [36] for related material). a description of hemispaces in rn, similar to that of theorem 5.18, was obtained by lassak [54]. theorem 5.21. ([54]) if q and q′ are complementary hemispaces in rn, then there is a point v ∈ rn, a (orthogonal) basis e1, . . . ,en for rn, and an integer r ≥ 1 such that one of the sets q and q′ consists of all vectors of the form v +αrer +· · ·+αnen, where α2r +· · ·+α2n > 0 and the first nonzero scalar in the sequence αr, . . . ,αn is positive. conversely, for any choice of a point v ∈ rn, a basis e1, . . . ,en for rn, and an integer r ≥ 1, the sets of vectors described above is a hemispace in rn. 274 v. soltan the next theorem shows that the above description of complementary hemispaces can be reformulated in terms of nested sequences of planes (5.6) and of their halfplanes e1, . . . ,en. theorem 5.22. ([76, theorem 10.28]) if q and q′ are complementary hemispaces in rn, then there is a sequence of planes of the form (5.6) and an integer 1 6 r 6 n such that either q = fr and q ′ = f ′r, or q = f ′ r and q′ = fr, with fr = er ∪·· ·∪en, f ′r = lr−1 ∪e ′ r ∪·· ·∪e ′ n, 1 6 r 6 n, (5.8) where ei,e ′ i are complementary open halfplanes of li determined by li−1. independently, mart́ınez-legaz [59] defined a hemispace in rn as the set of vectors x ∈ rn satisfying the condition ax ≺ v, where a is an arbitrary (not necessarily invertible) n×n matrix and v ∈ rn. later, mart́ınez-legaz and singer [60] proved that for any pair of disjoint convex sets k1 and k2 in rn, there exists an orthogonal n × n matrix a such that ax1 ≺ ax2 for all x1 ∈ k1 and x2 ∈ k2. the following description of hemispaces in terms of lexicographical order on r̄n is due to mart́ınez-legaz and singer[61] (here r̄n stands for the cartesian product of n extended lines r̄ = [−∞,∞]). theorem 5.23. 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[91] j.w. tukey, some notes on the separation of convex sets, portugal. math. 3 (1942), 95 – 102. introduction preliminaries minkowski's theorems support hyperplanes bounding and separating hyperplanes sufficient conditions for convexity of solid sets supports and bounds of convex sets support hyperplanes bounding hyperplanes and halfspaces separation of convex sets classification of separating hyperplanes geometric conditions on hyperplane separation sharp separation of convex cones penumbras and separation hemispaces e extracta mathematicae vol. 33, núm. 1, 11 – 32 (2018) spectral properties for polynomial and matrix operators involving demicompactness classes fatma ben brahim, aref jeribi, bilel krichen department of mathematics, faculty of sciences of sfax, university of sfax, road soukra km 3.5, b.p. 1171, 3000, sfax tunisia fatmaazzahraabenb@gmail.com aref.jeribi@fss.rnu.tn krichen bilel@yahoo.fr presented by jesús m.f. castillo received august 5, 2017 abstract: the first aim of this paper is to show that a polynomially demicompact operator satisfying certain conditions is demicompact. furthermore, we give a refinement of the schmoëger and the rakocević essential spectra of a closed linear operator involving the class of demicompact ones. the second aim of this work is devoted to provide some sufficient conditions on the inputs of a closable block operator matrix to ensure the demicompactness of its closure. an example involving the caputo derivative of fractional of order α is provided. moreover, a study of the essential spectra and an investigation of some perturbation results. key words: matrix operator, demicompact linear operator, fredholm and semi-fredholm operators, perturbation theory, essential spectra. ams subject class. (2010): 47a53, 47a55, 47a10. 1. introduction let x and y be two banach spaces. the set of all closed densely defined (resp. bounded) linear operators acting from x into y is denoted by c(x, y ) (resp. l(x, y )). we denote by k(x, y ) the subset of compact operators of l(x, y ). for t ∈ c(x, y ), we use the following notations: α(t) is the dimension of the kernel n(t) and β(t) is the codimension of the range r(t) in y . the next sets of upper semi-fredholm, lower semi-fredholm, fredholm and semi-fredholm operators from x into y are, respectively, defined by: φ+(x, y ) = {t ∈ c(x, y ) such that α(t) < ∞ and r(t) closed in y }, φ−(x, y ) = {t ∈ c(x, y ) such that β(t) < ∞ and r(t) closed in y }, φ(x, y ) := φ−(x, y ) ∩ φ+(x, y ) and φ±(x, y ) := φ−(x, y ) ∪ φ+(x, y ). 11 12 f. ben brahim, a. jeribi, b. krichen for t ∈ φ±(x, y ), the index is defined as i(t) := α(t) − β(t). a complex number λ is in φ+t , φ−t , φ±t or φt if λ − t is in φ+(x, y ), φ−(x, y ), φ±(x, y ) or φ(x, y ), respectively. if x = y , then l(x, y ), c(x, y ), k(x, y ), φ(x, y ), φ+(x, y ), φ−(x, y ) and φ±(x, y ) are replaced by l(x), c(x), k(x), φ(x), φ+(x), φ−(x) and φ±(x), respectively. if t ∈ c(x), we denote by ρ(t) the resolvent set of t and by σ(t) the spectrum of t . let t ∈ c(x). for x ∈ d(t), the graph norm ∥.∥t of x is defined by ∥x∥t = ∥x∥ + ∥tx∥. it follows from the closedness of t that xt := (d(t), ∥.∥t ) is a banach space. clearly, for every x ∈ d(t) we have ∥tx∥ ≤ ∥x∥t , so that t ∈ l(xt , x). a linear operator b is said to be t-defined if d(t) ⊆ d(b). if the restriction of b to d(t) is bounded from xt into x, we say that b is t-bounded. an operator t ∈ l(x, y ) is said to be weakly compact if t(b) is, relatively weakly compact in y for every bounded set b ⊂ x. the family of weakly compact operators from x into y is denoted by w(x, y ). if x = y , the family of weakly compact operators on x which is denoted by w(x) := w(x, x) is a closed two-sided ideal of l(x). definition 1.1. let x and y be two banach spaces and let f ∈ l(x, y ). the operator f is called: (a) fredholm perturbation if t + f ∈ φ(x, y ) whenever t ∈ φ(x, y ). (b) upper semi-fredholm perturbation if t + f ∈ φ+(x, y ) whenever t ∈ φ+(x, y ). (c) lower semi-fredholm perturbation if t + f ∈ φ−(x, y ) whenever t ∈ φ−(x, y ). the set of fredholm, upper semi-fredholm and lower semi-fredholm perturbations are denoted by f(x, y ), f+(x, y ) and f−(x, y ), respectively. the concept of demicompactness appeared in the literature since 1966 in order to discuss fixed points. it was introduced by w. v. petryshyn [16] as follows: definition 1.2. an operator t : d(t) ⊆ x −→ x is said to be demicompact if for every bounded sequence (xn)n in d(t) such that xn − txn → x ∈ x, there exists a convergent subsequence of (xn)n. the family of demicompact operators on x is denoted by dc(x). it is clear that the sum, the product of demicompact operators and the product of a complex number by a demicompact operator are not necessarily demicompact. w. v. petryshyn [16] and w. y. akashi [1] used the class spectral properties for polynomial and matrix operators 13 of demicompact operators to obtain some results on fredholm perturbation. in fact, in 1966 w. v. petryshyn [16] studied various conditions on a continuous 1-set-contractive map t of a real banach space x, which ensure the surjectivity. in the same paper, the author generalized to k-set-contractions the results obtained in [10] for lipschitzian pseudo-contractive maps. in 1983 w. y. akashi [1] generalized some known results in the classical theory of linear fredholm operators in which the compact operators played a fundamental role. for this the author introduced a new class of operators containing the class of compact operators. recently, w. chaker, a. jeribi and b. krichen [4] continued this study to investigate the essential spectra of densely defined linear operators. in the same work, it was proved that for a closed operator t , if t is demicompact, then i −t is an upper semi-fredholm operator and if µt is demicompact for all µ ∈ [0, 1], then i − t is a fredholm operator with index zero. in 2014, b. krichen [11] gave a generalization of this notion by introducing the class of relative demicompact linear operators with respect to a given linear operator. the theory of block operator matrices arise in various areas of mathematics and its applications: in systems theory as hamiltonians (see [7]), in the discretization of partial differential equations as large partitioned matrices due to sparsity patterns, in saddle point problems in non-linear analysis (see [3]), in evolution problems as linearization of second order cauchy problems and as linear operators describing coupled systems of partial differential equations. such systems occur widely in mathematical physics, e.g. in fluid mechanics (see [6]), magnetohydrodynamics (see [15]), and quantum mechanics (see [21]). in all these applications, the spectral properties of the corresponding block operator matrices are of vital importance as they govern for instance the time evolution and hence the stability of the underlying physical systems. from the most important works on the spectral theory of block operator matrices, we mention [9], in which the author developed the essential spectra of 2 × 2 and 3 × 3 block operator matrices. we also mention [22], in which it was presented a wide panorama of methods to investigate the essential spectra of block operator matrices. in this paper, we will study the demicompactness properties of the following matrix operator l0 acting on the banach space product x × x which is defined by: l0 = ( a b c d ) . in general, the entries of l0 are unbounded. the operator a acts on the banach space x and has the domain d(a), d acts on the same banach space 14 f. ben brahim, a. jeribi, b. krichen x and is defined on d(d) and the intertwining operator b (resp. d) is defined on d(b) (resp. d(d)) and acts from x into itself. below, we shall assume that d(a) ⊂ d(c) and d(b) ⊂ d(d). note in general that the operator l0 is neither closed nor closable operator even if its entries are closed operators. in [2] it was proved that under some conditions, l0 is closable and its closure is denoted by l. in the literature, many important results were obtained concerning the spectral theory of this type of operators. we mention from these works the paper [22] in which the authors investigated the essential spectra of the matrix operator by means of an abstract nonzero two-sided ideal. the central aim of this work is to use the concept of demicompactness to investigate the essential spectra of l, the closure of l0. more precisely, we are concerned with the following essential spectra: σe1(t) = {α ∈ c such that α − t /∈ φ+(x)} := c\φ+t , σe4(t) = {α ∈ c such that α − t /∈ φ(x)} := c\φt , σe5(t) = c\ρ5(t), σe7(t) = ∩ k∈k(x) σap(t + k), σe8(t) = ∩ k∈k(x) σδ(t + k), where ρ5(t) = {α ∈ c such that α ∈ φt and i(α − t) = 0}, σap(t) = { λ ∈ c such that inf x∈d(t);∥x∥=1 ∥(λ − t)x∥ = 0 } , and σδ(t) = {λ ∈ c such that λ − t is not surjective}. the subsets σe1(·) and σe4(·) are, respectively the gustafson and the wolf essential spectra [8]. σe5(·) is the schechter essential spectrum [18]. σe7(·) is the essential approximate point spectrum or the schmoëger essential spectrum and σe8(·) is the essential defect spectrum or the rakocević essential spectrum (see for instance [9, 17, 18, 19, 23]). note that for t ∈ c(x), we have: σe1(t) ⊂ σe4(t) ⊂ σe5(t) = σe7(t) ∪ σe8(t), and σe1(t) ⊂ σe7(t). let us recall the following lemma whose the proof can be found in [9]. spectral properties for polynomial and matrix operators 15 lemma 1.1. let t ∈ c(x), then (i) λ /∈ σe7(t) if, and only if, (λ − t) ∈ φ+(x) and i(λ − t) ≤ 0. (ii) λ /∈ σe8(t) if, and only if, (λ − t) ∈ φ−(x) and i(λ − t) ≥ 0. proposition 1.1. ([19]) let t ∈ c(x), then λ /∈ σe5(t) if, and only if, (λ − t) ∈ φ(x) and i(λ − t) = 0. this paper is organized in the following way. in section 2, we recall some definitions and results needed in the rest of the paper. in section 3, we show that under some conditions, a polynomially demicompact operator is demicompact and we give an example involving the caputo derivative of fractional of order α. in section 4, we give a fine description of the essential approximate point spectrum and the essential defect spectrum. in section 5, we prove in proposition 5.1 that under some conditions, µl is demicompact for each µ ∈ ρ(a) and we give, in theorem 5.3, a necessary condition for which i −l is an upper semi-fredholm operator on a banach space with the dunford-pettis property (see definition 2.1). in section 6, we investigate the essential spectra of the matrix operator l. 2. preliminary results we start this section by recalling some fredholm results related with demicompact operators. theorem 2.1. ([4]) let t ∈ c(x). if t is demicompact, then i − t is an upper semi-fredholm operator. theorem 2.2. ([4]) let t ∈ c(x). if µt is demicompact for each µ ∈ [0, 1], then i − t is a fredholm operator of index zero. theorem 2.3. ([4]) let t : d(t) ⊆ x −→ x be a closed linear operator. if t is a 1-set-contraction then µt is demicompact for each µ ∈ [0, 1). in the next, we give a lemma which shows that, in a special banach space x, the sum of demicompact and weakly compact operators is demicompact. to this end, we recall the following definition. 16 f. ben brahim, a. jeribi, b. krichen definition 2.1. a banach space x is said to have the dunford-pettis property (in short dp property) if every bounded weakly compact operator t from x into another banach space y transforms weakly compact sets on x into norm-compact sets on y . remark 2.1. it was proved in [13] that if x is banach space with dp property, then w(x) ⊂ f(x). lemma 2.1. let x be a banach space with dp property. if a ∈ dc(x) and b ∈ w(x), then a + b ∈ dc(x). proof. let (xn)n be a bounded sequence in d(a) such that ((i − a − b)xn)n converges. since b ∈ w(x), then there exists a subsequence of (xn)n, still denoted (xn)n, such that the operator (bxn)n is weakly convergent. we deduce from the fact that x has dp property, that (bxn)n has a convergent subsequence and therefore ((i − a)xn)n has also a convergent subsequence. using demicompactness of a, we infer that (xn)n has a convergent subsequence and we conclude that a + b is demicompact. 3. polynomially demicompact operators it was shown in [12] that a polynomially compact operator t , element of p(x) := {t ∈ l(x) such that there exists a nonzero complex polynomial p(z) = ∑p r=0 arz r satisfying p(1) ̸= 0, p(1) − a0 ̸= 0, and p(t) ∈ k(x)}, is demicompact. in this section, we show that this result remains valid for a broader class of polynomially demicompact operators on x. to this end we let pdc(x) be the set defined by pdc(x) := {t ∈ l(x) such that there exists a nonzero complex polynomial p(z) = ∑p r=0 arz r satisfying p(1) ̸= 0 and 1 p(1) p(t) ∈ dc(x)}. we note that pdc(x) contains the set p(x). theorem 3.1. if t ∈ pdc(x), then t is demicompact. proof. we first give the following relation that we will use in the proof. since i − t commutes with i, newton’s binomial formula allows us to write t j = i + j∑ i=1 (−1)icij(i − t) i. spectral properties for polynomial and matrix operators 17 by making some simple calculations, we may write p(t) = p(1)i + p∑ j=1 aj ( j∑ i=1 (−1)icij(i − t) i ) . (3.1) we start now proving our theorem. to this end we let t ∈ pdc(x), then there exists a nonzero complex polynomial p such that p(1) ̸= 0. we shall prove that t is a demicompact operator. to do so, it suffices from theorem 2.1 in [5] to establish that i − t is an upper semi-fredholm operator. first, we prove that α(i − t) < ∞. we let x ∈ n(i − t), then tx = x and therefore p(t)x = p∑ j=0 ajt jx = p(1)x. hence, x ∈ n(i − 1 p(1) p(t)) which implies that n(i −t) ⊂ n(i − 1 p(1) p(t)). since 1 p(1) p(t) is demicompact, we deduce that α(i − 1 p(1) p(t)) < ∞ and as consequence, α(i − t) < ∞. in order to complete the proof, we will check that r(i − t) is closed. indeed, since n(i − t) is finite dimensional, then there exists from lemma 5.1 in [19] a closed subspace x0 of x such that x = n(i − t) ⊕ x0. next, we let t0 be the restriction of i − t to x0. then, t0 is continuous and we shall see that n(t0)= {0}. since x0 is also closed and (i − t)(x0) = t0(x0) = (i − t)(x) = r(i − t), we need only to prove that t0(x0) is closed. to this end, we shall prove that t −10 : t0(x0) −→ x0 is continuous. by linearity, it is equivalent to that t −10 is continuous at 0. assume the contrary, for every n ∈ n, there exists a sequence (xn)n in x0 which does not converge to 0 such that (i − t)(xn) converges to 0. then, we can find ε > 0 such that ∥xn∥ ≥ ε > 0 for all n ∈ n. then, 1 ∥xn∥ ≤ 1 ε for all n ∈ n. it is clear that yn := xn/∥xn∥ has a norm equal to 1 and (i − t)(yn) → 0. this together with the relation (3.1) leads to( p(t) − p(1) ) yn → 0. since p(1) ̸= 0, then ( i − 1 p(1) p(t) ) yn → 0. 18 f. ben brahim, a. jeribi, b. krichen using the demicompactness of 1 p(1) p(t), we deduce that (yn)n admits a converging subsequence to an element y in x0, verifying ∥y∥ = 1. using the closedness of i − t , we get (i − t)y = 0, which implies that y ∈ n(i − t). this contradict the fact x0 ∩ n(i − t) = {0} and ∥y∥ = 1, which achieves the proof. remark 3.1. the converse of theorem 3.1 is not true, in fact if we take a demicompact operator t such that −t is not demicompact, then t 2 is not demicompact. example. before giving the example we recall the following definition and theorems. definition 3.1. the caputo derivative of fractional order α of function x ∈ cm is defined as cd (α) 0,t x(t) = d (−m+α) 0,t dm dtm x(t) = 1 γ(m − α) ∫ t 0 (t − τ)m−α−1x(m)(τ)dτ, in which m−1 < α < m ∈ n and γ is the well-known euler gamma function. theorem 3.2. [14] if x(t) ∈ c1[0, t ], for t > 0 then cd (α2) 0,t cd (α1) 0,t x(t) = cd (α1) 0,t cd (α2) 0,t x(t) = cd (α1+α2) 0,t x(t); t ∈ [0, t ], where α1 and α2 ∈ r+ and α1 + α2 ≤ 1. theorem 3.3. [14] if x(t) ∈ cm[0, t], for t > 0 then cd (α) 0,t x(t) = cd (αn) 0,t · · · cd (α2) 0,t cd (α1) 0,t x(t); t ∈ [0, t], where α = ∑n i=1 αi; αi ∈ (0, 1], m − 1 ≤ α < m ∈ n and there exists ik < n, such that ∑ik j=1 αj = k, and k = 1, 2, . . . , m − 1. let cω be the space of continuous ω-periodic functions x : r −→ r and c′ω the space of continuously differentiable ω-periodic functions x : r −→ r. cω equipped with the maximum norm ∥ · ∥∞ and c′ω with the norm given by ∥·∥1∞ = max{∥u∥∞, ∥u∥′∞} for u ∈ c′ω are banach spaces. let us consider the following differential equation: x′(t) = a(t)x′(t − h1) + b(t)x(t − h2) + f(t). spectral properties for polynomial and matrix operators 19 here, a and b are continuous ω-periodic functions such that |a(t)| < k, (k < ∞), where k < 1 ω if ω > 2 or k < 1 2 if ω ≤ 2; f ∈ cω is a given function and x ∈ c′ω is an unknown function. this equation can be rewritten in the operator from gx − ax = f, where g : c′ω −→ cω is given by the formula (gx)(t) = x′(t), and the operator a : c′ω → cω by the formula (ax)(t) = a(t)x′(t − h1) + b(t)x(t − h2). let us consider the polynomial p(x) = xn and the operator t = cd ( 1 n ); n ∈ n\{0}, where cd( 1 n ) is the caputo derivative of fractional order 1 n . applying theorem 3.3, we get p(t) = t n(x) = [cd ( 1 n )]nx(t) = x′(t). clearly, p(t) is bounded linear operator with ∥p(t)∥ = 1 and therefore, p(t) is 1-set-contractive. hence, using theorem 2.3, we get µcd ( 1 n ) ∈ dc(x) ∀ µ ∈ [0, 1[. 4. characterization of schmoëger and rakocević essential spectra the aim of this section is to give a refinement of the essential approximate point spectrum and the essential defect spectrum. for this, let x be a banach space and t ∈ c(x). let us consider the following sets λx, υt (x), and ψt (x), respectively, defined by: λx = { j ∈ l(x) such that µj is demicompact for all µ ∈ [0, 1] } , υt (x) = { k ∈ l(x) such that ∀λ ∈ ρ(t + k), −(λ − t − k)−1k ∈ λx } , ψt (x) = { k is t-bounded such that ∀λ ∈ ρ(t + k), − k(λ − t − k)−1 ∈ λx } . we also denote: σr(t) := ∩ k∈υt (x) σap(t + k) and σl(t) := ∩ k∈ψt (x) σδ(t + k). 20 f. ben brahim, a. jeribi, b. krichen theorem 4.1. let t ∈ c(x), we have σe7(t) = σr(t), and σe8(t) = σl(t). proof. we first should remark that λ − t = (λ − t − k)[i + (λ − t − k)−1k], (4.1) and λ − t = [i + k(λ − t − k)−1](λ − t − k). (4.2) let us notice that for t ∈ c(x), and k be a t-bounded operator such that λ ∈ ρ(t + k), then, according to closed graph theorem (lemma 2.1 in [18]), k(λ−t −k)−1 is a closed linear operator defined on x and then bounded. we start by showing that σe7(t) ⊂ σr(t) (resp. σe8(t) ⊂ σl(t)). for λ /∈ σr(t) (resp. λ /∈ σl(t)), there exists k ∈ υt (x)(resp. k ∈ ψt (x)) such that λ − t − k is injective (resp. surjective). it follows that λ − t − k ∈ φ+(x), (resp. φ−(x)) and i(λ − t − k) ≤ 0, (resp. i(λ − t − k) ≥ 0). now, since k ∈ υt (x), (resp. k ∈ ψt (x)), −(λ − t − k)−1k ∈ λx, (resp. −k(λ−t −k)−1 ∈ λx), whenever λ ∈ ρ(t +k). using theorem 2.2 we show that i+(λ−t −k)−1k, (resp. i+k(λ−t −k)−1) is a fredholm operator and i(i +(λ−t −k)−1k) = 0, (resp. i(i +k(λ−t −k)−1) = 0). which implies that (i + (λ − t − k)−1k) ∈ φ+(x), (resp. (i + k(λ − t − k)−1 ∈ φ−(x) and i(i + (λ − t − k)−1k) ≤ 0, (resp. i(i + k(λ − t − k)−1) ≥ 0). hence, applying theorem 5.26 (resp. 5.30) in [19] on (4.1) (resp. (4.2)), we obtain λ − t ∈ φ+(x) (resp. φ−(x)) and i(λ − t) ≤ 0 (resp. i(λ − t) ≥ 0). thanks to lemma 1.1, we conclude that λ /∈ σe7(t) (resp. λ /∈ σe8(t)). conversely, remark that k(x) ⊂ υ(x) (resp. k(x) ⊂ ψ(x)). in fact, if k ∈ k(x) and λ ∈ ρ(t + k), then −µ(λ − t − k)−1k ∈ k(x) ⊂ dc(x) (resp. −µk(λ − t − k)−1 ∈ k(x) ⊂ dc(x)). hence, σr(t) ⊂ σe7(t) (resp. σl(t) ⊂ σe8(t)). corollary 4.1. let t ∈ c(x) and let γ(x) be a subset of x containing k(x). then, (i) if γ(x) ⊂ υt (x), then σe7(t) = ∩ k∈γ(x) σap(t + k) spectral properties for polynomial and matrix operators 21 (ii) if γ(x) ⊂ ψ(x), then σe8(t) = ∩ k∈γ(x) σδ(t + k). proof. since k(x) ⊂ γ(x) ⊂ υt (x) (resp. k(x) ⊂ γ(x) ⊂ ψ(x)), we obtain∩ k∈υt (x) σap(t + k) ⊂ ∩ k∈γ(x) σap(t + k) ⊂ ∩ k∈k(x) σap(t + k) := σe7(t), (resp.∩ k∈ψt (x) σδ(t + k) ⊂ ∩ k∈γ(x) σδ(t + k) ⊂ ∩ k∈k(x) σδ(t + k) := σe8(t) ). the use of theorem 4.1 allows us to conclude that σe7(t) = ∩ k∈γ(x) σap(t + k), and σe8(t) = ∩ k∈γ(x) σδ(t + k). hence, we get the desired result. 5. demicompactness results for operator matrices in this section, we are concerned with some new results which can be used to determinate the essential spectra of the matrix operator l, the closure of l0, on the space x × x, where x is a banach space. in the product space x × x, we consider an operator which is formally defined by a matrix l0 := ( a b c d ) , (5.1) where the operator a acts on x and has domain d(a), d is defined on d(d) and acts on the banach space x, and the intertwining operator b (resp. c) is defined on the domain d(b) (resp., d(d)) and acts on x. in the following, it is always assumed that the entries of this matrix satisfy the following conditions, introduced in [20]. 22 f. ben brahim, a. jeribi, b. krichen (h1) a is closed, densely defined linear operator on x with nonempty resolvent set ρ(a). (h2) the operator b is a densely defined linear operator on x and for (hence all) µ ∈ ρ(a), the operator (a − µ)−1b is closable. (in particular, if b is closable, then (a − µ)−1b is closable). (h3) the operator c satisfies d(a) ⊂ d(c), and for some (hence all) µ ∈ ρ(a), the operator c(a−µ)−1 is bounded. (in particular, if c is closable, then c(a − µ)−1 is bounded). (h4) the lineal d(b)∩d(d) is dense in x and for some (hence all) µ ∈ ρ(a), the operator d − c(a − µ)−1b is closable. we will denote by s(µ) its closure. remark 5.1. (i) under the assumptions (h1) and (h2), we infer that for each µ ∈ ρ(a) the operator g(µ) := (a − µ)−1b is bounded on x. (ii) from the assumption (h3), it follows that the operator: f(µ) := c(a − µ)−1 is bounded on x. we recall the following result which describes the operator l0. theorem 5.1. ([2]) let conditions (h1)-(h3) be satisfied and the lineal d(b)∩d(d) be dense in x. then, the operator l0 is closable and the closure l of l0 is given by: l = µ − ( i 0 f(µ) i )( µ − a 0 0 µ − s(µ) )( i g(µ) 0 i ) . (5.2) or, spelled out, l : d(l) ⊂ (x × x) −→ x × x( x y ) −→ l ( x y ) = ( a ( x + g(µ)y ) − µg(µ) c ( x + g(µ)y ) − s(µ)y ) , with d(l) = {( x y ) ∈ x × x such that x + g(µ)y ∈ d(a) and y ∈ d ( s(µ) )} . note that the description of the operator l does not depend on the choice of the point µ ∈ ρ(a). spectral properties for polynomial and matrix operators 23 remark 5.2. let λ ∈ c. it follows from (5.2) that λ − l = ( i 0 f(µ) i )( λ − a 0 0 λ − s(µ) )( i g(µ) 0 i ) − (λ − µ)m(µ) := uv (λ)w − (λ − µ)m(µ), (5.3) where m(µ) = ( 0 g(µ) f(µ) f(µ)g(µ) ) . proposition 5.1. let l0 the matrix operator defined in (5.1) satisfies (h1)-(h4) and let l be its closure. suppose that there is µ ̸= 0 such that 1 µ ∈ ρ(a). if the operator µs( 1 µ ) is demicompact, then µl is a demicompact operator. proof. let ( xn yn ) n ∈ d(l) be a bounded sequence such that ( x′n y′n ) := (i − µl) ( xn yn ) → ( x0 y0 ) . recalling the factorization (5.2), one has l = 1 µ i − ( i 0 f ( 1 µ ) i )( 1 µ − a 0 0 1 µ − s ( 1 µ ))(i g(1µ) 0 i ) . then,( x′n y′n ) = ( i 0 f ( 1 µ ) i )( i − µa 0 0 i − µs ( 1 µ ))(i g(1µ) 0 i )( xn yn ) . it follows that( i 0 −f ( 1 µ ) i )( x′n y′n ) = ( i − µa 0 0 i − µs ( 1 µ ))(i g(1µ) 0 i )( xn yn ) . therefore, we get the following system:  (i − µa)−1x′n = xn + g ( 1 µ ) yn. −f ( 1 µ ) x′n + y ′ n = ( i − µs ( 1 µ )) yn. (5.4) 24 f. ben brahim, a. jeribi, b. krichen the use of the second equation of the system (5.4) allows us to conclude that (i − µs( 1 µ ))yn is convergent. this together with the demicompactness of µs( 1 µ ) show that (yn)n has a convergent subsequence. since g( 1 µ ) and (i − µa)−1 are bounded operators, we infer that (xn)n has a convergent subsequence, which proves the demicompactness of µl and this shows our claim. for more generalization, we give the following result. theorem 5.2. let l0 the operator defined in (5.1) satisfies (h1)-(h4) and let l be its closure. suppose that for a certain µ ∈ ρ(a), there is λ ∈ c\{0} such that λa ∈ dc(x). then, if f(µ) ∈ k(x) and λs(µ) ∈ dc(x), we have that λl ∈ dc(x × x). proof. take the following bounded sequence ( xn yn ) ∈ d(l) such that( x′n y′n ) := (i − λl) ( xn yn ) → ( x0 y0 ) . let µ ∈ ρ(a) be such that there is a complex nonzero number λ verifying λa ∈ dc(x). thanks to remark 5.2, one has 1 λ − l = ( i 0 f(µ) i )1λ − a 0 0 1 λ − s(µ)  (i g(µ) 0 i ) − ( 1 λ − µ ) m(µ), where m(µ) = ( 0 g(µ) f(µ) f(µ)g(µ) ) . thus, i − λl = ( i 0 f(µ) i )( i − λa 0 0 i − λs(µ) )( i g(µ) 0 i ) − (1 − λµ)m(µ). therefore,( x′n y′n ) = ( i 0 f(µ) i )( i − λa 0 0 i − λs(µ) )( i g(µ) 0 i )( xn yn ) − (1 − λµ)m(µ) ( xn yn ) . (5.5) spectral properties for polynomial and matrix operators 25 observe that (5.5) is equivalent to( i 0 −f(µ) i )( x′n y′n ) + (1 − λµ) ( i 0 −f(µ) i ) m(µ) ( xn yn ) = ( i − λa 0 0 i − λs(µ) )( i g(µ) 0 i )( xn yn ) . moreover, by making some simple calculations, we may show that( x′n −f(µ)x′n + y′n ) + ( (i − λµ)g(µ)yn (i − λµ)f(µ)xn ) = ( (i − λa)xn + (i − λa)g(µ)yn( i − λs(µ) ) yn ) , in equivalent way,  x′n − λ(µ − a)g(µ)yn = (i − λa)xn. −f(µ)x′n + y′n + (i − λµ)f(µ)xn = (i − λs(µ))yn. (5.6) we deduce from the fact that f(µ) ∈ k(x) and (xn)n is bounded, that (1 − λµ)f(µ)xn has a convergent subsequence. hence, from the second equation of system (5.6), we infer that (i − λs(µ))yn has a convergent subsequence. using the demicompactness of λs(µ), we deduce that there exists a convergent subsequence of (yn)n. now, since g(µ) and µ − a are bounded, we conclude from the first equation of system (5.6) that (i − λa)xn has a convergent subsequence. this together with the fact that λa is demicompact allows us to conclude that (xn)n has a convergent subsequence. therefore, there exists a subsequence of ( xn yn ) n which converges on d(l). thus, λl is demicompact. theorem 5.3. let x be a banach space with dp property. assume that the operator l0 defined in (5.1) and acting on x × x satisfies (h1)(h4) and denote l its closure. suppose that µ ∈ ρ(a), g(µ) ∈ w(x) and f(µ) ∈ f+(x). if the operators a and s(µ) are demicompact, then i − l is an upper semi-fredholm operator. proof. let µ ∈ ρ(a) be such that g(µ) ∈ w(x). since f(µ) is bounded, then the product f(µ)g(µ) ∈ w(x). therefore, we can deduce from remark 2.1 that f(µ)g(µ) ∈ f+(x). this together with the fact that f(µ) ∈ 26 f. ben brahim, a. jeribi, b. krichen f+(x) and g(µ) ∈ w(x) ⊂ f+(x) give us m(µ) ∈ f+(x × x). next, according to (5.3), we have for λ = 1: i − l = ( i 0 f(µ) i )( i − a 0 0 i − s(µ) )( i g(µ) 0 i ) − (1 − µ)m(µ) := uv (1)w − (1 − µ)m(µ). since a and s(µ) are demicompact and thanks to theorem 2.1, the operators i − a and i − s(µ) are upper semi-fredholm, hence v (1) ∈ φ+(x × x). the boundedness of the operators u and w and their inverses gives us that uv (1)w is an upper semi-fredholm operator. owing to the fact that m(µ) ∈ f+(x × x), it follows that i − l is an upper semi-fredholm operator. theorem 5.4. let x be a banach space with dp property. assume that the operator l0 defined in (5.1) acting on the product space x × x satisfies (h1)-(h4) and denote l its closure. suppose that [1, +∞[⊂ ρ(a). then, if there exists a complex number λ such that λd ∈ dc(x) and c(i −λa)−1b ∈ w(x), we have that λl ∈ dc(x × x). proof. we assume that the assumption holds and we take ( xn yn ) n a bounded sequence in d(l) which verifies( x′n y′n ) := (i − λl) ( xn yn ) → ( x0 y0 ) , where [1, +∞[⊂ ρ(a). according to the frobenius-schur factorization, one has λl = i − ( i 0 fλ(1) i )( i − λa 0 0 i − sλ(1) )( i gλ(1) 0 i ) , where fλ(1) = λc(λa − i)−1 , sλ(1) = λd − λ2c(λa − i)−1b and gλ(1) = λ(λa − i)−1b. it follows that( x′n y′n ) = ( i 0 fλ(1) i )( i − λa 0 0 i − sλ(1) )( i gλ(1) 0 i )( xn yn ) , thus,( i 0 −fλ(1) i )( x′n y′n ) = ( i − λa 0 0 i − sλ(1) )( i gλ(1) 0 i )( xn yn ) , spectral properties for polynomial and matrix operators 27 which allows us to get the following system   x′n = (i − λa)xn + (i − λa)gλ(1)yn. −fλ(1)x′n + y′n = ( i − sλ(1) ) yn. (5.7) since, λd ∈ dc(x) and c(λa−i)−1b ∈ w(x), we infer by the use of lemma 2.1 that the operator λd − λ2c(λa − i)−1b is demicompact. now, it is easy to show that if a closable operator is demicompact, then its closure is also demicompact. consequently, sλ(1) is a demicompact operator. moreover, it should be observed that the second equation of the system (5.7) implies the convergence of ((i − sλ(1))yn)n, hence (yn)n has a convergent subsequence. next, since gλ(1) is bounded and (i − λa) is invertible and has a bounded inverse, the first equation of the system (5.7) implies that (xn)n has a convergent subsequence. therefore, there exists a convergent subsequence of ( xn yn ) n which converges in d(l). hence, the demicompactness of λl is proved. the following corollary gives a sufficient condition to guarantee the demicompactness of l, the closure of the closable matrix operator l0. corollary 5.1. let x be a banach space with dp property. assume that the operator l0 defined in (5.1) and acting on x ×x satisfies (h1)-(h4) and denote l its closure. suppose that [1, +∞[⊂ ρ(a). then, if d ∈ dc(x) and c(i − a)−1b ∈ dc(x), we have that l ∈ dc(x × x). proof. the proof is a direct application of theorem 5.4 for λ = 1. 6. essential spectra of matrix operators by means of demicompactness we start this section by giving some notations that we will need in the proof. let l0 be the matrix operator defined in (5.1). assume that l0 satisfies (h1)-(h4) and denote l its closure. let α ∈ c\{0} and we suppose that [1, +∞[⊂ ρ(a). applying remark 5.2 on the operator 1 α l and for the case λ = 1, one has 28 f. ben brahim, a. jeribi, b. krichen i − 1 α l = ( i 0 f 1 α (µ) i )( i − 1 α a 0 0 i − s 1 α (µ) )( i g 1 α (µ) 0 i ) − (1 − µ)m 1 α (µ) := u 1 α v 1 α w 1 α − (1 − µ)m 1 α (µ), (6.1) where m 1 α (µ) := ( 0 g 1 α (µ) f 1 α (µ) f 1 α (µ)g 1 α (µ) ) , f 1 α (µ) := 1 α c ( 1 α a − µ )−1 , g 1 α (µ) := 1 α ( 1 α a − µ )−1 b and s 1 α (µ) := 1 α d − 1 α2 c ( 1 α a − µ )−1 b. theorem 6.1. let x be a banach space with dp property. assume that the matrix operator l0 defined in (5.1) satisfies (h1)-(h4) and denote l its closure. suppose that [1, +∞[⊂ ρ(a), then we have: (i) if for all α ∈ c\{0}, the operators 1 α d ∈ dc(x), 1 α2 c ( i − 1 α a )−1 b ∈ w(x) and m 1 α (µ) ∈ f+(x × x), then σe1(l)\{0} = σe1(a)\{0} ∪ σe1(αs 1 α (µ))\{0}. (ii) if for all λ ∈ [0, 1] and α ∈ c\{0} the operators λ α d ∈ dc(x), c(i − λ α a)−1b ∈ w(x) and m(µ) ∈ f(x × x), then σei(l)\{0} = σei(a)\{0} ∪ σei ( αs 1 α (µ) ) \{0}, where i ∈ {4, 5}, and σei(l)\{0} ⊆ σei(a)\{0} ∪ σei ( αs 1 α (µ) ) \{0}, where i ∈ {7, 8}. proof. (i) let α ∈ c\{0} be such that α /∈ σe1(l). then, α − l = α ( i − 1 α l ) ∈ φ+(x × x). (6.2) spectral properties for polynomial and matrix operators 29 clearly, αi ∈ φ+(x × x). we get then the following equivalence α − l ∈ φ+(x × x) ⇐⇒ ( i − 1 α l ) ∈ φ+(x × x). since 1 α d ∈ dc(x) and 1 α2 c ( i − 1 α a )−1 b ∈ w(x), it follows from corollary 5.1 that the operator 1 α l is demicompact. hence, thanks to theorem 2.1, the operator i − 1 α l ∈ φ+(x × x). using the fact that m 1 α (µ) ∈ f+(x × x), we infer that i − 1 α l ∈ φ+(x × x) if, and only if, the operator u 1 α v 1 α (µ)w 1 α is such too. now, observe that u 1 α and w 1 α are invertible and have bounded inverses, hence i − 1 α l ∈ φ+(x × x) if, and only if, v 1 α (µ) has this property, if and only if, i − 1 α a ∈ φ+(x) and i − s 1 α (µ) ∈ φ+(x). which is equivalent to that α − a ∈ φ+(x) and α − αs 1 α ∈ φ+(x). thus, σe1(l)\{0} = σe1(a)\{0} ∪ σe1 ( αs 1 α (µ) ) \{0}. (ii) we claim that σe4(l)\{0} = σe4(a)\{0} ∪ σe4 ( αs 1 α (µ) ) \{0}. for this purpose, take α ∈ c\{0}. since αi ∈ φ(x), then α − l ∈ φ(x × x) if, and only if, the operator (i − 1 α l) ∈ φ(x × x). next, since λ α d ∈ dc(x) and c(i − λ α a)−1b ∈ w(x) for all λ ∈ [0, 1], we deduce from theorem 5.4 that the operator λ α l is demicompact. hence, according to theorem 2.2, we have i − 1 α l ∈ φ(x × x). using (6.1) and the fact that m 1 α (µ) ∈ f(x × x), we infer that i − 1 α l is a fredholm operator if, and only if, the operator u 1 α v 1 α (µ)w 1 α is such too. now, observe that u 1 α and w 1 α are invertible and have bounded inverses, hence i − 1 α l ∈ φ(x × x) if, and only if, v 1 α (µ) has this property if, and only if, i − 1 α a ∈ φ(x) and i − s 1 α (µ) ∈ φ(x). thus the desired result follows. now, we prove the same equality for the schechter’s essential spectrum. to this end, we take α ∈ c\{0}. it is easy to see that α − l ∈ φ(x × x) and i(α−l) = 0 if, and only if, the operator (i − 1 α l) ∈ φ(x ×x) and i(i − 1 α l) = 0. since λ α d ∈ dc(x) and c(i − λ α a)−1b ∈ w(x) for all λ ∈ [0, 1], it follows from theorem 5.4 that the operator λ α l is demicompact. hence, according to theorem 2.2, the operator i − 1 α l ∈ φ(x ×x) and i(i − 1 α l) = 0. using (6.1) and the fact that m 1 α (µ) ∈ f(x × x), we infer that i − 1 α l is a fredholm 30 f. ben brahim, a. jeribi, b. krichen operator with index zero if, and only if, the operator u 1 α v 1 α (µ)w 1 α is such too. note that u 1 α and w 1 α are invertible and have bounded inverses, then i − 1 α l is fredholm with index zero if, and only if, v 1 α (µ) has this property, if and only if, i − 1 α a and i − 1 α s(µ) are fredholm operator with index zero. therefore, α − a ∈ φ(x) and i(α − a) = 0 and α − αs 1 α (µ) ∈ φ(x) and i(α − αs 1 α (µ)) = 0. hence α /∈ σe5(a)\{0} ∩ σe5(αs 1 α (µ))\{0}. thus, σe5(a)\{0} ∪ σe5(αs 1 α (µ))\{0} ⊆ σe5(l)\{0}. (6.3) conversely, let 0 ̸= α /∈ σe5(a) ∩ σe5(αs 1 α (µ)), then α − a ∈ φ(x) and i(α − a) = 0 and α − αs 1 α (µ) ∈ φ(x) and i(α − αs 1 α (µ)) = 0. which is equivalent to write i − 1 α a ∈ φ(x) and i(i − 1 α a) = 0 and i − s 1 α (µ) ∈ φ(x) and i(i−s 1 α (µ)) = 0. the boundedness of the operators u 1 α and w 1 α and their inverses and the fact that m 1 α (µ) ∈ f(x ×x) give us that i − 1 α l ∈ φ(x ×x) and i(i − 1 α l) = 0. therefore, α − l ∈ φ(x × x) and i(α − l) = 0, hence α /∈ σe5(l)\{0}. this immediately shows that σe5(l)\{0} ⊆ σe5(a)\{0} ∪ σe5 ( αs 1 α (µ) ) \{0}. (6.4) now, the use of (6.3) and (6.4) makes us to conclude that σe5(l)\{0} = σe5(a)\{0} ∪ σe5 ( αs 1 α (µ) ) \{0}. we give now the proof for i = 7. note that the case i = 8 can be checked in the same manner. let α ∈ c\{0}, we have proved for i = 5 that i − 1 α l ∈ φ(x × x) and i(i − 1 α l) = 0. this implies that i − 1 α l ∈ φ+(x × x) and i(i − 1 α l) ≤ 0. if α /∈ σe7(a) ∩ σe7(αs 1 α (µ)), then α − a ∈ φ+(x) and i(α−a) ≤ 0 and α−s 1 α (µ) ∈ φ+(x) and i(α−αs 1 α (µ)) ≤ 0. it remains to get i− 1 α a ∈ φ+(x) and i(i− 1αa) ≤ 0 and i−s 1 α (µ) ∈ φ+(x) and i(i−s 1 α (µ)) ≤ 0. since u 1 α and w 1 α are invertible and have bounded inverses and using the fact that m 1 α (µ) ∈ f+(x × x), we infer that i − 1αl ∈ φ+(x × x) and i(i − 1 α l) ≤ 0. therefore, α − l ∈ φ+(x × x) and i(α − l) ≤ 0. now, by applying lemma 1.1, we conclude that α /∈ σe7(l)\{0} and then, σe7(l)\{0} ⊆ σe7(a)\{0} ∪ σe7 ( αs 1 α (µ) ) \{0}. hence, the theorem is proved. spectral properties for polynomial and matrix operators 31 references [1] w.y. akashi, on the perturbation theory for fredholm operators, osaka j. math. 21 (3) (1984), 603 – 612. [2] f.v. atkinson, h. langer, r. mennicken, a.a. shkalikov, the essential spectrum of some matrix operators, math. nachr. 167 (1994), 5 – 20. [3] m. benzi, g.h. golub, j. liesen, numerical solution of saddle point problems, acta numer. 14 (2005), 1 – 137. [4] w. chaker, a. jeribi, b. krichen, demicompact linear operators, essential spectrum and some perturbation results, math. nachr. 288 (13) (2015), 1476 – 1486. [5] w. chaker, a. jeribi, b. krichen, some fredholm theory results around relative demicompactness concept, preprint, 2017. [6] s. chandrasekhar, “hydrodynamic and hydromagnetic stability”, the international series of monographs on physics, clarendon press, oxford, 1961. [7] r.f. curtain, h. zwart, an introduction to infinite-dimensional linear systems theory, texts in applied mathematics 21, springer-verlag, new york, 1995. [8] k. gustafson, j. weidmenn, on the essential spectrum, j. math. anal. appl. 25 (1969), 121 – 127. [9] a. jeribi, “spectral theory snd applications of linear operators and block operators matrices”, springer, cham, 2015. [10] w.a. kirk, remarks on pseudo-contractive mappings, proc. amer. math. soc. 25 (1970), 820 – 823. [11] b. krichen, relative essential spectra involving relative demicompact unbounded linear operators, acta math. sci. ser. b engl. ed. 34 (2) (2014), 546 – 556. [12] k. latrach, a. jeribi, some results on fredholm operators, essential spectra, and application, j. math. anal. appl. 225 (2) (1998), 461 – 485. [13] k. latrach, essential spectra on spaces with the dunford-pettis property, j. math. anal. appl. 223 (2) (1999), 607 – 622. [14] c. li, w. deng, remarks on fractional derivatives, appl. math. comput. 187 (2) (2007), 777 – 784. [15] a.e. lifschitz, “magnetohydrodynamics and spectral theory”, developments in electromagnetic theory and applications, 4, kluwer academic publishers group, dordrecht, 1989. [16] w.v. petryshyn, remarks on condensing and k-set contractive mappings, j. math. anal. appl. 39 (1972), 717 – 741. [17] v. rakocević, on one subset of m. schechter’s essential spectrum, mat. vesnik 5(18)(33) (4) (1981), 389 – 391. [18] m. schechter, on the essential spectrum of an arbitrary operator, i, j. math. anal. appl. 13 (1966), 205 – 215. 32 f. ben brahim, a. jeribi, b. krichen [19] m. schechter, “principles of functional analysis”, academic press, new york-london, 1971. [20] a.a. shkalikov, on the essential spectrum of matrix operators, math. notes 58 (6) (1995), 1359 – 1362. [21] b. thaller, “the dirac equation”, texts and monographs in physics, springer-verlag, berlin, 1992. [22] c. tretter, “spectral theory of block operator matrices and applications”, imperial college press, london, 2008 [23] f. wolf, on the invariance of the essential spectrum under a change of boundary conditions of partial differential boundary operators, nederl. akad. wetensch. proc. ser. a = indag. math. 62 (1959), 142 – 147. e extracta mathematicae vol. 33, núm. 1, 67 – 108 (2018) partial differential equations and strictly plurisubharmonic functions in several variables jamel abidi department of mathematics, faculty of sciences of tunis 1060 tunis, tunisia abidijamel1@gmail.com presented by manuel maestre received september 22, 2014 abstract: using algebraic methods, we prove that there exists a fundamental relation between partial differential equations and strictly plurisubharmonic functions over domains of cn (n ≥ 1). key words: analytic convex and plurisubharmonic functions, harmonic function, maximal plurisubharmonic, differential equation, analysis, inequalities. ams subject class. (2010): primary 32a10, 32a60, 32a70, 32u05; secondary 32w50. 1. introduction we investigate in this paper the relation between strictly plurisubharmonic functions and partial differential equations in domains of cn, (n ≥ 1). various related results are obtained in this context. several papers developed by lelong [14, 15, 16, 17, 18], sadullaev [20], oka [19], bremermann [4, 5],siciak [21], abidi [1], cegrell [6] and others studied plurisubharmonic functions and related topics are of particular importance in this context. for example we can state that strictly plurisubharmonic functions and analytic subsets are related in domains of cn as follows. let a = {z ∈ c : f(z) = 0} and b = {z ∈ c : g(z) = 0} two analytic subsets of c, where f,g : c → c be 2 analytic functions, fg ̸= 0. put f1 and g1 some analytic primitives of f and g respectively over c. then a ∩ b = ∅ if and only if the function u, u(z,w) = |w − f1(z)|2 + |w − g1(z)|2 , (z,w) ∈ c2 , is strictly psh in c2. some good references for the study of convex functions are [11, 13, 3]. for the study of analytic functions we cite the references [12, 10, 13]. for the study of the extension problem of analytic and plurisubharmonic functions we cite the references [7, 9, 6, 18, 8, 22, 23]. 67 68 j. abidi as usual, n := {1,2, . . .}, r and c are the sets of all natural, real and complex numbers, respectively. let u be a domain of rd, (d ≥ 2); md is the lebesgue measure on rd. let f : u → c be a function; |f| is the modulus of f, re(f) and im(f) are respectively the real and imaginary parts of f. let g : d → c be an analytic function, d is a domain of c. we denote by g(0) = g, g(1) = g′ is the holomorphic derivative of g over d. g(2) = g′′, g(3) = g′′′. in general g(m) = ∂mg ∂zm is the derivative of g of order m for all m ∈ n. let z ∈ cn, z = (z1, . . . ,zn), n ≥ 2. for j ∈ {1, . . . ,n}, we write z = (zj,zj) = (z1, . . . ,zj−1,zj,zj+1, . . . ,zn) where zj = (z1, . . . ,zj−1,zj+1, . . . ,zn) ∈ cn−1. ck(u) = {φ : u → c : φ is of class ck in u}, k ∈ n ∪ {∞}. let φ : u → c be a function of class c2. ∆(φ) is the laplacian of φ. let d be a domain of cn, (n ≥ 1); psh(d) and prh(d) are respectively the class of plurisubharmonic and pluriharmonic functions on d. for all a ∈ c, |a| is the modulus of a; re(a) and im(a) are the real and the imaginary parts of a respectively. 2. main results we begin this study by the next result. theorem 2.1. let h1, . . . ,hn : c → r be n harmonic functions, n ≥ 1. for the function u(z,w) = |w − h1(z)|2 + · · · + |w − hn(z)|2, (z,w) ∈ c2, the following conditions are equivalent: (a) u is strictly psh in c2; (b) { z ∈ c : ∂h1 ∂z (z) = 0 } ∩ · · · ∩ { z ∈ c : ∂hn ∂z (z) = 0 } = ∅. proof. (a) ⇒ (b) because |w − h1|2 + · · · + |w − hn|2 = n|w|2 + (h21 + · · · + h2n) − w(h1 + · · · + hn) − w(h1 + · · · + hn), u is a function of class c ∞ in c2. now let (z,w) ∈ c2, ∂2u ∂z∂z (z,w) = 2 [∣∣∣∣∂h1∂z (z) ∣∣∣∣2 + · · · + ∣∣∣∣∂hn∂z (z) ∣∣∣∣2 ] , ∂2u ∂w∂w (z,w) = n , ∂2u ∂z∂w (z,w) = − ( ∂h1 ∂z (z) + · · · + ∂hn ∂z (z) ) . strictly plurisubharmonic functions 69 the levi hermitian form associated to u is now l(u)(z,w)(α,β) = ∂2u ∂z∂z (z,w)αα + ∂2u ∂w∂w (z,w)ββ + 2 re[ ∂2u ∂z∂w (z,w)αβ] = 2 [∣∣∣∣∂h1∂z (z) ∣∣∣∣2 + · · · + ∣∣∣∣∂hn∂z (z) ∣∣∣∣2 ] αα + nββ + 2 re [ − ( ∂h1 ∂z (z) + · · · + ∂hn ∂z (z) ) αβ ] > 0 , for all (z,w) ∈ c2 and (α,β) ∈ c2\{(0,0)}. thus∣∣∣∣∂h1∂z (z) + · · · + ∂hn∂z (z) ∣∣∣∣2 < 2n [∣∣∣∣∂h1∂z (z) ∣∣∣∣2 + · · · + ∣∣∣∣∂hn∂z (z) ∣∣∣∣2 ] for each z ∈ c. now we use the following lemma. lemma 2.2. let a1, . . . ,an ∈ c and n ≥ 1. we have (i) n(|a1|2 + · · · + |an|2) ≥ |a1 + · · · + an|2 ; (ii) m(|a1|2 + · · · + |an|2) > |a1 + · · · + an|2 if m > n and there exists j0 such that aj0 ̸= 0. proof.( n∑ j=1 aj )( n∑ k=1 ak ) = n∑ j,k=1 ajak ≤ n∑ j,k=1 |aj||ak| ≤ n∑ j,k=1 ( |aj|2 2 + |aj|2 2 ) = 2 n∑ j,k=1 |aj|2 2 = n∑ k=1 n∑ j=1 |aj|2 = n n∑ j=1 |aj|2. now we complete the proof of the theorem. put a = [∣∣∣∣∂h1∂z ∣∣∣∣2 + · · · + ∣∣∣∣∂hn∂z ∣∣∣∣2 ] + [ (2n − 1) (∣∣∣∣∂h1∂z ∣∣∣∣2 + · · · + ∣∣∣∣∂hn∂z ∣∣∣∣2 ) − ∣∣∣∣∂h1∂z + · · · + ∂hn∂z ∣∣∣∣2 ] ; 70 j. abidi a > 0 over c. a = b + c, where b ≥ 0, c ≥ 0. then a = 0 if and only if b = c = 0. thus if z ∈ c such that b(z) = ∣∣∣∂h1∂z (z)∣∣∣2 + · · · + ∣∣∣∂hn∂z (z)∣∣∣2 = 0, then ∂h1 ∂z (z) = · · · = ∂hn ∂z (z) = 0. therefore c(z) = (2n −1) (∣∣∣∂h1∂z (z)∣∣∣2 +· · ·+∣∣∣∂hn∂z (z)∣∣∣2) − ∣∣∣∂h1∂z (z) + · · · + ∂hn∂z (z)∣∣∣2 = 0. the converse is also true. we conclude that a(z) > 0 if and only if b(z) > 0, for all z ∈ c. then ∣∣∣∂h1∂z ∣∣∣2 + · · · + ∣∣∣∂hn∂z ∣∣∣2 > 0 over c if and only if u is strictly psh in c2. but ∣∣∣∂h1∂z ∣∣∣2 + · · · + ∣∣∣∂hn∂z ∣∣∣2 > 0 in c if and only if{ z ∈ c : ∂h1 ∂z (z) = 0 } ∩ · · · ∩ { z ∈ c : ∂hn ∂z (z) = 0 } = ∅ . by this proof we deduce also (b) ⇒ (a). for analytic functions, we have now. theorem 2.3. let g1, . . . ,gn : d → c be n analytic functions, n ∈ n\{1}, d is a domain of c. put u(z,w) = |w − g1(z)|2 + · · · + |w − gn(z)|2, (z,w) ∈ d × c. then u is strictly psh in d × c if and only if n∑ j,k=1 g′jg ′ kδjk > 0 over d, where δjk = (n − 1 if j = k and − 1 if j ̸= k), j,k ∈ {1, . . . ,n}. proof. the function u = n|w|2 + ( |g1|2 + · · · + |gn|2 ) −w(g1 + · · · +gn) − w(g1 + · · · + gn) is of class c∞ over d × c. let (z,w) ∈ c2, ∂2u ∂z∂z (z,w) = |g′1(z)| 2 + · · · + |g′n(z)| 2 , ∂2u ∂w∂w (z,w) = n , ∂2u ∂z∂w (z,w) = − ( g′1(z) + · · · + g ′ n(z) ) . assume that u is strictly psh in d × c. the levi hermitian form of u is now l(u)(z,w)(α,β) = [ |g′1(z)| 2 + · · · + |g′n(z)| 2 ] αα + nββ + 2 re [ − (g′1(z) + strictly plurisubharmonic functions 71 · · · + g′n(z))αβ ] > 0 for all (α,β) ∈ c2\{(0,0)}. thus |g′1 + · · · + g ′ n| 2 < n[|g′1| 2 + · · · + |g′n| 2] over d. then n∑ j,k=1(j ̸=k) g′jg ′ kδjk + (n − 1) [ |g′1| 2 + · · · + |g′n| 2 ] = n∑ j,k=1 g′jg ′ kδjk > 0 on d. now assume that n∑ j,k=1 g′jg ′ kδjk > 0 on d. by the above proof, |g ′ 1 + · · · + g′n| 2 < n [ |g′1| 2 + · · · + |g′n| 2 ] . it follows that l(u)(z,w)(α,β) > 0 for each (α,β) ∈ c2\{0}. the theorem below gives a fundamental part of this paper and the study of the relation between partial differential equations and strictly plurisubharmonic functions over domains of cn, (n ≥ 1). theorem 2.4. let g : d → c be a function, d is a domain of c. put v(z,w) = |w − g(z)|2, for (z,w) ∈ d × c. the following assertions are equivalent: (a) v is strictly psh in d × c ; (b) g is harmonic in d and { z ∈ d : ∂g ∂z (z) = 0 } = ∅. proof. (a) ⇒ (b) v is strictly psh in d × c, then v is psh in d × c. therefore g is harmonic in d by abidi [1]. it follows that v is a function of class c∞ in d×c. let (z,w) ∈ d×c. write v(z,w) = |w|2+|g(z)|2−wg(z)−wg(z). we have ∂2v ∂z∂z (z,w)αα + ∂2v ∂w∂w (z,w)ββ + 2 re [ ∂2v ∂z∂w (z,w)αβ ] > 0 for all (α,β) ∈ c2\{0}, and ∂2v ∂z∂z = ∣∣∣∣∂g∂z ∣∣∣∣2 + ∣∣∣∣∂g∂z ∣∣∣∣2 , ∂2v∂w∂w = 1 , ∂ 2v ∂z∂w = − ∂g ∂z . therefore ∣∣ − ∂g ∂z ∣∣2 < ∣∣∂g ∂z ∣∣2 + |∂g ∂z ∣∣2, and consequently ∣∣∂g ∂z ∣∣2 > 0 over d. (b) ⇒ (a) since g is harmonic in d, then v is a function of class c∞ in d × c. we have ∂2v ∂z∂z = ∣∣∣∣∂g∂z ∣∣∣∣2 + ∣∣∣∣∂g∂z ∣∣∣∣2 , ∂2v∂w∂w = 1 , ∂ 2v ∂z∂w = − ∂g ∂z . 72 j. abidi the levi hermitian form of v is l(v)(z,w)(α,β) = ∂2v ∂z∂z (z,w)αα + ∂2v ∂w∂w (z,w)ββ + 2 re [ ∂2v ∂z∂w (z,w)αβ ] = [∣∣∣∣∂g∂z(z) ∣∣∣∣2 + ∣∣∣∣∂g∂z(z) ∣∣∣∣2 ] αα + ββ + 2 re [ − ∂g ∂z (z)αβ ] , for (z,w) ∈ d × c, (α,β) ∈ c2. since ∣∣∂g ∂z (z) ∣∣2 > 0, for every z ∈ d, then∣∣∣∣ − ∂g∂z(z) ∣∣∣∣2 < ∣∣∣∣∂g∂z(z) ∣∣∣∣2 + ∣∣∣∣∂g∂z(z) ∣∣∣∣2 for each z ∈ d. therefore, l(v)(z,w)(α,β) > 0 , ∀ (z,w) ∈ d × c , ∀ (α,β) ∈ c2\{0}. consequently, v is strictly psh in d × c. observe that if k = k1 + k2, where k1,k2 : d → c be 2 analytic functions in the domain d ⊂ c, and u(z,w) = |w − k(z)|2, (z,w) ∈ d × c, then the strict plurisubharmonicity of u is independent of the function k1. on the other hand if we replace the strict inequality < by the large inequality ≤, then the above theorem is false. remark 2.5. let k : d → c be an analytic function, d is a domain of c. put u(z,w) = |w − k(z)|2, v(z,w) = |w − k(z)|2, where (z,w) ∈ d × c. then u, log(u) and log(v) are not strictly psh functions on any not empty domain of d × c; v is strictly psh in d × c if and only if ∣∣∣∂k∂z∣∣∣ = ∣∣∣∂k∂z∣∣∣ > 0 in d. example. let k(z) = exp(z), z ∈ c, and v1(z,w) = |w − exp(z)|2, v2(z,w) = |w − exp(z)|2, for (z,w) ∈ c2; v1 is not strictly psh on any open of c2, but v2 is strictly psh in all c2. note that log(v2) is not strictly psh on any domain of { (z,w) : |w − exp(z)|2 > 0 } . on the other hand, g1(z) = z and g2(z) = 1 − z (z ∈ c) are analytic functions over c. set v(z,w) = |w − g1(z)|2 + |w − g2(z)|2, (z,w) ∈ c2. let (α,β) ∈ c2. the levi hermitian form of v is l(v)(z,w)(α,β) = αα + ββ + 2 re [ − αβ ] + αα + ββ + 2 re [ αβ ] = 2(αα + ββ) > 0 , ∀ (z,w) ∈ c × c , ∀ (α,β) ∈ c2\{0}. strictly plurisubharmonic functions 73 then v is strictly psh in c2. observe that in this case if we put u1(z,w) = |w − g1(z)|2, u2(z,w) = |w − g2(z)|2, then u1 and u2 are plurisubharmonic over c2 but not strictly psh functions on any domain of c2. but v = (u1 +u2) is strictly psh in c2. in fact we have the following result. claim 2.6. let g1,g2 : d → c be 2 analytic functions, d is a domain of c and v(z,w) = |w − g1(z)|2 + |w − g2(z)|2, where (z,w) ∈ d × c. then v is strictly psh in d × c if the function re [ g′1g ′ 2 ] < 0 over d. if d = c, then v is strictly psh in c2 if for example (g′1g ′ 2) is equal a constant c over c and re(c) < 0. according to the paper abidi [1], we can prove the following extension. claim 2.7. let a,b ∈ c. put v(z,w) = |(w − z)2 − (a + b)(w − z) + ab|, where (z,w) ∈ c2. then v is strictly psh on c2 if and only if a = b. in general we can state the following result: for all g : c → c be analytic, if we put u(z,w) = ∣∣(w − g(z))2 − (a + b)(w − g(z)) + ab∣∣ , where (z,w) ∈ c2, then u is strictly psh on c2 if and only if (a = b and∣∣∂g ∂z (z) ∣∣ > 0 for all z ∈ c). theorem 2.8. let d be a domain of c and g : d → c be an analytic function. the following statements are equivalent: (a1) |w − g|2 is strictly psh in d × c; (a2) |w − g|2 + |w − g|2 is strictly psh in d × c; (a3) |∂g∂z | > 0 in d; (a4) |w − cg − g|2 is strictly psh in d × c, where c ∈ c\{0}; (a5) |w1 − g|2 + |w2 − g|2 is strictly psh in d × c × c; (a6) for all n ∈ n, ( |w1 − g|2 + · · · + |wn − g|2 + |wn+1 − g|2 ) is strictly psh in d × cn+1. proposition 2.9. let g : d → c be analytic, d is a domain of c. g = h + ik, h = re(g), k = im(g). let a,b ∈ c, (a ̸= 0 or b ̸= 0). put u(z,w) = |w − g(z)|2, v(z,w) = |w − ah(z)|2 + |w − bk(z)|2, u1(z,w) = |w − h(z)|2, u2(z,w) = |w − k(z)|2, where (z,w) ∈ d × c. we have the equivalents: 74 j. abidi (a) u is strictly psh in d × c; (b) u1 is strictly psh in d × c; (c) u2 is strictly psh in d × c; (d) v is strictly psh in d × c. observe that in general we can not compare the structure strictly psh of the functions v1 and v2 where v1(z,w) = |w − g(z)|2, v2(z,w) = |w − g(z)|2, g : c → c be analytic and (z,w) ∈ c2. but if we add another function constructed according to the expression of g we have the following extension. claim 2.10. let g : cn → c be analytic g = h + ik, h = re(g), n ∈ n. denote by φ(z,w) = |w−g(z)|2, φ1(z,w) = |w−h(z)|2+|w−g(z)|2, φ2(z,w) = |w − h(z)|2 + |w − g(z)|2, φ3(z,w) = |w − g(z)|2, where (z,w) ∈ cn × c. we have the equivalents: (a) φ1 is strictly psh in cn × c; (b) φ2 is strictly psh in cn × c; (c) n = 1 and φ3 is strictly psh in c2. note that φ is not strictly psh on all not empty domain of c2. at this stage of the development, observe that if f : cn → r is pluriharmonic (n ≥ 1), and f(z,w) = |w − f(z)|2, where (z,w) ∈ cn × c, then f is not strictly psh on any not empty domain of cn × c if and only if (a1) n = 1 and f is constant in c, or (a2) n ≥ 2 and f is an arbitrary prh function over cn. the function f have real valued is of great importance in this subject. some fundamental remarks concerning strictly psh functions. at the beginning of this statements we observe the following assertions: let h : d → c be a function, d is a convex domain of c. if |w − h|2 is psh (resp. convex) in d × c, then ∣∣w − h∣∣2 is psh (resp. convex) in d × c and conversely. but we can obtain |w − h|2 is strictly psh (resp. strictly psh and convex) in d × c and ∣∣w − h∣∣2 is not strictly psh (resp. not strictly psh and convex) on any domain subset of d × c. this is one of the great differences between the classes of functions psh, convex, of the first part and the classes of strictly psh, (strictly psh and convex) functions for the second part. consequently, if strictly plurisubharmonic functions 75 we replace the large inequality ≤ by the strict inequality < the above result is not true. now let g1, . . . ,gn : cn → c be n analytic functions, where n,n ≥ 1. put u1(z,w) = |w − g1(z)|2 + · · · + |w − gn(z)|2, v1(z,w) = ∣∣w − g1(z)∣∣2 + · · · + ∣∣w − gn(z)∣∣2, (z,w) ∈ cn × c. if u1 is strictly psh in cn × c, then{ ∂ ∂z1 (g1, . . . ,gn), . . . , ∂ ∂zn (g1, . . . ,gn) } is linearly independent over cn and n < n (by using the hermitian levi form of the function u1). if u1 is strictly psh in cn × c, then v1 is strictly psh in cn × c. but not conversely. example. the functions k1(z) = z, k2(z) = z 2 (z ∈ c) are analytic over c. let v1(z,w) = ∣∣w −k1(z)∣∣2 + ∣∣w −k2(z)∣∣2, where (z,w) ∈ c2; v1 is strictly psh on c2. put u1(z,w) = |w−z|2+|w−z2|2, where (z,w) ∈ c2. let α,β ∈ c. the levi hermitian form of u1 is l(u1)(z,w)(α,β) = |β − α|2 + |β − 2zα|2. if z = 1 2 , then we have l(u1) ( 1 2 ,w ) (α,α) = 0 for each α ∈ c\{0}. therefore u1 is not strictly psh in c2. put u2(z,w) = |w1 − g1(z)|2 + · · · + |wn − gn(z)|2, z ∈ cn, w = (w1, . . . ,wn) ∈ cn; u2 is not strictly psh in any not empty domain of cn × cn. now put v2(z,w) = ∣∣w1 − g1(z)∣∣2 + · · · + ∣∣wn − gn(z)∣∣2. if for all fixed z in cn, the system   ∂g1 ∂z1 (z)α1 + · · · + ∂g1∂zn (z)αn = 0 ... ∂gn ∂z1 (z)α1 + · · · + ∂gn∂zn (z)αn = 0 (α1, . . . ,αn ∈ c) has only the solution (α1, . . . ,αn) = (0, . . . ,0), then v2 is strictly psh in cn × cn. therefore u2 and v2 do not have the same structure in the theory of the strictly plurisubharmonic functions. put u3(z,w) = ∣∣w − φ1(z)∣∣2 + ∣∣w − φ2(z)∣∣2, where (z,w) ∈ c2, φ1,φ2 : c → c are analytic functions, u4(z,w) = ∣∣w − φ1(z)∣∣2 + ∣∣w − φ2(z)∣∣2. then u3 is strictly psh in c2 if and only if u4 is strictly psh in c2. question 2.11. an original problem of the theory of functions in several complex variables is now the following. let f0, . . . ,fk−1 : cn → c be k 76 j. abidi analytic functions, (n,k ≥ 1). set u(z,w) = ∣∣wk + fk−1(z)wk−1 + · · · + f1(z)w + f0(z)∣∣ , v(z,w) = ∣∣wk + fk−1(z)wk−1 + · · · + f1(z)w + f0(z)∣∣ , where (z,w) ∈ cn × c. u is convex in cn × c if and only if v is convex in cn × c. now note that u is psh in cn × c, but v is not in general (example take v1(z,w) = ∣∣w2 + zw∣∣ is not psh in c2). find the condition described by the functions f0, . . . ,fk−1 such that v is psh in cn × c. (observe that we can consider in this study the question of a power series). remark 2.12. the above proposition is not true if g : d → c is harmonic. for example, if g : c → r, g(z) = x1, z = (x1 + ix2) ∈ c, where x1,x2 ∈ r, then ∣∣w − g∣∣2 is strictly psh in c2. but im(g) = 0 and |w − 0|2 = |w|2 is not strictly psh on any domain of c2. theorem 2.13. let g1, . . . ,gn : c → c, u(z,w) = |w1 − g1(z)|2 + · · · + |wn − gn(z)|2, where (z,w) = (z,w1, . . . ,wn) ∈ c × cn,n ∈ n. u is strictly psh in c × cn if and only if g1, . . . ,gn are harmonic functions in c and∣∣∂g1 ∂z ∣∣2 + · · · + ∣∣∂gn ∂z ∣∣2 > 0 on c. proof. assume that u is strictly psh on c×cn. note that u is a function of class c∞ on c×cn. let (z,w) = (z,w1, . . . ,wn) ∈ c×cn. fix w2, . . . ,wn ∈ c. then the function u(., .,w2, . . . ,wn) is strictly psh on c2. by abidi [1], g1 is harmonic on c. consequently, g1, . . . ,gn are harmonic functions on c. put gj = fj + kj, where fj,kj : c → c be two analytic functions and j ∈ {1, . . . ,n}. let (α,β) = (α,β1, . . . ,βn) ∈ c × cn\{(0,0)}. the levi hermitian form of u is now l(u)(z,w)(α,β) = ∣∣∣∣β1 − ∂f1∂z (z)α ∣∣∣∣2 + ∣∣∣∣∂k1∂z (z)α ∣∣∣∣2 + · · · + ∣∣∣∣βn − ∂fn∂z (z)α ∣∣∣∣2 + ∣∣∣∣∂kn∂z (z)α ∣∣∣∣2. assume that α ̸= 0. put β1 = ∂f1∂z (z)α,. . . ,βn = ∂fn ∂z (z)α. then l(u)(z,w)(α,β) = (∣∣∣∣∂k1∂z (z)α ∣∣∣∣2 + · · · + ∣∣∣∣∂kn∂z (z)α ∣∣∣∣2 ) strictly plurisubharmonic functions 77 for each z ∈ c. thus ∣∣∣∣∂g1∂z (z) ∣∣∣∣2 + · · · + ∣∣∣∣∂gn∂z (z) ∣∣∣∣2 > 0 . the converse is trivial. observe that the notion u is strictly psh in c × cn on the above theorem is independent of f1, . . . ,fn, where gj = fj + kj, fj,kj : c → c are analytic functions (1 ≤ j ≤ n). proposition 2.14. for every g : d → c analytic, d is a domain of cn, (n ≥ 2), u = |g|2 is not strictly psh on any domain d1 ⊂ d. indeed e|g| 2 , |g|2e|g| 2 , |g|2e|g| 2 ee |g|2 are not strictly psh functions in any domain d2 ⊂ d. for example let v = |g1|2 + · · · + |gn|2, where g1, . . . ,gn : cn → c are analytic functions. then v is strictly psh in cn if and only if the determinant det ( ∂gj ∂zk (z) ) j,k ̸= 0, for all z ∈ cn. note that we have the assertion. let g1, . . . ,gn : d → c be n analytic functions, d is a domain of cn, n ≥ 2, n ≥ 1. if n < n, then u = |g1|2 + · · · + |gn|2 is not strictly psh on any domain d1 ⊂ d. in fact u is a function of class c∞ in d. the levi hermitian form of u is l(u)(z)(α) = n∑ j,k=1 ∂2u ∂zj∂zk (z)αjαk = ∣∣∣∣∣ n∑ j=1 ∂g1 ∂zj (z)αj ∣∣∣∣∣ 2 + · · · + ∣∣∣∣∣ n∑ j=1 ∂gn ∂zj (z)αj ∣∣∣∣∣ 2 for each z = (z1, . . . ,zn) ∈ d and α = (α1, . . . ,αn) ∈ cn. suppose that u is strictly psh in d. then for all z ∈ d, for all α1, . . . ,αn ∈ c, l(u)(z)(α1, . . . ,αn) = 0 if and only if  ∂g1 ∂z1 (z)α1 + · · · + ∂g1∂zn (z)αn = 0 ... ∂gn ∂z1 (z)α1 + · · · + ∂gn∂zn (z)αn = 0. then α1 ( ∂g1 ∂z1 (z), . . . , ∂gn ∂z1 (z) ) + · · · + αn ( ∂g1 ∂zn (z), . . . , ∂gn ∂zn (z) ) = 0 78 j. abidi implies that α1 = · · · = αn = 0. therefore the subset of vectors{( ∂g1 ∂z1 (z), . . . , ∂gn ∂z1 (z) ) , . . . , ( ∂g1 ∂zn (z), . . . , ∂gn ∂zn (z) )} is a free family of n vectors of cn, and n < n. this is a contradiction. consequently, u is not strictly psh on any domain d1 ⊂ d. but we have the following result: for all n ∈ n, there exists u1, . . . ,un : c2n → c be n pluriharmonic functions such that v = ( |u1|2 + · · · + |un|2 ) is strictly psh in c2n. example. put uj(z) = zj + zn+j, 1 ≤ j ≤ n, where z = (z1, . . . ,z2n) ∈ c2n. uj is in fact prh in c2n; |u1(z)|2 = |z1+zn+1|2 = |z1|2+|zn+1|2+z1zn+1+ z1zn+1. note that the function k1(z) = z1zn+1 +z1zn+1, k1 is pluriharmonic in c2n and therefore the levi hermitian form of k1 is equal 0 over c2n × c2n. then l ( |u1|2)(z)(α1, . . . ,α2n) = |α1|2 + |αn+1|2. then l(v)(z)(α1, . . . ,α2n) = 2n∑ j=1 |αj|2 > 0 if (α1, . . . ,α2n) ∈ c2n\{0}. then v is strictly psh in c2n, but n < 2n. in fact for all n ≥ 1, there exists a function u : cn → r pluriharmonic such that |u|2 is not strictly psh in cn, u is not constant. observe that we have if h : c3 → c is pluriharmonic, then |h|2 is not strictly psh in c3. exactly we have for all h1, . . . ,hs : cn → c prh, if s < n2 , then (|h1| 2 + · · · + |hs|2) is not strictly psh in cn. now if one of the function have real valued, one of the above result is not true. for example, if u : c2 → r is a pluriharmonic function, then u2 is not strictly psh on c2. theorem 2.15. let u1, . . . ,un : c2n → r be n pluriharmonic functions, n ∈ n. set u = u21 + · · · + u 2 n. then u is not strictly psh on any domain of c2n. proof. the functions u21, . . . ,u 2 n and u are of class c ∞ in c2n. denote by l(u)(z)(α1, . . . ,α2n) = 2n∑ j,k=1 ∂2u ∂zj∂zk (z)αjαk for all z = (z1, . . . ,z2n) ∈ c2n and for all α = (α1, . . . ,α2n) ∈ c2n. we have l(u)(z)(α1, . . . ,α2n) = l(u 2 1)(z)(α1, . . . ,α2n) + · · · + l(u 2 n)(z)(α1, . . . ,α2n) strictly plurisubharmonic functions 79 and l(u21)(z)(α1, . . . ,α2n) = 2n∑ j,k=1 ∂2(u21) ∂zj∂zk (z)αjαk = 2 2n∑ j,k=1 ∂u1 ∂zj (z) ∂u1 ∂zk (z)αjαk = 2 ( 2n∑ j=1 ∂u1 ∂zj (z)αj )( 2n∑ k=1 ∂u1 ∂zk (z)αk ) = 2 ∣∣∣∣∣ 2n∑ j=1 ∂u1 ∂zj (z)αj ∣∣∣∣∣ 2 . consequently, l(u)(z)(α1, . . . ,α2n) = 2 ∣∣∣∣∣ 2n∑ j=1 ∂u1 ∂zj (z)αj ∣∣∣∣∣ 2 + · · · + 2 ∣∣∣∣∣ 2n∑ j=1 ∂un ∂zj (z)αj ∣∣∣∣∣ 2 . fix z = (z1, . . . ,z2n) ∈ c2n. l(u)(z)(α1, . . . ,α2n) = 0 if and only if 2n∑ j=1 ∂u1 ∂zj (z)αj = 0 , . . . , 2n∑ j=1 ∂un ∂zj (z)αj = 0. then   ∂u1 ∂z1 (z)α1 + · · · + ∂u1∂z2n (z)α2n = 0 ... ∂un ∂z1 (z)α1 + · · · + ∂un∂z2n (z)α2n = 0. thus α1 ( ∂u1 ∂z1 (z), . . . , ∂un ∂z1 (z) ) + · · · + α2n ( ∂u1 ∂z2n (z), . . . , ∂un ∂z2n (z) ) = (0, . . . ,0) ∈ cn, where α1, . . . ,α2n ∈ c. we have 2n vectors of cn (considered a vector space). therefore the subset of the above 2n vectors is not a linearly independent family in the c-vector space cn of dimension n. then there exists (α1, . . . ,α2n) ∈ c2n\{0} such that l(u)(z)(α1, . . . ,α2n) = 0. consequently, u is not strictly psh on any not empty domain of c2n. definition 2.16. (klimek [12]) let u : d → r be a psh function, where d is an open of cn, n ≥ 1. u is maximal psh on d if for all relatively compact open g subset of d and for each upper semi continuous function v on g such that v is psh on g and v ≤ u on ∂g, we have v ≤ u on g. 80 j. abidi remark 2.17. (a) let n ∈ n, n ≥ 2. given u1, . . . ,un−1 : d → r be n − 1 pluriharmonic functions, where d is a domain of cn. then u =( u21 + · · · + u 2 n−1 ) is not strictly psh on any domain d1 ⊂ d. (b) let n ∈ n and d a domain of cn. consider h1, . . . ,hn : d → r be n pluriharmonic functions and put u = h21 + · · · + h 2 n. u is psh on d. then u is strictly psh on d if and only if det ( ∂hj ∂zk (z) ) 1≤j, k≤n ̸= 0 for all z ∈ d. (c) let g : d → c be analytic, d is a domain of cn, (n ≥ 2). u = |g|2 is maximal plurisubharmonic (in the sense of klimek [12] or sadullaev [20]). but if k : c → c is analytic not constant, then |k|2 = v is not maximal subharmonic because v is not harmonic. this is one of the great differences between the theory of functions of one complex variable and the same theory in several complex variables. in one complex variable, the sum of 2 maximal subharmonic functions is maximal subharmonic. if now g1,g2 : c2 → c be 2 analytic functions such that |g1|2 + |g2|2 = φ is strictly psh in c2, then |g1|2 + |g2|2 is psh but not maximal plurisubharmonic on any open of c2. in this case |g1|2 and |g2|2 are maximal plurisubharmonic functions on c2. but the sum |g1|2 + |g2|2 = φ is not maximal psh on any not empty open of c2. but we have the following result. let g1, . . . ,gn : d → c be n analytic functions (n ≥ 1). then if n < n, u = |g1|2 + · · · + |gn|2 is maximal plurisubharmonic on d. proposition 2.18. there exists a function u : c2 → r, u real analytic on c2, u is maximal plurisubharmonic on c2, but eu is plurisubharmonic on c2 and not maximal plurisubharmonic on any not empty domain of c2. moreover, for all v : d → r prh, (d is a domain of cn, n ≥ 2) the function ev is maximal plurisubharmonic on d. proof. let u(z1,z2) = x 2 1 + x2, where z1 = (x1 + ix3), z2 = (x2 + ix4) ∈ c (x1,x2,x3,x4 ∈ r). u is plurisubharmonic in c2 and real analytic. we have the determinant det ( ∂2u ∂zj∂zk (z) ) j,k = 0 for each z ∈ c2. by klimek [12], u is maximal plurisubharmonic in c2. now det ( ∂2(eu) ∂zj∂zk (z) ) j,k = 1 8 ̸= 0 for every z ∈ c2. by klimek [12, proposition 3.1.6], eu is not maximal psh on any domain of c2. strictly plurisubharmonic functions 81 lemma 2.19. let u : d → r be plurisubharmonic, where d is a domain of cn, n ≥ 1. if eu is maximal psh on d, then u is maximal psh on d. proof. let g be a relatively compact open subset of d and v : g → [−∞,+∞[ be an upper semi continuous function such that v is psh on g and v ≤ u on ∂g. then ev ≤ eu on ∂g and consequently, ev ≤ eu on g. it follows that v ≤ u on g. remark 2.20. for all n ≥ 1, for all domain d of cn, there exists u : d → r be c∞ psh such that eu is strictly psh on d but u is not strictly psh on any domain d1 ⊂ d. in general we have the following lemma. lemma 2.21. let a,b two hermitian matrix of type (n,n) with coefficients in c. suppose that a and b are positive semi-definite. (a) if a is positive definite then a + b is positive definite on cn. (b) if the determinant det(a) ̸= 0 then a is positive definite on cn. (c) if a+b is positive definite, we can not conclude that a or b is positive definite on cn if n ≥ 2. example. let d be a domain of c2. let f = {(g1,g2) / g1,g2 : d → c be analytic functions such that (|g1|2 + |g2|2) is strictly psh in c2}. let (g1,g2) ∈ f. fix z = (z1,z2) ∈ d. put a = ( ∂2|g1|2 ∂zj∂zk (z) ) j, k , b = ( ∂2|g2|2 ∂zj∂zk (z) ) j, k . a and b are hermitian matrix positive semi definite on c2. then a+b is an hermitian matrix positive definite, but a and b are not positive definite over c2. now we can prove the following result. theorem 2.22. let u : d → r be a function of class c2, d is a domain of cn, n ≥ 1. suppose that u is psh on d. then eu is maximal psh on d if and only if ee u is maximal psh on d. therefore if eu is maximal psh on d, then fs(u) is maximal psh on d, for each s ∈ n, where fs = exp ◦ exp ◦ · · · ◦ exp (s times). 82 j. abidi this theorem have good and several applications in problems and exercises. proof. if eu is maximal psh on d. since eu is psh and of class c2 in d, then det ( ∂2(eu) ∂zj∂zk (z) ) j,k = 0 for all z ∈ d. fix z ∈ d. thus the matrix a = ( ∂2(u) ∂zj∂zk (z) + ∂u ∂zj (z) ∂u ∂zk (z) ) j,k is not an injection. hence there exists α = (α1, . . . ,αn) ∈ cn\{0} such that aα = 0. if < .,. > is the hermitian habitual product on cn, then < α,aα >= 0, n∑ j,k=1 ∂2(u) ∂zj∂zk (z)αjαk + n∑ j,k=1 ∂u ∂zj (z) ∂u ∂zk (z)αjαk = 0 . as a consequence n∑ j,k=1 ∂2(u) ∂zj∂zk (z)αjαk + ∣∣∣∣∣ n∑ j=1 ∂u ∂zj (z)αj ∣∣∣∣∣ 2 = 0 . since u is psh and of class c2 in d, thus n∑ j,k=1 ∂2(u) ∂zj∂zk (z)αjαk ≥ 0. now since ∣∣∣∣∣ n∑ j=1 ∂u ∂zj (z)αj ∣∣∣∣∣ 2 ≥ 0, it follows that n∑ j,k=1 ∂2(u) ∂zj∂zk (z)αjαk = 0 and∣∣∣∣∣ n∑ j=1 ∂u ∂zj (z)αj ∣∣∣∣∣ 2 = n∑ j,k=1 ∂u ∂zj (z) ∂u ∂zk (z)αjαk = 0, and thus ( 1 + eu(z) ) n∑ j,k=1 ∂u ∂zj (z) ∂u ∂zk (z)αjαk = 0 . consequently, n∑ j,k=1 ∂2(u) ∂zj∂zk (z)αjαk + ( 1 + eu(z) ) n∑ j,k=1 ∂u ∂zj (z) ∂u ∂zk (z)αjαk = n∑ j,k=1 ( ∂2(u) ∂zj∂zk (z) + ( 1 + eu(z) ) ∂u ∂zj (z) ∂u ∂zk (z) ) αjαk = 0 . strictly plurisubharmonic functions 83 now the matrix b = ( ∂2(u) ∂zj∂zk (z) + (1 + eu(z)) ∂u ∂zj (z) ∂u ∂zk (z) ) j, k is an hermitian matrix positive semi definite because ee u is psh on d. if det(b) ̸= 0, then b is positive definite on cn. but there exists α ∈ cn\{0} such that < α,bα >= 0. then b is not definite positive in cn. consequently, det(b) = 0 and we have ee u is maximal psh on d. the converse is trivial. example. let h : d → r be prh, where d is a domain of cn, n ≥ 2. we denote by fs = exp ◦ exp ◦ · · · ◦ exp (s times), for s ∈ n (and f0 is the identity operator). then fs(h) is maximal plurisubharmonic in d. now let s,t ∈ n. thus the function fs(h) − ft(h) is maximal plurisubharmonic in d in the case s ≥ t. (we prove that det ( ∂2(fs(h)−ft(h)) ∂zj∂zk ) 1≤j,k≤n = 0. by klimek [12, corollary 3.1.8] we conclude the required property). the following two theorems have several applications in the theory of functions. theorem 2.23. let f : d → r be a function, d is a domain of cn, n ≥ 1. put u(z,w) = |w − f(z)|2, where (z,w) ∈ d × c. the following two conditions are equivalent: (a) u is strictly psh in d × c; (b) n = 1, f is harmonic in d and ∂f ∂z (z) ̸= 0 for each z ∈ d. proof. (a) ⇒ (b) since u is strictly psh in d × c, then u is psh in d × c and consequently, f is pluriharmonic in d. therefore u is a function of class c∞ in d. suppose that n ≥ 2. let z0 = (z01, . . . ,z 0 n) ∈ d. consider now r > 0 such that p(z0,r) = d(z01,r) × d(z 0 2,r) × · · · × d(z 0 n,r) ⊂ d. we consider the function f(., .,z03, . . . ,z 0 n) defined and prh in d(z 0 1,r) × d(z 0 2,r) = a. let f1 = f(., .,z 0 3, . . . ,z 0 n) and u1(z1,z2,w) = u(z1,z2,z 0 3, . . . ,z 0 n,w) = ∣∣w − f(z1,z2,z03, . . . ,z0n)∣∣2 = |w − f1(z1,z2)|2 , 84 j. abidi where (z1,z2,w) ∈ d(z01,r) × d(z 0 2,r) × c. note that f1 is prh in a and u1 is strictly psh in a × c, u1(z1,z2,w) = |w|2 + |f1(z1,z2)|2 − wf1(z1,z2) − wf1(z1,z2) . fix w0 = 0 ∈ c. the levi hermitian form of u1(., .,0) is l(u1)(z1,z2,0)(α1,α2) = ∂2u1 ∂z1∂z1 (z1,z2,0)α1α1 + ∂2u1 ∂z2∂z2 (z1,z2,0)α2α2 + 2 re [ ∂2u1 ∂z1∂z2 (z1,z2,0)α1α2 ] > 0 , for all (z1,z2) ∈ a and for all (α1,α2) ∈ c2\{(0,0)}. moreover 2 ∣∣∣∣∂f1∂z1 ∣∣∣∣2α1α1 + 2 ∣∣∣∣∂f1∂z2 ∣∣∣∣2α2α2 + 2 re [ 2 ∂f1 ∂z1 ∂f1 ∂z2 α1α2 ] > 0 , on a for all (α1,α2) ∈ c2\{0}. then ∣∣∣2∂f1∂z1 ∂f1∂z2 ∣∣∣2 < 4∣∣∣∂f1∂z1 ∣∣∣2∣∣∣∂f1∂z2 ∣∣∣2 over a. but we have ∣∣∣∂f1∂z1 ∂f1∂z2 ∣∣∣ = ∣∣∣∂f1∂z1 ∣∣∣∣∣∣∂f1∂z2 ∣∣∣ = ∣∣∣∂f1∂z1 ∣∣∣∣∣∣∂f1∂z2 ∣∣∣ < ∣∣∣∂f1∂z1 ∣∣∣∣∣∣∂f1∂z2 ∣∣∣ in a (because f1 has real valued). a contradiction. then n = 1. now the levi hermitian form of u is l(u)(z,w)(α,β) = ∂2u ∂z∂z (z,w)αα + ∂2u ∂w∂w (z,w)ββ + 2 re [ ∂2u ∂z∂w (z,w)αβ ] = 2 ∣∣∣∣∂f∂z (z) ∣∣∣∣2αα + ββ + 2 re [ − ∂f ∂z (z)αβ ] > 0 , for all (z,w) ∈ d × c and for all (α,β) ∈ c2\{0}. then∣∣∣∣∂f∂z (z) ∣∣∣∣2 < 2 ∣∣∣∣∂f∂z (z) ∣∣∣∣2 for each z ∈ d. thus ∂f ∂z (z) ̸= 0 for every z ∈ d. consequently, n = 1, f is harmonic in d and { z ∈ d : ∂f ∂z (z) = 0 } = ∅. (b) ⇒ (a) the levi hermitian form of u is l(u)(z,w)(α,β) = 2 ∣∣∣∣∂f∂z (z) ∣∣∣∣2αα + ββ + 2 re [ − ∂f ∂z (z)αβ ] > 0 strictly plurisubharmonic functions 85 for each (z,w) ∈ d × c and (α,β) ∈ c2\{0}. we have l(u)(z,w)(α,β) > 0 ∀ (z,w) ∈ d × c , ∀ (α,β) ∈ c2\{0} if and only if ∣∣∣∣∂f∂z (z) ∣∣∣∣2 < 2 ∣∣∣∣∂f∂z (z) ∣∣∣∣2. but this is equivalent to ∂f ∂z (z) ̸= 0 for all z ∈ d. now the case where the function is complex valued, we prove the following extension. theorem 2.24. let g : d → c be a function, d is a domain of cn, n ≥ 1. put v(z,w) = |w−g(z)|2, where (z,w) ∈ d×c. the following two conditions are equivalent: (a) v is strictly psh in d × c; (b) n = 1, g is harmonic in d and { z ∈ d : ∂g ∂z (z) = 0 } = ∅. proof. (a) ⇒ (b) since v is strictly plurisubharmonic in d × c, then v is plurisubharmonic in d × c. consequently, g is pluriharmonic in d. let z0 = (z01, . . . ,z 0 n) ∈ d, r > 0 such that d(z01,r) × · · · × d(z 0 n,r) = a ⊂ d. put g = g1 +g2 in the convex domain a, where g1,g2 : a → c be two analytic functions. now we use the following fundamental decomposition v(z,w) = |w − g1(z) − g2(z)|2 = |w − g1(z)|2 + |g2(z)|2 − (w − g1(z))g2(z) − (w − g1(z))g2(z) for each (z,w) ∈ a × c. suppose that n ≥ 2. case 1: n = 2. we have v1(z,w) = |w − g1(z)|2, v2(z,w) = |g2(z)|2, v3(z,w) = −(w − g1(z))g2(z) − (w − g1(z))g2(z), where (z,w) ∈ a × c; v1,v2 and v3 are c ∞ functions in the domain a × c, and v3 is pluriharmonic in a × c. then the levi hermitian form of v3 is l(v3)(z1,z2,w)(α1,α2,β) = 0 86 j. abidi for all (z1,z2,w) ∈ a × c and for all (α1,α2,β) ∈ c3. the levi hermitian form of v2 is l(v2)(z1,z2,w)(α1,α2,β) = ∣∣∣∣∂g2∂z1 (z) ∣∣∣∣2α1α1 + ∣∣∣∣∂g2∂z2 (z) ∣∣∣∣2α2α2 + 2 re [ ∂g2 ∂z1 (z) ∂g2 ∂z2 (z)α1α2 ] = ∣∣∣∣∂g2∂z1 (z)α1 + ∂g2∂z2 (z)α2 ∣∣∣∣2 for each (z,w) = (z1,z2,w) ∈ a× c and (α1,α2,β) ∈ c3. the levi hermitian form of v1 is l(v1)(z1,z2,w)(α1,α2,β) = ∣∣∣∣∂g1∂z1 (z) ∣∣∣∣2α1α1 + ∣∣∣∣∂g1∂z2 (z) ∣∣∣∣2α2α2 + ββ + 2 re [ ∂g1 ∂z1 (z) ∂g1 ∂z2 (z)α1α2 ] + 2 re [ − ∂g1 ∂z1 (z)α1β ] + 2 re [ − ∂g1 ∂z2 (z)α2β ] = ∣∣∣∣∂g1∂z1 (z)α1 + ∂g1∂z2 (z)α2 ∣∣∣∣2 + |β|2 + 2 re [ − ( ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 ) β ] = ∣∣∣∣β − [ ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 ]∣∣∣∣2, where (z,w) = (z1,z2,w) ∈ a × c. now we have l(v)(z,w)(α1,α2,β) = ∣∣∣∣β− [ ∂g1 ∂z1 (z)α1+ ∂g1 ∂z2 (z)α2 ]∣∣∣∣2+ ∣∣∣∣∂g2∂z1 (z)α1+∂g2∂z2 (z)α2 ∣∣∣∣2 where (z,w) = (z1,z2,w) ∈ a × c and (α1,α2,β) ∈ c3. let z ∈ a. choose (α1,α2) ∈ c2 \{(0,0)} such that ∂g2∂z1 (z)α1 + ∂g2 ∂z2 (z)α2 = 0. now let β = ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2. we have (α1,α2,β) ∈ c3\{(0,0,0)} and l(v)(z,w)(α1,α2,β) = 0. this proves for example that v is not strictly psh on any open of d × c. a contradiction. strictly plurisubharmonic functions 87 case 2: n ≥ 3. we deduce by in fact the formula l(v)(z,w)(α1, . . . ,αn,β) = ∣∣∣∣∣β − n∑ j=1 ∂g1 ∂zj (z)αj ∣∣∣∣∣ 2 + ∣∣∣∣∣ n∑ j=1 ∂g2 ∂zj (z)αj ∣∣∣∣∣ 2 , where (z,w) = (z1, . . . ,zn,w) ∈ a × c and (α1, . . . ,αn,β) ∈ cn+1. let z ∈ a. now it is possible to choose (α1, . . . ,αn) ∈ cn\{0} such that n∑ j=1 ∂g2 ∂zj (z)αj = 0 (because n ≥ 3) . let β = n∑ j=1 ∂g1 ∂zj (z)αj. we have (α1, . . . ,αn,β) ∈ cn+1\{0} and l(v)(z,w)(α1, . . . ,αn,β) = 0 . therefore in fact v is not strictly psh on any domain of a×c. a contradiction. consequently, n = 1. by the above theorem, g is harmonic in d and∣∣∂g ∂z ∣∣ > 0 in d. (b) ⇒ (a) by the above theorem we deduce this assertion in fact. theorem 2.25. let g1, . . . ,gn : c2 → c, v(z,w) = |w1 − g1(z)|2 + · · · + |wn − gn(z)|2, where z ∈ c2, w = (w1, . . . ,wn) ∈ cn and n ∈ n. the following conditions are equivalent: (a1) v is strictly psh in c2 × cn; (a2) gj is pluriharmonic in c2, gj = fj + kj, where fj,kj : c2 → c are analytic functions, for all 1 ≤ j ≤ n (n ≥ 2). the functions k1, . . . ,kn satisfies an algebraic condition, that is for each z ∈ c2, the set {s1, . . . ,sn} is a generating family of the c-vector space c2, s1 = ( ∂k1 ∂z1 (z), ∂k1 ∂z2 (z) ) , . . . ,sn = ( ∂kn ∂z1 (z), ∂kn ∂z2 (z) ) ; (a3) gj is pluriharmonic in c2, gj = fj + kj, where fj,kj : c2 → c are analytic functions for all j ∈ {1, . . . ,n}, n ≥ 2 and for all z ∈ c2, there exist r > 0 and there exists s,t ∈ {1, . . . ,n} (s ̸= t) such that v1 is strictly psh in b(z,r)×c2, where v1(z,t) = |ws −gs(z)|2 +|wt −gt(z)|2, for (z,w) ∈ b(z,r) × c2 and w = (ws,wt); 88 j. abidi (a4) gj is pluriharmonic on c2, gj = fj + kj, where fj,kj : c2 → c are analytic functions for all 1 ≤ j ≤ n. k1, . . . ,kn satisfies {( ∂k1 ∂z1 (z), . . . , ∂kn ∂z1 (z) ) , ( ∂k1 ∂z2 (z), . . . , ∂kn ∂z2 (z) )} is a free family in the c-vector space cn, for all fixed z ∈ c2. proof. (a1) ⇒ (a2) firstly we prove that g1, . . . ,gn are continuous functions over c2. let z0 ∈ c2. put ζ1 = g1(z0), . . . ,ζn = gn(z0) ∈ c; v(z0,ζ1, . . . ,ζn) = 0. let ϵ > 0. since v is upper semi-continuous in the point (z0,ζ1, . . . ,ζn) then there exists δ > 0 such that ∥z − z0∥ + |w1 − ζ1| + · · · + |wn − ζn| < δ implies that |w1 − g1(z)|2 + · · · + |wn − gn(z)|2 ≤ ϵ2. let j ∈ {1, . . . ,n}. if we put w1 = ζ1, . . . ,wj−1 = ζj−1,wj+1 = ζj+1, . . . ,wn = ζn, then we have ∥z − z0∥ + |wj − ζj| < δ implies that |wj − gj(z)|2 < ϵ2. let wj = ζj = gj(z 0). then ∥z−z0∥ < δ implies that |gj(z)−gj(z0)| < ϵ. then gj is continuous in the point z0 ∈ c2. consequently, g1, . . . ,gn are continuous functions on c2. we have v is strictly psh in c2 × cn, therefore v is psh in c2 × cn. therefore the function of two variables v(., .,0, . . . ,0) is psh in c2 × c, where v(z,w1,0, . . . ,0) = v1(z,w1) = |w1 − g1(z)|2 + |g2(z)|2 + · · · + |gn(z)|2. let φ : c2 → r+, φ is of class c∞ and have a compact support in c2. let ∆ = 4 ( ∂2 ∂z1∂z1 + ∂2 ∂z2∂z2 ) the laplace operator on c2. then we have∫ |w1 − g1(z)|2∆φ(z) dm4(z) + n∑ j=2 ∫ |gj(z)|2∆φ(z) dm4(z) ≥ 0 for each w1 ∈ c. let w1 ∈ r. then we have −w1 ∫ [g1(z) + g1(z)]∆φ(z) dm4(z) + n∑ j=1 ∫ |gj(z)|2∆φ(z) dm4(z) ≥ 0 for all w1 ∈ r. if ∫ [g1(z) + g1(z)]∆φ(z) dm4(z) > 0, then we obtain a contradiction by letting w1 to +∞. if ∫ [g1(z) + g1(z)]∆φ(z) dm4(z) < 0, then we have a contradiction by letting w1 go to (−∞). consequently,∫ [g1(z) + g1(z)]∆φ(z) dm4(z) = 0. strictly plurisubharmonic functions 89 since g1 + g1 is a continuous function in c2, then g1 + g1 is harmonic in c2. let w1 ∈ ir. then w1 = −w1. in this case we prove that (g1 − g1) is harmonic in c2. now since g1 = 12 [ (g1 + g1) + (g1 − g1) ] , then g1 is harmonic in c2. let t1 : c2 → c2 be a c-linear bijective transformation. consider now t(z,w1) = (t1(z),w1), where z ∈ c2 and w1 ∈ c. note that t : c3 → c3 is a c-linear bijective transformation. v1 is psh in c2 × c, then v1 ◦ t is psh in c2 × c, v1 ◦ t(z,w1) = |w1 − g1 ◦ t1(z)|2 + |g2 ◦ t1(z)|2 + · · · + |gn ◦ t1(z)|2 , (z,w1) ∈ c2 × c. by the above development we have g1 ◦ t1 is harmonic in c2. consequently, g1 is a pluriharmonic function on c2. therefore g1, . . . ,gn are pluriharmonic functions on c2; gj = fj + kj,fj,kj : c2 → c are analytic functions , 1 ≤ j ≤ n. consider now a1(z,w) = |w1 −g1(z)|2, where (z,w1) ∈ c2 ×c; a1(z,w1) = |w1−f1(z)−k1(z)|2. we consider now the following decomposition a1(z,w1) = |w1 −f1(z)|2 +|k1(z)|2 −k1(z)(w1 −f1(z))−k1(z)(w1 − f1(z)). a1 is a function of class c∞ in c2 × c. let h1(z,w1) = k1(z)(w1 − f1(z)) + k1(z)(w1 − f1(z)), where (z,w1) ∈ c2 × c; h1 is pluriharmonic in c2 × c. therefore the levi hermitian form of h1 is l(h1)(z,w1)(α1,α2,β1) = 0, for all (z,w1) ∈ c2 × c and for all (α1,α2,β1) ∈ c3. then the levi hermitian form of a1 is l(a1)(z,w1)(α1,α2,β1) = l(b1)(z,w1)(α1,α2,β1) + l(c1)(z,w1)(α1,α2,β1) , where b1(z,w1) = |w1 − f1(z)|2, c1(z,w1) = |k1(z)|2 and (z,w1) ∈ c2 × c. b1 and c1 are in particular functions of class c ∞ in c2 × c. we have l(b1)(z,w1)(α1,α2,β1) = ∂2b1 ∂z1∂z1 (z,w1)α1α1 + ∂2b1 ∂z2∂z2 (z,w1)α2α2 + 2 re [ ∂2b1 ∂z1∂z2 (z,w1)α1α2 ] + ∂2b1 ∂w1∂w1 (z,w1)β1β1 + 2 re [ ∂2b1 ∂z1∂w1 (z,w1)α1β1 + ∂2b1 ∂z2∂w1 (z,w1)α2β1 ] = ∣∣∣∣∂f1∂z1 (z) ∣∣∣∣2α1α1 + ∣∣∣∣∂f1∂z2 (z) ∣∣∣∣2α2α2 + 2 re [ ∂f1 ∂z1 (z) ∂f1 ∂z2 (z)α1α2 ] + β1β1 + 2 re [ − ∂f1 ∂z1 (z)α1β1 − ∂f1 ∂z2 (z)α2β1 ] 90 j. abidi = ∣∣∣∣∂f1∂z1 (z)α1 + ∂f1∂z2 (z)α2 ∣∣∣∣2 + ∣∣β1∣∣2 − 2 re [( ∂f1 ∂z1 (z)α1 + ∂f1 ∂z2 (z)α2 ) β1 ] = ∣∣∣∣β1 − [ ∂f1 ∂z1 (z)α1 + ∂f1 ∂z2 (z)α2 ]∣∣∣∣2. since c1(z,w1) = |k1(z)|2, then l(c1)(z,w1)(α1,α2,β1) = ∂2c1 ∂z1∂z1 (z,w1)α1α1 + ∂2c1 ∂z2∂z2 (z,w1)α2α2 + 2 re [ ∂2c1 ∂z1∂z2 (z,w1)α1α2 ] = ∣∣∣∣∂k1∂z1 (z) ∣∣∣∣2α1α1 + ∣∣∣∣∂k1∂z2 (z) ∣∣∣∣2α2α2 + 2 re [ ∂k1 ∂z1 (z) ∂k1 ∂z2 (z)α1α2 ] = ∣∣∣∣∂k1∂z1 (z)α1 + ∂k1∂z2 (z)α2 ∣∣∣∣2. consequently, l(a1)(z,w1)(α1,α2,β1) = ∣∣∣β1 − [∂f1∂z1 (z)α1 + ∂f1∂z2 (z)α2]∣∣∣2 +∣∣∣∂k1∂z1 (z)α1 + ∂k1∂z2 (z)α2∣∣∣2 for each (z,w1) ∈ c2 × c and (α1,α2,β1) ∈ c3. since v is a function of class c∞ in c2 × cn, then we have for each (z,w1, . . . ,wn) ∈ c2 × cn, z = (z1,z2) ∈ c2 and all (α1,α2,β1, . . . ,βn) ∈ cn+2, the levi hermitian form of v is l(v)(z,w1, . . . ,wn)(α1,α2,β1, . . . ,βn) = ∣∣∣∣β1 − [ ∂f1 ∂z1 (z)α1 + ∂f1 ∂z2 (z)α2 ]∣∣∣∣2 + ∣∣∣∣∂k1∂z1 (z)α1 + ∂k1∂z2 (z)α2 ∣∣∣∣2 + · · · + ∣∣∣∣βn − [ ∂fn ∂z1 (z)α1 + ∂fn ∂z2 (z)α2 ]∣∣∣∣2 + ∣∣∣∣∂kn∂z1 (z)α1 + ∂kn∂z2 (z)α2 ∣∣∣∣2. fix z ∈ c2. if l(v)(z,w1, . . . ,wn)(α1,α2,β1, . . . ,βn) = 0, then  ∂k1 ∂z1 (z)α1 + ∂k1 ∂z2 (z)α2 = 0 ... ∂kn ∂z1 (z)α1 + ∂kn ∂z2 (z)α2 = 0. therefore if α1,α2 ∈ c such that α1 ( ∂k1 ∂z1 (z), . . . , ∂kn ∂z1 (z) ) + α2 ( ∂k1 ∂z2 (z), . . . , ∂kn ∂z2 (z) ) = (0, . . . ,0) ∈ cn, strictly plurisubharmonic functions 91 then α1 = α2 = 0. thus n ≥ 2 and there exists s,t ∈ {1, . . . ,n} (s ̸= t) such that {( ∂ks ∂z1 (z), ∂ks ∂z2 (z) ) , ( ∂kt ∂z1 (z), ∂kt ∂z2 (z) )} is a basis of the c-vector space c2. then {( ∂k1 ∂z1 (z), ∂k1 ∂z2 (z) ) , . . . , ( ∂kn ∂z1 (z), ∂kn ∂z2 (z) )} is a generating family of the c-vector space c2. observe that locally (s,t) is independent of z ∈ c2, but not globally if n ≥ 3. (a2) ⇒ (a1) let z ∈ c2. since {( ∂k1 ∂z1 (z), ∂k1 ∂z2 (z) ) , . . . , ( ∂kn ∂z1 (z), ∂kn ∂z2 (z) )} is a generating family of the c-vector space c2, then n ≥ 2 and we can exhibit a family of 2 vectors which is a basis of c2. without loss of generality we suppose that {( ∂k1 ∂z1 (z), ∂k1 ∂z2 (z) ) , ( ∂k2 ∂z1 (z), ∂k2 ∂z2 (z) )} is a basis of c2. therefore the matrix (λµν)1≤µ, ν≤2 have a determinant det(λµν)1≤µ, ν≤2 = φ(z) ̸= 0, where λµν = ∂kµ ∂zν (z). since the function φ is analytic in c2, then |φ| > 0 on a neighborhood b(z,r) of the point z (r > 0). then for all ξ ∈ b(z,r) and (α1,α2) ∈ c2, we have   ∂k1 ∂z1 (ξ)α1 + ∂k1 ∂z2 (ξ)α2 = 0 ∂k2 ∂z1 (ξ)α1 + ∂k2 ∂z2 (ξ)α2 = 0 if and only if α1 = α2 = 0. thus if (α1,α2,β1,β2) ∈ c4, ξ ∈ b(z,r),∣∣∣β1 − [∂f1∂z1 (ξ)α1 + ∂f1∂z2 (ξ)α2]∣∣∣2 + ∣∣∣∂k1∂z1 (ξ)α1 + ∂k1∂z2 (ξ)α2∣∣∣2 + ∣∣∣β2 − [∂f2∂z1 (ξ)α1 + ∂f2 ∂z2 (ξ)α2 ]∣∣∣2 + ∣∣∣∂k2∂z1 (ξ)α1 + ∂k2∂z2 (ξ)α2∣∣∣2 = 0, then  ∂k1 ∂z1 (ξ)α1 + ∂k1 ∂z2 (ξ)α2 = 0 ∂k2 ∂z1 (ξ)α1 + ∂k2 ∂z2 (ξ)α2 = 0. it follows that α1 = α2 = 0. thus β1 = β2 = 0. consequently, φ1(ξ,w1,w2) = |w1 − g1(ξ)|2 + |w2 − g2(ξ)|2 is strictly psh in b(z,r) × c × c. in fact we can prove that φ1 is strictly psh in (c2\a) × c2, where a is an analytic subset of c2. now the above proof implies that the assertions (a1), (a3) and (a4) are equivalent. corollary 2.26. let g1,g2 : c2 → c be two analytic functions. put u(z,w1,w2) = |w1 − g1(z)|2 + |w2 − g2(z)|2, where (z,w1,w2) ∈ c2 × c × c. let a ⊂ c2, a closed and bounded in c2. suppose that u is strictly psh in c2 × (c2\a). then u is strictly psh in c2 × c2. 92 j. abidi proof. note that u is a function of class c∞ on c2 × c2. assume that u is not strictly psh at the point (z0,w0) ∈ c2 × c2. then there exists( (α1,α2),(β1,β2) ) ∈ c2 × c2\{(0,0)} such that the levi hermitian form of u verify l(u)(z0,w0) ( (α1,α2),(β1,β2) ) = ∣∣∣∣∣β1 − 2∑ j=1 ∂g1 ∂zj (z0)αj ∣∣∣∣∣ 2 + ∣∣∣∣∣β2 − 2∑ j=1 ∂g2 ∂zj (z0)αj ∣∣∣∣∣ 2 = 0 . let b0 ∈ c2\a. since u is strictly psh on c2 × (c2\a), then u is strictly psh at the point (z0,b0). but we have l(u)(z0,b0) ( (α1,α2),(β1,β2) ) = ∣∣∣∣∣β1 − 2∑ j=1 ∂g1 ∂zj (z0)αj ∣∣∣∣∣ 2 + ∣∣∣∣∣β2 − 2∑ j=1 ∂g2 ∂zj (z0)αj ∣∣∣∣∣ 2 = 0 . and ( (α1,α2),(β1,β2) ) ∈ c2 × c2\{(0,0)}. a contradiction. consequently, u is strictly psh on c2 × c2. corollary 2.27. let g1,g2 : c2 → c be two analytic functions. set u(z,w) = |w1 −g1(z)|2 + |w2 −g2(z)|2, v(z,w) = |w1 −g1(z)|2 + |w2 −g2(z)|2, φ(z,ζ) = |ζ − g1(z)|2 + |ζ − g2(z)|2, where z ∈ c2, w = (w1,w2) ∈ c2 and ζ ∈ c. then u and v are not strictly plurisubharmonic functions in c2 × c2. we have, φ is strictly psh in c2×c if and only if |g1|2+|g2|2 (or |g1+g2|2) is strictly psh on c2. proof. we have the fundamental decomposition (complex structure) u(z,w) = |w1 − g1(z)|2 + |w2|2 + |g2(z)|2 − w2g2(z) − w2g2(z), for any (z,w) = (z,w1,w2) ∈ c2 × c × c where z = (z1,z2) ∈ c2. put u1(z,w) = w2g2(z)+w2g2(z); u1 is a pluriharmonic function in c2×c2. therefore the levi hermitian form of this function is equal to 0 over c4. let u2(z,w) = |w1 − g1(z)|2; u2 is a function of class c∞ in c2 × c, u2(z,w) = |w1|2 +|g1(z)|2 −w1g1(z)−w1g1(z). then the levi hermitian form of u2 is now l(u2)(z,w)(α1,α2,β1,β2) = ∣∣∣∣β1 − [ ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 ]∣∣∣∣2 strictly plurisubharmonic functions 93 for each z = (z1,z2) ∈ c2, w = (w1,w2) ∈ c2 and (α1,α2,β1,β2) ∈ c4. let u3(z,w) = |w2|2 + |g2(z)|2; u3 is a function of class c∞ in c4. the levi hermitian form of u3 is l(u3)(z,w)(α1,α2,β1,β2) = ∣∣β2∣∣2 + ∣∣∣∣∂g2∂z1 (z)α1 + ∂g2∂z2 (z)α2 ∣∣∣∣2. the function u is of class c∞ in c2 × c2. we have l(u)(z,w)(α1,α2,β1,β2) = −l(u1)(z,w)(α1,α2,β1,β2) + l(u2)(z,w)(α1,α2,β1,β2) + l(u3)(z,w)(α1,α2,β1,β2) = ∣∣∣∣β1 − [ ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 ]∣∣∣∣2 + ∣∣β2∣∣2 + ∣∣∣∣∂g2∂z1 (z)α1 + ∂g2∂z2 (z)α2 ∣∣∣∣2. case 1: |g1|2 + |g2|2 (or equivalently |g1 + g2|2) is not strictly psh on c2. note that |g1|2 and |g2|2 are functions of class c∞ in c2. the levi hermitian form (in c2) of |g1|2 is l ( |g1|2 ) (z)(δ1,δ2) = 2∑ j,k=1 ∂2 ( |g1|2 ) ∂zj∂zk δjδk = 2∑ j,k=1 ∂g1 ∂zj (z) ∂g1 ∂zk (z)δjδk = ( 2∑ j=1 ∂g1 ∂zj (z)δj )( 2∑ k=1 ∂g1 ∂zk (z)δk ) = ∣∣∣∣∣ 2∑ j=1 ∂g1 ∂zj (z)δj ∣∣∣∣∣ 2 , where z = (z1,z2) ∈ c2 and (δ1,δ2) ∈ c2. therefore l ( |g1|2+|g2|2 ) (z1,z2)(α1,α2) = ∣∣∣∣∂g1∂z1 (z)α1 + ∂g1∂z2 (z)α2 ∣∣∣∣2 + ∣∣∣∣∂g2∂z1 (z)α1 + ∂g2∂z2 (z)α2 ∣∣∣∣2 for each (α1,α2) ∈ c2. now fix z = (z1,z2) ∈ c2. since |g1|2 + |g2|2 is not strictly psh in c2, then there exists (α1,α2) ∈ c2\{0} such that  ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 = 0 ∂g2 ∂z1 (z)α1 + ∂g2 ∂z2 (z)α2 = 0. 94 j. abidi fix w = (w1,w2) ∈ c2. take now β1 = β2 = 0 ∈ c. then we have l(u)(z,w)(α1,α2,β1,β2) = 0 but (α1,α2,β1,β2) ∈ c4\{0}. consequently, u is not strictly psh in c2 × c2. case 2: |g1|2 + |g2|2 (or equivalently |g1 + g2|2) is strictly psh in c2. l(u)(z,w)(α1,α2,β1,β2) = 0 if and only if β1 = ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2,β2 = 0 and ∂g2 ∂z1 (z)α1 + ∂g2 ∂z2 (z)α2 = 0. fix (α1,α2) ∈ c2\{0} such that ∂g2∂z1 (z)α1 + ∂g2 ∂z2 (z)α2 = 0. define β1 ∈ c by β1 = ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 (β2 = 0). then (α1,α2,β1,β2) ∈ c4\{0} and we have l(u)(z,w)(α1,α2,β1,β2) = 0. consequently, u is not strictly psh on any domain d ⊂ c4. concerning the function v, we have v is defined on c4 and of class c∞. the levi hermitian form of v is l(v)(z,w)(α1,α2,β1,β2) = ∣∣∣∣β1 − [ ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 ]∣∣∣∣2 + ∣∣∣∣β2 − [ ∂g2 ∂z1 (z)α1 + ∂g2 ∂z2 (z)α2 ]∣∣∣∣2, where z = (z1,z2) ∈ c2, w = (w1,w2) ∈ c2 and (α1,α2,β1,β2) ∈ c4. fix (z,w) ∈ c2 × c2. let (α1,α2) ∈ c2\{0} such that ∂g1∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 = 0. put β1 = 0, β2 = [ ∂g2 ∂z1 (z)α1+ ∂g2 ∂z2 (z)α2 ] . then (α1,α2,β1,β2) ∈ c4\{0} and l(v)(z,w)(α1,α2,β1,β2) = 0. consequently, v is not strictly psh on any open of c4. now we have the decomposition φ(z,ζ) = |ζ − g1(z)|2 + |ζ − g2(z)|2 = |ζ − g1(z)|2 + |ζ|2 + |g2(z)|2 − ζg2(z) − ζg2(z) , for every ζ ∈ c and z = (z1,z2) ∈ c2; φ is a function of class c∞ in c3. put φ1(z,ζ) = ζg2(z) + ζg2(z). then φ1 is pluriharmonic in c3 and consequently, the levi hermitian form of this function is 0. let φ2(z,ζ) = |ζ − g1(z)|2; φ2 is a function of class c∞ in c3 and l(φ2)(z,ζ)(α1,α2,β) = ∣∣∣∣β − [ ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 ]∣∣∣∣2 for each (z,ζ) ∈ c2 × c and (α1,α2,β) ∈ c3. let φ3(z,ζ) = |ζ|2 + |g2(z)|2; φ3 is a function of class c∞ in c3 and l(φ3)(z,ζ)(α1,α2,β) = ∣∣β∣∣2 + ∣∣∣∣∂g2∂z1 (z)α1 + ∂g2∂z2 (z)α2 ∣∣∣∣2 strictly plurisubharmonic functions 95 for every (z,ζ) ∈ c2 × c and (α1,α2,β) ∈ c3. it follows that l(φ)(z,ζ)(α1,α2,β) = l(φ2)(z,ζ)(α1,α2,β) + l(φ3)(z,ζ)(α1,α2,β) = ∣∣∣∣β − [ ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 ]∣∣∣∣2 + ∣∣β∣∣2 + ∣∣∣∣∂g2∂z1 (z)α1 + ∂g2∂z2 (z)α2 ∣∣∣∣2. therefore l(φ)(z,ζ)(α1,α2,β) = 0 if and only if β = 0, ∂g2 ∂z1 (z)α1 + ∂g2 ∂z2 (z)α2 = 0 and ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 = 0. observe now that φ is strictly psh in c3 if and only if |g1|2 + |g2|2 is strictly psh in c2. corollary 2.28. let g1,g2 : c2 → c be two pluriharmonic functions. put g1 = f1 + k1, where g2 = f2 + k2, f1,f2,k1,k2 : c2 → c be four analytic functions. let u(z,w1,w2) = |w1 − g1(z)|2 + |w2 − g2(z)|2 , v(z,w1,w2) = ∣∣w1 − k1(z)∣∣2 + ∣∣w2 − k2(z)∣∣2, where (z,w1,w2) ∈ c2 × c × c. the following conditions are equivalent (a) u is strictly psh in c4; (b) v is strictly psh in c4. that is the strict plurisubharmonicity of u is independent of the choice of the analytic functions f1 and f2. corollary 2.29. let gj,kj : d → c be analytic functions, where 1 ≤ j ≤ n and d is a domain of cn, n,n ≥ 1. put u = n∑ j=1 ∣∣gj + kj∣∣2 and v = n∑ j=1 ∣∣gj∣∣2 + n∑ j=1 ∣∣kj∣∣2. then u is strictly psh in d if and only if v is strictly psh in d. corollary 2.30. let g1, . . . ,gn : d → c be n analytic functions, where n ≥ 1 and d is a domain of cn (n ≥ 1). set u(z,w) = n∑ j=1 |w−gj(z)|2, where (z,w) ∈ d × c. if n ≤ n, then u is not strictly psh on any domain of d × c. 96 j. abidi proof. fix z = (z1, . . . ,zn) ∈ d and w ∈ c. let uj(z,w) = |w − gj(z)|2, 1 ≤ j ≤ n. then uj is a function of class c∞ in d×c. if now (α1, . . . ,αn) ∈ cn and β ∈ c, we have the levi hermitian form of uj is l(uj)(z,w)(α1, . . . ,αn,β) = ∣∣∣∣∣β − n∑ s=1 ∂gj ∂zs (z)αs ∣∣∣∣∣ 2 . therefore, the levi form of u is l(u)(z,w)(α1, . . . ,αn,β) = n∑ j=1 l(uj)(z,w)(α1, . . . ,αn,β) = n∑ j=1 ∣∣∣∣∣β − n∑ s=1 ∂gj ∂zs (z)αs ∣∣∣∣∣ 2 . let v = [|g1|2 + · · · + |gn|2]; v is a function of class c∞ on d. case 1: v is strictly psh on d. we have l(v)(z)(α1, . . . ,αn) = 0 imply that (α1, . . . ,αn) = (0, . . . ,0). the levi form of v is l(v)(z)(α1, . . . ,αn) = ∣∣∣∣∣ n∑ s=1 ∂g1 ∂zs (z)αs ∣∣∣∣∣ 2 + · · · + ∣∣∣∣∣ n∑ s=1 ∂gn ∂zs (z)αs ∣∣∣∣∣ 2 . since l(v)(z)(α1, . . . ,αn) = 0 then (α1, . . . ,αn) = 0. thus the system of equations in (α1, . . . ,αn) ∈ cn satisfies  ∂g1 ∂z1 (z)α1 + · · · + ∂g1∂zn (z)αn = 0 ... ∂gn ∂z1 (z)α1 + · · · + ∂gn∂zn (z)αn = 0 if and only if (α1, . . . ,αn) = (0, . . . ,0). since n ≤ n, then n = n. thus the matrix (λjk)1≤j,k≤n is invertible; where λjk = ∂gj ∂zk (z). now we have l(u)(z,w)(α1, . . . ,αn,β) = 0 if and only if  ∂g1 ∂z1 (z)α1 + · · · + ∂g1∂zn (z)αn = β ... ∂gn ∂z1 (z)α1 + · · · + ∂gn∂zn (z)αn = β. strictly plurisubharmonic functions 97 fix β ∈ c\{0}; the above system has a unique solution (α1, . . . ,αn) ∈ cn\{0}. consequently, (α1, . . . ,αn,β) ∈ cn+1\{0} and l(u)(z,w)(α1, . . . ,αn,β) = 0. case 2: v is not strictly psh on d. then there exists (α1, . . . ,αn) ∈ cn\{0} such that l(v)(z)(α1, . . . ,αn) = 0. take β = 0 ∈ c. then (α1, . . . ,αn,β) ∈ cn+1\{0} and l(u)(z,w)(α1, . . . ,αn,β) = 0 . consequently, u is not strictly psh in d × c. example. let (z1,z2) ∈ c2 and w ∈ c. put g1(z) = z1, g2(z) = z2, g3(z) = z1 + z2; g1,g2,g3 are analytic functions in c2. put u(z,w) = 3∑ j=1 |w − gj(z)|2. then u is a function of class c∞ and strictly psh in c2 × c. if (w1,w2,w3) ∈ c3, we put v(z1,z2,w1,w2,w3) = 3∑ j=1 aj|wj − gj(z)|2, where (a1,a2,a3 ∈ r+\{0}). then v is not strictly psh on any domain of c2 × c3. in fact v is a function of class c∞ in c2 × c3 and the levi form of v is l(v)(z,w1,w2,w3)(α1,α2,β1,β2,β3) = a1 ∣∣∣∣β1 − [ ∂g1 ∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 ]∣∣∣∣2 + a2 ∣∣∣∣β2 − [ ∂g2 ∂z1 (z)α1 + ∂g2 ∂z2 (z)α2 ]∣∣∣∣2 + a3 ∣∣∣∣β3 − [ ∂g3 ∂z1 (z)α1 + ∂g3 ∂z2 (z)α2 ]∣∣∣∣2, for every (α1,α2,β1,β2,β3) ∈ c5. let (α1,α2) ∈ c2\{0} such that ∂g1∂z1 (z)α1 + ∂g1 ∂z2 (z)α2 = 0. put β1 = 0, β2 = [ ∂g2 ∂z1 (z)α1 + ∂g2 ∂z2 (z)α2 ] , β3 = [ ∂g3 ∂z1 (z)α1 + ∂g3 ∂z2 (z)α2 ] . then (α1,α2,β1,β2,β3) ∈ c5\{0} and l(v)(z,w1,w2,w3)(α1,α2,β1,β2,β3) = 0 . therefore v is not strictly psh on any not empty open of c2 × c3. observe that here in fact we have for all k1, . . . ,kn : cn → c analytic functions, where n,n ∈ n, if n ≥ n, then v1 is not strictly psh in any domain of cn × cn, where v1(z,w1, . . . ,wn) = n∑ j=1 bj|wj − kj(z)|2 ( b1, . . . ,bn ∈ r+\{0} ) , z ∈ cn and (w1, . . . ,wn) ∈ cn. 98 j. abidi a fundamental application concerning analytic functions and the complex structure is now the following extension. theorem 2.31. let g1, . . . ,gn : d → c be n analytic functions, d is a domain of cn, (n ≥ 1) and (n ≥ 1). put u(z,w) = |w − g1(z)|2 + · · · + |w − gn(z)|2, v(z,w) = |w − g1(z)|2 + · · · + |w − gn(z)|2, v1(z,w) = |w − h1(z)|2 + · · · + |w − hn(z)|2, where (z,w) ∈ d × c, and hj = re(gj), for 1 ≤ j ≤ n. (a) suppose that u is strictly psh in d × c. then v and v1 are strictly psh in d × c and n ≥ n + 1, but the converse is false. (b) in fact, v is strictly psh in d×c if and only if v1 is strictly psh in d×c. for the proof of this theorem, we use lemma 2.2. example. let g1(z) = z, g2(z) = z 2, where z ∈ c. u1(z,w) = |w − z|2 + |w − z2|2, u2(z,w) = |w − z|2 + |w − z2|2, for (z,w) ∈ c2; u1 is not strictly psh on any domain of the form d ( 1 2 ,r ) × c (for every r > 0); u2 is strictly psh on c2. on the other hand, the minimal number n of analytic functions k1, . . . ,kn : cn → c (n ≥ 1) such that if u1(z,w) = ∣∣w − k1(z)∣∣2 + · · · + ∣∣w − kn(z)∣∣2 is strictly psh on cn × c is in fact n = n. but for all φ1, . . . ,φn : cn → c be n analytic functions, u2(z,w) = |w − φ1(z)|2 + · · · + |w − φn(z)|2 satisfies u2 is not strictly psh on cn × c if n ≤ n. now there are a great differences between the class of functions defined analogues to u1 and the class of functions defined similar of u2. now we are in position to prove the following result. theorem 2.32. let g1, . . . ,gn : d → c, d is a domain of cn, n ∈ n. set v(z,w) = |w1 − g1(z)|2 + · · · + |wn − gn(z)|2, where (z,w) ∈ d × cn, w = (w1, . . . ,wn). the following conditions are equivalent: (a) v is strictly psh in d × cn; strictly plurisubharmonic functions 99 (b) g1, . . . ,gn are prh functions in d and for all z = (z1, . . . ,zn) ∈ d (fixed), the system   ∂g1 ∂z1 (z)α1 + · · · + ∂g1∂zn (z)αn = 0 ... ∂gn ∂z1 (z)α1 + · · · + ∂gn∂zn (z)αn = 0 has only the solution (α1, . . . ,αn) = (0, . . . ,0). that is strictly plurisubharmonic functions and partial differential equations have a rigid relation to discover here for example. question 2.33. let g1 : c2 → c be a prh function. find a condition satisfied by g1 such that there exists g2 : c2 → c prh and satisfying u is strictly psh on c2 × c2, where u(z,w) = a1|w1 − g1(z)|2 + a2|w2 − g2(z)|2, z ∈ c2, w = (w1,w2) ∈ c2 and a1,a2 ∈ r+\{0}. in general this problem have no solution and an affirmative answer is given by the following result. proposition 2.34. let g : c2 → c, g(z1,z2) = k1(z1)k2(z2), where (z1,z2) ∈ c2, k1,k2 : c → c be two analytic not constant functions, k1(0) = k2(0) = 0. for all a1,a2 ∈ r+\{0}, there does not exists a function k : c2 → c be analytic such that v = a1|g|2 + a2|k|2 is strictly psh on c2. proof. let k : c2 → c be a analytic function. put v = a1|g|2 + a2|k|2. v, |g|2 and |k|2 are functions smooth of class c∞ in c2. the levi hermitian form of |g|2 is l(|g|2)(z1,z2)(α1,α2) = ∣∣∣∣ ∂g∂z1 (z1,z2)α1 + ∂g∂z2 (z1,z2)α2 ∣∣∣∣2 for each z = (z1,z2) and (α1,α2) ∈ c2. therefore, the levi hermitian form of v is l(v)(z1,z2)(α1,α2) = a1 ∣∣∣∣k′1(z1)k2(z2)α1 + k1(z1)k′2(z2)α2 ∣∣∣∣2 + a2 ∣∣∣∣ ∂k∂z1 (z)α1 + ∂k∂z2 (z)α2 ∣∣∣∣2. take z1 = z2 = 0. l(v)(0,0)(α1,α2) = a2 ∣∣∣ ∂k∂z1 (0)α1 + ∂k∂z2 (0)α2∣∣∣2. now take (α1,α2) ∈ c2\{0} such that ∂k∂z1 (0)α1 + ∂k ∂z2 (0)α2 = 0. then l(v)(0,0)(α1,α2) = 0. consequently, v is not strictly psh on c2. 100 j. abidi it follows that, for all k : c2 → c be analytic, for all a1,a2 ∈ r+\{0}, if u1(z,w) = a1|w1 − g(z)|2 + a2|w2 − k(z)|2, where z = (z1,z2) ∈ c2,w = (w1,w2) ∈ c2. then u1 is not strictly psh on c2 × c2. consequently, the above question globally has a negative answer. but locally we have a positive answer. because in fact, by using all the notation of the question 2.33, we have if g2 exists, then |g1|2 + |g2|2 is strictly psh on c2. by the above proposition, there exists a function g : c2 → c be analytic such that a1|g|2 + a2|k|2 is not strictly psh on c2, for any k : c2 → c be analytic, for every a1,a2 ∈ r+\{0}. now locally, if z0 = (z01,z 0 2) ∈ c 2 we can write g1 = f1 + k1, g2 = f2 + k2, where f1,f2,k1,k2 : c2 → c be 4 analytic functions. in fact we can prove that, the functions f1 and f2 do not have any role on the subject of the strict plurisubharmonicity of u; u is a function of class c∞ in c2 × c2. the levi hermitian form of u is l(u) ( z0,w1,w2 ) (α1,α2,β1,β2) = a1 ∣∣∣∣β1 − [ ∂f1 ∂z1 ( z0 ) α1 + ∂f1 ∂z2 ( z0 ) α2 ]∣∣∣∣2 + a1 ∣∣∣∣∂k1∂z1 (z0)α1 + ∂k1∂z2 (z0)α2 ∣∣∣∣2 + a2 ∣∣∣∣β2 − [ ∂f2 ∂z1 ( z0 ) α1 + ∂f2 ∂z2 ( z0 ) α2 ]∣∣∣∣2 + a2 ∣∣∣∣∂k2∂z1 (z0)α1 + ∂k2∂z2 (z0)α2 ∣∣∣∣2, where (w1,w2) ∈ c2, (α1,α2,β1,β2) ∈ c4. if u is strictly psh on a neighborhood g of (z0,w0),w0 ∈ c2, then l(u)(z,w)(α1,α2,β1,β2) = 0 implies that (α1,α2,β1,β2) = 0, for every (z,w) ∈ g = g1 × g2, g1 and g2 are convex domains of c2, where z0 ∈ g1, w0 ∈ g2. but l(u)(z,w)(α1,α2,β1,β2) = 0 has only the solution (α1,α2,β1,β2) = 0 (for every (z,w) ∈ g), if and only if the system   ∂k1 ∂z1 (z)α1 + ∂k1 ∂z2 (z)α2 = 0 ∂k2 ∂z1 (z)α1 + ∂k2 ∂z2 (z)α2 = 0 (where z is fixed on g1 and (α1,α2) is the variable in c2) has only the solution (α1,α2) = (0,0). observe that this condition is independent of w0 ∈ c2. therefore if( ∂k1 ∂z1 ( z0 ) , ∂k1 ∂z2 ( z0 )) ̸= (0,0), there exists a ball b(z0,r) ⊂ c2 (r > 0) such that ( ∂k1 ∂z1 (z), ∂k1 ∂z2 (z) ) ̸= (0,0) for each z ∈ b(z0,r). suppose for example that ∂k1 ∂z1 (z) ̸= 0, for every z ∈ b(z0, t), where 0 < t < r. let k2(z1,z2) = z2, where (z1,z2) ∈ c2; k2 is analytic on c2. put g2 = k2; g2 is pluriharmonic on strictly plurisubharmonic functions 101 c2. we have ∂k2 ∂z1 (z) = 0, ∂k2 ∂z2 (z) = 1. the above system has only the solution (α1,α2) = (0,0). then u is strictly psh on b(z 0, t) × c2. proposition 2.35. let g1,g2 : c2 → c. put u(z,w1,w2) = a1|w1 − g1(z)|2 + a2|w2 − g2(z)|2, where z ∈ c2, (w1,w2) ∈ c2, a1,a2 ∈ r+\{0}; u1(z,w1) = |w1 − g1(z)|2 + |g2(z)|2, u2(z,w2) = |w2 − g2(z)|2 + |g1(z)|2. the following conditions are equivalent: (a) u is strictly psh on c2 × c2; (b) u1 and u2 are strictly psh functions on c2 × c; (c) g1 and g2 are prh functions over c2, g1 = f1 + k1, g2 = f2 + k2 (f1,k1,f2,k2 : c2 → c be analytic) and the antiholomorphic parts of g1 and g2 satisfies |k1|2 + |k2|2 is stictly psh on c2. moreover observe that if the holomorphic parts of g1 and g2 satisfies |f1|2+|f2|2 is strictly psh on c2 (therefore here |g1|2 + |g2|2 is strictly psh on c2) but we can not conclude that u is strictly psh on c2. example. let g1 : c2 → c be a prh function and let n ∈ n, n ≥ 2. prove that there exists g2, . . . ,gn+1 : c2 → c be n prh functions such that if u(z,w1,w2, . . . ,wn+1) = n+1∑ j=1 |wj − gj(z)|2, where z ∈ c2, then u is strictly psh on c2 ×cn+1. in fact, the answer is very simple, if we consider the family of prh functions g2(z) = z1, g3(z) = z2, g4(z) = · · · = gn+1(z) = 0, where z = (z1,z2) ∈ c2. we have in this case |w2 − g2|2 + |w3 − g3|2 is strictly psh in c2 × c2. then u is strictly psh in c2 × cn+1. 3. convex and strictly plurisubharmonic functions we consider in this section a classical family of psh functions, that is the class of convex and strictly psh functions. theorem 3.1. let g1,g2 : c → c be two analytic functions. assume that u(z,w,w1,w2) = a|w1 − g1(w − z)|2 + b|w2 − g2(w − z)|2, v(z,w1,w2) = a|w1 − g1(z)|2 + b|w2 − g2(z)|2, where (z,w,w1,w2) ∈ c4, a,b ∈ r+\{0}. the following statements are equivalent: 102 j. abidi (a) u is psh on c4; (b) g1 and g2 are analytic affine functions; (c) v is convex on c3. proof. (a) ⇒ (b) fix w1,w2 ∈ c. put u1(z,w) = |w1 − g1(w − z)|2 + |w2 − g2(w − z)|2, where (z,w) ∈ c2. since u1 is of class c∞ and psh on c2, then the levi hermitian form of u1 is l(u1)(z,w)(α,β) = ( |g′1(z)| 2 + |g′2(z)| 2 ) αα + ( |g′1(z)| 2 + |g′2(z)| 2 ) ββ + 2 re ([ (w1 − g1(z))g′′1(z) + (w2 − g2(z))g ′′ 2(z) ] αβ ) ≥ 0 , for all (α,β) ∈ c2. thus∣∣(w1 − g1(z))g′′1(z) + (w2 − g2(z))g′′2(z)∣∣ ≤ |g′1(z)|2 + |g′2(z)|2, for all z ∈ c and all (w1,w2) ∈ c2. now fix z ∈ c. if g′′1(z) ̸= 0. fix w2 = g2(z) ∈ c. then |(w1 − g1(z))g ′′ 1(z)| ≤ |g ′ 1(z)| 2+|g′2(z)| 2, for any w1 ∈ c. it follows that c is bounded. a contradiction. consequently, g′′1 = 0, g ′′ 2 = 0 over c. therefore g1 and g2 are analytic affine functions over c. comparison theorems. we prove in this context that there exists an infinite number f1 of c ∞ functions defined on c2, such that for each f ∈ f1, the function f satisfy f has a fixed type, f is convex and strictly psh on c2, but f is not strictly convex on c2. denote by < .,. > the habitual hermitian product over cn in all of this section. let f : cn → c be a analytic function. set u(z,w) = |w−f(z)|2, v(z,w) = |w−f(z)|2, u1(z,w) = a1|w−(< z,a > +b)|2 +a2|w−(< z,a >+b)|2, where (z,w) ∈ cn × c, a ∈ cn, b ∈ c, a1,a2 ∈ r + \{0}. we study now the structure of the functions u, v and u1. we have the following 3 assertions: (a) u is psh in cn × c, but u is not strictly psh on any domain of cn × c. (b) v is strictly psh on cn × c if and only if n = 1 and |f ′| > 0 over c. but v is not strictly convex in all not empty convex domain of cn × c for every n ≥ 1 and for any f : cn → c be analytic. (c) u1 is not strictly convex in all not empty euclidean open ball subset of cn × c, for a1,a2 ∈ r+\{0} and (a,b) ∈ cn × c. strictly plurisubharmonic functions 103 but if we consider u2(z,w) = |w − f(z)|2 + |w − f(z)|2 + |w − g(z)|2, where g : c → c be analytic, n = 1, (z,w) ∈ c2, we have the following result. proposition 3.2. u2 is strictly convex in c2 if and only if f and g are analytic affine functions, f(z) = a1z + b1, g(z) = a2z + b2, for z ∈ c, where a1,a2,b1,b2 ∈ c such that ((a1,a2 ∈ c\{0} and a2a1 ̸= 1) or (a1 = 0, a2 ̸= 0) or (a1 ̸= 0, a2 = 0)). proof. suppose that u2 is strictly convex in c2. recall that if φ : cm → r be a function of class c2 (m ≥ 1), then φ is strictly convex in cm if and only if ∣∣∣∣∣ m∑ j,k=1 ∂2φ ∂zj∂zk (z)αjαk ∣∣∣∣∣ < m∑ j,k=1 ∂2φ ∂zj∂zk (z)αjαk for each z ∈ cm and all (α1, . . . ,αm) ∈ cm\{0}. we have u2(z,w) = ww + f(z)f(z) − wf(z) − wf(z) + ww + f(z)f(z) − wf(z) − wf(z) + ww + g(z)g(z) − wg(z) − wg(z) , where (z,w) ∈ c2. let (α,β) ∈ c2; u2 is strictly convex in c2, then u2 is convex in all c2. now since u2 is of class c∞ in c2, then we have∣∣[f ′′f − wf ′′ + f ′′f − wf ′′ + g′′g − wg′′]α2 − 2αβf ′∣∣ ≤ |β − f ′α|2 + |β|2 + |f ′α|2 + |β − g′α|2 is valid over c for each (α,β) ∈ c2 and w ∈ c. if w ∈ r, then |w(2f ′′(z) + g′′(z)) + φ(z)| ≤ φ1(z), where φ : c → c and φ1 : c → r+ be two functions. the condition 2f ′′+g′′ ̸= 0, imply that r is bounded, which is a contradiction. thus 2f ′′ + g′′ = 0 over c. now put w = it, where t ∈ r. therefore for each t ∈ r, |tg′′ + θ| ≤ θ1, where θ : c → c and θ1 : c → r+ be two functions. then g′′ = 0 in c. it follows that f ′′ = 0. consequently, f and g are analytic affine functions over c; f(z) = a1z + b1, g(z) = a2z + b2 for z ∈ c, where a1,a2,b1,b2 ∈ c. case 1: a1 = 0. in this situation u2(z,w) = |w − b1|2 + |w − b1|2 + |w − g2(z)|2; u2 is a smooth function over c2. let (z,w) ∈ c2 and (α,β) ∈ c2\{0}. we have ∣∣∣∣∂2u2∂z2 (z,w)α2 + ∂ 2u2 ∂w2 (z,w)β2 + 2 ∂2u2 ∂z∂w (z,w)αβ ∣∣∣∣ = 0 , 104 j. abidi ∂2u2 ∂z∂z (z,w)|α2| + ∂2u2 ∂w∂w (z,w)|β|2 + 2 re [ ∂2u2 ∂z∂w (z,w)αβ ] = |β − a2α|2 + 2|β|2, and then 0 < |β − a2α|2 + 2|β|2 for each (α,β) ∈ c2\{0}. if β = 0, then α ̸= 0 and 0 < |a2α|2. it follows that a2 ̸= 0. in this case we have 2|β|2 + |β − a2α|2 > 0 for each (α,β) ∈ c2\{0}. case 2: a2 = 0. in this situation u2(z,w) = |w − f(z)|2 + |w − f(z)|2 + |w − b2|2. let (z,w) ∈ c2 and (α,β) ∈ c2\{0}; u2 is a function of class c∞ in c2. we have∣∣∣∣∂2u2∂z2 (z,w)α2 + ∂ 2u2 ∂w2 (z,w)β2 + 2 ∂2u2 ∂z∂w (z,w)αβ ∣∣∣∣ = | − 2αβf ′(z)| , ∂2u2 ∂z∂z (z,w)|α2| + ∂2u2 ∂w∂w (z,w)|β|2 + 2 re [ ∂2u2 ∂z∂w (z,w)αβ ] = |β − a1α|2 + |β|2 + |a1α|2 + |β|2 > |2αβa1| . assume that a1 = 0. we take β = 0 and α = 1. we obtain 0 > 0, which is a contradiction. it follows that a1 ̸= 0. in this case we have |2αβa1| ≤ |β|2 + |a1α|2. but also we have|β − a1α|2 + |β|2 > 0 for each (α,β) ∈ c2\{0}. thus |β − a1α|2 + 2|β|2 + |a1α|2 > 2|αβa1| for every (α,β) ∈ c2\{0}. case 3: a1 ̸= 0 and a2 ̸= 0. by the above development it follows that |2αβa1| < |β − a1α|2 + |β|2 + |a1α|2 + |β − a2α|2, for all (α,β) ∈ c2\{0}. thus t1(α,β) = |β − a1α|2 + |β − a2α|2 + (|β| − |a1α|)2 > 0 . strictly plurisubharmonic functions 105 assume that αa1 = βδ, where δ ∈ ∂d(0,1). then |αa1| = |β|. in this case, |1 − δ|2 + ∣∣1 − a2 a1 δ ∣∣2 > 0. consequently, t1(α,β) > 0 for each (α,β) ∈ c2\{0} if and only if |1−δ|2 + ∣∣1− a2 a1 δ ∣∣2 > 0 for every δ ∈ ∂d(0,1). thus t1(α,β) > 0 for each (α,β) ∈ c2\{0} if and only if a2 a1 ̸= 1. finally, we resume the above development by the following result. let φ1,φ2 : c → c be 2 analytic functions. set φ(z,w) = a1|w − φ1(z)|2 + a2|w − φ1(z)|2 + a3|w − φ2(z)|2, where (z,w) ∈ c2, a1,a2,a3 ∈ r+\{0}. then φ is strictly convex in c2 if and only if φ1(z) = a1z + b1, φ2(z) = a2z + b2, for z ∈ c and a1,a2,b1,b2 ∈ c with a1 ̸= a2 (the stictly convexity of φ is independent of b1 and b2). note that, for all b1,b2 ∈ r+\{0}, ψ is not strictly convex in c2, where ψ(z,w) = b1|w−φ1(z)|2 +b2|w−φ1(z)|2, (z,w) ∈ c2. but there exists several possible cases (of the analytic function φ1 defined over c) such that ψ is strictly psh over all c2. question 3.3. prove that there exists an analytic function g : c → c such that for all a0,a1,a2,a3 ∈ r+\{0}, the function u = a0|g|2 + a1|g′|2 + a2|g′′|2 + a3|g′′′|2 is not convex over c. we can in fact generalize this question for every fixed order m of the derivative of g denoted ∂mg ∂zm or over analytic functions defined on cn, where n ≥ 2. remark 3.4. let g1, . . . ,gn : cn → c be n analytic functions, where n,n ∈ n. assume that |g1|2 + · · · + |gn|2 = u is convex and strictly psh in cn. we can not conclude that u is strictly convex in cn. but we have the next statement. theorem 3.5. let g1, . . . ,gn : cn → c be n analytic functions and n,n ∈ n. put u(z,w) = |g1(w1 − z1)|2 + · · · + |gn(wn − zn)|2, v(w1, . . . ,wn) = |g1(w1)|2 + · · · + |gn(wn)|2, where (zj,wj) ∈ cn × cn, 1 ≤ j ≤ n and (z,w) = (z1, . . . ,zn,w1, . . . ,wn). the following conditions are equivalent: 106 j. abidi (a) u is strictly psh in (cn × cn)n; (b) n = 1 and |g1|2, . . . , |gn|2 are strictly convex functions over c; (c) n = 1 and v is strictly convex in cn. proof. recall that by abidi [2], we have for every function k : cn → c be analytic, if we put φ(z,w) = |k(w − z)|2, where (z,w) ∈ cn × cn. then φ is psh on cn × cn if and only if (k(z) = (< z,a > +b)m for each z ∈ cn, where a ∈ cn, b ∈ c and m ∈ n ∪ {0}) or (k(z) = e(+µ), for every z ∈ cn, where λ ∈ cn and µ ∈ c). note that φ is psh on cn × cn if and only if |k|2 is convex on cn. (a) ⇒ (b) let u1(z,w) = |g1(w1 − z1)|2, . . . ,un(z,w) = |gn(w − z)|2, where (z,w) = ((z1,w1), . . . ,(zn,wn)) ∈ (cn × cn)n. u is strictly psh on (cn ×cn)n if and only if u1, . . . ,un are strictly psh on cn ×cn. for example by abidi [2], u1 is strictly psh on cn × cn if and only if n = 1 and g1 is an affine bijective function over c. therefore |g1|2 is strictly convex on c. it follows that |g1|2, . . . , |gn|2 are strictly convex functions over c. the remainder of the proof of this theorem follows from the above development. claim 3.6. let k1, . . . ,kn : d → c be n analytic functions, d is a domain of cn, n ≥ 1. the system  α1 ∂k1 ∂z1 (z) + · · · + αn ∂k1∂zn (z) = 0 ... α1 ∂kn ∂z1 (z) + · · · + αn ∂kn∂zn (z) = 0 has only the solution (α1, . . . ,αn) = 0 ∈ cn (for all z fixed in d), if and only if u is strictly psh in d × cn, where u(z,w) = a1 ∣∣w1 − k1(z)∣∣2 + · · · + an∣∣wn − kn(z)∣∣2, for z = (z1, . . . ,zn) ∈ d, w = (w1, . . . ,wn) ∈ cn and a1, . . . ,an ∈ r+\{0}. now fix f1, . . . ,fn : d → c be n arbitrary analytic functions. the above system has only the solution (α1, . . . ,αn) = (0, . . . ,0) for all z ∈ d if and only if v is strictly psh in d × cn, where v(z,w) = a1 ∣∣w1 − f1(z) − k1(z)∣∣2 + · · · + an∣∣wn − fn(z) − kn(z)∣∣2, for z = (z1, . . . ,zn) ∈ d, w = (w1, . . . ,wn) ∈ cn and a1, . . . ,an ∈ r+\{0}. strictly plurisubharmonic functions 107 that is we have a rigid relation between strictly plurisubharmonic functions and holomorphic or antiholomorphic partial differential equations in cn, n ≥ 1. observe that we have a good relation between the algebraic method for the resolution of a system of holomorphic partial differential equations and the study of the strictly plurisubharmonic of a only one function in complex analysis and conversely. in the case of a power of analytic equations, we have the following result. theorem 3.7. let g1,g2, ,k1,k2 : c2 → c be four analytic functions, and let m1,s1,m2,s2 ∈ n. the system  ( α1 ∂k1 ∂z1 (z) + α2 ∂k1 ∂z2 (z) )2m1 + ( α1 ∂g1 ∂z1 (z) + α2 ∂g1 ∂z2 (z) )2s1 = 0( α1 ∂k2 ∂z1 (z) + α2 ∂k2 ∂z2 (z) )2m2 + ( α1 ∂g2 ∂z1 (z) + α2 ∂g2 ∂z2 (z) )2s2 = 0 has only the solution (α1,α2) = (0,0) for each z = (z1,z2) ∈ c2, if and only if u is strictly psh on c2 × c, where u(z,w) = ∣∣w − k1(z)∣∣2 + ∣∣w − k2(z)∣∣2 + ∣∣w − g1(z)∣∣2 + ∣∣w − g2(z)∣∣2 for each (z,w) ∈ c2 × c. proof. define v by v(z,w) = 4|w|2 +|k1(z)|2 +|k2(z)|2 +|g1(z)|2 +|g2(z)|2, where (z,w) ∈ c2 ×c; u and v are functions of class c∞ on c2 ×c. in fact u and v have the same hermitian levi form over c2 × c. now the proof is easy to describe. references [1] j. abidi, sur quelques problèmes concernant les fonctions holomorphes et plurisousharmoniques, rend. circ. mat. palermo (2) 51 (2002), 411 – 424. 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[23] a. zeriahi, pluricomplex green functions and the dirichlet problem for the complex monge-ampère operator, michigan math. j. 44 (1997), 579 – 596. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 1 (2022), 1 – 56 doi:10.17398/2605-5686.37.1.1 available online december 3, 2021 the hitchhiker guide to categorical banach space theory. part ii∗ j.m.f. castillo instituto de matemáticas imuex, universidad de extremadura avenida de elvas s/n, 06006 badajoz, spain castillo@unex.es received march 1, 2021 presented by f. cabello accepted october 20, 2021 abstract: what has category theory to offer to banach spacers? in this second part survey-like paper we will focus on very much needed advanced categorical and homological elements, such as kan extensions, derived category and derived functor or abelian hearts of banach spaces. key words: categorical banach space theory, derived functors, kan extension. msc (2020): 46m15, 46m18, 18-02, 18a40. hmbst. so it goes the hitchhiker guide to categorical banach space theory. part i [extracta math. 25 (2010) 103–149], from now on referred to as hhi was written to display the basic elements from categorical algebra transplanted to banach space theory: • the definition of category, functor and natural transformation. • pullbacks and pushout construction. • limits and colimits. • adjoint functors. • duality as it was promised there “the study of derived functors conforms what is called homology theory, and will be treated in part ii”. ten years later, it is showtime! in the span between hhi and the present moment, quite a few ∗ this research was supported in part by mincin project pid2019-103961gb-c21 and project ib20038 de la junta de extremadura. issn: 0213-8743 (print), 2605-5686 (online) c©the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.37.1.1 mailto:castillo@unex.es https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 2 j.m.f. castillo things happened: one of them is the writing and we hope imminent publication of the book [f. cabello sánchez, j.m.f. castillo, homological methods in banach space theory, cambridge studies in advanced mathematics, cambridge univ. press] [8], from now on referred to as hmbst. this book contains, as the title clearly states, the lion’s share of what (the authors know that) is currently known about homological methods in banach space theory. more precisely, the following topics, always with the focus set on banach and quasi banach spaces, are treated there: the definition of category, functor and natural transformation, pullback and pushout constructions, exact sequences, the functor ext, derivation of functors and why ext is the derived functor of l in the category of p-banach spaces, the construction of the long homology sequence up to ext terms, limits and colimits, adjoint functors, natural equivalences of ext and fräıssé limits. since it is absurd to ignore that, but also because we will not be able to provide a better exposition than the one presented in hmbst, this survey continues where hmbst stopped. we have also made every possible effort not to continue where [2] stopped. a point hmbst attempted to make clear is that the study exact sequences of banach spaces needs —actually, necessarily, peremptorily needs— dealing with exact sequences of quasi banach spaces. there is no escape to that. thus, if one accepts that homology deals with exact sequences, whatever attempt to study homology in the category of banach spaces requires coming to terms with the idea of making homology in the category of quasi banach spaces. but not all things that can be done in banach spaces can be done in quasi banach spaces. and this is, categorically speaking, a catastrophe. throughout this paper it is our plan to display several categorical tools that could help to overcome, surround or circumvent such difficulties. categories will be labeled a,b,c,d,e,f, . . . , and functors a,b,c,d,e, f, . . . . those of banach and quasi banach spaces (and operators) will be called ban and qban; and that of vector spaces and linear maps that will be called vect. the categories of banach (resp. quasi banach) spaces and contractive operators will be called ban1 (resp. qban1). 7. the categories cd given a category c and a small category (say, one that can be understood as a diagram; or else, one whose objects and morphisms are sets) d, the product category cd has as objects the functors d → c and the corresponding natural transformations as morphisms. there is an obvious covariant diagonal categorical banach space theory ii 3 functor ∆ : c → cd: the image of an object is the functor d → c sending all objects to x and all arrows to the identity of x. which is the image of an arrow is clear. this uplifting of the action from c to cd can be enriching. think about ban. a conceptually valid reason to jump from ban to banban is to provide a solid support to the eilenberg-maclane program [22] (which, as explained in hhi, means that functors and natural transformations is all there is in a mathematical theory). thus, banach spaces must be interpreted as functors and operators would then be natural transformations. let us attempt to identifying the category ban of [banach spaces + operators] as a full subcategory of the category banban of [banach functors + banach natural transformations] (see below for unexplained notation). recall that a banach functor is a functor f : ban → ban. it is often required that f is linear, with the meaning that f(λt + s) = λf(t) + f(s); it will also be sometimes required to be norm decreasing, with the meaning ‖f(t)‖≤‖t‖. a number of reasonably interesting banach functors were presented in hhi: • given a banach space y the contravariant ly functor defined by ly (x) = l(x,y ) and ly (t)(s) = st. the choice y = r gives the duality functor. • given a banach space x the covariant lx functor defined by lx(y ) = l(x,y ) and lx(t)(s) = ts. the choice x = r gives the identity. • given a banach space x the covariant ⊗x functor defined by ⊗x(y ) = x⊗̂πy and ⊗x(t) = 1x ⊗t. • semadeni’s covariant banach-mazur functor [55] that assigns to a banach space x the space c(bx∗); and to a contractive operator t : x −→ y the operator f → ft∗. • the ultraproduct functor x → xu. • the remainder functor x → x∗∗/x. • the covariant functors assigning to a banach space x the space `p(x) of p-summable sequences or the space c0(x) of norm null sequences and the naturally induced operators. • the grothendieck-pietsch functors that assign to a banach space x the space `wp (x) of weakly p-summmable sequences on x or the space c w 0 (x) of weakly null sequences. in addition to that, there are two of distinguished functors between ban and qban: 4 j.m.f. castillo • the forgetful functor � : ban −→ qban that simply forgets the banach structure and just leaves the quasi banach structure. • the banach envelope functor co : q −→ b that associates to each quasi banach space q with unit ball bq its banach envelope co(q), which is the completion of q endowed with the norm whose unit ball is the convex envelope of bq (if the dual of q does not separate points, making the necessary quotient). this is a functor [8, 1.1]: the canonical map ıq : q −→ co(q) has the property that whenever y is a banach space every operator q −→ y factors through ıq. a natural transformation η : f −→g is called a banach natural transformation if ‖η‖ = supx ‖ηx : fx −→gx‖ < ∞. the banach space of banach natural transformations between two banach functors f,g will be denoted [f,g]. there are however many ways to reinterpret ban inside banban. in a sense, everything one has to do is to replace banach spaces by banach functors, which can be done in many ways, and operators by banach natural transformations. more precisely, one has to define an interpretation functor δ : ban −→ banban. for instance (see hhi for details): • if δ(x) = ⊗x then [⊗a,⊗b] = l(a,b); which shows this is a faithful representation. • if δ(x) = lx then [la,lb] = l(b,a), so this is also a faithful representation. • if δ(x) = lx then [la,lb] = l(a,b), so this is also a faithful representation. • of course, one could set δ = ∆, in which case one also has l(x,y ) = [∆(x), ∆(y )]. in this way, when working on ban the action does actually takes place in banban, and one must thus be ready to transplant our notions to banban. how to do that? an idea is to trust in kan extensions. naively said, once an interpretation δ has been established, a kan extension of a banach functor f : ban → ban is a functor fk : banban −→ banban such that fkδ = δf. to understand what is going on here, we need to recall a couple of notions. categorical banach space theory ii 5 8. adjoint functors the adjointness notion, also due to kan [35], was explained in part i and will be explained now. given two covariant functors f : a → b and g : b → a one says that f is a left adjoint of g, written f |g (and, consequently, g is a right adjoint of f) when for every objects a of a and b of b there is a natural equivalence η : homb(f(·), ·) −→ homa(·,g(·)) between the functors a,b → homb(f(a),b) and a,b → homa(a,g(b)). as semadeni mentions [56, p. 217], sometimes during proofs it is more convenient to use an equivalent formulation of the adjunction relation: it turns out that f |g if and only if there are natural transformations ε : 1 → gf and δ : fg → 1 such that δf◦gε = 1 and gδ◦εg = 1 as natural transformations f fε−−−→fgf δf−−−→f , f εg−−−→gfg gδ−−−→g . in other words, if we decide to ignore the natural transformations involved and replace them by equalities, the idea behind this notion is that f |g when homb(f(a),b) = homa(a,g(b)). a second interpretation is only possible with care: one is certainly saying that g and f are, somehow, inverses, namely 1 = gf and fg = 1 in the sense that there are natural transformations 1 →gf and fg → 1 (but not gf = 1 or 1 = fg !). just to give an example at hand, let us recall from [8, 4.6.1 (b)] that co |�. 9. limits this topic is covered both in hhi and hmbst. briefly said, given a functor f ∈ cd, its limits lim → f and lim ← f, if they exist, are objects of c provided with arrows αd : f(d) → lim← f (resp. βd : lim→ f → f(d)) making a commutative diagram, and being universal with respect to this property; namely, given any other object c of c provided with arrows α′d : f(d) → c making the diagram commutative, there is a unique arrow α : lim → f → c making all diagrams commutative (resp. given any other object c of c provided with arrows α′d : c → f(d) making the diagram commutative, there is a unique arrow β : c → lim ← f making all diagrams commutative). enthusiastic readers will have surely observed that when limits exist then lim → and lim ← can 6 j.m.f. castillo be understood as functors cd −→ c, which turn out to be the adjoints of ∆ : c → cd; precisely: ∆ | lim ← f and lim → f | ∆ . a lively discussion about which one, lim → or lim ← , is the limit and which one the co-limit is in hhi. here we will call, when the occasion demands it, lim → the inductive limit and lim ← the projective limit. a simple example can be provided just for fun: proposition 9.1. in the category ban1, every banach space is an inductive limit of its finite dimensional subspaces. hint: if the reader feels that she (or he) really needs a proof, do it. or else, read it [21, i, 1.20]. 10. comma categories furnishing us with this tool is like putting a gun in a monkey’s hand. or, worse, a typewriter. assume that one has a category c and a larger category d and pick an object d of d. definition 10.1. the comma category (c � d) has as objects pairs (x,f) formed by an object x of c and an arrow f : d −→ x, and whose arrows ϕ : (x,f) −→ (y,g) are arrows ϕ : x −→ y so that ϕg = f. the comma category (c � d) has as objects pairs (x,f) formed by an object x of c and an arrow f : x −→ d, and whose arrows ϕ : (x,f) −→ (y,g) are arrows ϕ : x −→ y so that gϕ = f. any functor f : c −→ e induces functors fd : (c � d) −→ e given by fd(x,f) = f(x) and another functor fd : (c � d) −→ e given by fd(x,f) = f(x). paraphrasing roberto begnini, when a man with a gun meets a man with a pen, the man with the gun is done. let us show why the monkey was more dangerous with the comma category: proposition 10.2. every banach space x is an inductive limit x = lim −→ `n1 in ban1. proof. think of ban1 as “the larger category” inside which one will work. construct the comma category (`•1 � x) where ` • 1 represents here the (full categorical banach space theory ii 7 sub-) category (of ban1) whose objects are finite dimensional ` n 1 spaces and let f : (`•1,� x) −→ ban1 the obvious forgetful functor that associates to an object (`n1,ϕ) the space ` n 1 . let us show that x = lim−→ f(`d1,ϕd) = lim−→ ` d 1. this amounts showing that whenever one has a banach space y and another commuting family of arrows ψd : ` d 1 −→ y (i.e., when αd,d′ : ` d 1 −→ ` d′ 1 in the comma category —ϕd′αdd′ = ϕd— as in the diagram below) then it is possible to define a contractive operator t : x −→ y so that tϕd = ψd. of course that given a commuting diagram `d1 ϕd ��@ @@ @@ @@ @ αdd′ �� ψd **vvv vvvv vvvv vvvv vvvv vvvv vvvv x y `d ′ 1 ϕd′ ??~~~~~~~ ψd′ 44hhhhhhhhhhhhhhhhhhhhhhhhhh there is not, necessarily, a t : x −→ y making the whole diagram commutative. but that is on a one-by-one basis. when the families (ϕd) and (ψd) involve all choices, such t exist. to prove it let us show that whenever ϕd(u) = ϕd′(u ′) then ψd(u) = ψd′(u ′). if so, given x ∈ x such that x = ϕd(u) the point t(x) = ψd(u) is well defined. now, pick x ∈ x and φd : `d1 −→ x so that φd(u) = x. since φd is contractive, ‖x‖≤ ‖u‖1. let l be the one dimensional banach space. when it is not confusing we simplify `d1 to just d and define φ[d,φd,u,x] : l −→ x by φ[d,φd,u,x](1) = x/‖u‖1. the pair (l,φ[d,φd,u,x]) is an object in the comma category, and α : (l,φ[d,φd,u,x]) −→ (`d1,φd) given by α(1) = u/‖u‖1 defines a morphism of the comma category. so we are all friends here, right? and with this we want to say that when the family (ψd) enters the game then ψd = ψdα which, in particular, yields ψd(u) = ψdα(‖u‖1) = ‖u‖ψdα(1) = ‖u‖ψ(l,φ[d,φd,u,x])(1) . assume that ϕd(u) = ϕd′(u ′) with ‖u‖ ≥ ‖u′‖ (the other case is similar). the commutativity of the diagram l φ[d,u,x] ++vvv vvvv vvvv vvvv vvv x l β oo φ[d′,u′,x] 33hhhhhhhhhhhhhhhhhh 8 j.m.f. castillo in which β(1) = ‖u‖/‖u′‖ provides that also l ψ(l,φ[d,u,x]) ++vvv vvvv vvvv vvvv vvv y l β oo ψ(l,φ[d′,u′,x]) 33hhhhhhhhhhhhhhhhhh is commutative, so that ψd(u) = ‖u‖ψ(l,φ[d,φd,u,x])(1) = ‖u‖‖u′‖ ‖u‖ ψ(l,φ[d,φd,u,x]) (‖u‖ ‖u′‖ ) = ‖u‖‖u′‖ ‖u‖ ψ(l,φ[d,φd,u,x])(β(1)) = ‖u′‖ψ(l,φ[d′,φd′,u′,x])(1) = ψd′(u ′) . thus, t(x) = ψd(u) = ‖u‖ψ(l,φ[d,φd,u,x])(1) is a judicious choice. or would be, were this t a linear contractive operator. the contractive part is clear since, no matter what it seems, ‖t(x)‖≤ inf{‖u‖ : x = ϕd(u)}≤‖x‖ . the map t is homogeneous since t(λx) = ψd(λu) = λϕd(u) = λt(x), so it only remains the additivity: why t(x + y) = tx + ty? well, because given two independent x,y (otherwise . . . ) then we also have the element (`21,ϕ) with ϕ : `21 → x given by ϕ(λe1 + µe2) = λ x ‖x‖ + µ y ‖y‖ that, together with the canonical inclusions ı(1) = e1, (1) = e2 form a commutative diagram l ı // ϕı &&l ll ll ll ll ll ll ` 2 1 ϕ �� l oo ϕ xxrrr rr rr rr rr rr x if we call d1 = (` 1 1 = [x],ϕ1), where ϕ1 : ` 1 1 = [x] −→ x is given by ϕ1(x) = x and d2 = (` 1 1 = [y],ϕ2) with ϕ2 : ` 1 1 = [y] −→ x given by ϕ2(y) = y, then categorical banach space theory ii 9 ϕı = φ[d1,ϕı,x,x] and ϕ = φ[d2,ϕ,y,x]. thus, the diagram above becomes l ı // φ[1,ϕı,x,x] &&l ll ll ll ll ll ll ` 2 1 ϕ �� l oo φ[2,ϕ,y,y] xxrrr rr rr rr rr rr x then, by commutativity, ψ(`21,ϕ) ı = ψ(l,φ[1,x,x]) and ψ(`21,ϕ)  = ψ(l,φ[2,y,y]). let us assign the index d to (`21,ϕ) and call w = ‖x‖e1 + ‖y‖e2 to write t(x + y) = ‖w‖ψ(l,φ[d,ϕ,w,x+y])(1) t(x) = ‖x‖ψ(l,φ[d1,ϕı,x,x])(1) t(y) = ‖y‖ψ(l,φ[d2,ϕ,y,y])(1). now, ψdı = ψd1 and ψd = ψd2 by commutativity and thus t(x + y) = ‖w‖ψ(l,φ[d,ϕ,w,x+y])(1) = ψd(w) = ψd(‖x‖e1 + ‖y‖e2) = ‖x‖ψ(l,φ[1,ϕı,x,x])(1) + ‖y‖ψ(l,φ[1,ϕ,y,y])(1) = t(x) + t(y). and, consequently, proposition 10.3. every dual banach space x∗ is a projective limit x∗ = lim ← `n∞ in ban1. there are different ways to prove this, being the shortest one to appeal to ( lim → xi )∗ = lim ← x∗i , something that, at least for finite dimensional xi anyone can do by hand. a more sophisticated way could be to recall from [13, proposition 5.5] that contravariant functors adjoint on the right transform inductive limits into projective limits to then recall from [13, proposition 5.7] that the duality functor is adjoint to itself on the right. some of the most elegant applications of comma categories to banach space theory, the construction of universal operators, were obtained by kubís and his collaborators [12, 27] and have been no less elegantly exposed in hmbst, section 6.4. see also [38]. 10 j.m.f. castillo 11. kan extensions we are ready now to know what kan extensions are. extension theorems are central in mathematics, and twicely central in functional analysis. one has a mathematical object c “inside” another d and an arrow or function f : c → e and wants to obtain an extension f : d → e. even when this can be done (tietze’s theorem, hahn-banach theorem, . . . ) the extension is not, as a rule, unique and the way of performing the extension is not canonical. quoting maclane [41, p. 229] “however, if c is a subcategory of d each functor t : c −→ e has in principle two canonical (or extreme) “extensions” to functors d −→ e”. we will call lt and rt those extensions. to see how to proceed, observe that given a covariant functor δ : c −→ d between two categories and another category e there is a natural functor δ∗ : ed −→ ec which is just plain composition with δ; i.e., it associates to each functor s the composition functor sδ. this functor δ∗ may have or have not a left adjoint l |δ∗ or a right adjoint δ∗ |r. if it does, the functors l,r provide apparently good natured extensions of functors c −→ e to functors d −→ e through δ. 11.1. left kan extension definition 11.1. the left kan extension of f through δ : c −→ d is a functor lf : d −→ e together with a natural transformation ε : f −→ lfδ, with the property of being universal for this diagram; namely, given a functor g : d −→ e and a natural transformation η : f −→ gδ there is a unique natural transformation σ : lf −→g so that η = σε. to show its distinguished character, the functor lf : d −→ e comes accompanied with a natural transformation providing its uniqueness property. let us draw the situation: this is the extension functor c δ // f a aa aa aa d lf �� e and here it is the associated natural transformation categorical banach space theory ii 11 c f �� c δ // lfδ �� d lf �� ε // e e e (1) and here is the uniqueness lf σ //g lfδ σδ //gδ f ε bbdddddddd η >>}}}}}}}} one has: proposition 11.2. if δ∗ has a left adjoint l |δ∗ then lf is the left kan extension of f. thus, the left kan extension of f comes defined by the equation [lf,g] = [f,gδ] for every g : d −→ e. this proof we will make in the next section for the right kan extension; so the reader just have to dualize it. if δ∗ has a left adjoint then every functor has a left kan extension, but it could happen that a given functor admits left kan extension even when δ∗ has no left adjoint. the next method shows a way to compute lf. proposition 11.3. if the category e admits limits then lf(d) = lim −→ fd. proof. let f : c −→ e be a functor. pick an object d of d, form the comma category (c � d) and then the induced functor fd : (c � d) −→ e. the limit lim −→ fd exists by hypothesis and we can form the functor lf as above. our plan is to show that it defines the left kan extension of f. it is clear that lf(c) = lim −→ fc = fc for every object c of c. thus lfδ = f and we can set ε = 1 as the accompanying natural transformation ε : f −→ lfδ. let g be another functor d −→ e with accompanying 12 j.m.f. castillo natural transformation η : f −→gδ. namely, for given c : c −→ c′ there is a commutative diagram f(c) f(c) �� ηc //g(c) g(c) �� f(c′) ηc′ //g(c′) it is then guaranteed the existence of the natural transformation σ : lf −→g we need, namely that for given d : d −→ d′ one has a commutative diagram lim −→ fd lf(d) �� σd //g(d) g(d) �� lim −→ fd ′ σd′ //g(d′) to make appear the arrow σd recall that objects of the comma category (c � d) are pairs (c,f) with c an object of c and f : c → d an arrow. there exists therefore arrows g(f) : g(c) −→ g(d) and thus the universal property of limits yields σd. 11.2. right kan extension definition 11.4. the right kan extension of f through δ : c −→ d is a functor rf : d −→ e together with a natural transformation ε : rfδ −→f, with the property of being universal for this diagram; namely, given a functor g : d −→ e and a natural transformation η : gδ −→ f there is a unique natural transformation σ : g −→ rf so that η = εσδ. the existence of natural transformation accompanying the extension that enjoys some uniqueness property provides a clean way of saying that the functor rf : d −→ e is a distinguished extension of f. let us draw the required properties: here it is the extension functor c δ // f a aa aa aa d rf �� e categorical banach space theory ii 13 here it is the associated natural transformation c f �� c δ // rfδ �� d rf �� εoo e e e (2) and here it is the uniqueness rf g σoo rfδ ε ""d dd dd dd d gδ η ~~}} }} }} }} σδoo f one has the dual version of proposition 11.2, now with proof: proposition 11.5. if δ∗ has a right adjoint δ∗ |r then rf is the right kan extension of f. thus, the right kan extension of f comes defined by the equation [g,rf] = [gδ,f] for every g : d −→ e. proof. set g = rf in the equation [g,rf] = [gδ,f] that thus becomes [rf,rf] = [rfδ,f] and pick the natural transformation ε : rfδ →f corresponding to the identity rf → rf. now, if g is another functor accompanied with a natural transformation η ∈ [gδ,f] then obtain the corresponding σ ∈ [g,rf]. to check that εσδ = η as elements of [gδ,f] one just has to check that the corresponding elements in [g,rf] are the same: σ. and this is simple since σδ ∈ [gδ,rfδ] = [g,r(rfδ)] = [g,rf] corresponds to σ then εσδ corresponds to 1σ = σ. of course that if δ∗ has a right adjoint then every functor has a right kan extension, but it could happen that a given functor admits right kan extension even when δ∗ has no right adjoint. the following characterization and its proof are clear dualizations of proposition 11.3. proposition 11.6. if the category e admits colimits then rf(d) = lim ←− fd . 14 j.m.f. castillo 12. uses of kan extensions for we, banach spacers the category of banach spaces does not live alone in banach space affairs: it is intimately connected with the categories pban of p-banach spaces and with the more obscure one qban of quasi banach spaces. now, ban is not, categorically speaking, a good category because operators do not have the right cokernels, arbitrary products do not exist, let alone limits . . . . however, it has, at least, enough injective and projective objects (in a more restrictive sense: see section heart after all, the operators one can actually deal with are those with closed range (see section 19), and where enough means enough to present any banach space). the category pban is worse: in addition to that, it has no injectives at all [8]. luckily, it still has (in the same restricted sense) enough projectives. but, alas! there is no minimum in the slope of worst, and qban is awful: in addition to everything that was already wrong it does not have injective or projective objects. how operative ban is is a returning topic in this paper. let us focus now in how bad qban is. moving at bullet-time, skipping next section and arriving to section 14 we will encounter that there are two standard procedures to proceed with the derivation of functors: one uses injective objects and yields right-derivation and the other uses projective objects and yields left-derivation. the good news is that none of them is available in qban, so we need to rethink derivation in this category from scratch. in pban one at least can still perform leftderivation and that fosters the delusion of understanding. but not in qban. we could think this is important because homology is based on the idea of derivation (in approximately the same sense that calculus is based on the idea of derivative). but we could also adopt a broader view, not stick to derivation, and consider that homology is based on the idea that one can obtain an algebraic map of topological spaces. in which case to put in focus qban seems yet more important. to provide us with a something to take home: is ext the derived functor of l in qban? well, the first issue is: which functor is exactly l and ext? because the definition of a functor requires to specify the categories between which it acts. and l is deceitful. and it is deceitful precisely because it is a functor ban −→ ban; thus, if we trust that ext is going to be the derived functor of l then we must find a way to make ext a functor ban −→ ban. this, to the best of our current knowledge, is impossible: ext(`2,`2) is not and cannot be a banach space, see hmbst, section 4.5: it is not because the “natural norm” to be set on ext(`2,`2), which is moreover the one that makes continuous the maps appearing in the homology sequence, is not hausdorff; namely, there categorical banach space theory ii 15 are 0 6= α ∈ ext(`2,`2) for which ‖α‖ = 0 (hmbst, proposition 4.5.5). mathematical life is funny here, because when p < 1 the space ext(lp/`2,r) is a hilbert space, (hmbst, corollary 4.5.2). the part of me that believes that tyrion was a rightful candidate to rule the seven kingdoms grunts that having or not having a trivial dual cannot be the responsible for the difference of behaviours because ext(`2,`2) is the same in ban than in qban. while the part of me that believes that it should have been gandalf who sits in the iron throne is convinced that it is. be as it may, and summing up: we cannot support or endorse in any natural way that ext is the derived functor of the functor l : ban −→ ban. what we can do and do is to slip away and consider the functor l : ban −→ vect (the category of vector spaces and linear maps). now we can prove that ext is the derived functor of this l ! and who could be the derived functor or l : qban −→ vect? well, ext behaves as if it were: its construction can be done yoneda style, as well as the iterated extn functors and the long homology sequence that connects them exists in complete analogy with the ban case. is this enough to accept that ext is the derived functor of l in qban through a derivation process that could not exist? of course not. but the pot of gold at the end of this rainbow is whether, and how, derivation can be defined in qban. one idea we can explore in this section is to what extent extqban is a kan extension of extban. this would at least provide us some basis to believe that a kan extension of a derived functor can be called a derived functor. moving in that direction, let us gain first some confidence about obtaining kan extensions of banach functors. observe that since the semadeni-zidenberg theorem [54] establishes that ban1 admits limits and colimits, one has: corollary 12.1. let f : ban1 −→ ban1 be a banach functor. • if rf is a right kan extension through � to qban1 then rf� = f. • if lf is a left kan extension through � to qban1 then lf� = f. let us now test our abilities obtaining the kan extensions qban1 −→ ban1 of the identity 1 : ban1 −→ ban1: proposition 12.2. l1 = r1 = co. proof. it is obvious that co� = 1 and thus the natural transformation ε : co� −→ 1 we st is the identity. to show that co is the right kan extension of 1, let us assume first that one has another functor g : qban1 −→ ban1 16 j.m.f. castillo so that g� = 1, for which the natural transformation η is also the identity. to obtain the required natural transformation σ : g −→ co, pick a quasi banach space q, consider the canonical operator δq : q → co(q) and form g(δq) : g(q) → g(co(q)) = co(q). set σq = g(δq). in the general case, if we just assume that there is a natural transformation η : g� → 1 then set σq = ηco(q)g(δq). this shows that co is the right kan extension of the identity. to show it is also the left kan extension, fix a quasi banach space q and let us show that lim → 1q of the translation 1q of the identity functor to the comma category (ban1 � q) is co(q): consider the family of banach spaces x and contractive operators τ : x −→ q; so bx ⊂ τ−1(bq). this family includes the following elements: pick x ∈ bq and set the pair ([x], ıx) where ıx(x) = x. the existence of a contractive operator `x : ([x], ıx) −→ l implies ‖`x(x)‖≤‖x‖≤ 1. this yields the existence of a contractive operator q −→ lim 1q. now, given a banach space y and a contractive operator τ : q −→ y , the composition τϕx with ϕx : x −→ q provides contractive operators x −→ y , and consequently a contractive operator lim → 1q −→ y . all this means that lim → 1q = co(q). the following two diagrams are to illustrate the similitude between the two kan extensions (see also [39, theorem 2.6]). compare to diagrams (2) and (1) in the previous section: c δ // f !!c cc cc cc cc cc c d rf }}{{ {{ {{ {{ {{ {{ ← e c δ // f !!c cc cc cc cc cc c d lf }}{{ {{ {{ {{ {{ {{ → e keep also in mind that a kan extension is nowhere claimed to be an extension, in the sense that rfδ = f (when this makes sense, as it occurs with our � : ban −→ qban case). namely, even if g : qban −→ ban is a functor such that g� = f that does not guarantee that g is the right kan extension of f because there is no way to guarantee the existence of the natural transformation σ : g −→ rf. time to splash. what we did in proposition 12.2 was by no means showing that �∗ has adjoints, but to calculate that r1 = l1 = co, that is, to showing that one has the identities [g�, 1] = [g,co] and [co,g] = [1,g�] for every g : qban −→ ban. and those two things, getting an adjoint for �∗ and categorical banach space theory ii 17 getting some identities, are different things, by far. well, not that far. let us prepare the way: lemma 12.3. given functors u : c −→ d and v : d −→ c, if v |u then u∗ |v∗. proof. one claims that given g ∈ ed and f ∈ ec then [u∗g,f] = [g,v∗f] which is as obvious as obscure: given morphisms c : c −→ c′ and d : d −→ d′ one claims that natural transformations η,ν as in guc guc �� ηc //fc fc �� guc′ ηc′ //fc′ gd gd �� νd //fvd fvg �� gd′ νd′ //fvd′ correspond one to another, as it is clear they do (recall there are natural transformations ε : 1 →uv and ε′ : vu → 1): set νd = ηvd to get guvd guvd �� ηc //fvd fvd �� guvd′ ηvd′ //fvd′ =⇒ gd gd �� εd //guvd guvd �� ηvd //fvd fvg �� gd′ εd′ //guvd′ ηvd′ //fvd′ and analogously in the other case: set ηc = νuc and use ε ′. in particular (and probably simpler to prove) we get �∗ |co∗ as it follows from co |�. thus, by the uniqueness of adjoints, one gets: theorem 12.4. co∗ = r. thus, given a banach functor f, or a functor f : ban −→ qban or f : ban −→ vect, its right kan extension is co∗f. in particular: let x be a banach space and let q′,q be quasi banach spaces: • for lq′ : ban −→ ban or lq′ : ban −→ qban one has rlq′(q) = l(q ′,co(q)) . • for lx : ban −→ ban or lq ′ : ban −→ qban one has rlx(q) = l(co(q),x) = l(q,x) and rlq ′ (q) = l(co(q),q′) . 18 j.m.f. castillo • for extban(x, ·) : ban −→ vect one has r extban(x, ·)(q) = extban(x,co(q)). • for extqban(q′, ·) : ban −→ vect one has r extqban(q′, ·)(q) = extqban(q ′,co(q)). • for extban(·,x) : ban −→ vect one has r extban(·,x)(q) = extban(co(q),x). • for extqban(·,q′) : ban −→ vect one has r extban(·,q′)(q) = extban(co(q),q ′). what about the left kan extensions of lx,l x? well, a very much expected surprise is that our first hazardous guess l = r cannot actually be: l |�∗ and l = r yields co∗ |�∗, and this is at a teeth skin from saying � |co, which is false. and false on its own, anyway. by the pointwise formula in proposition 11.3 we get: theorem 12.5. let x be a banach space and q a quasi banach space. given lx taking values in either vect or ban1 one has • llx(q) = lim−→(lx) q, • llx(q) = lim −→ (lx)q. given extb(x, ·) taking values in vect one has • l extban(x, ·)(q) = lim−→ extban(x, ·) q, • l extban(·,x)(q) = lim−→ extban(·,x) q. always taking advantage from the fact that the categories vect and ban1 have limits and colimits. we have not a plausible guess about what occurs with those functors when taking values in qban1 since the semadeni-zidenberg theorem does not work in qban1 and the limit formulae are no longer available. is this what we expected? likely not: one would have wanted, perhaps, that the true natural extension of l to be the same l. but this is not how things are, as it is demonstrated by the work of herz and pelletier [29] and then pelletier [48, 49]. inspired, to some extent, by the work of cigler [20] as it is mentioned in [48, p. 486], they show that if b is a suitable subcategory of ban1 then the left kan extension of the functor l to ban1 is formed by the operators that factorize through an element of b. in particular, if categorical banach space theory ii 19 r is the category of reflexive spaces, its left kan extension are the weakly compact operators. and if f is the subcategory of finite dimensional spaces its left kan extension is, under some approximation property hypotheses, the compact operators. very nice. and showing that the extension of l should not be expected to be l. let alone that the natural extension of extban should be extqban. oh, well, worse things happen at sea. 13. homology after 540 pages of hmbst we (authors, readers, by-passers skimming the pages . . . ?) should have formed an idea about what banach space homology is and what is it for. the fulcrum point is the idea hinted above that a mathematical object is the object and its many representations, whatever that might mean. an analogue could be the (quite standard) idea that a real number r is not only the number r but also all sequences of rational numbers converging to r. in categorical jargon, the category of real numbers is derived from the category of rational numbers by: fixing as objects (certain) sequences of rational numbers plus an equivalence relation that suitably transforms morphisms of sequences into morphisms in the derived category. to do this uplifting from, say, ban to banban we need the core ideas of “complex” and “exact sequence”: definition 13.1. an exact sequence of quasi banach spaces is a diagram formed by quasi banach spaces and operators · · · −→ xi−1 −→ xi −→ xi+1 −→··· in which the kernel of each arrow coincides with the image of the preceding one. a morphism between two exact sequences is obviously a sequence (fi) of morphisms fi : xi → yi yielding commutative diagrams xi // fi �� xi+1 fi+1 �� yi // yi+1 let us call ex the category of exact sequences of banach spaces. an especially distinguished type of exact sequences is that of short exact sequences 0 −→ y −→ z −→ x −→ 0 . 20 j.m.f. castillo the arrows in the category are provided by triples of arrows (α,β,γ) making commutative diagrams 0 // · // α �� · // β �� · // γ �� 0 0 // · // · // · // 0 and we thus arrive to the category s of short exact sequences. two short sequences are thus isomorphic when there is a diagram as above with α,β,γ isomorphisms, which is the notion of isomorphic sequences used in hmbst. good examples of (long) exact sequences in ban are provided by the so-called projective presentations of a given space x; namely, exact sequences · · · −−→ p−n −−→··· d 2 −−−→ p−2 d 1 −−−→ p−1 d 0 −−−→ x −−→ 0 in which each space p−n is projective (i.e., some space `1(γ), since these are the only projective (in our restricted sense) banach spaces [37] (see also [24] for a close result); or else [45] where it is proved that `p(i) are the only projective (in restricted sense) p-banach spaces [45]. alternatively, injective presentations namely, exact sequences 0 −−→ x ∂ 0 −−−→ i1 ∂ 1 −−−→ i2 −−→···−−→ in −−→··· in which each space in is injective. examples of short exact sequences are provided by a banach space z, a subspace y and the corresponding quotient z/y in the form 0 −−→ y −−→ z −−→ z/y −−→ 0 . every short exact sequence of banach spaces is isomorphic to one of this type. but beyond the exactness wall there is much more life. the wildlings in this case are the complexes. a complex is, well, the “non exact” part of an exact sequence; that is, a sequence · · · −→ xi−1 −→ xi −→ xi+1 −→··· such that the composition of any two consecutive arrows is 0. so the reader may ask: why do we bother with complexes when we can have exact sequences? there are various reasons. one is that the notion of complex makes sense in every category having 0 object while that of exact sequence is much more delicate. intermission about the future. in a category such as abelian groups, an exact sequence is a complex in which the kernel of each arrow coincides with categorical banach space theory ii 21 the image of the preceding. the notion of kernel passes without difficulties to an arbitrary category but that of “image” does not. one has the cokernel notion, of course, and probably one would like to say that a sequence is exact (at least short exact) when each arrow is the kernel of the next one and the cokernel of the previous one. in an abelian category (see later) things go more or less straight since one can define (see definition 20.1) the image of an arrow as the kernel of its cokernel (see [66, p. 6]) and proceed. but that is because an abelian category is, by definition, a place where monics are kernels of their cokernels and epics are cokernels of their kernels. end of the future intermission. thus, having exact sequences requires a category where kernels and cokernels exist and behave well. it is curious that banach spaces, which is not actually one of those places, has a clean notion of exact sequence with it (this topic shall be discussed in section 19). another reason is that, even if one has exact sequences, general complexes could be needed (something similar to: to deal with exact sequences of banach spaces we need quasi banach spaces) to do things that cannot be done with only exact sequences. moving on, the typical complex one will consider will have the form · · · −→ c−n −→··· d 1 −−−→ c−1 d 0 −−−→ x ∂ 0 −−−→ c1 ∂ 1 −−−→···−→ cn −→··· projective presentations are complexes as above with cn = 0 for all n > 0 (and only nontrivial dn maps) and injective presentations are complexes with cn = 0 for all n < 0 (and only nontrivial ∂n maps). let us call com(c) the category of complexes of a category c. in particular com(b) is the category of banach space complexes. isomorphism in this category are what they are. let us reconsider if this we are groot situation is what we want. working in the short context, a projective presentation of a banach space x is a short exact sequence 0 −→ κ −→ `1(γ) −→ x −→ 0. a result that is half-classical and half-half-homological [8, section 2.7 and 2.11.6] is that if 0 −→ κ −→ `1(γ) −→ x −→ 0 and 0 −→ κ′ −→ `1(γ′) −→ x −→ 0 are two projective presentations of x then there is a commutative diagram 0 // κ× `1(γ′) // α �� `1(γ) × `1(γ′) // β �� x // 0 0 // κ′ × `1(γ) // `1(γ′) × `1(γ) // x // 0 (3) with α,β isomorphisms. in other words, the two projective presentations are isomorphic “up to some projective space”. thus, if we want that “all 22 j.m.f. castillo projective presentations are the same”, that is what we must get. even if the isomorphism notion of com(c) for exact sequences is useful and has provided many deep results in banach space theory, the right notion to work with projective presentations seems to be “equivalent up to some projective space”. let us give shape to this. grasping the right notion just requires to realize that the right objects in a category of short exact sequences are equivalence classes of exact sequences. two short exact sequences are said to be equivalent if there exists an operator β making a commutative diagram 0 // · // · // β �� · // 0 0 // · // · // · // 0 so beware that equivalence is a notion defined between exact sequences having the same initial and final spaces. the equivalence class of a sequence z will be denoted [z]. now, triples of arrows are no longer well suited to mingle with equivalence classes of exact sequences: after all, where should one define β? so, let us declare eex to be the category having as objects equivalence classes [z] of exact sequences and as morphisms couples (actually, equivalence classes of, see below) (α,γ) such that [αz] = [z′γ]. recall that this just means that the pushout sequence αz and the pullback sequence z′γ are equivalent. but, acting this way, the isomorphism notion has changed to a new isomorphism notion that we proceed to describe. given an exact sequence z and a quasi banach space e then will denote e ×z the sequence 0 −−→ e ×y 1e×−−−−→ e ×z 0⊕ρ −−−−→ x −−→ 0 and z×e the sequence 0 −−→ y (, 0) −−−−→ z ×e ρ×1e−−−−−→ x ×e −−→ 0 . in general, these sequences will be called multiples of z. since two objects [z] and [z′] are isomorphic if there exist arrows (α,γ) : [z] −→ [z′] and (α′,γ′) : [z′] −→ [z] such that (α,γ)(α′,γ′) = 1 = (α′,γ′)(α,γ), it is rather clear that [e × z] and [z ×e] are isomorphic objects (although, in general, the sequences e × z and z × e need not be either equivalent or isomorphic (for obvious reasons!). one has [16, proposition 3.1]: proposition 13.2. [z] and [z′] are isomorphic if and only if there exist e,e′ so that e ×z×e′ and e ×z′ ×e′ are isomorphic. categorical banach space theory ii 23 we still need to keep moving on to get home: to define our category sex with objects as in eex but with some identification of arrows: we declare (α,γ) to be 0 when αz = 0 = z′γ and, consequently, (α,γ) = (α′,γ′) when [(α − α′)z] = 0 = [z′(γ − γ′)]. to translate this to complexes, where no natural notion of equivalence disturbs us, we will return to an equivalence-free approach. peculiarities of short exact sequences imply that (α,γ) = (α′,γ′) in sex (namely, (α − α′)z = 0 = z′(γ − γ′)) if and only if there are arrows u,v 0 // y  // α−α′ �� z u �� ρ // �� x // γ−γ′ �� v ��~~ ~~ ~~ ~~ 0 (z) 0 // y ′ ′ // z′ ρ′ // x′ // 0 (z ′) such that u = α −α′, ρ′v = γ −γ′ and ′u + vρ = β −β′. and this means that the morphisms (α,β,γ) and (α′,β′,γ′) acting z −→ z′ are homotopic in the following sense: definition 13.3. two morphisms f,g : c −→ d between complexes are said to be homotopic when there is a morphism of complexes ui : c i −→ di−1 so that fn −gn = dnun + vndn. the conclusion of of all this is that there is an homological notion, homotopic maps, that extends the equality notion in sex to com. we can test this notion: pick a projective (short) presentation 0 −→ κ −→ `1(γ) −→ x −→ 0 of the banach space x. the identity of x can be lifted in many ways to `1(γ), and each lifting provides an arrow (α,β, 1). the point is that any two of these arrows (α,β, 1) and (α′,β′, 1) are homotopic. thus, the identification of a space x with its (short) projective presentations and operators with triples is faithful in the homotopic sense. move ahead with this idea, do yourself a favour and delete the word “short” in the last sentence (replacing triple by morphism of complexes) and show that whatever two projective presentations of a space x are homotopic (or whatever two injective presentations, for that matter). and we are ready to conform the category kom(b) having complexes of banach spaces as objects and homotopic equivalence classes of morphisms of complexes as morphisms. 13.1. exactness and how to measure it. exactness is however important. a short exact sequence 0 −→ y  −→ z ρ −→ x −→ 0 in ban is, 24 j.m.f. castillo by the virtues of the open mapping theorem, a banach space z, a subspace [y ] and the corresponding quotient x = ρ[z] = z/[y ]. a complex, even a short one, is . . . whatever it is. homology attempts to measure how large the deviation of a complex from exactness is. keep in mind that a complex could be exact at some point and not exact at another point. fix a complex (we do not distinguish here between i > 0 and i < 0) · · · −−→ ci+1 d i −−−→ ci d i−1 −−−−→ ci−1 −−→··· and define the ith-homology group (vector space in our case) hi(c) = ker di−1/ im di. of course that the complex c is an exact sequence if and only if hi(c) = 0 for all i. when exactness occurs only at position i one says that the complex c is acyclic at i or exact at i. a morphism of complexes f : c −→ d generates morphisms hif : hi(c) −→ hi(d) in a natural way hif(x+ im di+1) = fi+1(x) + im d i. it makes sense since x ∈ ker di ⇒ fi+1(x) ∈ ker di and since fi+1(x − y) = fi+1di+1(z) = di+1fi+1(z) then x − y ∈ im di+1 ⇒ fi+1x − fi+1y ∈ im di+1. now, different complex morphisms f,g induce different maps between the homology groups. when two morphisms of complexes f,g induce the same morphisms hif = hig? when they are homotopic, which is not hard to show [28, lemma iii.1.2]. thus, a morphism of complexes f : c −→ d is homotopic to 0 if and only if hi(f) = 0 for all i. the question of when a morphism f induces a morphism hf so that all hif are isomorphisms is rather more elusive. definition 13.4. a morphism of complexes f is said to be a quasiisomorphism when all hif are isomorphisms. for instance, every morphism of complexes between exact sequences is a quasi-isomorphism. this does not mean that a morphism homotopic to 0 has to be a quasi-isomorphism when the complexes are not exact! recapitulating, banach space homology means working in a setting in which banach spaces are identified with complexes —long, large, possibly endless complexes—. these could be exact (or not), they could have been obtained using either projectives (complexes with cn = 0 for n > 0) or injectives (complexes with cn = 0 for n < 0) or not. an example of non-acyclic complex? this one: pick a banach space x and form · · · −→ 0 −→ 0 −→ x −→ 0 −→ 0 −→··· categorical banach space theory ii 25 usually called x[n] when x is placed at position −n (yes, there is a reason for that that we will encounter later). so, the seed idea is to to represent banach spaces via complexes to then lift banach functors ban −→ ban to functors banban −→ banban, or at least to functors com(ban) −→ com(ban). if we try it, we will soon discover there is no natural way in which one can lift an operator τ : x → x to a morphism between arbitrary exact complexes such as, say, · · · −−−−→ 0 −−−−→ y y −−−−→ 0 −−−−→ x x −−−−→ 0 ? y y? τy yτ · · · −−−−→ 0 −−−−→ z z −−−−→ 0 −−−−→ x x −−−−→ 0 let alone between complexes. however, projective presentations are more faithful in their representation of banach spaces because of: proposition 13.5. every operator τ : x −→ y between two banach spaces induces a morphism between whatever projective presentations of those spaces. moreover, all morphism obtained that way are homotopic and quasiisomorphisms. in particular, two projective presentations of x are quasiisomorphic. and how could one make a banach functor f act between, say, projective presentations of two spaces x,y ? well, it induces a functor com −→ com acting term-by-term since it transforms complexes into complexes. since it also transforms homotopic maps into homotopic maps, the induced functor also acts kom −→ kom. a different thing is the behaviour of this induced map regarding quasi-isomorphisms. a functor f is said to be exact if it transforms short exact sequence into short exact sequences. it is clear that exact functors transform complexes quasi-isomorphic to 0 (i.e., acyclic) into complexes quasi-isomorphic to 0 (acyclic). not as easy [28, proposition iii.6.2] and [66, 10.5.2] is to show that: proposition 13.6. an exact functor preserves quasi-isomorphisms. this suggests that another way to measure the deviation of a functor from exactness could be to check its behaviour with respect to quasi-isomorphisms. we will explore both ways in the next two sections: section 14 explores how to measure the deviation from exactness; while section 16 explores how to measure its behaviour on quasi-isomorphisms. 26 j.m.f. castillo 13.2. stop and hear the sound of waves: homological dimension. the words still resonate in our minds: a long, large, possibly endless complex to represent a banach space x. yes, why not. with the only added problem that (almost) nothing, zero, zip, zilch, nada is known about even the simplest question: is actually infinite the projective presentation? or the injective presentation for that matter. to the best of our knowledge, only wodzicki [68] and recently cabello, castillo and garćıa [10] considered the topic. let us define the projective dimension p(x) of a banach space x as the smallest n for which ker dn is projective; analogously, the injective dimension i(x) is is the smallest n for which the cokernel space of dn is injective. it will take us no place to list all what is currently known: • p(x) = 0 if and only if x is projective. • there is a quotient b of c0 having p(b∗) = 1. • p(x) = ∞ if x is an l1-space not projective. • if k denotes the kadec space [31, 46, 47] then p(k) = ∞. as remarked in [68], it is expected these dimensions to be ∞ for most “classical” spaces. even if there is not a sniff of evidence (apart from many unsuccessful attempts) we conjecture that hilbert spaces have infinite homological (projective or injective) dimension. 14. derivation let us say it once again: exact sequences are important. thus, exact functors are interesting. a good number of current-life banach functors like • the duality functor x −→ x∗, which is exact thanks to the hahnbanach theorem, • functors of p-summable sequences: x −→ `p(x) for 1 ≤ p ≤∞, • the ultraproduct functor, • the residual functor, are exact, while several other important functors, such as • hom functors lx and lx, • in general, the functors a(·,x) and a(x, ·) for most operator ideals a, • tensor functors ⊗x, categorical banach space theory ii 27 • functors of weakly p-summable sequences: x −→ `wp (x) for 1 ≤ p < ∞, are not. what occurs when a functor f is not exact? well, banach functors tend to be “left-exact”, in the sense that the image of a short exact sequence 0 −→ y −→ z −→ x −→ 0 is an exact sequence 0 −→fy −→fz −→fx. what homology does in this case, or would like to do, is to obtain new functors f1,f2, . . . that provide an exact sequence 0 −→fy −→fz −→fx −→f1y −→f1z −→f1x −→f2y −→f2z −→f2x −→··· in which the functor fn+1 measures how much fn diverges from exactness, and for this reason fn+1 is called the derived functor of fn, and the process of passing from f to f1 is called derivation. how does one derive a functor? not so simple: only certain functors on certain categories can be derived; and, according to maclane [42, xii, §9]: “a standard method is: take a resolution, apply a covariant functor, take homology of the resulting complex. this gives a connected sequence of functors, called the derived functors”. let’s see this in action, with all due assumptions on a functor f to make sense of what follows: to obtain the left-derivation, pick an object x in a suitable category, form a projective resolution · · · −−→p3 d3−−−→p2 d2−−−→p1 d1−−−→ x −−→ 0 , apply the functor f and get a complex fx · · · −−→fp3 fd3−−−−→fp2 fd2−−−−→fp1 fd1−−−−→fx −−→ 0 . the homology group hn(fx) can be used to define the nth (left) derived functor lnf of f at x. l1f is a functor because arrows f : a −→ b yield a morphism hn(fa) −→ hn(fb) well defined since any two liftings of f are homotopic. a similar process worked out with injective presentations yields the right derived functors. the existence of a long homology exact sequence formed by derived functors still depends on the existence of connecting morphisms fnx −→ fn+1y . let us show how they appear in the construction above of left derivation: given an exact sequence 0 → y → z → x → 0, construct connected projective resolutions of the spaces y,z,x forming a commutative diagram 28 j.m.f. castillo ... �� ... �� ... �� 0 //py2 // �� pz2 // �� px2 // �� 0 0 //py1 // �� pz1 // �� px1 // �� 0 0 // y // �� z // �� x // �� 0 0 0 0 this yields an exact sequence of complexes 0 −→ y −→ z −→ x −→ 0. now, the necessary screw that supports the construction is provided by the observation that each map dn in a complex c induces a map dn : cokerdn+1 → ker dn−1 so that ker dn = h n(c) and cokerdn = h n−1(c) as it is easy to see [30, iv lemma 2.2.]. one thus gets a commutative diagram hn(y) // �� hn(z) // �� hn(x) �� cokerdn+1 // dn �� cokerdn+1 // dn �� cokerdn+1 // dn �� 0 0 // ker dn−1 // �� ker dn−1 // �� ker dn−1 �� hn−1(y) // hn−1(z) // hn−1(x) the snake lemma yields a connecting morphism hn(x) −→ hn−1(y). to conclude, when projective and injective presentations are available, right and left derivations are naturally equivalent, and thus the derived functor is unique. in p-banach spaces, at least projective derivation is possible. but that is no longer the case of quasi banach spaces. so, the question, to the best of my knowledge posed by félix cabello, is: how to derive quasi banach functors? 14.1. pedestrian, likely misleading, examples. let’s pick as template the case of one of the basic banach functors: la. to understand its derivation, we follow hmbst: let us make a strategic move and attempt to categorical banach space theory ii 29 guess how, given an exact sequence 0 −→ y −→ z −→ x −→ 0, we could continue the sequence 0 −−→ l(a,y ) −−→ l(a,z) −−→ l(a,x) keeping exactness. this entails finding a vector space v and a linear map l : l(a,x) −→ v whose kernel is the space of operators l(a,x) that lift to z. the simplest solution could be: l should send τ ∈ l(a,x) to the lower exact sequence in the pullback diagram 0 // y // z // x // 0 0 // y // pb // oo a // τ oo 0 because l(τ) = 0 precisely when τ can be lifted to an operator l(a,z). according with this, our best candidate for v is the space ext(a,y ) of exact sequences 0 −→ y −→ ♦ −→ a −→ 0, modulo the standard equivalence relation. the manifest implication is that the continuation can be provided by the functor x 7→ ext(a,x). in other words, it is reasonable to think that ext(a, ·) is a derived functor of l(a, ·) and ext(·,a) is a derived functor of l(·,a). the point to beware with is that all of this we have done works exactly the same in quasi banach spaces, even if no projective or injective elements exist in that category and thus there is no clear justification for the assertion “ext is the derived functor of l”. the way of looking at kan extensions of extban as reasonable candidates as derivations of l in qban was what we did, without a stratospheric success, in section 12. let us sum up known facts • the derived functor of an exact functor should be 0. • extban is the derived functor of l : ban −→ vect. • the derived functor of ⊗ is called tor. • the derivation of operator ideal functors a opens the topic of relative homology [15]. • derivation continues, and the derived functor of extban is called ext2ban. in general, the derived functor of extnban is denoted ext n+1 ban. the nature and properties of extnban are, by the times of writing these lines, a good mystery. • the paper [10] was written to uncover the yoneda approach to extn in banach and, to some extent, in quasi banach spaces. 30 j.m.f. castillo • the central result in the study of exact sequences and twisted sums of banach spaces is that ext(`2,`2) 6= 0 [23, 34]. in [9] it is shown that also ext2(`2,`2) 6= 0. we conjecture that extn(`2,`2) 6= 0 as well. 15. a necessary, maybe, step: localization one feels better off working in a banach space theory in which all one dimensional spaces are the same. this has effects even at the mathematics level since the set of all real one dimensional banach spaces up to isometries has just one element r while the set of all real one dimensional banach spaces is not even a set (since it essentially is the whole mathematical universe). after that one has grown accustomed to believe that all isometric banach spaces are the same. which, in combination with the outstanding result that all hilbert spaces of the same dimension are isometric, sooner or later makes one slip into the idea that all hilbert spaces are the same. let us continue from success to success until the final defeat: who has not dreamed of working in a banach space theory in which all reflexive spaces are the same? or all finite dimensional spaces are the same? from an operator ideal point of view, we could reformulate this as: pick an operator ideal a and create a banach space category in which all elements of a are isomorphisms. could we do that? of course not, but for a reason intrinsic to operator ideals: 0 is always an element of the ideal. thus, operator ideals are bad for creating isomorphisms; i.e., for localization, which means that other operator structures are needed. a similar phenomenon was pointed at in [14]: operator ideals are bad for “three-operator problems”; in this case, the semigroups of operators seem to be the adequate class to work with. fix a category b and a family a of morphisms of b. the localized category b[a−1] has as objects the same as b, while morphisms of b[a−1] are equivalence classes of arrows according to the following rules [26, 1.1]: • introduce for each a : x → y in a an arrow a−1 : y → x in b[a−1]. • each arrow f : x → y in b is an arrow f : x → y in b[a−1]. • a finite sequence of arrows in b[a−1] each of them ending where the previous one ends is an arrow in b[a−1]. • the arrows a−1a and aa−1 are equivalent to the identity in b[a−1]. • two consecutive arrows is equivalent to their composition. the localization b[a−1] has an associated functor q : b −→ b[a−1] sending an arrow f : x → y of b to the morphisms f : x → y of b[a−1]. categorical banach space theory ii 31 in this way, (the class of) a : x → y in a is an isomorphism in b[a−1] with inverse (the class of) a−1 : y → x. the localized category comes accompanied with the corresponding universal property: proposition 15.1. let c be a category and let q : b −→ c be a functor transforming morphisms in a into isomorphisms. then q factorizes through la in the form b q // q �� b[a−1] f wwnnn nnn nnn nnn n c proof. the functor f is cleanly defined as f(x) = x for objects, f(f) = f if f is an arrow of b and f(a−1) = q(a)−1. suitably chosen classes a allow nicer descriptions of the localized categories: definition 15.2. a class a of morphisms will be called (left) localizing if it enjoys the following stability properties: • is closed under composition. • 1x ∈a for every object x. • in a pullback diagram a a // b pb a // oo c oo if a ∈a then also a ∈a. • given two arrows τ,η : x → y there exists a ∈a such that aτ = aη if and only if there exists a ∈a such that τa = ηa. now, a localizing class a allows a rather visual approach to the localized category b[a−1]: objects are the same as in b while a morphism x → y in b[a−1] is an equivalence class of roofs (f,a) with f a morphism and a ∈a: a f ~~~~ ~~ ~~ ~ a ��@ @@ @@ @@ x y 32 j.m.f. castillo which corresponds to q(f)q(a)−1. (astute readers won’t get fooled: if one imagines composition of f with a then a should be on the left and f on the right, so that when f = a ten (a,a) is the identity of x (not “the opposite of the identity” as it occurs with our drawing). since aesthetically is nicer to have a on the right, my advice is simple: stop imagining.) two roofs (f,a) and (f ′,a′) are equivalent if and only id there exists another roof (f ′′,a′′) making a commutative diagram a′′ f′′ ~~}} }} }} }} a′′ b bb bb bb b a f ��~~ ~~ ~~ ~~ a **uuu uuuu uuuu uuuu uuuu uuuu a ′ a′ a aa aa aa f′ ttiiii iiii iiii iiii iiii iii x y the composition of two roofs (f,a) and (f ′,a′) is the roof pb f′ ~~|| || || || a !!c cc cc cc c a f ��~~ ~~ ~~ ~~ a b bb bb bb b a ′ a′ @ @@ @@ @@ f′ }}{{ {{ {{ {{ x y z the equivalence of roofs is actually an equivalence relation and composition respects this equivalence. to prove that is uncomplicated, but not exactly simple. let us call b̂ the category of roofs: claim. b̂ = b[a−1]. to prove it, it is enough to check that b̂ endowed with the functor ∧ : b −→ b̂ given by ∧(x) = x for objects and ∧(f) = (f, 1x) enjoys the same universal property: given a category c and a functor q : b −→ c transforming morphisms in a into isomorphisms then q factorizes through ∧ in the form b ∧ // q �� b̂ g xxqqq qq qq qq qq qq c categorical banach space theory ii 33 to construct g we set g(x) = x for objects and g(f,a) = q(f)q(a)−1 (check that it is well defined) which clearly makes g∧ = q. we could have worked with the dual construction just working in the opposite category: definition 15.3. a class a of morphisms will be called (right) localizing if it enjoys the following stability properties: • is closed under composition. • 1x ∈a for every object x. • in a pushout diagram a a // �� b �� c a // po if a ∈a then also a ∈a. • given two arrows τ,η : x → y there exists a ∈a such that aτ = aη if and only if there exists a ∈a such that τa = ηa. this yields the (right) localized category b[a−1]: objects are the same as in b while a morphism x → y in b[a−1] is an equivalence class of floors(f,a): x f @ @@ @@ @@ y a ��~~ ~~ ~~ ~ a all the rest goes as it should. another possible approach is to pick as a a class of what is called weak equivalences: definition 15.4. a class a of weak equivalences is a class of morphisms in a category c that contains the identity and satisfies the 2-out-of-6 property: whenever three arrows a u−−−→ b v−−−→ c w−−−→ d are such that wv and vu belong to a then u,v,w and wvu belong to a. 34 j.m.f. castillo a category c equipped with a class of weak equivalences is usually called a homotopical category and yields after localization a category hc called the homotopy category. there is an obvious localization functor hc : c −→ hc, and it has the universal property with respect to categories d endowed with a functor η : c −→ d that transforms weak equivalences into isomorphisms: η factorizes through hc. the identification of right or left localizing classes of operators in banach spaces is an untouched topic. on the other hand, isomorphisms form quite obviously a class of weak equivalences. when working in categories of complexes kom, quasi-isomorphisms usually conform the class of weak equivalences. 16. derived category the notion of derived category d(c) of a given a suitable category c is simple to describe but not as simple to handle: it is the localization of kom(c) with respect to quasi-isomorphisms. one could also have said that d(c) is the localization of com(c) with respect to quasi-isomorphisms: if q is the localization functor then whenever f is homotopic to g then q(f) = q(g). however the problem is not that, the problem is that the notion is not so easy to swallow, is it? be as it may, where we are pointing at is: does a functor f : c −→ c induce a functor d(c) −→ d(c)? we see that f induces, by plain term-byterm application to complexes, a functor c(f) : com(c) −→ com(c). this functor respects homotopy and thus it induces a functor k(f) : kom(c) −→ kom(c). now, given an object c, represented by a complex c if we apply f componentwise we obtain a complex fc that represents fc. if the complex c is acyclic and f is exact then fc is acyclic too. recall from proposition 13.6 that exact functors transform quasi-isomorphism into quasiisomorphisms, and thus it induces a functor d(f) : d(c) −→ d(c). in the general case, when f is not exact, then an induced functor d(c) −→ d(c) could exist or not. but, even when it does, it is not necessarily exact and, therefore, the homology groups hi(d(f)(c)) are not necessarily 0. the idea then is that hi(d(f)(c)) is the ith-derived functor of f at c. after all, if d(f) respects quasi-isomorphisms, these homology groups are isomorphic no matter which projective presentation of c has been chosen. according to this, the derived functors of an exact functor f are 0, as they should. the return journey would then be that the derived functors of a functor f (in the categorical banach space theory ii 35 standard sense earlier described) conform a suitable extension of f to d(c). of course that we have not arrived thus far to think that “a suitable extension” could be anything different from a kan extension. to see these ideas in action the problem is, of course, the nature of the derived category which, in turn, depends on the nature of the starting category. that is what makes so complicated the work of homologists. but we are just interested in what occurs with ban and qban so, it is enough for us to outdo the night king instead of worrying about what happens with the iron throne. some of the byways often followed in order to have a manageable representation of d(c) are: • instead of using all complexes, work only with complexes on the positive or negative range. this leads to com+ (complexes with ci = 0 for all i < 0) and then kom+. analogously with com− (complexes with ci = 0 for all i > 0) and then kom−. the functors induced by some f will be then either c+(f) or c−(f) and k+(f) or k+(f) as well. • that produces also the derived categories d− and d+ . • instead of using arbitrary complexes, use only a limited amount of them. for instance, complexes formed only by injective (or projective) objects (not necessarily acyclic!). this produces the categories kom(c, inj) or kom(c,proj). • using only injective or projective presentations yields a certain disruption: all morphism of complexes between exact complexes are quasiisomorphisms. keeping these strategies in mind one can obtain: 1. [28, theorem iii.4.4] the class of quasi-isomorphisms is localizing in kom(c). this means that the approach to d(c) via roofs is possible. it is also localizing in kom+(c, inj). 2. [66, theorem 10.4.8] when c has enough projectives then d−(c) exists and is equivalent to the (full) subcategory kom−(c,proj). 3. [66, theorem 10.4.8] [28, theorem iii.5.21] when c has enough injectives then d+(c) exists and is equivalent to the (full) subcategory kom+(c, inj). 4. (please go to section 19 to discuss about what “abelian category” means) an abelian category c can be seen as the full subcategory of d(c) formed by the complexes c with hi(c) = 0 for i 6= 0. 36 j.m.f. castillo the “exists” assertion coda in (2) and (3) may surprise devil-may-care readers. it means “it exists in our universe”: when c is a small category, localizations exist. otherwise . . . . see [66, remark 10.3.3] for a discussion on this topic when c is not small. once again, we have not arrived thus far to pretend believing that d(c) is not characterized by some universal property: (4) [28, definition theorem ii.2.1] if c is an abelian category there exists a category d(c) and a functor q : kom(c) −→ d(c) with the following properties • q transforms quasi-isomorphisms into isomorphisms. • any functor f : kom(c) −→ d transforming quasi-isomorphisms into isomorphisms factors through q. this universal property is at the root of assertions (2) and (3) above: if one is able to work only with some type of exact complexes in such a way that quasi-isomorphisms become isomorphisms then the corresponding kom is the derived category. but, as many times in life, in categorical life whatever is not too much is not even enough. 17. derived functor different things, in different context, see [67], are called derived functors. in the context of complexes and the derived category one has [28, iii.5.6]: theorem 17.1. let f : c −→ d be an additive functor acting between abelian categories. • the right derived functor rf of a left exact functor f is the left kan extension of q k+(f) : kom+(c) −→ d+(d) through q. this means that rf makes a commutative square kom+(c) k+(f) �� q // d+(d) rf �� >>}}}}}}} kom+(c) q // d+(d) categorical banach space theory ii 37 and there is a natural transformation � : q k+(f) −→ rf◦q yielding the uniqueness property with respect to that diagram. • the left derived functor of a right exact functor f is the right kan extension of q k−(f) : kom−(c) −→ d−(d) through q. this means that lf makes a commutative square kom−(c) k−(f) �� q // d−(d) lf �� η ~~}} }} }} } kom−(c) q // d−(d) and there is a natural transformation η : lf q −→ q k−(f) yielding the uniqueness property with respect to that diagram. in the context of homotopic categories (i.e., categories c endowed with weak equivalences so that one can construct the category hc) the formulation is similar [52, 6.4]: definition 17.2. let f : c −→ d be a functor between homotopical categories • the right kan extension of hdf through hc defines the so-called total left derived functor of f • the left kan extension of hdf through hc defines the so-called total right derived functor of f all this, however, leaves us with a bittersweet taste since our expectations were that derivation of a functor f : c −→ d would be a functor c −→ d. let us see two ways to proceed when the target category d is abelian, so that taking homology “groups” provides objects of d. 1. define as the nth-right derived functor of f : c −→ d the composition c [n] −−−→ kom(c) rf−−−−→ kom(d) h 0 −−−−→ d where 38 j.m.f. castillo • rf is the right derived functor as described above. • [n] is the functor x −→ x[n] described in section 13 in which t x[n] is the non-acyclic complex · · · −→ 0 −→ 0 −→ x −→ 0 −→ 0 −→··· with x placed at position −n. • h0 is just taking the homology group at position 0. and define the nth-left derived functor of f : c −→ d will be defined as the composition c [n] −−−→ kom(c) lf−−−→ kom(d) h 0 −−−−→ d where • lf is the right derived functor as described above. • [n] is the functor x −→ x[n] described in section 13 in which t x[n] is the non-acyclic complex · · · −→ 0 −→ 0 −→ x −→ 0 −→ 0 −→··· with x placed at position −n. • h0 is just taking the homology group at position 0. 2. given a right exact functor between suitable categories admitting enough projectives, a nth-left derived functor would be provided by the composition c −−→ kom−(c, proj) k−(f) −−−−−−→ kom−(d, proj) h n −−−−→ d while a nth-right derived of a left exact functor between suitable categories admitting enough injectives would be provided by the composition c −−→ kom+(c, inj) k+(f) −−−−−−→ kom+(d, inj) h n −−−−→ d . in those cases, the natural inclusions c −→ kom−(c, proj) and c −→ kom+(c, inj) are the obvious ones. in general, a category c can be seen in different ways inside kom(c), but the most natural is to represent is as the full subcategory of h0-complexes; i.e., complexes c for which hi(c) = 0 for all i 6= 0. further composition with the localization functor q : kom −→ d(c) yields a full subcategory equivalent to c. categorical banach space theory ii 39 18. back to ban how do all these ideas work in banach spaces? one must always keep in mind the underlying problem that ban is not an abelian category (section 19). since it has enough injectives and projectives, we can well use injectives and work with d+(ban) = kom+(ban) = kom+(ban, inj) or use projectives and work with d−(ban) = kom−(ban) = kom−(ban,proj). all this is fairly standard. or we could construct the derived category, which is not standard. let us present first the classical approaches, with a twist, to then exhibit the construction via the derived category. we will do all those things keeping the functor lx in focus. in all cases, a warning must be made right now: the “extraction” functors h0,hn do not take values in ban. we would very much like that were true but it is not. guess why? yes, because ban is not an abelian category and thus, as a rule, when one has an operator t : x −→ y , the quotient y/t [x] is not a banach space. this generates a crisis but, according with the original meaning of the word, gives us a non insignificant opportunity: homology groups (vector spaces) are not banach spaces, but they are semibanach spaces; namely, everything is ok except that we must work with a seminormed not-necessarily-hausdorff topology. therefore, if we call sban the category of semi banach spaces, the functors hn act between whatever category of complexes one is using and sban. 18.1. the standard approach, with a twist. the standard left derivation procedure of le is as follows: given the banach space x, pick a projective presentation 0 → κ → p → x → 0, then a projective presentation of κ, all of which we can assemble as κ′ → p ′ → κ → p → x → 0 and this forms an exact sequence κ′ → p ′ → p → x → 0 that we will call p (we do not care what comes next to get the first derivation). apply le to get le(p), namely 0 −→ l(x,e) −→ l(p,e) −→ l(κ,e) −→ l(p ′,e) that we reassemble as the complex 0 −→ l(x,e) −→ l(p,e) −→ l(p ′,e) that will be called le(p). therefore l(le)(x) = h 1(le(p)) = l(κ,e)/ ∼ , where ∼ is the equivalence relation τ ∼ η ⇐⇒ τ − η can be extended to an operator p → e. the reader can take some time to check that what has been done does not depend on the projective presentations (or read it in hmbst chapter 2). thus, the basic results about the pushout construction imply that precisely l(κ,e)/ ∼ = extban(x,e). this is the way we are certain that l(·,e)′ = extban(·,e) . 40 j.m.f. castillo the construction above has a weak point: what has been actually shown is that l(·,e)′ = l(κ,e)/ ∼, which is a bad formula because κ depends on the projective space p and the quotient map chosen. so, we would be much more happy if we could write l(·,e)′ = l(κ(x),e)/ ∼, where now κ is some banach functor. that can be done using the functor coz(·), different forms of whose construction can be found in [43, 18, 19, 11, 17], and in hmbst, section 3.10. the key point is that there is a banach functor coz(·) such that for each banach space there is an exact sequence ♦ with the form 0 −→ coz(x) −→ ♦ −→ x −→ 0 such that every element ♠ in extban(x,e) is equivalent to a pushout of ♦. in other words, there is a commutative diagram 0 // coz(x) // φ♠ �� ♦ �� // x // 0 0 // e //♠ // x // 0 in this way, again by the general properties of the pushout construction, the correspondence ♠−→ φ♠ establishes an identification extban(x,e) = l(coz(x),e)/ ∼ . this construction can be iterated to provide ext2ban(x,e) = extban(coz(x),e) = l(coz(coz(x)),e)/ ∼ and analogously for higher order derived extn spaces. see [10] for details. a kind of dual construction works in the same way for injective presentations and right derivation. 18.2. using the derived category. in the description above one gets ban [n] −−−→ kom(ban) rlx−−−−→ kom(ban) h0−−−→♣ where we would like to place ban instead of ♣, but we can’t. ban [n] −−−→ kom(ban) rlx−−−−→ kom(ban) h0−−−→ sban and thus what we can obtain is that if l (n) x = h 0rlx[n] is the n th-right derived functor of lx above then l (n) x (y ) is a semi-banach space. but, who is this l (n) x (y )? categorical banach space theory ii 41 proposition 18.1. extnban(x,y ) = homkom(ban)(x[0],y [n]) . instead of making a really annoying proof, let us explain why extban(x,y ) = homkom(ban)(x[0],y [1]): because exact sequences 0 → y → z → x → 0 correspond to morphisms in kom(ban) · · · −−−−→ 0 −−−−→ 0 −−−−→ x −−−−→ 0 −−−−→ 0 −−−−→ ···y y y y y · · · −−−−→ 0 −−−−→ y −−−−→ 0 −−−−→ 0 −−−−→ 0 −−−−→ ··· or, in the language of section 15, to roofs q(morphism of complexes)q(quasiisomorphism)−1 · · · −−−−→ 0 −−−−→ 0 −−−−→ x −−−−→ 0 −−−−→ ···x x x x quasi-isomorphism · · · −−−−→ 0 −−−−→ y −−−−→ z −−−−→ 0 −−−−→ ···y y y y morphism of complexes · · · −−−−→ 0 −−−−→ y −−−−→ 0 −−−−→ 0 −−−−→ ··· in fact, carrying away this idea, one could form a new category d(ban), whose objects are pairs (x,n) with x a banach space and n a natural number. the space hom((x,n), (y,m)) is declared to be (the usual) extn−m(x,y ). this yields a larger category into which ban can be embedded in the form x → (x, 0) since also l(x,y ) = hom((x, 0), (y, 0)). of course, extnban(x,y ) = homd(ban)(x[0],y [n]) . be as it may, from this we infer: proposition 18.2. l (n) x (y ) = ext n ban(x,y ) . according to what is currently known (see [8, 1.8.1, 4.5]) a semi banach structure on extban(x,y ) is the best one can get. the advantage of direct approaches, such as those in [8, 7, 6], is that they provide explicit forms for the seminorm in ext. however, they do not provide insights about what occurs with extn. what we have done in this paper shows that extnban(x,y ) carries a semi-banach topology too (semi-p-banach or semi-quasi-banach in the more general context of p-banach and quasi banach spaces). 42 j.m.f. castillo 19. the heart of banach spaces a category is called additive when the sets hom(a,b) have an abelian group structure compatible with composition, in the sense that f(g + g′)h = fgh + fg′h, that there is a 0 object that finite products and coproducts exist. an additive category is abelian when every arrow has kernel and cokernel and, moreover, each monic arrow is the kernel of its cokernel and every epic arrow is the cokernel of its kernel. banach and quasi banach spaces are additive categories but not abelian and the reason is that the cokernel of an operator τ : a → b is the quotient map b → b/τ[a] (the proof is easy and can be seen in hmbst, section 2.4). but that is not the cokernel in the underlying category vect of vector spaces! in banach spaces monic arrows are the injective operators and epic arrows are the dense range operators, and that is the reason why even r is not injective or projective, and therefore injective or projective objects do not exist (all this was observed by pothoven [50]). in particular, it is impossible that every monic (injective) operator is the kernel of its cokernel, as any injective operator with dense range shows. still, the reason is so subtle that categorical and homological constructions in ban and qban mostly work, one way or another (r, after all, does behave like an injective and projective space, doesn’t it?). to put a name to his phenomenon quillen [51] introduces the notion of exact category : an additive category in which one determines a class of “short exact sequences” satisfying some suitable conditions [4, 1.1.4], that he calls admissible. there are other notions of exact category: barr [3] introduces a notion of exactness so that (abelian) = (exact) + (additive) thus, barr’s notion is intrinsic, while quillen’s is extrinsic, in the sense that, first, one is already working in an additive category and then specifies which are the exact sequences. a detailed exposition of quillen’s notion can be enjoyed in [5]. so, banach and quasi banach spaces are exact categories in quillen’s sense; why? because the items to which we award the certification of short exact sequences are . . . well . . . the short exact sequences. such is the magic that the open mapping theorem brings in ban and qban. of course that the work of categorical people in defining the notions is much more delicate (and also that of functional analysis people, see [25]), because in a category no such magic exist (even if the hom(a,b) groups are made of morphisms). in quillen’s definition, admissible exact sequences are made with admissible monics (injective maps) and admissible (epics) subject to some rules. just to give you something to think about, try to specify which are the categorical banach space theory ii 43 admissible monic and epic in normed spaces, and which are the admissible short exact sequences. exactly: those. a detailed list of the rules to follow can be seen in bühler [5, definition 2.1], and it reads like this: admissible arrows form a closed class under composition, of course that identities are admissible, and pushouting (pullbacking) admissible monics (epics) yields admissible monics (epics). things become clearer and nicer if we formulate them as in bühler [5, proposition 2.19]. proposition 19.1. consider a commutative square a // �� b �� c // d in which the horizontal arrows are admissible monics. the following assertions are equivalent: 1. the diagram is a pushout. 2. the diagonal sequence 0 −→ a −→ c ⊕b −→ d −→ 0 is exact. 3. the square is a dolittle diagram: i.e., it is both a pushout and a pullback diagram. 4. the square is part of a commutative diagram 0 // a // �� b �� // e // 0 0 // c // d // e // 0 with exact rows. what we feel when looking at those conditions in ban or qban is: ok, exactly that is what happens! well, that is why ban is an exact category. there have been more efforts to formalize what makes ban good enough to be good and correct what makes it bad enough to be bad: that only some maps have “good” cokernel. the basic idea is: how to embed ban as a full subcategory of an abelian category? paraphrasing [4]: 44 j.m.f. castillo definition 19.2. a heart of a category is an abelian category that contains it as a full subcategory. thus, guided by tom waits let us look for the heart of banach spaces. the name of waelbroeck should have appeared first in this regard. his attempt was the construction of the abelian category of quotient banach spaces [58, 59, 63, 64] and his reasons to do so are described by himself in [60, 62]. i find a few difficulties to describe waelbroeck’s work, one of then that it does not help that definitions change from one paper to another. the final section 4 of [65] contains a respectful discussion on this topic. let me focus on the description provided in the paper [58] instead of the much more complete compendium [64]. waelbroeck’s definition in [58, definition 1] is “a quotient banach space e/f is the quotient of a banach space (e,‖·‖e) by a vector subspace f ⊂ e that is a banach space with a norm (f,‖ · ‖f ) finer than the restriction of ‖ · ‖e. in this definition, as far as i understand it, the vector space e/f is part of the definition. this is probably due to a kind of atavic pulsion to ensure that objects of the new category are sets. once this drive passes and one accepts that the objects of the new category are not sets, waelbroeck [61, definition 2.1] sets (see also, in [64, section 2.1]): definition 19.3. a q-space x|x′ is a couple (x,x′) where x is a banach space and x′ is a vector subspace of x that is a banach space with a norm ‖ ·‖x′ finer than ‖ ·‖x. acting this way things are ok, and we are on our way to defining wegner’s mon category [65]. a morphism u : x|x′ −→ y |y ′ between two quotient banach spaces is in [58] a linear map u : x/x′ −→ y/y ′ for which there is a bounded operator u : x → y fitting in a commutative diagram x′ // u′ �� x q1 // u �� x/x′ u �� y ′ // y q2 // y/y ′ (4) these morphisms have been called strict operators. in [64, definition 2.1.2] the author considers that the right equality notion has not been defined. this is important because we need to know what does exactly mean u = 0: that u = 0? that u = 0? and it is not explicitly defined probably because it is not easy to do, in those terms. indeed, what one is is desperately attempting to say is that homotopy is the right equivalence relation to be set. homotopy involving categorical banach space theory ii 45 linear maps is, however, a sticky affair. later in [64], where a quotient banach space is just an injective operator f : x′ −→ x between banach spaces, a strict morphism f −→ g is just a couple of operators x′ u′ // f �� y ′ g �� x u // y and (u,u′) = 0 if and only if there is an operator h : x −→ y ′ making the diagram x′ u′ // f �� y ′ g �� x u // h ==|||||||| y commutative. it is clear moreover that (u,u′) = (v,v′) if and only if (u − v,u′ −v′) = 0. waelbroeck goes on and defines in [58] a pseudo-isomorphism when the linear map b is so that the operator u in the diagram (4) is surjective and u−1[y ′] = x′ (to be precise, assuming that f,g are as before, u−1(g[y ′]) = f[x′]). let’s us continue being imprecise for the sake of clarity. a pseudo-isomorphism is necessarily bijective: 0 = u(x + x′) = ux + y ′ ⇒ ux ∈ y ′ ⇒ x ∈ x′ ⇒ x + x′ = y + x′ = 0. the category qb in [58] has quotient banach spaces as objects and as morphisms compositions s−1b of strict morphisms b and inverses of pseudo-isomorphisms s. in other words, it is the localization of (quotient banach spaces, strict morphisms) with respect to pseudo-isomorphisms. the localization is needed here because strict morphisms do not suffice to get an abelian category. as we have already mentioned, [71] clearly explains that qb is one of those categories whose objects are not sets and have no points and its morphisms are not maps. the crucial arrival point is [58, lemma 5]: lemma 19.4. every morphism in qb has kernel and cokernel. proof. as always, the cokernel is the tricky part. waelbroeck hints the proof like: use that if u : x|x′ −→ y |y ′ is a strict morphism then there exists a cokernel c : y/y ′ → (y/y ′)/b[x/x′] for u in vect. back to diagram (4), u[x/x′] = uq1[x] = q2u[x] = (u[x] + y ′)/y ′ so that (y/y ′)/u[x/x′] = (y/y ′)/(u[x] + y ′/y ′) = y/u[x] + y ′. the space y ′ + u[x] can be provided 46 j.m.f. castillo with a banach structure given by the norm ‖z‖ = inf{‖y′‖y ′ + ‖x‖x} where the infimum is taken over all representations z = y′ + u(x). thus, one can form the object y |(y ′ + u[x]) with associated strict morphism c : y |y ′ → y |(y ′ + u[x]) and form the commutative diagram y ′ −−−−→ y −−−−→ y/y ′y ∥∥∥ yc y ′ + u[x] −−−−→ y −−−−→ y/(y ′ + u[x]) (5) obviously cu = 0 and therefore cu = 0; and if w : y |y ′ −→ s|s′ is another strict morphism y ′ −−−−→ y q2−−−−→ y/y ′yw′ yw yw s′ −−−−→ s q −−−−→ s/s′ such that wu = 0 then w factors through c in the form w = mc. look at the diagram x′ // �� x u �� q1 // x/x′ b �� y ′ // �� 6 66 66 66 66 66 66 66 66 y q2 // w ��8 88 88 88 88 88 88 88 88 8 y/y ′ c �� w ��; ;; ;; ;; ;; ;; ;; ;; ;; ; y ′ + u[x] // $$i ii ii ii ii i y q2 // m %%k kk kk kk kk kk kk y/y ′ + u[x] m &&m mm mm mm mm m s′ // s q // s/s′ set m = w to get qm = qw = wq2 = mcq2 = mq2. making the general case s−1b in which b is a strict morphism and s a pseudo-isomorphism is now merry joy: its cokernel is cs. by construction it takes now little effort [58, proposition 6] to show that monic arrows are the kernels of their cokernels and epic arrows are cokernels of their kernels, which leads us to the inexorable conclusion: proposition 19.5. the category qb is abelian. categorical banach space theory ii 47 the natural inclusion of ban into qb is to regard x as x|0 and thus ban is a full subcategory of an abelian category. the category qb seems to be a kind of portable derived category of ban since, as we said before, it is a localization of the category with objects f|e and strict morphisms with respect to pseudo-isomorphism. in conclusion, waelbroeck’s category qb is a heart of banach spaces. wegner [65] cuts the problem straight to arrive to the heart of banach spaces: he defines the category mon whose objects are injective operators f : x′ → x between banach spaces while an arrow (α′,α) : f → g between two objects is a commutative diagram x′ α′ // f �� y ′ g �� x α // y with the following equality notion between morphisms: (α′,α) = 0 if and only if there exists an operator h making a commutative diagram x′ α′ // f �� y ′ g �� x α // h ==|||||||| y since f and g are injective it is enough to ask gh = α to get hf = α′ free of charge. and therefore, (α′,α) = (β′,β) if and only if (f ′−g′,f −g) = 0. if the process that brought us thus far has not vanished into oblivion the reader will realize that (α′,α) = (β′,β) if and only if (α′,α) and (β′,β), seen as morphisms of complexes are homotopic. the proof only requires to tilt your head (or rotate 90◦ the page), this way: // 0 // x′ f // α′ ��~~ ~~ ~~ ~~ x g �� h ~~}} }} }} }} // 0 // �� // 0 // y ′ α // y // 0 // the same rotation shows that wegner’s category is waelbroeck’s category qb prior to localization. the difference, or so it seems, between waelbroeck’s localized category qb and wegner’s localization of hmon[d] are that i) waelbroeck does not clearly sets the localizing class; ii) even if he does, it is not 48 j.m.f. castillo clear that the definition of such class is correct. wegner localizes using a larger system (pulations, see below) than waelbroeck (pulations whose induced linear application on the quotients is surjective (?)). indeed, while wegner shows that if something equivalent to a pulation is a pulation there remains the question of whether a strict morphism equivalent to a pseudoisomorphism a pseudo-isomorphism. keeping all that in mind, one of course has: lemma 19.6. every morphism in mon has kernel and cokernel proof. the cokernel stuff is crystal clear now: given an arrow (α′,α) : f −→ g form the object ı : g[y ′] + α[x] −→ y where g[y ′] + α[x] is endowed with the complete norm ‖z‖ = inf{‖y′‖y ′ + ‖x‖x : z = g(y′) + α(x)} and the arrow (g, 1) : g −→ ı x′ α′ // f �� y ′ g �� g // g[y ′] + α[x] ı �� x α // y 1 y one has (gα′, 1α) = 0 since there is the operator α : x −→ g[x] + α[x]. astute readers and who fans won’t get fooled again: this is lemma 19.4 verbatim. now, to proceed with localization, wegner makes a clever move, full of fun for us. to understand why it would be good to make a detour to review the kaijser-pelletier approach to interpolation (see a passing mention in hhi, pp. 118-119) as exposed in [32, 33]. the core idea in those papers is that of doolittle diagram; namely, a commutative diagram a −−−−→ by y c −−−−→ d that is simultaneously a pullback and a pushout diagram. wegner adopts in [65, definition 5] the name “pulation” for such square, introduced by j. adámek, h. herrlich, and g.e. strecker in [1, p. 205]. even if the joy of cats is a wonderful subtitle for a book on categories, pulation is an awful name at different levels and in different languages. the author, wegner, is aware that “other naming conventions (doolittle diagram, push-me pull-you diagram or bicartesian square) are mentioned in the literature.” categorical banach space theory ii 49 let’s leave this sad episode in the past and move ahead. wegner proceeds to localize with respects to the class d of maps (α′,α) that provide doolittle diagrams. after that, wegner forms the localized category ♥ = mon[d] and proves the inevitable result that ♥ is an abelian category in which ban embeds as a full subcategory in the obvious way: making x correspond to [0 → x]; namely: proposition 19.7. ♥ is a heart of banach spaces. indeed, categories, like people, may have many hearts. a nice one, intended to provide a solid foundation for waelbroeck’s category of q-banach spaces, was presented by noël [44]. we cannot resits to sketch it with its many virtues: noël’s category. as it is mentioned in [64, 2.1.2] “the abelian category of quotient spaces was defined by waelbroeck in 1962, but it was difficult to work with it. with the “miracle functor”, noël [64], defined a similar abelian category in 1969.” what is this miracle functor? well, as clear as ever, [64, 2.3.3] sets something between s 7→ l(`1(s),x) [64, 2.3.6] when x is a banach space and x|y 7→ l(`1(s),x)/l(`1(s),y ) [64, 2.3.11]. see also below. let k be the category whose objects are the banach spaces `1(s) with s any set (yes, there is some set-theoretic problem here that can somehow be circumvented [44]). a q-space is an additive contravariant functor f : k −→ vect. a morphism between two q-spaces is a natural transformation between the functors, and the category obtained this way is called qesp. theorem 19.8. qesp is a heart of banach spaces. this means that qesp is an abelian category and that there is a faithful representation functor δ• : ban −→ qesp, x � lx (we already mentioned in section 7 that this functor ban −→ banban was a faithful representation). as we have already mentioned, objects of qb are not sets with elements. noël explicitly mentions this in his final remarks (translated from french): “observe that a q-space is not a vector space endowed with an additional structure”. however, noël observes that there is a natural functor σ : qesp −→ vect given by f � f(c) (noël works with complex banach spaces), which makes perfect sense: σ(lx) = lx(c) = l(c,x) = x. the hardcore of noël ideas seems to be [44, theorem 2] which seems to say: 50 j.m.f. castillo proposition 19.9. every q-space f fits in an exact sequence 0 −→ ly −→ lx −→f −→ 0 in which both y,x are banach spaces. whose proof is not, unfortunately, in [44]. the proposition yields the identity fx = l(`1(s),x)/l(`1(s),y ) (the miracle functor) in vect and, in particular, f(c) = x)/y . once one believes that, the rest is easy. moreover, noël claims that δ• admits a left adjoint n |δ•, which means that there exists a covariant functor n : qesp −→ ban such that l(n(f),x) = [f,δ•(x)] = [f,lx] . since n(f) has to be a banach (not a mere vector) space, noël claims: n(f) = x/y . about the construction of clausen and scholze. this issue, that objects of a heart of banach spaces are not sets remains in the construction of clausen and scholze [53] we mention next. peter scholze mentioned to us in a private conversation that “embedding banach spaces into an abelian category with wonderful categorical properties has been something that clausen and i have spent a lot(!) of time thinking about, and we believe we have now found an excellent way. it is explained in my lectures notes on analytic geometry, https://www.math.uni-bonn.de/people/scholze/ analytic.pdf. we build a variant of ”complete locally convex topological vector space” that is an abelian category (so taking cokernels is fine, even if the image is not closed), closed under extensions (so we actually have to weaken local convexity, because of the ribe extension etc. – as you say, one feels the need to enlarge banach spaces to quasi-banachs), and with further really nice properties (e.g., with compact projective generators). to prove that it works, we have to prove a more general statement of ”arithmetic” flavour – to set up functional analysis over the reals, we have to work with the integers! see also https://xenaproject.wordpress.com/liquid-tensor-experiment/ and https://mathoverflow.net/questions/386796/nonconv”. we end this section with a quote from [64]: “mathematicians want to put their hands on elements of an object”. however, those hearts of banach spaces have no elements. in contrast, s. lubkin’s beautiful exact embedding theorem [40] (see also [36, 1.6]) says: theorem 19.10. every abelian category (whose objects form a set) admits an additive imbedding into the category of abelian groups which carries exact sequences into exact sequences. https://www.math.uni-bonn.de/people/scholze/analytic.pdf https://www.math.uni-bonn.de/people/scholze/analytic.pdf https://xenaproject.wordpress.com/liquid-tensor-experiment/ https://mathoverflow.net/questions/386796/nonconv categorical banach space theory ii 51 20. derivation in banach spaces, what if . . . ? we are ready to face the good, the bad and the ugly issue about derivation in banach spaces: the same reason why ban is not abelian is what makes derivation of l functors work . . . as functors ban → vect and not as functor ban → ban. let us explain this assertion: the definition of the homology group at step n of a complex is clear ker dn+1/im dn . here there is something we overlooked in all our previous discussions: when the functor one is trying to derive takes values in a reasonable category whose objects are sets with some additional structure and the arrows are functions, the meaning of im dn is clear. so, while deriving l functors in banach spaces, we are still on solid ground. but, at the cost of having to deal with the image of an operator, something that provides the “wrong” cokernel and being thus responsible for not having homology groups in the right category (ban). as we have already mentioned: definition 20.1. in an abelian category, the image of an arrow is the kernel of its cokernel. so things work again in an abelian category and, moreover, homology “groups” are objects of the category so that both the construction of the derived category and the final particularizations (1) and (2) after definition 10 of the derived functor to a functor between the original categories make sense. and, in general? this is a question for categorical people. in the search of a manageable definition for “image of an arrow” wegner [65] isolates the notion of range. definition 20.2. the range of a morphisms f : x → y is monomorphism r : r → y for which there is a morphism q : x → r in such a way that f = rq and such that for all monomorphism s : s → z and morphisms g : y → z,h : x → s such that gf = sh there exists a unique g′ : r → s such that gr = sg′. which is obviously adapted so that an operator t : x −→ y has range t[x]. then, wegner acknowledges the work of waelbroeck formulating: definition 20.3. an additive category a is a waelbroeck category if there exists an additive functor f : a −→ ab satisfying the following three conditions: 52 j.m.f. castillo (w1) every morphism has a kernel and f preserves kernels. (w2) every morphism has a range and f preserves ranges. (w3) the functor f preserves and reflects kernel-cokernel pairs, i.e., (f,g) is a kernel-cokernel pair in a if and only if (ff,fg) is a kernel-cokernel pair in ab. to finally arrive to [65, proposition 13]: proposition 20.4. ban is a waelbroeck category. that can be completed to coda. qban is a waelbroeck category. 21. where are we now? two questions have been wheeling around this survey: how to derive a banach functor? how to derive a quasi banach functor? with the elements displayed in this paper there are four possibilities to deal with the first of those questions: • accept that derivation is as it is, including that the derived functor of l is not a banach functor. there is however a reasonable consolation that they are semi banach functors. • carry the previous idea to its natural conclusion and derive the semi banach functor l : sban → sban. • carry the derivation process strictly (the cokernel of an operator t : y −→ x in ban is x/t[y ]) and be prepared to whatever new ext functors may appears since the first cohomology group of x at e would be now l(e,`∞/x)/ql(e,`∞) where 0 −→ x −→ `∞ q −→ `∞/x −→ 0 is a projective presentation of x. now live with that. • (suggested by félix cabello) embed ban as a full subcategory of an abelian category and derive there. namely, derive in a heart category for banach spaces. quoting again waelbroeck [64]: “it is not difficult to define categories of quotient spaces and prove that they are abelian. it is more difficult to develop functional analysis in them”. a fifth and a six possibilities, if they exist, will hopefully be developed elsewhere. categorical banach space theory ii 53 references [1] j. adámek, h. herrlich, g.e. strecke, abstract and concrete categories. the joy of cats. available at http://katmat.math.uni-bremen. de/acc. 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[68] m. wodzicki, homological dimensions of banach spaces, in “ linear and complex analysis problem book 3, part i ”, v.p. havin and n.k. nikolskii (eds), notes in mathematics, 1573, springer-verlag, berlin, 1994, 34 – 35. the categories cd adjoint functors limits comma categories kan extensions left kan extension right kan extension uses of kan extensions for we, banach spacers homology exactness and how to measure it. stop and hear the sound of waves: homological dimension. derivation pedestrian, likely misleading, examples. a necessary, maybe, step: localization derived category derived functor back to ban the standard approach, with a twist. using the derived category. the heart of banach spaces derivation in banach spaces, what if …? where are we now? � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 34, num. 2 (2019), 135 – 200 doi:10.17398/2605-5686.34.2.135 available online october 3, 2019 the ξ,ζ-dunford pettis property r.m. causey department of mathematics, miami university, oxford, oh 45056, usa causeyrm@miamioh.edu received may 30, 2019 presented by jesús m.f. castillo accepted july 31, 2019 and manuel gonzález abstract: using the hierarchy of weakly null sequences introduced in [2], we introduce two new families of operator classes. the first family simultaneously generalizes the completely continuous operators and the weak banach-saks operators. the second family generalizes the class dp. we study the distinctness of these classes, and prove that each class is an operator ideal. we also investigate the properties possessed by each class, such as injectivity, surjectivity, and identification of the dual class. we produce a number of examples, including the higher ordinal schreier and baernstein spaces. we prove ordinal analogues of several known results for banach spaces with the dunford-pettis, hereditary dunford-pettis property, and hereditary by quotients dunford-pettis property. for example, we prove that for any 0 ≤ ξ,ζ < ω1, a banach space x has the hereditary ωξ,ωζ-dunford pettis property if and only if every seminormalized, weakly null sequence either has a subsequence which is an `ω ξ 1 -spreading model or a c ωζ 0 -spreading model. key words: completely continuous operators, schur property, dunford pettis property, operator ideals, ordinal ranks. ams subject class. (2010): primary: 46b03, 47l20; secondary: 46b28. 1. introduction in [14], dunford and pettis showed that any weakly compact operator defined on an l1(µ) space must be completely continuous (sometimes also called a dunford-pettis operator). in [17], grothendieck showed that c(k) spaces enjoy the same property. that is, any weakly compact operator defined on a c(k) domain is also completely continuous. now, we say a banach space x has the dunford-pettis property provided that for any banach space y and any weakly compact operator a : x → y , a is completely continuous. a standard characterization of this property is as follows: x has the dunfordpettis property if for any weakly null sequences (xn) ∞ n=1 ⊂ x, (x ∗ n) ∞ n=1 ⊂ x ∗, limn x ∗ n(xn) = 0. generalizing this, one can study the class of operators a : x → y such that for any weakly null sequences (xn)∞n=1 ⊂ x and (y∗n) ∞ n=1 ⊂ y ∗, limn y ∗ n(axn) = 0. by the well-known mazur lemma, if x is a banach space and (xn) ∞ n=1 is issn: 0213-8743 (print), 2605-5686 (online) https://doi.org/10.17398/2605-5686.34.2.135 mailto:causeyrm@miamioh.edu https://www.eweb.unex.es/eweb/extracta/ https://creativecommons.org/licenses/by-nc/3.0/ 136 r.m. causey a weakly null sequence in x, then (xn) ∞ n=1 admits a norm null convex block sequence. of course, the simplest form of convex block sequences would be one in which all coefficients are equal to 1, in which case the convex block sequence of (xn) ∞ n=1 is actually a subsequence. the next simplest form of a convex block sequence is a sequence of cesaro means. a property of significant interest is whether the sequence (xn) ∞ n=1 has a subsequence (or whether every subsequence of (xn) ∞ n=1 has a further subsequence) whose cesaro means converge to zero in norm. a weakly null sequence (xn) ∞ n=1 having the property that for every ε > 0, there exists k = k(ε) ∈ n such that for any x∗ ∈ bx∗, |{n ∈ n : |x∗(xn)| ≥ ε} ≤ k is called uniformly weakly null. a weakly null sequence has the property that each of its subsequences has a further subsequence whose cesaro means converge to zero in norm if and only if it has the property that each of its subsequences has a further subsequence which is uniformly weakly null. schreier [21] produced an example of a weakly null sequence which has no uniformly weakly null subsequence. schreier’s example showed that the convex combinations required to witness weak nullity in mazur’s lemma cannot be assumed to be cesaro means, and must occasionally be more complex convex combinations. providing a quantification of the complexity of convex combinations required to witness weak nullity in mazur’s lemma, argyros, merkourakis, and tsarpalilas [2] defined the banach-saks index, which provides canonical coefficients which measure the complexity a given weakly null sequence requires to obtain norm null convex block sequences. as described above, norm null sequences are 0-weakly null, uniformly weakly null sequences are 1-weakly null, and for every countable ordinal ξ there exists a weakly null sequence which is ξ-weakly null and not ζ-weakly null for any ζ < ξ. by convention, we establish that a sequence is said to be ω1-weakly null if it is weakly null. consistent with this convention is the fact that for any 0 ≤ ξ ≤ ζ ≤ ω1, every sequence which is ξ-weakly null is ζ-weakly null. the ordinal quantification assigns to a given weakly null sequence some measure of how complex the convex coefficients of a norm null convex block sequence must be. our notation and terminology follows the standard reference of pietsch [20]. we denote classes of operators with fraktur letters, a,b,i, etc. we recall that for a given operator ideal i, the associated space ideal is the class of banach spaces x such that ix ∈ i. given an operator ideal a,b,i, . . . , the associated space ideal is denoted by the corresponding sans serif letter, a,b, i, . . . . the notion of quantified weak nullity defined in the preceding section yields a natural generalization of the class dp. given an operathe ξ,ζ-dunford pettis property 137 tor a : x → y , rather than asking that every weakly null sequence in (xn) ∞ n=1 ⊂ x and any weakly null sequence (y ∗ n) ∞ n=1 ⊂ y ∗, limn y ∗ n(axn) = 0, we may instead only require the weaker condition that every pair of sequences (xn) ∞ n=1 ⊂ x, (y ∗ n) ∞ n=1 ⊂ y ∗ which are “very” weakly null, limn y ∗ n(axn) = 0. formally, for any 0 ≤ ξ,ζ ≤ ω1, we let mξ,ζ denote the class of all operators a : x → y such that for every ξ-weakly null (xn)∞n=1 ⊂ x and every ζ-weakly null (y∗n) ∞ n=1 ⊂ y ∗, limn y ∗ n(axn) = 0. we let mξ,ζ denote the class of all banach spaces x such that ix ∈ mξ,ζ. then dp = mω1,ω1 and mω1,ω1 is the class of all banach spaces with the dunford-pettis property. note that every operator lies in mξ,ζ when min{ξ,ζ} = 0, since 0-weakly null sequences are norm null. thus we are interested in studying the classes mξ,η only for 0 < ξ,ζ. furthermore, one may ask for a characterization, as one does with the dunford-pettis property, of banach spaces all of whose subspaces, or all of whose quotients, enjoy a given property (in our case, membership in mξ,ζ). we note that the classes m1,ω1 were introduced and studied in [16], while the classes mω1,ξ, were introduced and studied in [1]. the study of classes of operators with these weakened dunford-pettis conditions rather than spaces with these conditions is new to this work. along these lines, we have the following results. theorem 1.1. for every 0 < ξ,ζ ≤ ω1, mξ,ζ is a closed ideal which is not injective, surjective, or symmetric. moreover, the ideals (mξ,ζ)0<ξ,ζ≤ω1 are distinct. in addition to generalizations of the dunford-pettis property, one may use the quantified weak nullity to generalize other classes of operators. two classes of interest are the classes v of completely continuous operators and wbs of weak banach-saks operators. also of interest are the associated space ideals v of schur spaces and wbs of weak banach-saks spaces. the concepts behind these classes are that weakly null sequences are mapped by the operator to sequences which are “very” weakly null (completely continuous operators send weakly null sequences to 0-weakly null sequences, and weak banach-saks operators send weakly null sequences to 1-weakly null sequences). in [12], the notions of ξ-completely continuous operators, the class of which is denoted by vξ, and ξ-schur banach spaces were introduced. these notions are weakenings of the notions of completely continuous operators and schur banach spaces, respectively. an operator is ξ-completely continuous if it sends ξweakly null sequences to norm null (0-weakly null) sequences. heuristically, this is an operator which sends sequences which are “not too bad” to se138 r.m. causey quences which are “good.” in [3], the notion of ξ-weak banach-saks was introduced. an operator is ξ-weak banach-saks if it sends weakly null sequences to ξ-weakly null sequences. heuristically, this is an operator which sends any weakly null sequence, regardless of how “bad” it is, to sequences which are “not too bad.” of course, there is a simultaneous generalization of both of these notions. for 0 ≤ ζ < ξ ≤ ω1, we let gξ,ζ denote the class of operators which send ξ-weakly null sequences to ζ-weakly null sequences. along these lines, we prove the following. theorem 1.2. for every 0 ≤ ζ < ξ ≤ ω1, gξ,ζ is a closed, injective ideal which fails to be surjective or symmetric. these ideals are distinct. we also recall the stratification (wξ)0≤ξ≤ω1 of the weakly compact operators. note that, by the eberlein-šmulian theorem, an operator a : x → y is weakly compact if and only if every sequence in abx has a subsequence which is weakly convergent. equivalently, a : x → y is weakly compact if and only if for any (xn) ∞ n=1 ⊂ bx, there exist a subsequence (x ′ n) ∞ n=1 of (xn) ∞ n=1 and y ∈ y such that (ax ′ n − y)∞n=1 is weakly null. the classes wξ, 0 ≤ ξ ≤ ω1, are analogously defined using our quantified weak nullity: the operator a : x → y lies in wξ if and only if for any (xn)∞n=1 ⊂ bx, there exist a subsequence (x′n) ∞ n=1 of (xn) ∞ n=1 and y ∈ y such that (ax ′ n − y)∞n=1 is ξ-weakly null. the class wξ appears in the literature under the names ξ-weakly compact operators and ξ-banach-saks operators. the former name is due to the fact that wω1 is the class of weakly compact operators, while the latter is due to the fact that w1 is the class of banach-saks operators. in this work, we use the former terminology. we recall the basic facts of these classes and basic facts about operator classes, including the quotients a ◦ b−1 and b−1 ◦ a, in section 3. we note that w0 is the class of compact operators, also denoted by k. the class of weakly compact operators is denoted by w and wω1 , and w1 denotes the class of banach-saks operators. it is a well-known identity regarding completely continuous operators that v = k ◦ w−1. it is also standard that dp = w−1◦v = w−1◦k◦w−1. rewriting theses identities using the ordinal notation for these classes gives vω1 = w0 ◦w −1 ω1 , mω1,ω1 = w −1 ω1 ◦k◦w−1ω1 . we generalize these identities in the following theorem. the ξ,ζ-dunford pettis property 139 theorem 1.3. for 0 ≤ ζ < ξ ≤ ω1, gξ,ζ = wζ ◦w−1ξ , gdualξ,ζ = (w dual ξ ) −1 ◦wdualζ . for 0 < ζ,ξ ≤ ω1, mξ,ζ = (w dual ζ ) −1 ◦vξ = (wdualζ ) −1 ◦k◦w−1ξ . the appearance of wdualξ , rather than simply wξ as it appeared in the identities preceding the theorem are due to the fact that w0 = k = k dual = wdual0 and wω1 = w = w dual = wdualω1 , while wξ 6= wdualξ for 0 < ξ < ω1. this duality is known to fail for all 0 < ξ < ω1. the failure for ξ = 1 is the classical fact that the banachsaks property is not a self-dual property, while the 1 < ξ < ω1 cases are generalizations of this. we say banach space x is hereditarily mξ,ζ if for every every closed subspace y of x, y ∈ mξ,ζ. we say x is hereditary by quotients mξ,ζ if for every closed subspace y of x, x/y ∈ mξ,ζ. in section 2, we define the relevant notions regarding ` ξ 1 and c ζ 0-spreading models. we also adopt the convention that a sequence which is equivalent to the canonical c0 basis will be called a cω10 -spreading model. we summarize our results regarding these hereditary and spatial notions in the following theorem. we note that item (i) of the following theorem generalizes a characterization of the hereditary dunfordpettis property due to elton, as well as a characterization of the hereditary ζ-dunford-pettis property defined by argyros and gasparis. theorem 1.4. fix 0 < ξ,ζ ≤ ω1. (i) x is hereditarily mξ,ζ if every ξ-weakly null sequence has a subsequence which is a c ζ 0-spreading model. (ii) x is hereditary by quotients mω1,ζ if and only if x ∗ is hereditarily mζ,ω1 . (iii) if ξ < ω1, then x is hereditarily mγ,ζ for some ω ξ < γ < ωξ+1 if and only if x is hereditarily mγ,ζ for every ω ξ < γ < ωξ+1. (iv) if ζ < ω1, then x is hereditarily mξ,γ for some ω ζ < γ < ωζ+1 if and only if x is hereditarily mξ,γ for every ω ζ < γ < ωζ+1. 140 r.m. causey we also study three space properties related to the ξ-weak banach-saks property, modifying a method of ostrovskii [19]. in [19], it was shown that the weak banach-saks property is not a three-space property. our final theorem generalizes this. in our final theorem, wbsξ denotes the class of banach spaces x such that ix ∈ wbsξ. theorem 1.5. for 0 ≤ ζ,ξ < ω1, if x is a banach space and y is a closed subspace such that y ∈ wbsζ and x/y ∈ wbsξ, then x ∈ wbsξ+ζ. for every 0 ≤ ζ,ξ < ω1, there exists a banach space x with a closed subspace y such that y ∈ wbsζ, x/y ∈ wbsξ, and for each γ < ξ + ζ, x fails to lie in wbsγ. 2. combinatorics regular families. througout, we let 2n denote the power set of n. we endow {0, 1}n with its product topology and endow 2n with the cantor topology, which is the topology making the identification 2n 3 f ↔ 1f ∈ {0, 1}n a homeomorphism. given a subset m of n, we let [m] (resp. [m] 0, there exist f ∈ sξ and y ∈ co(yn : n ∈ f) such that ‖y‖ < ε. we say (xn)∞n=1 is ξ-weakly convergent to x if (xn −x)∞n=1 is ξ-weakly null. we say (xn) ∞ n=1 is ξ-weakly convergent if it is ξ-weakly convergent to some x. we say a sequence is ω1-weakly null, ω1-weakly convergent to x, or ω1-weakly convergent if it is weakly null, weakly convergent to x, or weakly convergent, respectively. remark 2.5. note that if (xn) ∞ n=1 is a ξ-weakly null sequence in the banach space x, then there exist sets f1 < f2 < ... , fn ∈ sξ, and positive scalars (ai)i∈∪∞n=1fn such that for each n ∈ n, ∑ i∈fn ai = 1, and such that limn‖ ∑ i∈fn aixi‖ = 0. we will use this fact often. however, we will also often need a technical fact which states that the coefficients (ai)i∈fn can come from the repeated averages hierarchy. we make this precise below. remark 2.6. it follows from theorem c of [2] that a weakly null sequence fails to be ξ-weakly null if and only if it has a subsequence which is an ` ξ 1-spreading model. from this it follows that if (xn) ∞ n=1 is a weakly null ` ξ 1-spreading model, it can have no ξ-weakly convergent subsequence. indeed, since ξ-weak convergence to x implies weak convergence to x, the only x to which a subsequence of (xn) ∞ n=1 could be ξ-convergent is x = 0. but if (xn) ∞ n=1 is an ` ξ 1-spreading model, all of its subsequences are, and so no subsequence can be ξ-convergent to zero by the first sentence of the remark. let p denote the set of all probability measures on n. we treat each member p of p as a function from n into [0, 1], where p(n) = p({n}). we let supp(p) = {n ∈ n : p(n) > 0}. given a nice family p and a subset p = {pm,n : m ∈ [n],n ∈ n} of p, we say (p,p) is a probability block provided that (i) for each m ∈ [n], supp(pm,1) = mp,1, and (ii) for any m ∈ [n] and r ∈ n, if n = m \ ∪r−1i=1 supp(pm,i), then pn,1 = pm,r. remark 2.7. it follows from the definition of probability block that for any m ∈ [n], (mp,n)∞n=1 = (supp(pm,n)) ∞ n=1 and for any s ∈ n and m,n ∈ n, and r1 < · · · < rs such that ∪si=1 supp(pm,ri) is an initial segment of n, then pn,i = pm,ri for all 1 ≤ i ≤ s. this was proved in [12]. 148 r.m. causey suppose that q is nice. given l = (ln)∞n=1 ∈ [n], there exists a unique sequence 0 = p0 < p1 < ... such that (li) pn i=pn−1+1 ∈ max(q) for all n ∈ n. we then define l−1q,n = n∩ (pn−1,pn]. suppose we have probability blocks (p,p), (q,q). we define a collection q∗p such that (q∗p,q[p]) is a probability block. fix m ∈ n and for each n ∈ n, let ln = min supp(pm,n) and l = (ln)∞n=1. we then let om,n = ∑ i∈l−1q,n ql,n(li)pm,i and q∗p = {om,n : m ∈ [n],n ∈ n}. in [2], the repeated averages hierarchy was defined. this is a collection sξ, ξ < ω1, such that (sξ,sξ) is a probability block for every ξ < ω1. we will denote the members of sξ by s ξ m,n, m ∈ [n], n ∈ n. for ξ < ω1, we say a probability block (p,p) is ξ-sufficient provided that for any l ∈ [n], any ε > 0, and any regular family g with cb(g) ≤ ωξ, there exists m ∈ [n] such that sup { pn,1(e) : e ∈g,n ∈ [m] } < ε. it was shown in [2] that (sξ,sξ) is ξ-sufficient. the following facts were shown in [12]. item (ii) was shown in [2] in the particular case that (p,p) = (sξ,sξ). theorem 2.8. (i) for ξ,ζ < ω1, if (p,p) is ξ-sufficient and (q,q) is ζ-sufficient, then (q∗p,q[p]) is (ξ + ζ)-sufficient. (ii) if x is a banach space, ξ < ω1, (p,p) is ξ-sufficient, and cb(p) = ωξ + 1, then a weakly null sequence (xn) ∞ n=1 ⊂ x is ξ-weakly null if and only if for any l ∈ [n] and ε > 0, there exists m ∈ [l] such that for all n ∈ [m], ‖ ∑∞ i=1 pn,1(i)xi‖ < ε. remark 2.9. since for each ξ < ω1, at least one ξ-sufficient probability block (p,p) with cb(p) = ωξ + 1 exists, item (ii) of the preceding theorem yields that if x is a banach space and (xn) ∞ n=1, (yn) ∞ n=1 are ξ-weakly null in x, then (xn + yn) ∞ n=1 is also ξ-weakly null. this generalizes to sums of any number of sequences. the importance of this fact, which we will use often throughout, is that if for k = 1, . . . , l, if (xkn) ∞ n=1 ⊂ x is a ξ-weakly null sequence, then for any ε > 0, there exist f ∈ sξ and positive scalars (ai)i∈f the ξ,ζ-dunford pettis property 149 such that ∑ i∈f ai = 1 and for each 1 ≤ k ≤ l,∥∥∥∥∑ i∈f aix k i ∥∥∥∥ ≤ ε. that is, there is one choice of f and (ai)i∈f such that the corresponding linear combinations of the l different sequences are simultaneously small. note that the preceding implies that for two banach spaces x,y and ξweakly null sequences (xn) ∞ n=1 ⊂ x, (yn) ∞ n=1 ⊂ y , for any ε > 0, there exist f ∈sξ and positive scalars (ai)i∈f summing to 1 such that∥∥∥∥∑ i∈f aixi ∥∥∥∥ x , ∥∥∥∥∑ i∈f aiyi ∥∥∥∥ y < ε. this is because the sequences (xn, 0) ∞ n=1 ⊂ x⊕∞y and (0,yn) ∞ n=1 ⊂ x⊕∞y are also ξ-weakly null, as is their sum in x ⊕∞ y . remark 2.10. let x be a banach space and let (xn) ∞ n=1 be ξ-weakly null. let (p,p) be ξ-sufficient with cb(p) = ωξ + 1. then by theorem 2.8(ii), we may recursively select m1 ⊃ m2 ⊃ . . . , mn ∈ [n] such that for each n ∈ n, sup {∥∥∥∥ ∞∑ i=1 pn,1(i)xi ∥∥∥∥ : n ∈ [mn] } < 1/n. now choose mn ∈ mn with m1 < m2 < ... and let m = (mn)∞n=1. then for any n ∈ [m] and n ∈ n, if f1 < f2 < ... is a partition of n into consecutive, maximal members of p and nj = n \∪ j−1 i=1 fi for each j ∈ n, nn ∈ [mn]. by the permanence property mentioned in remark 2.7,∥∥∥∥ ∞∑ i=1 pn,n(i)xi ∥∥∥∥ = ∥∥∥∥ ∞∑ i=1 pnn,1(i)xi ∥∥∥∥ < 1/n. before proceeding to the following, we recall that for m ∈ [n] and a regular family f, we let m|f denote the maximal initial segment of m which lies in f. if f is nice, then m|f lies in max(f). lemma 2.11. let x be a banach space, (xn) ∞ n=1 ⊂ x a seminormalized, weakly null sequence, and f a nice family. 150 r.m. causey (i) (xn) ∞ n=1 admits a subsequence which is a c f 0 -spreading model if and only if there exists l ∈ [n] such that sup {∥∥∥∥ ∑ n∈m|f xn ∥∥∥∥ : m ∈ [l] } < ∞. (ii) if (xn) ∞ n=1 admits no subsequence which is a c f 0 -spreading model, then there exists l ∈ [n] such that for any h1 < h2 < ... , hn ∈ max(f) ∩ [l] n for each n ∈ n. proof. (i) assume there exists l ∈ [n] such that sup {∥∥∥∥ ∑ n∈m|f xn ∥∥∥∥ : m ∈ [l] } = c < ∞. by passing to an infinite subset of l, we may assume (xn)n∈l is 2-basic. if f ∈f∩ [l] rsn for all n ∈ n. let ln = rsn, l = (ln) ∞ n=1, and s = (sn) ∞ n=1. fix m ∈ [l] and note that m = (rtn) ∞ n=1 for some (tn) ∞ n=1 ∈ [s]. let m|f = (rtn) k n=1 ∈ f and note that (tn) k n=2 ∈ f. indeed, if tn−1 = si and tn = sj, then i < j and then tn = sj ≥ si+1 > rsi = rtn−1. thus e := (tn) k n=2 is a spread of (rtn) k−1 n=1 ⊂ (rtn) k n=1 ∈ f, and e ∈ f. therefore, with b = supn‖xn‖,∥∥∥∥ ∑ n∈m|f xn ∥∥∥∥ ≤ ∥∥∥∥xrt1 ∥∥∥∥ + ∥∥∥∥ k∑ n=2 xrtn ∥∥∥∥ = ∥∥∥∥xrt1 ∥∥∥∥ + ∥∥∥∥ ∑ n∈e xrtn ∥∥∥∥ ≤ b + c =: c . therefore we have shown that sup {∥∥∥∥ ∑ n∈m|f xn ∥∥∥∥ : m ∈ [l] } ≤ c. (ii) for each n ∈ n, let vn = { m ∈ [n] : ∥∥∥∥ ∑ i∈m|f xi ∥∥∥∥ ≤ n‖ } . it is evident that vn is closed, and in fact m 7→ ‖ ∑ i∈m|f xi‖ is locally constant on [n]. by the ramsey theorem, we may select m1 ⊃ m2 ⊃ . . . such that for all n ∈ n, either [mn] ⊂ vn or vn ∩ [mn] = ∅. by (i) and the hypothesis that (xn) ∞ n=1 admits no subsequence which is a c f 0 -spreading model, for each n ∈ n, vn ∩ [mn] = ∅. now fix l1 < l2 < ... , ln ∈ mn, and let l = (ln) ∞ n=1. fix ∅ 6= h1 < h2 < ... , hn ∈ max(f) ∩ [l] n. 152 r.m. causey for ordinals ξ,ζ < ω1 and any m ∈ [n], there exists n ∈ [m] such that sξ[sζ](n) ⊂ sζ+ξ and sζ+ξ(n) ⊂ sξ[sζ] ([18, proposition 3.2]). from this it follows that for a given sequence (xn) ∞ n=1 in a banach space x, there exist m1 < m2 < ... such that 0 < inf { ‖x‖ : f ∈sζ+ξ,x ∈ aco(xmn : n ∈ f) } if and only if there exist m1 < m2 < ... such that 0 < inf { ‖x‖ : f ∈sξ[sζ],x ∈ aco(xmn : n ∈ f) } . this fact will be used throughout to deduce that if (xn) ∞ n=1 is an ` ζ+ξ 1 -spreading model (or has a subsequence which is an ` ζ+ξ 1 -spreading model), then there exists a subsequence of (xn) ∞ n=1 which is an ` sξ[sζ] 1 -spreading model. similarly, if (xn) ∞ n=1 has a subsequence which is a c ζ+ξ 0 -spreading model, then it has a subsequence which is a c sξ[sζ] 0 -spreading model. corollary 2.12. fix α,β,γ < ω1. let x,y be banach spaces, a : x → y an operator, and let (xn)∞n=1 be a seminormalized, weakly null sequence in x. (i) if (axn) ∞ n=1 has a subsequence which is an ` α+β 1 -spreading model and (xn) ∞ n=1 has no subsequence which is an ` α+γ 1 -spreading model, then there exists a convex block sequence (zn) ∞ n=1 of (xn) ∞ n=1 which has no subsequence which is an ` γ 1 -spreading model and such that (azn) ∞ n=1 is an ` β 1 -spreading model. (ii) if (xn) ∞ n=1 has a subsequence which is a c α+β 0 -spreading model but no subsequence which is a c α+γ 0 -spreading model, then there exists a block sequence of (xn) ∞ n=1 which is a c β 0 -spreading model and has no subsequence which is a c γ 0 -spreading model. if 0 < β, the block sequence is also weakly null. proof. (i) we first assume supn‖xn‖ = 1. by passing to a subsequence, we may assume without loss of generality that 0 < ε = inf { ‖ax‖ : f ∈sβ[sα],x ∈ abs co(xn : n ∈ f) } . let p = sγ[sα], p = sγ ∗ sα = {pm,n : m ∈ [n],n ∈ n}. as mentioned in remark 2.10, we may also fix l ∈ [n] such that for all m ∈ [l] and n ∈ n,∥∥∥∥ ∞∑ i=1 pm,n(i)xi ∥∥∥∥ ≤ 1/n. the ξ,ζ-dunford pettis property 153 now fix f1 < f2 < ... , fn ∈ max(sα), l = ∪∞n=1fn and let yn =∑∞ i=1 s α l,n(i)xi = ∑ i∈fn s α l,n(i)xi. it follows from the second sentence of the proof that ε ≤ inf { ‖ay‖ : f ∈sβ,y ∈ abs co(yn : n ∈ f) } . that is, (ayn) ∞ n=1 is an ` β 1 -spreading model. it remains to show that (yn) ∞ n=1 has no subsequence which is an ` γ 1 -spreading model. to that end, assume r = (rn) ∞ n=1, δ > 0 are such that δ ≤ inf {∥∥∥∥ ∑ n∈f anyrn ∥∥∥∥ : f ∈sγ, ∑ n∈f |an| = 1 } . now let en = frn, n = ∪∞n=1en, s = (sn) ∞ n=1 = (min en) ∞ n=1 and note that, by the permanence property, zn := yrn = ∞∑ i=1 sαn,n(i)xi for all n ∈ n. now fix 1 = q1 < q2 < ... such that qn+1 > sqn. let m = ∪∞n=1eqn and note that there exist 0 = k0 < k1 < ... such that for all n ∈ n, pm,n = kn∑ j=kn−1+1 sγt,n(sqj )s α m,j and (sqj ) kn j=kn−1+1 ∈ sγ, where t = (sqj ) ∞ j=1. moreover s γ t,n(sqkn−1+1 ) → 0 since 0 < γ. we now observe that since sqj < qj+1, gn := (qj) kn j=kn−1+2 is a spread of (qj) kn−1 j=kn−1+1 , which is a subset of a member of sγ. therefore, for any n ∈ n, δ ( 1 −sγt,n(sqkn−1+1 ) ) ≤ ∥∥∥∥ kn∑ j=kn−1+2 sγt,n(sqj )zj ∥∥∥∥ ≤ ∥∥∥∥ kn∑ j=kn−1+1 sγt,n(sqj )zj ∥∥∥∥ + sγt,n(sqkn−1+1 ) ≤ ∥∥∥∥ ∞∑ i=1 pm,n(i)xi ∥∥∥∥ + sγt,n(sqkn−1+1 ) ≤ 1/n + sγt,n(sqkn−1+1 ). 154 r.m. causey since limn s γ t,n(sqkn−1+1 ) = 0, these inequalities yield a contradiction for sufficiently large n. (ii) we may assume without loss of generality that sup {∥∥∥∥ ∑ n∈f εnxn ∥∥∥∥ : f ∈sβ[sα], |εn| = 1 } = c < ∞ and that (xn) ∞ n=1 is basic. by lemma 2.11 applied with f = sγ[sα], there exists l ∈ [n] such that for all h1 < h2 < ... , hn ∈ max(sγ[sα]) ∩ [l] n. we claim that for any f1 < f2 < ... , fn ∈ max(sα) ∩ [l] min fsn. let t = ∪∞n=1fsn and let h1 < h2 < ... be such that hn ∈ max(sγ[sα]) and t = ∪∞n=1hn. note that ‖ ∑ i∈hn xi‖ > n for all n ∈ n. note also that there exist 0 = k0 < k1 < ... such that hn = ∪knj=kn−1+1fsj , and these numbers are uniquely determined by the property that (min fsj ) kn j=kn−1+1 ∈ max(sγ). as is now familiar, we note that for each n ∈ n, en := (sj)knkn−1+2 is a spread of a subset (min fsj ) kn−1 j=kn−1+1 , so that en ∈sγ. we note that for each n ∈ n, n < ∥∥∥∥ ∑ i∈hn xi ∥∥∥∥ ≤ ∥∥∥∥ ∑ i∈fkn−1+1 xi ∥∥∥∥ + ∥∥∥∥ kn∑ j=kn−1+2 ∑ i∈fsj xi ∥∥∥∥ ≤ c + ∥∥∥∥ ∑ j∈en xi ∥∥∥∥ ≤ c + d. this is a contradiction for sufficiently large n. schreier and baernstein spaces. if f is a nice family, we let xf denote the completion of c00 with respect to the norm ‖x‖f = sup { ‖ex‖`1 : e ∈f } . the ξ,ζ-dunford pettis property 155 in the case that f = sξ, we write ‖·‖ξ in place of ‖·‖sξ and xξ in place of xsξ. the spaces xξ are called schreier spaces. note that x0 = c0 isometrically. given 1 < p < ∞ and a nice family f, we let xf,p be the completion of c00 with respect to the norm ‖x‖f,p = sup {( ∞∑ i=1 ‖eix‖ p `1 )1/p : e1 < e2 < ... , ei ∈f } . for convenience, we let xξ,p and ‖·‖ξ,p denote xsξ,p and ‖·‖sξ,p, respectively. the spaces xξ,p are called baernstein spaces. for convenience, we let xξ,∞ denote xξ. remark 2.13. the schreier families sξ, ξ < ω1, possess the almost monotone property, which means that for any ζ < ξ < ω1, there exists m ∈ n such that if m ≤ e ∈sζ, then e ∈sξ. from this it follows that the formal inclusion i : xξ → xζ is bounded for any ζ ≤ ξ < ω1. in fact, there exists a tail subspace [ei : i ≥ m] of xξ such that the restriction of i : [ei : i ≥ m] → xζ is norm 1. we will use this fact throughout. it is also obvious that the formal inclusion from xξ,p to xζ,p is bounded for any ζ ≤ ξ < ω1, as is the inclusion from xξ,p to xξ,q whenever p < q ≤∞. combining these facts yields that the formal inclusion from xξ,p to xζ is bounded whenever ζ ≤ ξ. furthermore, the adjoints of all of these maps are also bounded. the following collects known facts about the schreier and baernstein spaces. throughout, we let ‖ · ‖ξ,p denote the norm of xξ,p as well as its first and second duals. theorem 2.14. fix ξ < ω1 and 1 < p ≤∞. (i) ‖ ∑n i=1 xi‖ξ,p = ‖ ∑n i=1 |xi|‖ξ,p for any disjointly supported x1, . . . ,xn ∈ xξ,p. (ii) the canonical basis of xξ,p is shrinking. (iii) the basis of xξ,p is boundedly-complete (and xξ,p is reflexive) if and only if p < ∞. (iv) if p < ∞ and 1/p + 1/q = 1,∥∥∥∥ n∑ i=1 xi ∥∥∥∥ ξ,p ≥ ( n∑ i=1 ‖xi‖ p ξ,p )1/p and ∥∥∥∥ n∑ i=1 x∗i ∥∥∥∥ ξ,p ≤ ( n∑ i=1 ‖x∗i‖ q ξ,p )1/q for any x1 < · · · < xn ∈ xξ,p and x∗1 < · · · < x ∗ n, x ∗ i ∈ x ∗ ξ,p. 156 r.m. causey (v) the canonical basis of xξ,p is a weakly null ` ξ 1-spreading model, while every normalized, weakly null sequence in xξ,p is ξ + 1-weakly null. (vi) the space xξ is isomorphically embeddable into c(sξ). remark 2.15. throughout, if e ∈ [n] 0, then (xn) ∞ n=1 has a subsequence which dominates the xδ,p basis. proof. (i) by passing to a subsequence, we may assume that (x∗n) ∞ n=1 is a block sequence and supn‖x∗n‖ < c1 < c. by scaling, we may assume c1 = 1. for each n ∈ n, let sn = supp(x∗n). for each n ∈ n, it follows from convexity and compactness arguments that for each n ∈ n, there exist dn, (x∗n,i) dn i=1, and (en,i) dn i=1 ⊂sγ ∩ [sn] j, since max ei < min esj ≤ max supp(xj) < min supp(xl). furthermore, since esjxj 6= 0, esj ∈ sγ with min esj ≤ mj. if γ is a limit ordinal, then esj ∈sγmj , which means that for any k > j, ‖esjxk‖`1 ≤ εk/mj ≤ εk. if γ is a successor, then γ = γmj +1 and min esj ≤ mj yield that esj = ∪ q i=1fi for some f1 < · · · < fq, q ≤ mj, and fi ∈sγmj . then for k > j, ‖esjxk‖`1 ≤ q∑ i=1 ‖fixk‖`1 ≤ mn‖xk‖γmj ≤ εk. in the case γ = 0, each ei is a singleton, so we have the trivial estimate that for i ∈ sj and l > j, eixl = 0. therefore in each of the γ = 0, γ a successor, and γ a limit ordinal cases, ∑ i∈sj ‖eix‖`1 ≤ |aj|‖exj‖`1 + ∞∑ k=j+1 ‖esjxk‖`1 ≤ |aj| + ∞∑ k=j+1 εk. summing over i yields that ‖ex‖`1 ≤ ∑ j∈j ∑ i∈sj ‖eix‖`1 ≤ ∑ j∈j |aj| + ∑ j∈j ∞∑ k=j+1 εk ≤ ∑ j∈j |aj| + ∞∑ j=m(e) ∞∑ k=j εk, where m(e) = min{j : exj 6= 0}. now for each j ∈ j, fix some ij ∈ {1, . . . ,d} such that j = jij . then j 7→ ij is an injection of j into {1, . . . ,d}, and (mij )j∈j is a spread of a subset of (min ei) d i=1. therefore t(e) := (rij )j∈j = (nmij )j∈j is a spread of a subset of (nmin ei) d i=1 ∈sδ, so t(e) ∈sδ. therefore ‖y‖δ ≥‖t(e)y‖`1 = ∑ j∈j |aj|. the ξ,ζ-dunford pettis property 159 collecting these estimates and recalling our assumption that (ai) ∞ i=1 ∈ s`∞, we deduce that ‖x‖γ+δ ≤ ∑ j∈j |ai| + ∞∑ j=m(e) ∞∑ k=j εk ≤ 2‖y‖δ. this completes the p = ∞ case. now assume 1 < p < ∞. fix e1 < e2 < ... , ei ∈ sγ+δ. let x =∑∞ i=1 aixi, y = ∑∞ i=1 aieri as in the previous paragraph. for each i ∈ n, let ji = { j ∈ n : (∀i 6= k ∈ n)(ejxk = 0) } . let j = ∪∞i=1ji and i = n \j. let us rename the sets (ei)i∈i as f1 < g1 < f2 < g2 < ... (ignoring this step if i is empty and with the appropriate notational change if i is finite and non-empty). by the properties of i, for each i such that fi (resp. gi) is defined, there exist at least two distinct indices j,k such that fixj,fixk 6= 0 (resp. at least two distinct indices j′,k′ such that gixj′,gixk′ 6= 0). from this it follows that, with ui = {j : fixj 6= 0} and vi = {j : gixj 6= 0}, the sets (ui)i are successive, as are (vi)i. in particular, fixj = gixj = 0 whenever j < i. observe that(∑ i∈j ‖eix‖ p `1 )1/p = ( ∞∑ j=1 |aj|p ∑ i∈jj ‖eixj‖ p `1 )1/p ≤‖(aj)∞j=1‖`p ≤‖y‖δ,p. now, arguing as in the p = ∞ case, for each i such that fi is defined, if m(fi) = min{j : fixj 6= 0}, there exists a set t(fi) ∈sδ such that ‖fix‖`1 ≤‖t(fi)y‖`1 + ∞∑ l=m(fi) ∞∑ k=l εk. furthermore, t(fi) ⊂ {nmj : j ∈ ui}, from which it follows that the sets t(fi) are successive, since the sets ui are. from this, the triangle inequality, and the fact that m(fi) ≥ i for each appropriate i, it follows that(∑ i ‖fix‖ p `1 )1/p ≤ ( ∞∑ i=1 ‖t(fi)y‖ p `1 )1/p + ∞∑ i=1 ∞∑ l=m(fi) ∞∑ k=l εk ≤ ‖y‖δ,p + ∞∑ i=1 ∞∑ l=i ∞∑ k=l εk = ‖y‖δ,p + 1 ≤ 2‖y‖δ,p. 160 r.m. causey a similar argument yields that (∑ i ‖gix‖ p `1 )1/p ≤ 2‖y‖δ,p. therefore ( ∞∑ j=1 ‖ejx‖ p `1 )1/p ≤ 5‖y‖δ,p. since e1 < e2 < ... , ei ∈sγ+δ were arbitrary, ‖x‖γ+δ,p ≤ 5‖y‖δ,p. (iii) by passing to a subsequence and perturbing, we may assume (xn) ∞ n=1 is a block sequence in xγ+δ,p and infn‖xn‖γ = ε > 0. we may fix a block sequence (x∗n) ∈ ε−1bx∗γ biorthogonal to (xn) ∞ n=1. by (i), after passing to a subsequence and using properties of the xγ+δ,p basis, assume that sup {∥∥∥∥ ∑ n∈g εnx ∗ n ∥∥∥∥ γ+δ : g ∈sδ, |εn| = 1 } ≤ 1/ε. if p = ∞, note that for any (ai)∞i=1 ∈ c00,∥∥∥∥ ∞∑ i=1 aiei ∥∥∥∥ γ = sup { ∑ n∈g |an| : g ∈sδ } ≤ sup { re ( ∑ n∈g εnx ∗ n )( ∞∑ n=1 anxn ) : g ∈sγ, |εn| = 1 } ≤ ε−1 ( ∞∑ n=1 anxn ) . now suppose that 1 < p < ∞. fix (ai)∞i=1 ∈ c00 and let x = ∑∞ i=1 aiei. fix e1 < e2 < · · · < en, ei ∈sδ and a sequence (bi)ni=1 ∈ s`nq , such that ‖x‖γ,p = ( n∑ i=1 ‖eix‖ p `1 )1/p = n∑ i=1 bi ( ∑ j∈ei |aj| ) . let y∗i = ∑ j∈ei εjx ∗ j , where εjaj = |aj|, and let y ∗ = ∑n i=1 biy ∗ i . since ‖y∗i‖γ+δ ≤ ε −1 and ∑n i=1 b q i = 1, ‖y ∗‖γ+δ,p ≤ ε−1. indeed, by hölder’s inthe ξ,ζ-dunford pettis property 161 equality, for any x ∈ c00, if i1 < · · · < in are such that supp(y∗i ) ⊂ ii, then |y∗(x)| ≤ n∑ i=1 bi|y∗i (x)| ≤ ε −1 n∑ i=1 bi‖iixi‖γ ≤ ε−1 ( n∑ i=1 b q i )( n∑ i=1 ‖iix‖pγ )1/p ≤ ε−1‖x‖γ,p. moreover, ε−1 ∥∥∥∥ ∞∑ i=1 aixi ∥∥∥∥ ξ,p ≥ y∗ ( ∞∑ i=1 aixi ) = n∑ i=1 bi ( ∑ j∈ei |aj| ) = ‖x‖γ,p. let us recall that for any ordinals γ,ξ with γ ≤ ξ, there exists a unique ordinal δ such that γ + δ = ξ. we denote this ordinal δ by ξ −γ. we also recall that any non-zero ordinal ξ admits a unique representation (called the cantor normal form) as ξ = ωε1n1 + · · · + ωεknk, where k,n1, . . . ,nk ∈ n and ε1 > · · · > εk. using the cantor normal form ξ = ωε1n1 + · · · + ωεknk, we define the least non-trivial part λ(ξ) of ξ by λ(ξ) = ωε1 . for completeness, we let λ(0) = 0. we also note that if ζ ≤ ξ, λ(ζ) ≤ λ(ξ). for 0 < ξ, let ωε1n1 + · · · + ωεknk be the cantor normal form of ξ. by writing ωεn = ωε + · · ·+ ωε, where the summand ωε appears n times, we may also uniquely represent ξ as ξ = ωδ1 + · · · + ωδl where l ∈ n and δ1 ≥ ···≥ δl. in this case, δ1 = ε1. theorem 2.17. fix ξ < ω1 and 1 < p ≤ ∞. fix a weakly null sequence (xn) ∞ n=1 ⊂ xξ,p. let γ = {ζ ≤ ξ : lim supn‖xn‖ζ > 0}. (i) if p = ∞, then γ = ∅ if and only if (xn)∞n=1 is norm null. (ii) if p < ∞ and γ = ∅, then either (xn)∞n=1 is norm null or (xn) ∞ n=1 has a subsequence equivalent to the canonical `p basis. (iii) if γ 6= ∅ and γ = min γ, then (xn)∞n=1 admits a subsequence which is equivalent to a subsequence of the xξ−γ,p basis. in particular, (xn) ∞ n=1 is ξ −γ + 1 weakly null and not ξ −γ weakly null. 162 r.m. causey (iv) if p = ∞, then every subsequence of (xn)∞n=1 has a further wuc subsequence if and only if γ ⊂{ξ}. (v) if 0 < ξ, a weakly null sequence (xn) ∞ n=1 is ξ-weakly null if and only if for every β < λ(ξ), limn‖xn‖β = 0. proof. first note that by the almost monotone property of the schreier families, if ζ ∈ γ, then [ζ,ξ] ⊂ γ. (i) it is evident that limn‖xn‖ξ = 0 if and only if ξ /∈ γ. (ii) if ξ /∈ γ, then let γ = ξ and δ = 0. by proposition 2.16(ii), any subsequence of (xn) ∞ n=1 has a further subsequence which is dominated by a subsequence of the xδ,p = `p basis. then since every seminormalized block sequence in xξ,p which dominates the `p basis, either limn‖xn‖ξ,p = 0, or (xn) ∞ n=1 has a seminormalized subsequence which dominates the `p basis, and this subsequence has a further subsequence equivalent to the `p basis. (iii) let δ = ξ − γ, so that γ + δ = ξ. proposition 2.16(ii) yields that every subsequence of (xn) ∞ n=1 has a further subsequence which is dominated by a subsequence of the xδ,p basis. since no subsequence of the xδ,p basis is an `δ+11 -spreading model, this yields that (xn) ∞ n=1 is δ + 1-weakly null. since γ ∈ γ, proposition 2.16(iii) yields the existence of a subsequence (yn)∞n=1 of (xn) ∞ n=1 which dominates the xδ,p basis, so (xn) ∞ n=1 is not δ-weakly null. now note that limn‖yn‖β = 0 for all β < γ, so by proposition 2.16(ii), there exists a subsequence (zn) ∞ n=1 of (yn) ∞ n=1 which is dominated by some subsequence (xni) ∞ i=1 of the canonical xδ,p basis. this sequence (zn) ∞ n=1 also dominates some subsequence (xmi) ∞ i=1 of the canonical xδ,p basis (where mi has the property that zi = ymi). now let us choose 1 = k1 < k2 < ... such that mki+1 > nki for all i ∈ n and let ui = zki. then (ui) ∞ i=1 is dominated by some subsequence (xri) ∞ i=1 of the xδ,p basis and dominates some subsequence (xsi) ∞ i=1 of the xδ,p basis, where s1 ≤ r1 < s2 ≤ r2 < ... . this is seen by taking si = mki and ri = nki. but it is observed in [10] that two such subsequences of the xδ,p basis must be 2-equivalent, so (ui) ∞ i=1 is equivalent to (eri) ∞ i=1 (and to (esi) ∞ i=1). (iv) if γ ⊂{ξ}, then by proposition 2.16(ii) applied with γ = ξ and δ = 0, every subsequence of (xn) ∞ n=1 has a further subsequence which is dominated by the xδ = c0 basis. conversely, if ξ > γ ∈ γ, then with δ = ξ − γ > 0, (xn) ∞ n=1 has a subsequence which is an ` δ 1-spreading model. no subsequence of this sequence can be wuc. (v) note that both conditions are satisfied if (xn) ∞ n=1 is norm null, so assume (xn) ∞ n=1 is not norm null. if γ = ∅, then p < ∞, and every subsequence of (xn) ∞ n=1 has a further subsequence which is equivalent to the `p the ξ,ζ-dunford pettis property 163 basis, which means (xn) ∞ n=1 is 1-weakly null, and therefore ξ-weakly null. thus both conditions are satisfied in this case as well. it remains to consider the case γ 6= ∅. let γ = min γ. let us write ξ = ωε1 + · · · + ωεk, where ε1 ≥ ···≥ εk. note that λ(ξ) = ωε1 . first assume that limn‖xn‖β = 0 for all β < λ(ξ), which means γ ≥ λ(ξ). then if γ +δ = ξ, δ ≤ ωε2 +· · ·+ωεk. by (iii), (xn) ∞ n=1 is δ + 1-weakly null, and δ + 1 ≤ ωε2 + · · · + ωεk + 1 ≤ ωε2 + · · · + ωεk + ωε1 ≤ ωε1 + · · · + ωεk = ξ yields that (xn) ∞ n=1 is ξ-weakly null. conversely, assume there exists β < λ(ξ) such that lim supn‖xn‖β > 0. then γ < λ(ξ). if γ + δ = ξ, then δ = ξ. by (iii), (xn) ∞ n=1 is not ξ-weakly null. corollary 2.18. for any 0 < ξ < ω1 and any seminormalized, weakly null sequence (xn) ∞ n=1 in xωξ, (xn) ∞ n=1 has a subsequence (yn) ∞ n=1 which is either equivalent to the canonical c0 basis or to a subsequence of the xωξ basis. proof. by theorem 2.17(iv), every subsequence of (xn) ∞ n=1 has a further wuc (and therefore equivalent to the c0 basis) subsequence if and only if limn‖xn‖β = 0 for every β < ξ = λ(ξ). if this condition fails, then there exists a minimum γ < ωξ such that lim supn‖xn‖γ > 0. then if γ + δ = ωξ, δ = ωξ, and (xn) ∞ n=1 has a subsequence equivalent to a subsequence of the xωξ basis. corollary 2.19. fix 0 < ξ < ω1, 1 < p ≤ ∞, and let (xn)∞n=1 ⊂ xξ,p be weakly null. then (xn) ∞ n=1 is ξ-weakly null in xξ,p if and only if for every γ < λ(ξ), limn‖xn‖γ = 0 if and only if every subsequence of (xn)∞n=1 has a further subsequence which is wuc in xλ(ξ). proof. this follows from combining theorem 2.17 (iv) – (v). in the sequel, we will need the following standard duality argument. as it involves some non-trivial computation, we isolate it. proposition 2.20. suppose that f is a spreading set of finite subsets of n, (xn)∞n=1 ⊂ x is weakly null, (x ∗ n) ∞ n=1 is weakly null, inf |x ∗ n(xn)| ≥ ε > 0. 164 r.m. causey (i) if sup {∥∥∥∥ ∑ n∈f anx ∗ n ∥∥∥∥ : f ∈f, |an| ≤ 1 } = c < ∞, then there exists a subsequence (xkn) ∞ n=1 of (xn) ∞ n=1 such that inf {∥∥∥∥ ∑ n∈f bnxkn ∥∥∥∥ : f ∈f, ∑ n∈f |bn| = 1 } ≥ ε 2c . (ii) if sup {∥∥∥∥ ∑ n∈f anxn ∥∥∥∥ : f ∈f, |an| ≤ 1 } = c < ∞, then there exists a subsequence (x∗kn) ∞ n=1 of (x ∗ n) ∞ n=1 such that inf {∥∥∥∥ ∑ n∈f bnx ∗ kn ∥∥∥∥ : f ∈f, ∑ n∈f |bn| = 1 } ≥ ε 2c . proof. (i) first note that the condition sup {∥∥∥∥ ∑ n∈f anx ∗ n ∥∥∥∥ : f ∈f, |an| ≤ 1 } ≤ c passing to subsequences, since f is spreading. fix (εn)∞n=1(0,ε) such that∑∞ n=1 ∑∞ m=n+1 εm < ε/4. we may recursively choose 1 = k1 < k2 < ... such that for all n < m, |x∗n(xm)|, |x∗m(xn)| < εm. then for any f ∈ f and (bn)n∈f , fix (an)n∈f such that |an| = 1 for all n ∈ f and∑ n∈f anbnx ∗ kn (xkn) = ∑ n∈f |anbnx∗kn(xkn)| ≥ ε ∑ n∈f |bn|. since ‖ ∑ n∈f anx ∗ kn ‖≤ c by the first sentence of the proof, c ∥∥∥∥ ∑ n∈f bnxkn ∥∥∥∥ ≥ ∣∣∣∣ ( ∑ n∈f anx ∗ kn )( ∑ n∈f bnxkn )∣∣∣∣ ≥ ∑ n∈f anbnx ∗ kn (xkn) − ∞∑ n=1 ∞∑ m=n+1 |bn||x∗km(xkn)| − ∞∑ n=1 ∞∑ m=n+1 |bm||x∗kn(xkm)| ≥ ε ∑ n∈f |bn|− 2 max n∈f |bn| ∞∑ n=1 ∞∑ m=n+1 εm ≥ ε 2 ∑ n∈f |bn|. the ξ,ζ-dunford pettis property 165 (ii) this follows from (i) by considering (xn) ∞ n=1 as a sequence in x ∗∗. lemma 2.21. fix 0 < ξ < ω1 and 1 < p ≤∞. (i) if (x∗∗n ) ∞ n=1 ⊂ x ∗∗ ξ,p is ξ-weakly null, then limn‖x ∗∗ n ‖γ = 0 for every γ < λ(ξ). (ii) if (x∗∗n ) ∞ n=1 ⊂ x ∗∗ ξ,p is ξ-weakly null and (x ∗ n) ∞ n=1 ⊂ x ∗ λ(ξ) is weakly null, then limn x ∗∗ n (x ∗ n) = 0. proof. (i) suppose not. then for some γ < λ(ξ) and ε > 0, we may pass to a subsequence and assume infn‖x∗∗n ‖γ > ε. we may choose a sequence (x∗n) ∞ n=1 ⊂ bx∗γ ∩ c00 such that infn |x ∗∗ n (xn)| > ε. since limn x∗∗n (e∗i ) = 0 for all i ∈ n, we may, by passing to a subsequence and replacing the functionals x∗n by tail projections thereof, assume that (x ∗ n) ∞ n=1 is a block sequence in bx∗γ ∩c00. then by standard properties of ordinals, if δ is such that γ +δ = ξ, δ = ξ. by proposition 2.16(i), we may pass to a subsequence once more and asssume (x∗n) ∞ n=1 is a c ξ 0-spreading model in x ∗ ξ , and therefore weakly null in x∗ξ . by passing to a subsequence one final time and appealing to proposition 2.20, assume (x∗∗n ) ∞ n=1 is an ` ξ 1-spreading model. therefore (x ∗∗ n ) ∞ n=1 is not ξ-weakly null. this contradiction finishes (i). (ii) also by contradiction. assume (x∗∗n ) ∞ n=1 ⊂ x ∗∗ ξ,p is ξ-weakly null, (x∗n) ∞ n=1 ⊂ x ∗ λ(ξ) is weakly null, and infn |x∗∗n (x∗n)| > ε > 0. by perturbing, we may assume (x∗n) ∞ n=1 is a block sequence and there exist i1 < i2 < ... such that inx ∗ n = x ∗ n for all n ∈ n. let (γk)∞k=1 ⊂ [0,λ(ξ)) be a sequence (possibly with repitition) such that [0,λ(ξ)) = {γk : k ∈ n}. by (i), limn‖x∗∗n ‖γk = 0 for all k ∈ n. by passing to a subsequence and relabeling, we may assume that for each 1 ≤ k ≤ n, ‖x∗∗n ‖γk < 1/n. let xn = inx ∗∗ n ∈ xξ and note that for each γ < λ(ξ), limn‖xn‖γ = 0. indeed, if γ = γk, then for all n ≥ k, ‖xn‖γ ≤‖x∗∗n ‖γk ≤ 1/n. since inx ∗ n = x ∗ n, |x∗n(xn)| = |x∗∗n (x∗n)| > ε. but by corollary 2.19, some subsequence of (xn) ∞ n=1, which we may assume is the entire sequence after relabeling, is wuc in xλ(ξ). but now we reach a contradiction by combining the facts that (xn) ∞ n=1 is wuc in xλ(ξ), (x ∗ n) ∞ n=1 ⊂ x ∗ λ(ξ) is weakly null, and infn |x∗n(xn)| > 0. 166 r.m. causey 3. ideals of interest basic definitions. we recall that ban is the class of all banach spaces and l denotes the class of all operators between banach spaces. for each pair x,y ∈ ban, l(x,y ) is the class of all operators from x into y . given a subclass i of l, we let i(x,y ) = i∩l(x,y ). we say that a class i of operators is an operator ideal (or just an ideal) provided that (i) for any w,x,y,z ∈ ban, c ∈ l(w,x), b ∈ i(x,y ), and a ∈ l(y,z), abc ∈ i(w,z), (ii) ik ∈ i, (iii) for each x,y ∈ ban, i(x,y ) is a vector subspace of l(x,y ). given an operator ideal i, we define the (i) closure i of i to be the class of operators such that for every x,y ∈ ban, i(x,y ) = i(x,y ), (ii) injective hull iinj of i to be the class of all operators a : x → y such that if there exists z ∈ ban and an isometric (equivalently, isomorphic) embedding j : y → z such that ja ∈ i(x,z), (iii) surjective hull isur of i to be the class of all operators a : x → y such that there exist w ∈ ban and a quotient map (equivalently, a surjection) q : w → x such that aq ∈ i(w,y ), (iv) dual idual to be the class of all operators a : x → y such that a∗ ∈ i(y ∗,x∗). we also let {i denote the class of operators such that for each pair x,y of banach spaces, {i(x,y ) = l(x,y ) \i(x,y ). each of i, iinj, isur is also an ideal. given two ideals i,j, we let (i) i ◦ j−1 denote the class of all operators a : x → y such that for all w ∈ ban and r ∈ j(w,x), ar ∈ i(w,y ), (ii) i−1 ◦ j denote the class of all operators a : x → y such that for all z ∈ ban and all l ∈ i(y,z), la ∈ j(x,z). we remark that for any three ideals i1,i2,j, (i−11 ◦j) ◦i −1 2 = i −1 1 ◦ (j◦i −1 2 ), so that the symbol i−11 ◦j◦i −1 2 is unambiguous. the ξ,ζ-dunford pettis property 167 we say an operator ideal is (i) closed if i = i, (ii) injective if i = iinj, (iii) surjective if i = isur, (iv) symmetric if i = idual. with each ideal, we will associate the class of banach spaces the identity of which lies in the given ideal. our ideals will be denoted by fraktur lettering (a,b,i, . . . ) and the associated space ideal will be denoted by the same sans serif letter (a,b, i, . . . ). we next list some ideals of interest. we let k,w, and v denote the class of compact, weakly compact, and completely continuous operators, respectively. for the remaining paragraphs in this subsection, ξ will be a fixed ordinal in [0,ω1]. we let wξ denote the class of operators a : x → y such that any bounded sequence in x has a subsequence whose image under a is ξweakly convergent in y (let us recall that a sequence (yn) ∞ n=1 ⊂ y is said to be ξ-weakly convergent to y ∈ y if (yn −y)∞n=1 is ξ-weakly null). note that w0 = k and wω1 = w. we refer to wξ as the class of ξ-weakly compact operators. this class was introduced in this generality in [6]. we let wbsξ denote the class of operators a : x → y such that for any weakly null sequence (xn) ∞ n=1, (axn) ∞ n=1 is ξ-weakly convergent to 0 in y . note that wbs0 = v, wbsω1 = l, and wbs1 is the class of weak banach-saks operators. for this reason, we refer to wbsξ as the class of ξ-weak banach-saks operators. these classes were introduced in this generality in [4]. we let vξ denote the class of operators a : x → y such that for any ξ-weakly null sequence (xn) ∞ n=1, (axn) ∞ n=1 is norm nul. it is evident that vω1 = v and v0 = l. these classes were introduced in this generality in [12]. for 0 ≤ ζ ≤ ω1, we let gξ,ζ denote the class of operators a : x → y such that whenever (xn) ∞ n=1 is ξ-weakly null, (axn) ∞ n=1 is ζ-weakly null. we isolate this class because it is a simultaneous generalization of the two previous paragraphs. indeed, vξ = gξ,0, while wbsξ = gω1,ξ. it is evident that gξ,ζ = l whenever ξ ≤ ζ. these classes are newly introduced here. for 0 ≤ ζ ≤ ω1, we let mξ,ζ denote the class of all operators a : x → y such that for any ξ-weakly null (xn) ∞ n=1 ⊂ x and any ζ-weakly null (y ∗ n) ∞ n=1 ⊂ y ∗, limn y ∗ n(axn) = 0. the class mω1,ω1 (sometimes denoted by dp) is a 168 r.m. causey previously defined class of significant interest, most notably because the associated space ideal mω1,ω1 is the class of banach spaces with the dunford-pettis property. as a class of operators, mξ,ζ has not previously been investigated, but the space ideals m1,ω1 and mω1,ξ have been investigated in [16] and [1], respectively. remark 3.1. let us recall that the image of a ξ-weakly null sequence under a continuous, linear operator is also ξ-weakly null, for any 0 ≤ ξ ≤ ζ ≤ ω1, any sequence which is ξ-weakly null is also ζ-weakly null, and the 0-weakly null sequences are the norm null sequences. from this we deduce the following: (i) gξ,ζ = l for any ξ ≤ ζ ≤ ω1. (ii) mξ,ζ = l if min{ξ,ζ} = 0. (iii) for ζ ≤ α ≤ ω1 and β ≤ ξ ≤ ω1, gξ,ζ ⊂ gβ,α. (iv) if α ≤ ζ ≤ ω1 and β ≤ ξ ≤ ω1, then mξ,ζ ⊂ mβ,α. we next record an easy consequence of corollary 2.12. corollary 3.2. for any 0 ≤ ζ,ξ ≤ ω1, gξ,ζ ⊂ ⋂ α<ω1 gα+ξ,α+ζ. proof. suppose x,y are banach spaces, a : x → y is an operator, α < ω1, and 0 ≤ ζ,ξ ≤ ω1 are such that a ∈ {gα+ξ,α+ζ. then there exists a sequence (xn) ∞ n=1 ⊂ x which is α + ξ-weakly null and such that (axn) ∞ n=1 is not α +ζ-weakly null. note that ζ < ω1, since otherwise α +ζ = α +ω1 = ω1, and (axn) ∞ n=1 would be a non-weakly null image of a weakly null sequence. if ξ = ω1, we deduce that a ∈ {gξ,ζ, since (xn)∞n=1 is a ξ-weakly null sequence the image of which under a is not α+ζ-weakly null, and therefore not ζ-weakly null. if ξ < ω1, we use corollary 2.12 to deduce the existence of some convex blocking (zn) ∞ n=1 of (xn) ∞ n=1 which is ξ-weakly null and the image of which under a is an ` ζ 1-spreading model. thus a ∈ {gξ,ζ. therefore {gα+ξ,α+ζ ⊂ {gξ,ζ. taking complements and noting that α < ω1 was arbitrary, we are done. remark 3.3. we remark that adding α on the left in the previous corollary is necessary. the analogous statement fails if we add α on the right. for example, for any 0 < ξ < ω1 and ζ < ω ξ, the formal identity i : xωξ → xζ lies in gωξ,0 ∩{gωξ+1,ζ. the ξ,ζ-dunford pettis property 169 examples. in this subsection, we provide examples to show the richness of the classes of interest, wbsξ, gξ,ζ, and mξ,ζ. we note that wbs0 = v, gξ,ζ = l whenever ξ ≤ ζ, and mξ,ζ = l whenever min{ξ,ζ} = 0. we typically omit reference to these trivial cases. proposition 3.4. fix 0 < ξ < ω1. then for any subset s of [0,ξ) with sup s = ξ, (⊕ζ∈sxζ)`1(s) ∈ wbsξ ∩ ⋃ ζ<ξ {wbsζ. proof. by theorem 2.14(v), if ζ < ξ, xζ ∈ wbsξ. we will prove in proposition 3.15 that the `1 direct sum of members of wbsξ also lies in wbsξ. theorem 3.5. for 0 ≤ ζ < ξ < ω1, the formal inclusion i : xξ → xζ lies in gξ,ζ ∩{gξ+1,ζ. proof. fix (xn) ∞ n=1 ⊂ xξ ξ-weakly null. then by theorem 2.17 (v), limn‖xn‖β = 0 for every β < λ(ξ). if ζ = 0, then ζ < λ(ξ) and limn‖xn‖ζ = 0. therefore (ixn) ∞ n=1 is ζ-weakly null. if ζ > 0, then since λ(ζ) ≤ λ(ξ), limn‖ixn‖β = 0 for every β < λ(ζ), and theorem 2.17(v) yields that (ixn)∞n=1 is ζ-weakly null in this case. in either case, (ixn) ∞ n=1 is ζ-weakly null, and i ∈ gξ,ζ. however, the canonical basis is ξ + 1-weakly null in xξ and not ζ-weakly null in xζ, so i ∈ {gξ+1,ζ. it is well-known and obvious that every schur space and every space whose dual is a schur space has the dunford-pettis property. the generalization of this fact to operators is v,vdual ⊂ dp. the ordinal analogues are also obvious: for any 0 < ξ ≤ ω1, vξ ⊂ mξ,ω1 and v dual ξ ⊂ mω1,ξ. thus it is of interest to come up with examples of members of mξ,ω1 , or more generally mξ,ζ, which do not come from vξ or v dual ζ . theorem 3.6. for 0 < ξ < ω1 and 1 < p ≤ ∞, the formal inclusion i : xξ,p → xλ(ξ) lies in mξ,ω1 ∩ {mξ+1,1 ∩ {vξ and the formal inclusion j : x∗ λ(ξ) → x∗ξ,p lies in mω1,ξ ∩{m1,ξ+1 ∩{v dual ξ . proof. it follows from lemma 2.21(ii) that i ∈ mξ,ω1 and j ∈ mω1,ξ. since the canonical basis of xξ,p ⊂ x∗∗ξ,p is ξ + 1-weakly null and the canonical basis of x∗ λ(ξ) is a c10-spreading model, and therefore 1-weakly null, i ∈ {mξ+1,1 and j ∈ {m1,ξ+1. now if (γk)∞k=1 ⊂ [0,λ(ξ)) is such that [0,λ(ξ)) = {γk : k ∈ n}, we may select f1 < f2 < ... , fi ∈ sλ(ξ), and positive scalars (ai)i∈∪∞n=1fn 170 r.m. causey such that for each 1 ≤ k ≤ n, ∑ i∈fn ai = 1 and ‖ ∑ i∈fn aiei‖γk < 1/n. then with xn = ∑ i∈fn aiei, theorem 2.17(v) yields that (xn) ∞ n=1 is ξ-weakly null in xξ,p ⊂ x∗∗ξ,p. evidently (xn) ∞ n=1 is normalized in xλ(ξ), so that i ∈ {vξ and j ∈ {vdualξ . corollary 3.7. for any 0 ≤ α,β,ζ,ξ ≤ ω1, gβ,α = gξ,ζ if and only if one of the two exclusive conditions holds: (i) ξ ≤ ζ and β ≤ α (in which case gβ,α = l = gξ,ζ). (ii) α = ζ < ξ = β. proof. it is obvious that (i) and (ii) are exclusive and either implies equality. now suppose that neither (i) nor (ii) holds. suppose ξ ≤ ζ and β > α. then ixα ∈ l∩{gβ,α = gξ,ζ∩{gβ,α, and gξ,ζ 6= gβ,α. similarly, gξ,ζ 6= gβ,α if β ≤ α and ζ < ξ. for the remainder of the proof, suppose that α < β and ζ < ξ. now suppose α < ζ. then ixα ∈ wbsα+1 ∩{wbsα ⊂ gξ,ζ ∩{gβ,α. similarly, gξ,ζ 6= gβ,α if ζ < α. next assume ζ = α < ξ < β. then if i : xξ → xζ is the formal inclusion, i ∈ gξ,ζ ∩ {gβ,α. if ζ = α < β < ξ, we argue similarly with the inclusion i : xβ → xα. since this is a complete list of the possible ways for (i) and (ii) to simultaneously fail, we are done. corollary 3.8. for any 0 ≤ α,β,ζ,ξ ≤ ω1, mβ,α ⊂ mξ,ζ if and only if one of the two exclusive conditions holds: (i) 0 = min{ζ,ξ} (in which case mβ,α = l = mξ,ζ). (ii) 0 < ζ ≤ α and 0 < ξ ≤ β. in particular, mβ,α = mξ,ζ if and only if min{β,α} = 0 = min{ξ,ζ} or 0 < α = ζ and 0 < β = ξ. proof. it is obvious that (i) and (ii) are exclusive, and either implies that mβ,α ⊂ mξ,ζ. now assume that min{ζ,ξ} > 0. if min{α,β} = 0, mβ,α = l 6⊂ mξ,ζ, since i`2 ∈ {m1,1 ⊂ {mξ,ζ. if 0 < α,β and β < ξ, then let i : xβ → xλ(β) be the formal inclusion. then i ∈ mβ,ω1 ∩{mβ+1,1 ⊂ mβ,α ∩{mξ,ζ. the ξ,ζ-dunford pettis property 171 now if 0 < α,β and α < ζ, let j : x∗ λ(α) → x∗α be the formal inclusion. then j ∈ mω1,α ∩{m1,α+1 ⊂ mβ,α ∩{mξ,ζ. the last statement follows from the fact that if mβ,α = mξ,ζ, then either both classes must equal l, which happens if and only if min{β,α} = 0 = min{ξ,ζ}, or neither class is l, in which case min{β,α}, min{ξ,ζ} > 0. in the latter case, using the previous paragraph and symmetry, α = ζ and β = ξ. general properties. we will need the following fact, shown in [12]. proposition 3.9. if x is a banach space and (xn) ∞ n=1 ⊂ x isξ-weakly null, then there exists a subsequence (xni) ∞ i=1 of (xn) ∞ n=1 such that the operator φ : `1 → x given by φ ∑∞ i=1 aiei = ∑∞ i=1 aixni lies in wξ(`1,x). remark 3.10. it follows that if y is a banach space and (y∗n) ∞ n=1 ⊂ y ∗ is ξ-weakly null, there exist a subsequence (y∗ni) ∞ i=1 of (y ∗ n) ∞ n=1 such that the operator given by ψ : y → c0 given by ψy = (y∗ni(y)) ∞ i=1 lies in w dual ξ (y,c0). this follows immediately from proposition 3.9, since ψ∗ : `1 → y ∗ is given by ψ∗ ∑∞ i=1 aiei = ∑∞ i=1 aiy ∗ ni . remark 3.11. in the following results, we will repeatedly use the previously stated fact that a weakly null ` ζ 1-spreading model can have no ζ-weakly convergent subsequence. theorem 3.12. fix 0 ≤ ζ < ξ ≤ ω1. then gξ,ζ = wζ ◦w−1ξ and g dual ξ,ζ = (w dual ξ ) −1 ◦wdualζ . consequently, gξ,ζ is a closed, two-sided ideal containing all compact operators. moreover, gξ,ζ is injective but not surjective. finally, gdual dualξ,ζ ( gξ,ζ, while neither of gξ,ζ, g dual ξ,ζ is contained in the other. proof. fix x,y ∈ ban and a ∈ l(x,y ). first suppose that a ∈ gξ,ζ(x,y ). fix a banach space w and r ∈ wξ(w,x). fix a bounded sequence (wn) ∞ n=1. by passing to a subsequence, we may assume there exists 172 r.m. causey x ∈ x such that (x − rwn)∞n=1 is ξ-weakly null, from which it follows that (ax−arwn)∞n=1 is ζ-weakly null. since this holds for an arbitrary bounded sequence in (wn) ∞ n=1, ar ∈ wζ. since w ∈ ban and r ∈ wξ(w,x) were arbitrary, a ∈ wζ ◦w−1ξ (x,y ). now suppose that a ∈ {gξ,ζ. then there exists a ξ-weakly null sequence (xn) ∞ n=1 in x such that (axn) ∞ n=1 is an ` ζ 1-spreading model. by proposition 3.9, after passing to a subsequence and relabeling, we may assume the operator r : `1 → x given by r ∑∞ i=1 aiei = ∑∞ i=1 aixi lies in wξ(`1,x). but since (arei) ∞ i=1 = (axi) ∞ i=1 has no ζ-weakly convergent subsequence, a ∈ {wζ ◦ w−1ξ (x,y ). next, suppose that a ∈ gdualξ,ζ (x,y ). fix z ∈ ban and an operator l ∈ wdualξ (y,z). then a ∗ ∈ gξ,ζ(y ∗,x∗) = wζ ◦ w−1ξ (y ∗,x∗) and l∗ ∈ wξ(z ∗,y ∗), and (la)∗ = a∗l∗ ∈ wζ(z∗,x∗). thus la ∈ wdualζ (x,z). since this holds for any z ∈ ban and l ∈ wdualξ (y,z), a ∈ (w dual ξ ) −1 ◦ wdualζ (x,y ). now if a ∈ {gdualξ,ζ (x,y ), there exists (y ∗ n) ∞ n=1 ⊂ y ∗ which is ξ-weakly null and (a∗y∗n) ∞ n=1 is an ` ζ 1-spreading model. by the remarks preceding the theorem, by passing to a subsequence and relabeling, we may assume the operator l : y → c0 given by ly = (y∗n(y))∞n=1 lies in w dual ξ (y,c0). but since (a∗l∗ei) ∞ i=1 = (a ∗y∗i ) ∞ i=1 is a weakly null ` ζ 1-spreading model, (la) ∗ = a∗l∗ ∈ {wζ(`1,x∗). thus la ∈ {((wdualξ ) −1 ◦wdualζ )(x,y ). this yields the first two equalities. it follows from the fact that wζ,wξ are closed, two-sided ideals containing the compact operators that gξ,ζ is also. it is evident that gξ,ζ is injective, since a given sequence is ζ-weakly null if and only if its image under some (equivalently, every) isomorphic image of that sequence is ζ-weakly null. the ideal gξ,ζ is not surjective, since xζ ∈ {gξ,ζ, while xζ is a quotient of `1 ∈ v ⊂ gξ,ζ. it is also easy to see that if a∗∗ ∈ gξ,ζ, then a ∈ gξ,ζ, so gdual dualξ,ζ ⊂ gξ,ζ. if ζ = 0, note that `1 ∈ v ⊂ gξ,ζ, but `∗∗1 contains an isomorphic copy of `2, so that ` ∗∗ 1 ∈ {gξ,0. this yields that g dual dual ξ,0 6= gξ,0. now if ζ > 0, c0 ∈ wbs1 ⊂ gξ,ζ. but `∞ = c∗∗0 ∈ {gξ,ζ. in order to see that `∞ ∈ {gξ,ζ, simply note that `∞ contains a sequence equivalent to the xζ basis, which is ξ-weakly null and not ζ-weakly null. finally, let us note that if ζ = 0, `1 ∈ v ⊂ gξ,ζ, while c0,`∞ ∈ {gξ,0. thus neither of gξ,0,g dual ξ,0 is contained in the other. now suppose that ζ > 0. then the ξ,ζ-dunford pettis property 173 since x∗ζ,2 ∈ wbs1 ⊂ gξ,ζ, xζ,2 ∈ gdualξ,ζ ∩{gξ,ζ and x ∗ ζ,2 ∈ gξ,ζ ∩{g dual ξ,ζ . here we recall that xζ,2 is reflexive. this yields that if 0 < ζ < ξ ≤ ω1, neither of gξ,ζ,g dual ξ,ζ is contained in the other. theorem 3.13. fix 0 < ζ,ξ ≤ ω1. then mξ,ζ = (w dual ζ ) −1 ◦vξ = (wdualζ ) −1 ◦k◦w−1ξ . consequently, mξ,ζ is a closed, two-sided ideal containing all compact operators. moreover, mξ,ζ is neither injective nor surjective. finally, mdualξ,ζ ( mζ,ξ and m dual dual ξ,ζ ( mξ,ζ. proof. it follows from the fact that vξ = k ◦ w−1ξ , which was shown in [12], that (wdualζ ) −1 ◦ vξ = (wdualζ ) −1 ◦ k ◦ w−1ξ . we will show that mξ,ζ = (w dual ζ ) −1 ◦ k ◦ w−1ξ . to that end, fix banach spaces x,y and a ∈ l(x,y ). suppose that a ∈ l(x,y ). fix banach spaces w,z and operators r ∈ wξ(w,x) and l ∈ wdualζ (y,z). we will show that lar ∈ k(w,z). seeking a contradiction, suppose lar ∈ {k. note that there exists a bounded sequence (wn) ∞ n=1 ⊂ w such that infm 6=n‖larwm −larwn‖≥ 4. by passing to a subsequence, we may assume there exist x ∈ x such that (x−rwn)∞n=1 is ξ-weakly null. since ‖larwm −larwn‖≥ 4 for all m 6= n, there is at most one n ∈ n such that ‖lax−larwn‖ < 2. by passing to a subsequence, we may assume ‖lax−larwn‖≥ 2 for all n ∈ n. for each n ∈ n, fix z∗n ∈ bz∗ such that |z∗n(lax − larwn)| ≥ 2. by passing to a subsequence one final time, we may assume there exists y∗ ∈ y ∗ such that (y∗−l∗z∗n)∞n=1 is ζ-weakly null and, since (ax−arwn)∞n=1 is weakly null, |y ∗(ax−arwn)| < 1 for all n ∈ n. then (y∗−l∗z∗n)∞n=1 ⊂ y ∗ is ζ-weakly null, (x−rwn)∞n=1 is ξ-weakly null, and inf n |(y∗ −l∗z∗n)(ax−arwn)| ≥ inf n |l∗z∗n(ax−arwn)|− 1 = inf n |z∗n(lax−larwn)|− 1 ≥ 1. this contradiction yields that mξ,ζ ⊂ (wdualζ ) −1 ◦k◦w−1ξ . 174 r.m. causey now suppose that a ∈ {mξ,ζ(x,y ). then there exist a ξ-weakly null sequence (xn) ∞ n=1 ⊂ x and a ζ-weakly null sequence (y ∗ n) ∞ n=1 ⊂ y ∗ such that infn |y∗n(axn)| = 1. using proposition 3.9 and the remark following it, after passing to subsequences twice and relabling, we may assume the operators r : `1 → x given by r ∑∞ i=1 aiei = ∑∞ i=1 aixi and l : y → c0 given by ly = (y∗n(y)) ∞ n=1 lie in wξ(`1,x) and w dual ζ (y,c0), respectively. but lar : `1 → c0 is not compact, since |e∗n(laren)| = |y ∗ n(axn)| ≥ 1 for all n ∈ n. this yields that mξ,ζ = (wdualζ ) −1 ◦k◦w−1ξ . since `2 ∈ {m1,1 ⊂ {mξ,ζ is a subspace of `∞ ∈ mω1,ω1 ⊂ mξ,ζ and a quotient of `1 ∈ mω1,ω1 ⊂ mξ,ζ, mξ,ζ is neither injective nor surjective. now suppose a ∈ mdualξ,ζ (x,y ). now if (xn) ∞ n=1 ⊂ x is ζ-weakly null, (y∗n) ∞ n=1 is ξ-weakly null, and j : x → x ∗∗ is the canonical embedding, then (jxn) ∞ n=1 ⊂ x ∗∗ is ζ-weakly null. since a ∈ mdualξ,ζ (x,y ), lim n y∗n(axn) = lim n a∗y∗n(xn) = lim n jxn(a ∗y∗n) = 0. thus a ∈ mζ,ξ(x,y ). this yields that mdualξ,ζ ⊂ mζ,ξ. to see that m dual ξ,ζ 6= mζ,ξ, we cite stegall’s example [22], x = `1(` n 2 ). this space has the schur property, and therefore lies in mω1,ω1 ⊂ mζ,ξ, while x∗ contains a complemented copy of `2. the fact that x ∗ contains a complemented copy of `2 is stated explicitly in [8]. thus x ∈ {mdual1,1 ⊂ {m dual ξ,ζ . next, we note that mdual dualξ,ζ = (m dual ξ,ζ ) dual ⊂ mdualζ,ξ ⊂ mξ,ζ. to see that mdual dualξ,ζ 6= mξ,ζ, we make yet another appeal to stegall’s example and let y = c0(` n 2 ). then y ∗ = x has the schur property, and therefore y ∈ mω1,ω1 ⊂ mξ,ζ. but y ∗∗ = x∗ ∈ {m1,1 ⊂ {mξ,ζ. therefore y ∈ {mdual dualξ,ζ . direct sums. for 1 ≤ p ≤∞ and classes i,j, we say j is closed under i-`p sums provided that for any set i and any collection (ai : xi → yi)i∈i ⊂ i such that supi∈i ‖ai‖ < ∞, the operator a : (⊕i∈ixi)`p(i) → (⊕i∈iyi)`p(i) lies in j. the notion of an ideal being closed under i-c0 sums is defined similarly. we will use the following well-known fact about weakly null sequences in `1 sums of banach spaces. the ξ,ζ-dunford pettis property 175 fact 3.14. let i be a set, (xi)i∈i a collection of banach spaces, and (xn) ∞ n=1 = ( (xi,n)i∈i )∞ n=1 a weakly null sequence in (⊕i∈ixi)`1(i). then for any ε > 0, there exists a subset j ⊂ i such that |i\j| < ∞ and for all n ∈ n,∑ i∈j ‖xi,n‖ < ε. proposition 3.15. fix 0 ≤ ζ < ξ ≤ ω1. (i) the class gξ,ζ is closed under gξ,ζ-`1 sums. (ii) the class gξ,ζ is closed under gξ,ζ-`p sums for 1 < p < ∞ if and only if ζ > 0. (iii) the class gξ,ζ+1 is closed under gξ,ζ-c0 sums. (iv) the class gξ,ζ is not closed under gξ,ζ-c0 sums. (v) the class gξ,ζ is not closed under v-`∞ sums. proof. throughout, let i be a set, (ai : xi → yi)i∈i a collection of operators such that supi∈i ‖ai‖ = 1. let xp = (⊕i∈ixi)`p(i), yp = (⊕i∈iyi)`p(i), and ap : xp → yp the operator such that ap|xi = ai. as usual, p = 0 will correspond to the c0 direct sum. (i) assume ai ∈ gξ,ζ for all i ∈ i. fix (xn)∞n=1 ⊂ x1 ξ-weakly null. write xn = (xi,n)i∈i and note that for each i ∈ i, (xi,n)∞n=1 is ξ-weakly null, so (aixi,n) ∞ i=1 is ζ-weakly null. fix ε > 0 and m ∈ [n]. using fact 3.14, there exists a subset j of i such that |i\j| < ∞ and supn ∑ i∈j ‖xi,n‖ < ε/2. since (aixi,n) ∞ n=1 is ζ-weakly null, then there exists f ∈ sζ ∩ [m] 0 and m ∈ [n] were arbitrary, (a1xn)∞n=1 is ζ-weakly null. (ii) fix 1 < p < ∞. since `p ∈ {gξ,0 and k ∈ gξ,0, gξ,0 is not closed under `p sums. it follows by an inessential modification of work from [3] that for 0 < ζ < ω1, gξ,ζ is closed under gξ,ζ-`p sums. more specifically, let 176 r.m. causey (xn) ∞ n=1 ⊂ bxp be ξ-weakly null and let vn = (‖xi,n‖xi)i∈i ∈ b`p(i). assume (apxn) ∞ n=1 satisfies 0 < ε ≤ inf { ‖apx‖ : f ∈sζ,x ∈ co(xn : n ∈ f) } . by passing to a subsequence, we may assume vn → v = (vi)i∈i ∈ b`p(i) weakly, and that vn is a small perturbation of v + bn, where the sequence (bn) ∞ n=1 consists of disjointly supported vectors in bxp. we may fix a subset j of i such that |i\j| < ∞ and (∑ i∈j v p i ) 1/p < ε/3. for k ∈ n, we may first choose m = (mi) ∞ i=1 ∈ [n] such that sζ[ak](m) ⊂sζ and let un = 1 k (n+1)k∑ j=nk+1 xmj. if k was chosen sufficiently large, then sup n (∑ i∈j ‖ui,n‖ p xi )1/p < ε/2. by our choice of m, (apun) ∞ n=1 also satisfies ε ≤ inf { ‖aun‖ : f ∈sζ,x ∈ co(xn : n ∈ f) } . since (aixi,n) ∞ n=1 is ζ-weakly null, there exist f ∈ sζ and positive scalars (an)n∈f summing to 1 such that∥∥∥∥ ∑ n∈f anaixi,n ∥∥∥∥ yi < ε/2 1 + |i \j| for each i ∈ i \j. we reach a contradiction as in (i). (iii) fix (xn) ∞ n=1 = ((xi,n)i∈i) ∞ n=1 ⊂ bx0 ξ-weakly null. fix (εn) ∞ n=1 such that ∑∞ n=1 εn < 1. since for each i ∈ i, (aixi,n) ∞ n=1 is ζ-weakly null, we may recursively select f1 < f2 < ... , fn ∈ sζ, positive scalars (aj)j∈∪∞n=1fn, and finite subsets ∅ = i0 ⊂ i2 ⊂ . . . of i such that for each n ∈ n, ∑ j∈fn aj = 1 , max i∈in−1 ∥∥∥∥ai ∑ j∈fn ajxi,j ∥∥∥∥ < εn and max i∈i\in ∥∥∥∥ ∑ j∈fn ajxi,j ∥∥∥∥ < εn. the ξ,ζ-dunford pettis property 177 then since for each n ∈ n, ∪2nm=n+1fm ∈sζ+1 for each n ∈ n, we deduce that sup i∈i ∥∥∥∥a0 1n 2n∑ m=n+1 ∑ j∈fm ajxi,j ∥∥∥∥ ≤ max { max i∈i\i2n 2n∑ m=n+1 ∥∥∥∥ ∑ j∈fn ajxi,j ∥∥∥∥, max n 0. by passing to a subsequence, we may assume (yn) ∞ n=1 is a c ζ 0-spreading model. by proposition 2.20 applied with f = sζ if ζ < ω1 and f = [n] 2ε. by passing to a subsequence and relabeling, we may assume that for all m < n, |x∗n(xm)| < ε. since `1 6↪→ x, we may also assume that (xn)∞n=1 is weakly cauchy. then with y∗n = x ∗ 2n and yn = x2n − x2n−1, (y ∗ n) ∞ n=1 is γ-weakly null, (yn) ∞ n=1 is weakly null, and infn |y∗n(yn)| ≥ ε. distinctness of space ideals. we recall that, given an operator ideal i, the associated space ideal i consists of all banach spaces x such that ix ∈ i. we showed in section 3 that for any 0 ≤ ζ < ξ ≤ ω1 and 0 ≤ α < β ≤ ω1, gξ,ζ = gβ,α if and only if ζ = α and ξ = ζ. our next goal is to show that this is not true for the space ideals, due to the idempotence of identity operators. we recall the result from [12] that a banach space x lies in vζ for some ω ξ < ζ < ωξ+1 if and only if x lies in vζ for every ω ξ < ζ < ωξ+1, which is a consequence of considering blocks of blocks. we prove analogous results below. we need the following result for blocks of blocks. proposition 4.6. let x,y,z be operators, α,β,ζ countable ordinals, and assume b ∈ gβ+ζ,ζ and a ∈ gα+ζ,ζ. then ab ∈ gα+β+ζ,ζ. proof. by corollary 3.1, b ∈ gα+β+ζ,α+ζ. thus if (xn)∞n=1 is α + β + ζweakly null, it is sent by b to a sequence which is α + ζ-weakly null, which is sent by a to a sequence which is ζ-weakly null. corollary 4.7. for a banach space x and ζ < ω1, let g ζ(x) = ω1 if x ∈ gω1,ζ, and otherwise let gζ(x) be the minimum ordinal ξ < ω1 such that x ∈ {g ξ+ζ,ζ (noting that such a ξ must exist). then there exists γ ≤ ω1 such that gζ(x) = ω γ. proof. note that gζ(x) > 0. fix α,β < gζ(x). then ix ∈ gβ+ζ,ζ and ix ∈ gα+ζ,ζ. by proposition 4.6, ix ∈ gα+β+ζ,ζ. thus we have shown that if α,β < gζ(x), α + β < gζ(x). since 0 < gζ(x) ≤ ω1, standard facts about ordinals yield that there exists γ ≤ ω1 such that gζ(x) = ωγ. for the following theorem, note that if ωξ < λ(ζ), then ωξ + ζ = ζ, so gωξ+ζ,ζ = l. this is the reason for the omission of this trivial case. 182 r.m. causey theorem 4.8. fix 0 ≤ ζ < ω1 and ξ < ω1 such that ωξ ≥ λ(ζ). then ∅ 6= {gωξ+ζ,ζ ∩ ⋂ η<ωξ gη+ζ,ζ. proof. it was shown in [12] that for any banach space y with a normalized, bimonotone basis and 0 < ξ < ω1, there exists a banach space z (there denoted by zξ(ey )) such that z has a normalized, bimonotone basis, y is a quotient of z, z ∈ ∩η<ωξvη, and if (yn)∞n=1 is an ω ξ-weakly null sequence in y , then there exists an ωξ-weakly null sequence (zn) ∞ n=1 in z such that qzn = yn for all n ∈ n. if ζ = 0, we consider z as above with y = c0. this space lies in {vωξ ∩ ⋂ η<ωξ vη = {gωξ,0 ∩ ⋂ η<ωξ gη,0. this completes the ζ = 0 case. for the remainder of the proof, we consider ζ > 0. suppose that ξ = 0. then since 1 = ωξ ≥ λ(ζ) ≥ 1, ζ is finite. futhermore, η + ζ = ζ for any η < λ(ζ), since the only such η is 0. then x = xζ is easily seen to satisfy the conclusions. for the remainder of the proof, we assume 0 < ξ < ω1. if λ(ζ) = ωξ, then for every η < ωξ, η +ζ = ζ. in this case, membership in⋂ η<ωξ gη+ζ,ζ = ban is automatic. in this case, xζ ∈ {gωξ+ζ,ζ is the example we seek. we consider the remaining case, 0 < ζ,ξ and λ(ζ) < ωξ. note that this implies that ζ < ωξ. we use a technique of ostrovskii from [19]. if λ(ζ) is finite, then it is equal to 1. in this case, let y = c0. if λ(ζ) is infinite, then it is a limit ordinal. in this case, let (λn) ∞ n=1 be the sequence used to define sλ(ζ). let y be the completion of c00 with respect to the norm ‖x‖ = sup n∈n 2−n‖x‖λn. note that the formal inclusions i1 : xζ → xλ(ζ), i2 : xλ(ζ) → y are bounded. the first is bounded by the almost monotone property. for n ∈ n and e ∈ sλn, f = e ∩ [n,∞) ∈sλ(ζ). therefore for x ∈ c00, 2−n‖ex‖`1 ≤ 2 −n((n− 1)‖x‖c0 + ‖fx‖`1) ≤ n2−n‖x‖λ(ζ) ≤ 2−1‖x‖λ(ζ). let us also note that a bounded block sequence (xn) ∞ n=1 in xζ is ζ-weakly null if and only if limn‖x‖β = 0 for every β < λ(ζ) if and only if (i2i1xn)∞n=1 is the ξ,ζ-dunford pettis property 183 norm null in y . we have already established the equivalence of the first two properties. let us explain the equivalence of the last two properties. first, if (i2i1xn) ∞ n=1 is norm null in y , then for any β < λ(ζ), we can fix k such that β < λk and note that lim n ‖xn‖β ≤ c lim n ‖xn‖λk ≤ c2 k lim n ‖xn‖y = 0. here, c is the norm of the formal inclusion of xλk into xβ. for the reverse direction, suppose (xn) ∞ n=1 ⊂ bxζ and limn‖xn‖β = 0 for all β < λ(ζ). then lim sup n ‖i2i1y‖ ≤ inf { max { lim sup n k∑ m=1 ‖xn‖λm, sup n>k 2−n‖i2‖‖i1‖ } : k ∈ n } = 0. let i = i2i1 and let z be as described in the first paragraph with this choice of y . let q : z → y be the quotient map the existence of which was indicated above. let w = z ⊕1 xζ and let t : w → y be given by t(z,x) = ix− qz. let x = ker(t). since we are in the case ζ < ωξ, standard properties of ordinals yield that for η < ωξ, η + ζ < ωξ. suppose that (zn,xn) ∞ n=1 ⊂ x is η + ζ-weakly null. then since z ∈ vη, ‖zn‖ → 0. from this it follows that (ixn) ∞ n=1 = (qzn) ∞ n=1 is norm null. therefore (ixn) ∞ n=1 is norm null, and (xn) ∞ n=1 is ζ-weakly null in xζ. therefore (zn,xn) ∞ n=1 is ζ-weakly null in x. we last show that x ∈ {gωξ+ζ,ζ. to that end, let us first note that the basis of y is λ(ζ)-weakly null. this is obvious if λ(ζ) = 1 and y = c0. for the case in which λ(ζ) is infinite, the space y is a mixed schreier space as defined in [12], where it was shown that the basis of y is λ(ζ)-weakly null. by the properties of z and q, since λ(ζ) ≤ ωξ, there exists an ωξ-weakly null sequence (zn) ∞ n=1 in z such that qzn = en, where (en) ∞ n=1 simultaneously denotes the bases of y and xζ. also note that (en) ∞ n=1 is ζ + 1-weakly null in xζ. since ωξ + ζ ≥ ζ + ωξ ≥ ζ + 1, (en) ∞ n=1 is ω ξ + ζ-weakly null in xζ. therefore (zn,en) ∞ n=1 is ω ξ + ζ-weakly null in x. however, since (en) ∞ n=1 is not ζ-weakly null in xζ, (zn,en) ∞ n=1 is not ζ-weakly null in x. therefore x ∈ {gωξ+ζ,ζ. corollary 4.9. the classes wbsζ,gωγ+ζ,ζ, ζ,γ < ω1,ω γ ≥ λ(ζ), are distinct. 184 r.m. causey theorem 4.10. the classes gζ+ωγ,ζ, 0 ≤ ζ < ω1, 0 ≤ γ ≤ ω1, are distinct. proof. we first recall that if ζ < ω1 and γ ≤ γ1 ≤ ω1, gζ+ωγ1,ζ ⊂ gζ+ωγ,ζ. thus the statement that these two classes are distinct is equivalent to saying that the former is a proper subset of the latter. we will show that the classes are distinct. fix 0 ≤ ζ,ζ1 < ω1 and 0 ≤ γ,γ1 ≤ ω1. if ζ < ζ1, xζ ∈ wbsζ1 ∩{gζ+ωγ,ζ ⊂ gζ1+ωγ1,ζ1 ∩{gζ+ωγ,ζ. by symmetry, if ζ1 < ζ, gζ+ωγ,ζ 6= gζ1+ωγ1,ζ1 . thus if ζ 6= ζ1, gζ+ωγ,ζ 6= gζ1+ωγ1,ζ1 . in order to complete the proof that the classes are distinct, it suffices to assume that γ1 < γ ≤ ω1 and exhibit some banach space z ∈ gζ+ωγ1,ζ ∩ {gζ+ωγ,ζ. we first claim that it is sufficient to prove the case γ < ω1. this is because if we prove that gζ+ωγ,ζ ( gζ+ωγ1,ζ whenever 0 ≤ γ1 < γ < ω1, then for any 0 ≤ γ1 < ω1, gζ+ωω1,ζ = gω1,ζ ⊂ gζ+ωγ1+1,ζ ( gζ+ωγ1,ζ. fix 0 < γ < ω1 and let (γn) ∞ n=1 be the sequence defining sωγ . fix a sequence (ϑn) ∞ n=1 such that ϑ := ∑∞ n=1 ϑn < 1. given a banach space e with normalized, 1-unconditional basis, we define norm on [·] on c00 by letting | · |0 = ‖ ·‖e, |x|k+1,n = sup { ϑn d∑ i=1 |eix|k : n ∈ n,e1 < · · · < ed, (min ei)di=1 ∈sγn } , |x|k+1 = max { |x|k, ( ∞∑ n=1 |x|2k+1,n )1/2} , [x] = lim k |x|k and [x]n = lim k |x|k,n. let us denote the completion of c00 with respect to this norm by zγ(e). the norm [·] on zγ(e) satisfies the following [z] = max { ‖z‖e, ( ∞∑ n=1 [z]2n )1/2} . the ξ,ζ-dunford pettis property 185 this construction is a generalization of a odell-schlumprecht construction. we will apply the construction with e = xζ. it is a well known fact of such constructions that, since the basis of xζ is shrinking, so is the basis of zγ(xζ) (see, for example, [12]). it was shown in [12] that if (zn) ∞ n=1 is any seminormalized block sequence in zγ(xζ), then (a) (zn) ∞ n=1 is not β-weakly null for any β < ω γ, (b) (zn) ∞ n=1 is ω γ-weakly null in zγ(xζ) if and only if it is ω γ-weakly null in xζ. we will show that zγ(xζ) ∈ ∩β<ωγgζ+β,ζ, and in particular zγ(xζ) ∈ gζ+ωγ1,ζ, while zγ(xζ) ∈ {gζ+ωγ,ζ. this will complete the proof of the distinctness of the classes. we prove that zγ(xζ) ∈ {gζ+ωγ,ζ. as remarked above, the basis is shrinking and normalized, and so it is weakly null. if it were not ζ +ωγ-weakly null, there would exist some (mn) ∞ n=1 ∈ [n] and ε > 0 such that ε ≤ inf { [z] : f ∈sωγ [sζ],z ∈ co(emn : n ∈ f) } . but by theorem 2.17, we may choose f1 < f2 < ... , fi ∈ sζ, and positive scalars (ai)i∈∪∞n=1fn such that ∑ i∈fn ai = 1 and the sequence (zn) ∞ n=1 defined by zn = ∑ i∈fn aiemi is equivalent to the c0 basis in xζ. but since ε ≤ inf { [z] : f ∈sωγ [sζ],z ∈ co(emn : n ∈ f) } , (zn) ∞ n=1 is an ` ωγ 1 -spreading model in zγ(xζ), contradicting item (b) above. therefore the canonical zγ(xζ) basis is ζ +ω γ-weakly null. but it is evidently not ζ-weakly null, and zγ(xζ) ∈ {gζ+ωγ,ζ. now let us show that zγ(xζ) ∈ ∩β<ωγgζ+β,ζ. first consider the case λ(ζ) < ωγ, which is equivalent to ζ + β < ωγ for all β < ωγ. in this case,{ ζ + β : β < ωγ } = [0,ωγ). it therefore follows from property (a) above that zγ(xζ) ∈ ⋂ β<ωγ gβ,0 = ⋂ β<ωγ gζ+β,0 ⊂ ⋂ β<ωγ gζ+β,ζ. let us now treat the case λ(ζ) ≥ ωγ. write ζ = λ(ζ) + µ 186 r.m. causey and note that µ + ωγ ≤ µ + λ(ζ) ≤ λ(ζ) + µ = ζ. we claim that if (zn) ∞ n=1 is a seminormalized block sequence in zγ(xζ) which is not ζ-weakly null in zγ(xζ), then there exists β < λ(ζ) such that lim sup n ‖zn‖β > 0. to see this, suppose that for every β < λ(ζ), limn‖zn‖β = 0, but (zn)∞n=1 is not ζ-weakly null in zγ(xζ). then, by proposition 2.16(ii), by passing to a subsequence and relabeling, we may assume (zn) ∞ n=1, when treated as a sequence in xζ, is dominated by a subsequence (emi) ∞ i=1 of the xµ basis, and (zn) ∞ n=1, when treated as a sequence in zγ(xζ), is an ` ζ 1-spreading model. since ζ ≥ µ + ωγ, we may, after passing to a subsequence again, assume 0 < ε ≤ inf { [z] : f ∈sωγ [sµ],z ∈ co(zn : n ∈ f) } . we may select f1 < f2 < ... , fi ∈ sµ, and positive scalars (ai)i∈∪∞n=1fn such that ∑ i∈fn ai = 1 and ( ∑ i∈fn aiemi) ∞ n=1 ⊂ xµ is equivalent to the canonical c0 basis (again using theorem 2.17 as in the previous case). since (zn) ∞ n=1 ⊂ xζ is dominated by (emi) ∞ i=1 ⊂ xµ, ( ∑ i∈fn aizi) ∞ n=1 is wuc in xζ. but since 0 < ε ≤ inf { [z] : f ∈sωγ [sµ],z ∈ co(zn : n ∈ f) } , ( ∑ i∈fn aizi) ∞ n=1 must be an ` ωγ 1 -spreading model in zγ(xζ), contradicting (b) above. this proves the claim from the beginning of the paragraph. now suppose that (zn) ∞ n=1 is a weakly null sequence in zγ(xζ) which is not ζweakly null. then by the claim combined with corollary 2.19, (zn) ∞ n=1 is not ζ-weakly null in xζ. after passing to a subsequence, we may assume (zn) ∞ n=1 is an ` ζ 1-spreading model in xζ. assume that 0 < ε ≤ inf { [z] : f ∈sζ,z ∈ co(zn : n ∈ f) } . now fix n ∈ n and f ∈sγn[sζ] and scalars (ai)i∈f . by definition of sγn[sζ], there exist f1 < · · · < fd such that f = ∪dj=1fj, ∅ 6= fj ∈ sζ, and (min fj) d j=1 ∈sγn. let ei = supp(zi) and let ij = ∪i∈fjei. since min ii = min supp(zmin fj ) ≥ min fj, the ξ,ζ-dunford pettis property 187 (min ij) d j=1 is a spread of (min fj) d j=1, so that (min ij) d j=1 ∈sγn. therefore [∑ i∈f aizi ] ≥ ( ∞∑ k=1 [∑ i∈f aizi ]2 k )1/2 ≥ [∑ i∈f aizi ] n ≥ ϑn d∑ j=1 [ ij ∑ i∈f aizi ] = ϑn d∑ j=1 [ ∑ i∈fj aizi ] ≥ εϑn d∑ j=1 ∑ i∈fj |ai| = εϑn ∑ i∈f |ai| . thus 0 < inf { [z] : f ∈sγn[sζ],x ∈ co(zn : n ∈ f) } . from this it follows that (zi) ∞ i=1 is not ζ + γn-weakly null. since this holds for any n ∈ n and supn γn = ωγ, (zi)∞i=1 is not ζ + β-weakly null for any β < ω γ. thus by contraposition, for any β < ωγ, any ζ + β-weakly null sequence in zγ(xζ) is ζ-weakly null, from which it follows that zγ(xζ) ∈ ∩β<ωγgζ+β,ζ. this completes the proof of the distinctness of these classes. remark 4.11. for ξ,η < ω1 and δ,ζ ≤ ω1 with η 6= ζ, the classes gωξ+ζ,ζ, gη+ωδ,η are not equal. indeed, if η < ζ, xη ∈ gωξ+ζ,ζ ∩ {gη+ωδ,η. this is because every sequence in xη is η + 1-weakly null, and therefore ζ-weakly null. however, the basis of xη is η + 1-weakly null, and therefore η + ω δweakly null, but not η-weakly null. now if ζ < η, either ωξ + ζ > ζ or ωξ + ζ = ζ. if ωξ + ζ > ζ, xζ ∈ gη+ωδ,η ∩ {gωξ+ζ,ζ. if ωξ + ζ = ζ, then gωξ+ζ,ζ = ban 6= gη+ωδ,η. we next wish to discuss how the classes gωξ+ζ,ζ can be compared to the classes gζ+ωδ,ζ. in particular, we will show that they are equal if and only if ωξ + ζ = ζ + ωδ. if ζ = 0, then gωξ+ζ,ζ = vωξ and gζ+ωδ,ζ = vωδ . then vmax{ωξ,ωδ} ⊂ vmin{ωξ,ωδ}, with proper containment if and only if ξ 6= δ. now for 0 < ζ < ω1, write ζ = ω α1n1 + · · · + ωαlnl l,n1, . . . ,nl ∈ n, α1 > · · · > αl. let us consider several cases. for convenience, let α = α1 and n = n1. case 1: ξ < α. then ωξ + ζ = ζ and gωξ+ζ,ζ = ban 6= gζ+ωδ,δ. for the remaining cases, we will assume ξ ≥ α, which implies that ωξ + ζ > ζ. case 2: ωξ +ζ < ζ+ωδ. then there exists β < ωδ such that ωξ +ζ = ζ+β. then the space zδ(xζ) from theorem 4.10 lies in {gζ+ωδ,ζ ∩gζ+β,ζ = {gζ+δ,ζ ∩gωξ+ζ,ζ. 188 r.m. causey case 3: ωξ + ζ = ζ + ωδ. in this case, of course gωξ+ζ,ζ = gζ+ωδ,ζ. by considering the cantor normal forms of ωξ + ζ and ζ + ωδ, it follows that equality can only hold in the case that ξ = δ = α and ζ = ωαn, in which case ωξ + ζ = ωα(n + 1) = ζ + ωδ. for the remaining cases, we will assume ωξ + ζ > ζ + ωδ. note that this implies δ ≤ ξ. indeed, if δ > ξ, then since we are in the case ξ ≥ α, it follows that ωδ > ωξ,ζ. by standard properties of ordinals, ωξ > ωξ + ζ. therefore for the remaining cases, ωξ + ζ > ζ + ωδ and α,δ ≤ ξ. case 4: δ = ξ > α. then the space xωδ lies in {gωξ+ζ,ζ ∩gζ+ωδ,ζ. to see this, note that since δ > α, ζ+ωδ = ωδ. moreover, we have already shown that any ωδ-weakly null sequence in xωδ has the property that every subsequence has a further wuc subsequence. thus any ωδ-weakly null sequence in xωδ is 1-weakly null, and xωδ ∈ gζ+ωδ,ζ. but of course the basis of xωδ shows that it does not lie in gωξ+ζ,ζ ⊂ gωδ+1,ωδ . case 5: ξ = α > δ. the space zξ(xζ), as defined in theorem 4.10, lies in {gωξ+ζ,ζ ∩gζ+ωδ,ζ. to see this, let us note that zξ(xζ) ∈ {gζ+ωξ,ζ ∩ ⋂ γ<ωξ gζ+γ,ζ. since ξ ≥ α, ωξ +ζ ≥ ζ +ωξ, and gωξ+ζ,ζ ⊂ gζ+ωξ,ζ and zξ(xζ) ∈ {gζ+ωξ,ζ ⊂ {gωξ+ζ,ζ.since ω δ < ωξ, zξ(xζ) ∈ gζ+ωδ,ζ. case 6: ξ > α,δ. then the space zξ(c0), as shown in [12], lies in wbsωξ ∩⋂ γ<ωξ vγ. furthermore, the basis of the space is normalized, weakly null. therefore the basis is ωξ-weakly null but not γ-weakly null for any γ < ωξ. therefore zξ(c0) ∈ {gωξ,ζ ⊂ {gωξ+ζ,ζ. however, since α,δ < ξ, ζ + ωδ < ξ, and zξ(c0) ∈ vζ+ωδ ⊂ gζ+ωδ,ζ. therefore zξ(c0) lies in {gωξ+ζ,ζ ∩gζ+ωδ,ζ. case 7: ξ = α = δ. in this case, we can write ζ = ωαn + µ, where µ = ωα2n2 + · · · + ωαlnl. note that in this case, µ > 0, since otherwise we would be in the case ωξ + ζ = ζ + ωδ. then the space xωα(n+1) lies in {gωα+ζ,ζ ∩gζ+ωα,ζ. to see this, note that the canonical basis of xωα(n+1) is ωα(n + 1) + 1 ≤ ωα(n + 1) + µ = ωα + ζ weakly null, but it is not ωα(n+ 1) = ωαn+ωα-weakly null, and therefore not ζ-weakly null. thus xωα(n+1) ∈ {gωα+ζ,ζ. however, if (xn)∞n=1 is ω α(n + 1)weakly null, then by theorem 2.17, every subsequence of (xn) ∞ n=1 has a further subsequence which is dominated by a subsequence of the xωαn basis. this means (xn) ∞ n=1 is ω αn + 1-weakly null. since ωαn + 1 ≤ ωαn + µ = ζ and ζ + ωα = ωα(n + 1), xωα(n+1) ∈ gζ+ωα,ζ. the ξ,ζ-dunford pettis property 189 our next goal will be to prove a fact regarding the distinctness of the space ideals mξ,ζ analogous to those proved above for the classes gξ,ζ. remark 4.12. if ξ,η are ordinals such that ωξ + 1 < η < ωξ+1, then there exist ordinals α,γ < η such that γ > 1 and α + γ = η. this is obvious if ξ = 0, since since η > 2 is finite and we may take η = 1 + (η− 1) in this case. assume 0 < ξ. then there exist n ∈ n and δ < ωξ such that η = ωξn + δ. if n > 1, we may take α = ωξ(n−1) and γ = ωξ. now if n = 1, then δ > 1, and we may take α = ωξ and γ = δ. theorem 4.13. fix 0 ≤ ξ < ω1 and 0 < ν ≤ ω1. let x be a banach space. (i) x is hereditarily mµ,ν for some ω ξ < µ < ωξ+1 if and only if x is hereditarily mµ,ν for every ω ξ < µ < ωξ+1. (ii) x is hereditarily mν,µ for some ω ξ < µ < ωξ+1 if and only if x is hereditarily mν,µ for every ω ξ < µ < ωξ+1. proof. (i) seeking a contradiction, suppose that x is hereditarily mµ,ν for some but not all µ ∈ (ωξ,ωξ+1). let η be the minimum ordinal µ such that x is not hereditarily mµ,ν. note that, since the classes mµ,ν are decreasing with µ and x is hereditarily mµ,ν for some ω ξ < µ < ωξ+1, it follows that ωξ + 1 < η. we can write η = α + γ for some α,γ < η with γ > 1. since x is not hereditarily mη,ν, there exists a seminormalized, η-weakly null sequence (xn) ∞ n=1 in x which has no subsequence which is a c ν 0 -spreading model. since α + 1 < α + γ, the minimality of η implies that x is hereditarily mα+1,ν, which means (xn) ∞ n=1 has a subsequence which is an ` α+1 1 -spreading model. by corollary 2.12(i), there exists a convex block sequence (yn) ∞ n=1 of (xn) ∞ n=1 which is an `11-spreading model and which is γ-weakly null. but since (yn) ∞ n=1 is an `11-spreading model, it can have no subsequence which is a c ν 0 -spreading model. since γ < η, (yn) ∞ n=1 witnesses that x is not hereditarily mγ,ν, contradicting the minimality of η. (ii) arguing as in (i), let us suppose we have ωξ + 1 < η < ωξ+1 such that x is hereditarily mν,µ for every µ < η but x is not hereditarily mν,η. then there exists a ν-weakly null (xn) ∞ n=1 ⊂ x which has no subsequence which is a c η 0-spreading model. write η = α + γ, α,γ < η, γ > 1. by passing to a subsequence, we may assume (xn) ∞ n=1 is a c α+1 0 -spreading model. by corollary 2.12(ii), there exists a blocking (yn) ∞ n=1 of (xn) ∞ n=1 which is a c 1 0spreading model and has no subsequence which is a c γ 0 -spreading model. since 190 r.m. causey (yn) ∞ n=1 is a c 1 0-spreading model, it is 1-weakly null, and therefore ν-weakly null. but (yn) ∞ n=1 has no subsequence which is a c γ 0 -spreading model. since γ < η, this contradicts the minimality of η. remark 4.14. the previous theorem yields that for a fixed 0 < ζ ≤ ω1 and 0 ≤ ξ < ω1, a given banach space x may lie in {mωξ,ζ ∩ ⋂ η<ωξ mη,ζ. that is, the first ordinal η for which x fails to lie in mη,ζ is of the form ω ξ, 0 ≤ ξ < ω1. but it also allows for x to lie in mωξ,ζ and fail to lie in mωξ+1,ζ. let us make this precise: for 1 ≤ ζ ≤ ω1, let mζ(x) = ω1 if x ∈ mω1,ζ and otherwise let mζ(x) be the minimum η such that x ∈ {mη,ζ. let m∗ζ(x) = ω1 if x ∈ mζ,ω1 , and otherwise let m∗ζ(x) be the minimum η such that x ∈ {mζ,η. then the preceding theorem yields that for any 1 ≤ ζ ≤ ω1 and any banach space x, there exists 0 ≤ ξ ≤ ω1 such that either mζ(x) = ωξ or mζ(x) = ωξ + 1, and a similar statement holds for m∗ζ. contrary to the gξ,ζ case, both alternatives can occur for both mζ and m ∗ ζ. for example, for 0 < ξ < ω1, our spaces zξ(c0) lie in ⋂ η<ωξ vη, and therefore lie in ⋂ η<ωξ mη,ω1 ⊂ ⋂ ζ≤ω1 ⋂ η<ωξ mη,ζ. however, the basis of this space is ωξ-weakly null, and the dual basis is 1weakly null, so zξ(c0) ∈ {mωξ,1 ⊂ ⋂ 1≤ζ≤ω1 mωξ,ζ. thus for every 1 ≤ ζ ≤ ω1, mζ(zξ(c0)) = ωξ. since these spaces have a shrinking, asymptotic `1 basis, they are reflexive. from this it follows that for all 1 ≤ ζ ≤ ω1, m∗ζ(zξ(c0) ∗) = ωξ. for the ξ = 0 case, mζ(`2) = m ∗ ζ(`2) = 1 = ω0 for every 1 ≤ ζ ≤ ω1. however, as we have already seen, for any 0 ≤ ξ < ω1, mζ(xωξ) = m∗ζ(x ∗ ωξ ) = ωξ + 1. this completely elucidates the examples with ξ < ω1. for the ξ = ω1 case, we note that mζ(x) = ω1 if and only if x ∈⋂ η<ω1 mη,ζ = mω1,ζ, and a similar statement holds for m ∗ ζ. three-space properties. in [19], a banach space x with subspace y was exhibited such that y,x/y have the weak banach-saks property, while x does not. in [7], it was shown that y,x/y have the hereditary dunfordpettis property, while x does not. in [9], it was shown that any banach space is a complemented subspace of a twisted sum of two banach spaces with the the ξ,ζ-dunford pettis property 191 dunford-pettis property. therefore there exists a banach space x containing a complemented copy of `2 and a subspace y of x such that y and x/y both lie in mω1,ω1 . since `2 ∈ {m1,1 and x contains a complemented copy of `2, x ∈ {m1,1. thus y,x/y ∈ mω1,ω1 , while x ∈ {m1,1. this implies that for any 1 ≤ ξ,ζ ≤ ω1, the property z ∈ mξ,ζ is not a three space property. we modify ostrovskii’s example to provide a sharp solution to the three space properties of the classes wbsξ. theorem 4.15. for any 0 ≤ ζ,ξ < ω1, any banach space x, and any subspace y such that y ∈ wbsξ, and x/y ∈ wbsζ, x ∈ wbsζ+ξ. for any 0 ≤ ζ,ξ < ω1, there exist a banach space x with a subspace y such that y ∈ wbsξ, x/y ∈ wbsζ, and x ∈∩γ<ζ+ξ{wbsγ. proof. assume y ∈ wbsξ and x/y ∈ wbsζ. fix a weakly null sequence (xn) ∞ n=1 ⊂ x and, seeking a contradiction, assume 0 < ε = inf { ‖x‖ : f ∈sζ+ξ,x ∈ co(xn : n ∈ f) } . by passing to a subsequence, we may assume ε ≤ inf { ‖x‖ : f ∈sξ[sζ],x ∈ co(xn : n ∈ f) } . since (xn + y ) ∞ n=1 is weakly null in x/y , it is ζ-weakly null. thus there exist f1 < f2 < ... , fi ∈ sζ, and positive scalars (ai)i∈∪∞n=1fn such that∑ i∈fn ai = 1 and ‖ ∑ i∈fn aixi + y‖ < min{ε/2, 1/n}. for each n ∈ n, we fix yn ∈ y such that ‖yn− ∑ i∈fn aixi‖ < min{ε/2, 1/n}. since (xn) ∞ n=1 is weakly null, so are ( ∑ i∈fn aixi) ∞ n=1 and (yn) ∞ n=1. since y ∈ wbsξ, there exist g ∈sξ and positive scalars (bn)n∈g such that ∑ n∈g bn = 1 and ‖ ∑ n∈g bnyn‖ < ε/2. since ∪n∈gfn ∈sξ[sζ], ε ≤ ∥∥∥∥ ∑ n∈g ∑ i∈fn bnaixi ∥∥∥∥ ≤ ∥∥∥∥ ∑ n∈g bnyn ∥∥∥∥ + ∑ n∈g bn ∥∥∥∥yn − ∑ i∈fn aixi ∥∥∥∥ < ε/2 + ε/2 = ε, and this contradiction finishes the first statement. now if ζ = 0 = ξ, let x be any finite dimensional space and let y = x. if ζ = 0 and ξ > 0, let (ξn) ∞ n=1 be any sequence such that supn ξn + 1 = ξ. let x = (⊕∞n=1xξn)`1 and let y = x. if ξ = 0 and ζ > 0, let (ζn) ∞ n=1 be any 192 r.m. causey sequence such that supn ζn + 1 = ζ. let x = (⊕∞n=1xζn)`1 and let y = {0}. each of these choices is easily seen to be the example we seek in these trivial cases. we now turn to the non-trivial case, ξ,ζ > 0. fix (ξn) ∞ n=1 such that if ξ is a successor, ξn + 1 = ξ for all n ∈ n. otherwise let (ξn)∞n=1 be the sequence such that sξ = { e ∈ [n] ζn. let am,n = ‖im,n‖−1. for each m ∈ n, let zm = (⊕∞n=1xζn)`1 and let z = (⊕ ∞ m=1zm)`1 . define jm : xζ+ξm → zm by jm(w) = (2 −nam,nim,nw) ∞ n=1. note that ‖jm‖ ≤ 1. now let w = (⊕∞m=1xζ+ξm)`1 and define s : w → z by letting s|xζ+ξm = jm. note that ‖s‖≤ 1. let q : `1 → z be a quotient map. let x = `1 ⊕1 w and define t : x → z by t(x,w) = qx + sw. then t is also a quotient map, and, with y = ker(t), x/y = z. since ζn < ζ, xζn ∈ wbsζ. since wbsζ is closed under `1 sums, zm and z lie in wbsζ. fix γ < ζ +ξ and note that there exists m ∈ n such that γ ≤ ζ + ξm. since x contains an isomorph of xζ+ξm, the basis of which is not ζ + ξm-weakly null, x in{wbsγ. it remains to show that y ∈ wbsξ. to that end, fix a weakly null sequence ((xn,wn))∞n=1 ⊂ bker(t). then xn → 0, and txn → 0. from this it follows that swn → 0. seeking a contradiction, assume that 0 < ε = inf { ‖z‖ : f ∈sξ,z ∈ co((xn,wn) : n ∈ f) } . by passing to a subsequence, we may assume ‖xn‖ < ε/2 for all n, so that ε/2 ≤ inf { ‖w‖ : f ∈sξ,w ∈ co(wn : n ∈ f) } . since (wn) ∞ n=1 ⊂ w is weakly null, there exists k ∈ n such that for all n ∈ n, ∞∑ m=k+1 ‖wn,m‖xζ+ξm < ε/4, where wn = (wn,m) ∞ m=1. since swn → 0, it follows that for all m ∈ n, jmwn,m → n 0. in particular, for every β < ζ and m ∈ n, limn‖wn,m‖β = 0. by passing to a subsequence k times, once for each 1 ≤ m ≤ k, we may assume (wn,m) ∞ n=1 is dominated by a subsequence of the xξm basis. for this we are using proposition 2.16(ii). since ξm < ξ, (wn,m) ∞ n=1 is ξ-weakly null for each 1 ≤ m ≤ k. from this it follows that there exist f ∈ sξ and the ξ,ζ-dunford pettis property 193 positive scalars (an)n∈f such that ∑ n∈f an = 1 and for each 1 ≤ m ≤ k, ‖ ∑ n∈f anwm,n‖ζ+ξm < ε/4k. then ε/2 ≤ ∥∥∥∥ ∑ n∈f anwn ∥∥∥∥ ≤ k∑ m=1 ∥∥∥∥ ∑ n∈f anwm,n ∥∥∥∥ ζ+ξm + ∑ n∈f an ∞∑ m=k+1 ‖wm,n‖ζ+ξm < ε/4 + ε/4 = ε/2 , a contradiction. 5. partial unconditionality in this section, we give the promised modification in the complex case of the cited result of elton required for our proof of proposition 4.1. lemma 5.1. fix k ∈ n and suppose we have vectors (u1, . . . ,uk−1,v1, v2, . . . ) ⊂ sx forming a normalized, weakly null, monotone basic sequence. for any c,ε > 0, there exists a subsequence (wj) ∞ j=1 of (vj) ∞ j=1 such that for any t ⊂ {1, . . . ,k − 1}, any n ∈ n, and m0 < · · · < mn, any functional x∗ ∈ bx∗ such that ∣∣∣∣x∗ (∑ j∈t uj ) + x∗ ( n∑ j=1 wmj )∣∣∣∣ ≥ c, there exists y∗ ∈ bx∗ such that∣∣∣∣x∗ (∑ j∈t uj ) + x∗ ( n∑ j=1 wmj )∣∣∣∣ ≥ c, |y∗(uj) −x∗(uj)| ≤ ε for all j ≤ k, and |y∗(wm0 )| ≤ ε. proof. we prove only the k > 1 case, with the k = 1 case following by omitting superfluous parts of the k > 1 case. for l ∈ [n], u ⊂ b `k−1∞ , t ⊂ {1, . . . ,k − 1}, and n ∈ n, let a(t,u,n,l) (resp. b(t,u,n,l)) denote the set of x∗ ∈ bx∗ such that, with l = (l0, l1, l2, . . . ), ∣∣∣∣x∗ (∑ j∈t uj ) + x∗ ( n∑ j=1 vlj )∣∣∣∣ ≥ c 194 r.m. causey and (x∗(uj)) k−1 j=1 ∈ u (resp.∣∣∣∣x∗ (∑ j∈t uj ) + x∗ ( n∑ j=1 vlj )∣∣∣∣ ≥ c, (x∗(uj)) k−1 j=1 ∈ u, and |x ∗(vl0 )| ≤ ε). now for a fixed t ⊂ {1, . . . ,k − 1} and u ⊂ b `k−1∞ , let an denote the set of those l ∈ [n] such that if a(t,u,n,l) 6= ∅, then b(t,u,n,l) 6= ∅. let an = ∩n∈nan. we claim that for any n ∈ [n], there exists l ∈ [n] such that [l] ⊂a. we prove this by contradiction. note that since membership in an is determined by properties of the n + 1-element subsets of a given set, an is closed. since a is an intersection of closed sets, it is also closed, and therefore ramsey. therefore if the claim were to fail, there would exist some l ∈ [n] such that [l] ∩a = ∅. write l = (l1, l2, . . . ). for 1 ≤ q ≤ p, let lp,q = (lq, lp+1, lp+2, . . . ) and note that, since lp,q ∈ [l] ⊂ [n] \ a, there exists np,q ∈ n such that a(t,u,np,q,lp,q) 6= ∅ but b(t,u,np,q,lp,q) = ∅. for each such p,q, fix x∗p,q ∈ a(t,u,np,q,lp,q). fix np = min{np,q : q ≤ p} and qp ≤ p such that np,qp = np. by monotonicity of the basis, there exists x∗p ∈ bx∗ such that x∗p(uj) = x∗p,qp(uj) for all j < k, x∗p(vj) = x ∗ p,qp (vj) for all j ≤ lnp, and x∗p(vj) = 0 for all j > lnp. note that ∣∣∣∣x∗p (∑ j∈t uj ) + x∗p ( np∑ j=1 vlp+j )∣∣∣∣ = ∣∣∣∣x∗p,qp (∑ j∈t uj ) + x∗p,qp (np,qp∑ j=1 vlp+j )∣∣∣∣ ≥ c and ( x∗p(uj) )k−1 j=1 = ( x∗p,qp(uj) )k−1 j=1 ∈ u. now note that since np,q ≥ np = np,qp for all 1 ≤ q ≤ p and x∗p(vlj ) = 0 for any j > np, for each 1 ≤ q ≤ p,∣∣∣∣x∗p (∑ j∈t uj ) + x∗p (np,q∑ j=1 vlj )∣∣∣∣ = ∣∣∣∣x∗p (∑ j∈t uj ) + x∗p ( np∑ j=1 vlp+j )∣∣∣∣ ≥ c, and (x∗(uj)) p−1 j=1 ∈ u. since b(t,u,np,q,lp,q) = ∅, it must be the case that |x∗p(vlq )| ≥ ε. now if x ∗ is any weak∗-cluster point of (x∗p) ∞ p=1, |x ∗(vlq )| ≥ ε for all q ∈ n, contradicting the weak nullity of (vj)∞j=1. this gives the claim. now let t1, . . . ,tr be an enumeration of the subsets of {1, . . . ,k− 1} and let u1, . . . ,us be a partition of b`k−1∞ into sets of diameter not more than ε. by repeated applications of the claim from the preceding paragraph, we the ξ,ζ-dunford pettis property 195 may choose n = l0 ⊃ ··· ⊃ lrs = l such that if j = (k − 1)r + (i − 1) with 1 ≤ k ≤ s and 1 ≤ i ≤ r, then for any m ∈ [lj], if for some n ∈ n, a(tk,ui,n,m) 6= ∅, then b(tk,ui,n,m) 6= ∅. then l has the property that for any m ∈ [l], if a(tk,ui,n,m) 6= ∅, then b(tk,ui,n,m). write l = (lj) ∞ j=1 and let wj = vlj . now suppose t ⊂{1, . . . ,k−1}, x ∗ ∈ bx∗, and m0 < m1 < · · · < mn are such that∣∣∣∣x∗ (∑ j∈t uj ) + x∗ ( n∑ j=1 vlmj )∣∣∣∣ = ∣∣∣∣x∗ (∑ j∈t uj ) + x∗ ( n∑ j=1 wmj )∣∣∣∣ ≥ c. pick k such that t = tk and i such that (x ∗(uj)) k−1 j=1 ∈ ui. fix any mn+1 < mn+2 < ... such that mn+1 > mn and let m = (lmj ) ∞ j=0 ∈ [l]. then x ∗ ∈ a(tk,ui,n,m), so that b(tk,ui,n,m) 6= ∅. now fix y∗ ∈ b(tk,ui,n,m). by definition of b(tk,ui,n,m),∣∣∣∣y∗ (∑ j∈t uj ) + y∗ ( n∑ j=1 wmj )∣∣∣∣ = ∣∣∣∣y∗ (∑ j∈t uj ) + y∗ ( n∑ j=1 vlmj )∣∣∣∣ ≥ c and |y∗(wm0 )| = |y∗(vlm0 )| ≥ ε. since (y ∗(uj)) k−1 j=1, (x ∗(uj)) k−1 j=1 ∈ ui, it follows that max 1≤j 0, there exists a subsequence (yj) ∞ j=1 of (xj) ∞ j=1 such that for any pairwise disjoint, finite subsets g,h of n and scalars (aj)j∈h such that ‖ ∑ j∈g yj‖≥ c,∥∥∥∥∑ j∈g yj + ∑ j∈h ajyj ∥∥∥∥ ≥ c −ε maxj∈h |aj|. (ii) for any sequences (cn) ∞ n=1, (εn) ∞ n=1 of positive numbers, there exists a subsequence (yj) ∞ j=1 of (xj) ∞ j=1 such that for any n ∈ n, any pairwise 196 r.m. causey disjoint subsets g,h of n such that ‖ ∑ j∈g yn‖ ≥ cn + 2n, and any scalars (aj)j∈h,∥∥∥∥∑ j∈g yj + ∑ j∈h aj ∥∥∥∥ ≥ cn − (n + εn) maxj∈h |aj|. proof. (i) fix positive numbers (εj) ∞ j=1 such that ∑∞ j=1 ∑∞ k=j εk < ε. let l0 = n and apply the k = 1 case of lemma 5.1 with (vj)∞j=1 = (xj) ∞ j=1, c = c, and ε = ε1 to find m1 ∈ [n] satisfying the conclusions of lemma 5.1. let r1 = min m1 and l1 = mn \ {r1}. now suppose that for some k > 1, r1 < · · · < rk−1 and l0 ⊃ ··· ⊃ lk−1 ∈ [n] with min lk−1 > rk−1 have been chosen. apply the k case of lemma 5.1 with uj = xrj , (vj) ∞ j=1 = (xj)j∈lk−1 , c = c, and ε = εk to find mk ∈ [lk−1] satisfying the conclusions of lemma 5.1. let rk = min mk and lk = mk \ {rk}. this completes the recursive construction of r1 < r2 < ... . let yj = xrj . now fix a finite subset g of n such that ‖ ∑ j∈g yj‖ ≥ c. fix x∗0 ∈ bx∗ such that ∣∣∣∣x∗ (∑ j∈g yj )∣∣∣∣ ≥ c. we may use the conclusions of lemma 5.1 to find x∗1,x ∗ 2, . . . such that for each k ∈ n and for each j < k, |x∗k(yj)−x ∗ k−1(yj)| ≤ εk, ∣∣∣x∗k(∑j∈g yj)∣∣∣ ≥ c, and if k ∈ n\g, |x∗k(yk)| ≤ εk. we explain how to choose x ∗ k assuming x ∗ k−1 is chosen. if k ∈ g, we simply let x∗k = x ∗ k−1. if k = 1 + max g, we use monotonicity to deduce the existence of x∗1+max g such that x ∗ 1+max g(yj) = x ∗ max g(yj) for all j ≤ max g and x∗1+max g(yj) = 0 for all j > max g. we then let x ∗ k = x ∗ 1+max g for all k > 1 + max g. now suppose that k /∈ g and k < max g. fix n ∈ n and some m1 < · · · < mn such that g ∩ (k,∞) = {m1, . . . ,mn}. fix any mn < mn+1 < ... . now note that, since (rk,rm1,rm2, . . . ) ∈ [mk] and∣∣∣∣x∗k−1 (∑ j∈g yj )∣∣∣∣ = ∣∣∣∣x∗k−1 ( ∑ j∈g∩[1,k] yj ) + x∗k−1 ( n∑ j=1 xrmn )∣∣∣∣ ≥ c, the properties of mk yield the existence of some x ∗ k ∈ bx∗ such that∣∣∣∣x∗k (∑ j∈g yj )∣∣∣∣ = ∣∣∣∣x∗k ( ∑ j∈g∩[1,k] yj ) + x∗k ( n∑ j=1 xrmn )∣∣∣∣ ≥ c, |x∗k(yj) −x ∗ k(yj)| ≤ εk for all j < k, and |x ∗ k(yk)| ≤ εk. the ξ,ζ-dunford pettis property 197 now note that the previous recursion yields x∗ = x∗1+max g ∈ bx∗ such that ∣∣∣∣x∗ (∑ j∈g yj )∣∣∣∣ ≥ c. furthermore, for any j < max g such that j /∈ g, |x∗(yj)| ≤ |x∗j (yj)| + 1+max g∑ k=j+1 |x∗k(yj) −x ∗ k−1(yj)| ≤ ∞∑ k=j εk. for j > max g, x∗(yj) = 0. now fix any set disjoint from h and any scalars (aj)j∈h. then∥∥∥∥∑ j∈g yj + ∑ j∈h ajyj ∥∥∥∥ ≥ ∣∣∣∣x∗ (∑ j∈g yj )∣∣∣∣− maxj∈h |aj| ∞∑ j=1 ∞∑ k=j εk ≥ c −ε max j∈h |aj|. (ii) recursively select l1 ⊃ l2 ⊃ . . . such that (xj)j∈ln is the sequence obtained by applying (i) with c = cn + n and ε = εn. fix l1 < l2 < ... , ln ∈ ln, and l = (ln)∞n=1. let yj = xlj . suppose that n ∈ n, g ⊂ n are such that g is finite and ‖ ∑ j∈g yj‖≥ cn + 2n. fix h ⊂ n \g finite and scalars (aj)j∈h. note that∥∥∥∥ ∑ j∈g∩(n,∞) yj ∥∥∥∥ ≥ cn + 2n− n∑ j=1 ‖yj‖≥ cn + n. by the properties of (yn+j) ∞ j=1 obtained from the conclusions of (i),∥∥∥∥ ∑ j∈g∩(n,∞) yj + ∑ j∈h∩(n,∞) ajyj ∥∥∥∥ ≥ cn + n−εn maxj∈h |aj|. now∥∥∥∥∑ j∈g yj + ∑ j∈h ajyj ∥∥∥∥ ≥ cn + n−εn maxj∈h |aj|− n∑ j=1 ‖yj‖− max j∈h |aj| n∑ j=1 ‖yj‖ ≥ cn − (n + εn) max j∈h |aj|. 198 r.m. causey proposition 5.3. (johnson) if (xn) ∞ n=1 is a normalized, weakly null sequence having no subsequence equivalent to the canonical c0 basis, then there exists a subsequence (yn) ∞ n=1 of (xn) ∞ n=1 such that for any r1 < r2 < ... , sup t ∥∥∥∥ t∑ j=1 yrj ∥∥∥∥ = ∞. since the complex version of the preceding result can be easily obtained from the real part by splitting coefficients into real and imaginary parts, we omit the proof. corollary 5.4. let (xj) ∞ j=1 be a normalized, weakly null sequence with no subsequence equivalent to the canonical c0 basis. then there exists a subsequence (yj) ∞ j=1 of (xj) ∞ j=1 such that for any (bj) ∞ j=1 ∈ `∞ \ c0, sup t ∥∥∥∥ n∑ j=1 bjyj ∥∥∥∥ = ∞. proof. by passing to a subsequence and passing to an equivalent norm, we may assume that (xj) ∞ j=1 is monotone basic. we may pass to subsequences twice and assume that for any r1 < r2 < ... , sup t ∥∥∥∥ t∑ j=1 yrj ∥∥∥∥ = ∞, a property which is retained by all subsequences. we may also let cn = n 2 and εn = 1 and assume that for any n ∈ n and pairwise disjoint, finite subsets g,h of n such that ‖ ∑ j∈g yj‖≥ cn + 2n and scalars (aj)j∈h,∥∥∥∥∑ j∈g yj + ∑ j∈h ajyj ∥∥∥∥ ≥ cn − (n + εn) maxj∈h |aj|. we prove that this sequence (yj) ∞ j=1 has the desired property. fix (aj) ∞ j=1 ∈ b`∞\c0. we may select r1 < r2 < ... and a non-zero number a with |a| ≤ 1 such that ∑∞ j=1 |a − arj| < 1. by multiplying the sequence (aj) ∞ j=1 by a unimodular scalar, we may assume a is a positive real number. by monotonicity, supt‖ ∑t j=1 ajyj‖ = limt‖ ∑t j=1 ajyj‖ = limt‖ ∑rt j=1 ajyj‖. the ξ,ζ-dunford pettis property 199 in order to reach the conclusion, it is sufficient to define gt = {r1, . . . ,rt} and ht = {1, . . . ,rt}\gt and show that ∞ = lim t ∥∥∥∥ ∑ j∈gt yj + ∑ j∈ht aj a yj ∥∥∥∥. indeed, from this it follows that ∞ = −1 + lim t ∥∥∥∥ ∑ j∈gt ayj + ∑ j∈ht ajyj ∥∥∥∥ ≤−1 + lim t ∥∥∥∥ rt∑ j=1 ajyj ∥∥∥∥ + t∑ j=1 |arj −a| ≤ lim t ∥∥∥∥ rt∑ j=1 ajyj ∥∥∥∥. note that for each t, maxj∈ht | aj a | ≤ 1/a. for each n ∈ n, by the properties of (yj) ∞ j=1, there exists t0 so large that for all t ≥ t0,∥∥∥∥ ∑ j∈gt yj ∥∥∥∥ > cn + 2n, so that ∥∥∥∥ ∑ j∈gt yj + ∑ j∈ht aj a yj ∥∥∥∥ ≥ cn − (n + εn)/a = n2 − n + 1a . since this holds for any n ∈ n and limn n2 − n+1a = ∞, we are done. references [1] s.a. argyros, i. gasparis, unconditional structures of weakly null sequences, trans. amer. math. soc. 353 (5) (2001), 2019 – 2058. 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[22] c. stegall, duals of certain spaces with the dunford-pettis property, notices amer. math. soc. 19 (1972), a799. introduction combinatorics ideals of interest space ideals partial unconditionality � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 36, num. 2 (2021), 157 – 239 doi:10.17398/2605-5686.36.2.157 available online november 8, 2021 homotopy theory of moore flows (ii) p. gaucher université de paris, cnrs, irif, f-75006, paris, france http://www.irif.fr/~gaucher received august 21, 2021 presented by a.m. cegarra accepted september 29, 2021 abstract: this paper proves that the q-model structures of moore flows and of multipointed d-spaces are quillen equivalent. the main step is the proof that the counit and unit maps of the quillen adjunction are isomorphisms on the q-cofibrant objects (all objects are q-fibrant). as an application, we provide a new proof of the fact that the categorization functor from multipointed d-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories. the new proof sheds light on the internal structure of the categorization functor which is neither a left adjoint nor a right adjoint. it is even possible to write an inverse up to homotopy of this functor using moore flows. key words: enriched semicategory, semimonoidal structure, combinatorial model category, quillen equivalence, locally presentable category, topologically enriched category, moore path. msc (2020): 18c35, 18d20, 55u35, 68q85. contents 1 introduction 158 2 multipointed d-spaces 162 3 moore composition and ω-final structure 170 4 from multipointed d-spaces to moore flows 177 5 cellular multipointed d-spaces 184 6 chains of globes 198 7 the unit and the counit of the adjunction on q-cofibrant objects 207 8 from multipointed d-spaces to flows 220 a the reedy category pu,v(s): reminder 230 b an explicit construction of the left adjoint mg! 231 c the setting of k-spaces 235 issn: 0213-8743 (print), 2605-5686 (online) © the author(s) released under a creative commons attribution license (cc by-nc 3.0) https://doi.org/10.17398/2605-5686.36.2.157 http://www.irif.fr/~gaucher https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 158 p. gaucher 1. introduction presentation. this paper is the companion paper of [14]. the purpose of these two papers is to exhibit, by means of the q-model category of moore flows (cf. definition 4.6), a zig-zag of quillen equivalences between the qmodel structure of multipointed d-spaces introduced in [11] and the q-model structure of flows introduced in [7]. the only known functor which was a good candidate for a quillen equivalence from multipointed d-spaces to flows (definition 8.7) has indeed a total left derived functor in the sense of [4] which induces an equivalence of categories between the homotopy categories ( [11, theorem 7.5]). however, this functor is neither a left adjoint nor a right adjoint by theorem 8.8. multipointed d-spaces and flows can be used to model concurrent processes. for example, the paper [10] shows how to model all process algebras for any synchronization algebra using flows. there are many geometric models of concurrency available in the literature [17–20, 28] (the list does not pretend to be exhaustive). most of them are used to study the fundamental category of a concurrent process or any derived concept. it is something which can be also carried out with the formalisms of flows and multipointed d-spaces. the fundamental category functor is easily calculable indeed, at least for flows since it is a left adjoint, and for cellular multipointed d-spaces by using corollary 8.12, and it interacts very well with the underlying simplicial structures. the quillen equivalence between flows and moore flows is proved in [14, theorem 10.9]. the quillen equivalence between multipointed d-spaces and moore flows is proved in theorem 8.1. the latter theorem is a consequence of the structural properties of the adjunction between multipointed d-spaces and moore flows which can be summarized as follows: theorem 1.1. (theorem 7.6, corollary 7.9 and theorem 8.1) the adjunction mg! a m g : gflow �gdtop between moore flows and multipointed d-spaces is a quillen equivalence. the counit map and the unit map of this quillen adjunction are isomorphisms on q-cofibrant objects (recall that all objects are q-fibrant). another standard example of the situation of theorem 1.1 is the quillen equivalence between the q-model structures of ∆-generated spaces and of kspaces (cf. appendix c). homotopy theory of moore flows (ii) 159 this paper is the first use in a real practical situation of the closed semimonoidal category of g-spaces (definition 4.1 and theorem 4.3). it illustrates the interest of this structure already for calculating spaces of execution paths of cellular multipointed d-spaces. the interest of this structure is beyond directed homotopy theory, as remarked in the introduction of [14] where possible connections with type theory are briefly discussed. the potential of this semimonoidal structure is visible in the proofs of proposition 6.3, theorem 7.2, theorem 7.3 and corollary 7.4. the moore flows enable us to write explicitly an “inverse up to homotopy” of the categorization functor of definition 8.7 in definition 8.13. two applications of the existence of this inverse up to homotopy are given. the first one is a new proof of [11, theorem 7.5] provided in theorem 8.14 which is totally independent from [8, 11]. the second one is a concise and very natural definition of the underlying homotopy type of a flow in proposition 8.16. as a curiosity, it is also proved in passing a kind of second dini theorem for spaces of execution paths of finite cellular multipointed d-spaces without loops in corollary 6.12. this paper is written in the setting of ∆-hausdorff or not ∆-generated spaces. the setting of weakly hausdorff or not k-spaces is of very little interest for the study of multipointed d-spaces and flows not only because all concrete examples coming from computer science are cellular objects of the q-model structures, and also because it is not known how to left bousfield localize because of the cofibration identifying two states. the locally presentable setting has many other advantages like the existence of adjoints [15, theorem 5.10], the smallness conditions of [25, definition 2.1.3] always satisfied and the existence of left-determined model categories in the tractable cases [22]. it might be interesting anyway to make some comments about k-spaces to emphasize some topological arguments of this paper. these comments are postponed to appendix c. outline of the paper. • section 2 is a reminder about multipointed d-spaces and about their qmodel structure. it also contains new results about the topology of the space of execution paths. the section starts with a short reminder about ∆-generated spaces. the notion of ∆-inclusion is introduced to clarify some topological arguments: they are for ∆-generated spaces what k-inclusions are for k-spaces. • the functor ω which forgets the set of execution paths of a multipointed d-space is topological. section 3 gives an explicit description of the ω-final 160 p. gaucher structure in term of moore composition. it culminates with theorem 3.9. the calculations are a bit laborious but some of them are used further in the paper. • section 4, after a reminder about moore flows and their q-model structure, describes the adjunctions between multipointed d-spaces and moore flows. the right adjoint from multipointed d-spaces to moore flows is quite easy to define. the existence of the left adjoint is straightforward. appendix b provides an explicit construction of this left adjoint mg! : gflow →gdtop. it uses results dating back to [7] obtained for flows, i.e., small semicategories enriched over topological spaces, and adapted to moore flows, i.e., small semicategories enriched over g-spaces. this explicit construction is not necessary to establish the results of the main part of the paper. it is the reason why it is postponed to an appendix. • section 5 gathers some geometric properties of cellular multipointed dspaces concerning their underlying topologies, the topologies of their spaces of execution paths and some of their structural properties like theorem 5.18 which has important consequences. the main tools are the notion of carrier of an execution path (definition 5.10) and the notion of achronal slice of a globular cell (definition 5.15) studied in proposition 5.16 and proposition 5.17. it also contains theorem 5.20 which provides a kind of normal form for the execution paths of a cellular multipointed d-space obtained as a pushout along a generating q-cofibration. • section 6 studies chains of globes. it is an important geometric object for the proofs of this paper. it enables us to understand what happens locally in the space of execution paths of a cellular multipointed d-space. the main theorem is theorem 6.11 which can be viewed as a workaround of the fact that the space g(1, 1) of nondecreasing homeomorphisms from [0, 1] to itself equipped with the compact-open topology is not sequentially compact. as a byproduct, it is also proved in corollary 6.12 a second dini theorem for finite cellular multipointed d-spaces without loops. • section 7 is the core of the paper. it proves that the unit and the counit of the adjunction are isomorphisms on q-cofibrant objects in theorem 7.6 and in corollary 7.9. the main tool of this part is corollary 7.4 which proves that the right adjoint constructed in section 4 preserves pushouts of cellular multipointed d-spaces along q-cofibrations. it relies on theorem 7.2 whose proof performs an analysis of the execution paths in a pushout along a qcofibration and on theorem 7.3 whose proof carries out a careful analysis of the underlying topology of the spaces of execution paths involved. homotopy theory of moore flows (ii) 161 • section 8 is the concluding section. it establishes that the adjunction between multipointed d-spaces and moore flows yields a quillen equivalence between the q-model structures. it provides, as an application, a more conceptual proof of the fact that the categorization functor cat from multipointed d-spaces to flows has a total left derived functor which induces a category equivalence between the homotopy categories of the q-model structures of multipointed d-spaces and flows. and finally, it is shown how to recover the underlying homotopy type of a flow in a very intuitive way. prerequisites. we refer to [1] for locally presentable categories, to [33] for combinatorial model categories. we refer to [24,25] for more general model categories. we refer to [27] and to [2, chapter 6] for enriched categories. all enriched categories are topologically enriched categories: the word topologically is therefore omitted. what follows is some notations and conventions: • a := b means that b is the definition of a. • ∼= denotes an isomorphism, ' denotes a weak equivalence. • f �a denotes the restriction of f to a. • set is the category of sets. • top is the category of general topological spaces together with the continuous maps. • kop denotes the opposite category of k. • obj(k) is the class of objects of k. • mor(k) is the category of morphisms of k with the commutative squares for the morphisms. • ki is the category of functors and natural transformations from a small category i to k. • ∆i(z) is the constant diagram over the small category i with unique value z. • ∅ is the initial object, 1 is the final object, idx is the identity of x. • k(x,y ) is the set of maps in a set-enriched, i.e., locally small, category k. • k(x,y ) is the space of maps in an enriched category k. the underlying set of maps may be denoted by k0(x,y ) if it is necessary to specify that we are considering the underlying set. 162 p. gaucher • the composition of two maps f : a → b and g : b → c is denoted by gf or, if it is helpful for the reader, by g.f; the composition of two functors is denoted in the same way. • the notations `,`′,`i,l, . . . mean a strictly positive real number unless specified something else. • [`,`′] denotes a segment. unless specified, it is always understood that ` < `′. • a cellular object of a combinatorial model category is an object x such that the canonical map ∅ → x is a transfinite composition of pushouts of generating cofibrations. • the notation (−)cof denotes a cofibrant replacement functor of a combinatorial model structure; note that all model categories of this paper contain only fibrant objects. • a compact space is a quasicompact hausdorff space. • a sequentially compact space is a space such that each sequence has a limit point. • the set of rational numbers is denoted by q, the set of real numbers by r. • the complement of a ⊂ b is denoted by ac if there is no ambiguity. • let n > 1. denote by dn = {b ∈ rn, |b| 6 1} the n-dimensional disk, and by sn−1 = {b ∈ rn, |b| = 1} the (n−1)-dimensional sphere. by convention, let d0 = {0} and s−1 = ∅. acknowledgments. i am indebted to tyrone cutler for drawing my attention to the paper [3]. i thank the anonymous referee for reading this very technical paper. 2. multipointed d-spaces throughout the paper, we work with the category, denoted by top, either of ∆-generated spaces or of ∆-hausdorff ∆-generated spaces (cf. [15, section 2 and appendix b]) equipped with its q-model structure (we use the terminology of [30]). we summarize some basic properties of top for the convenience of the reader: • top is locally presentable. • all objects of top are sequential topological spaces. homotopy theory of moore flows (ii) 163 • a closed subset of a ∆-generated space equipped with the relative topology is not necessarily ∆-generated (e.g., the cantor set), but it is always sequential. • all locally path-connected first-countable topological spaces are ∆-generated by [3, proposition 3.11], in particular all locally path-connected metrizable topological spaces are ∆-generated. • the inclusion functor from the full subcategory of ∆-generated spaces to the category of general topological spaces together with the continuous maps has a right adjoint called the ∆-kelleyfication functor. the latter functor does not change the underlying set. • let a ⊂ b be a subset of a space b of top. then a equipped with the ∆-kelleyfication of the relative topology belongs to top. • the colimit in top is given by the final topology in the following situations: – a transfinite compositions of one-to-one maps. – a pushout along a closed inclusion. – a quotient by a closed subset or by an equivalence relation having a closed graph. in these cases, the underlying set of the colimit is therefore the colimit of the underlying sets. in particular, the cw-complexes, and more generally all cellular spaces are equipped with the final topology. • cellular spaces are weakly hausdorff. it implies that the image by any continuous application of any compact is closed and compact, i.e., closed, quasicompact and hausdorff. cellular spaces are also ∆-hausdorff and therefore has unique sequential limits by [15, proposition b.17]. • top is cartesian closed. the internal hom top(x,y ) is given by taking the ∆-kelleyfication of the compact-open topology on the set top(x,y ) of all continuous maps from x to y . definition 2.1. a one-to-one map of ∆-generated spaces i : a → b is a ∆-inclusion if for all ∆-generated spaces z, the set map z → a is continuous if and only if the composite set map z → a → b is continuous. proposition 2.2. let i : a → b be a one-to-one continuous map. the following assertions are equivalent: (1) i is a ∆-inclusion; (2) a is homeomorphic to i(a) equipped with the ∆-kelleyfication of the relative topology; 164 p. gaucher (3) a set map [0, 1] → a is continuous if and only if the composite set map [0, 1] → a → b is continuous. proof. the proof is similar to the same statement for k-inclusions of k-spaces. corollary 2.3. a continuous bijection f : u → v of top is a homeomorphism if and only if it is a ∆-inclusion. notation 2.4. the notation [0,`1] ∼=+ [0,`2] for two real numbers `1,`2 > 0 means a nondecreasing homeomorphism from [0,`1] to [0,`2]. it takes 0 to 0 and `1 to `2. the enriched small category g is defined as follows: • the set of objects is the open interval ]0,∞[. • the space g(`1,`2) is the set {[0,`1] ∼=+ [0,`2]} for all `1,`2 > 0 equipped with the ∆-kelleyfication of the relative topology induced by the set inclusion g(`1,`2) ⊂ top([0,`1], [0,`2]). in other terms, a set map [0, 1] → g(`1,`2) is continuous if and only if the composite set map [0, 1] →g(`1,`2) ⊂ top([0,`1], [0,`2]) is continuous. • for every `1,`2,`3 > 0, the composition map g(`1,`2) ×g(`2,`3) −→g(`1,`3) is induced by the composition of continuous maps. it induces a continuous map since the composite set map g(`1,`2) ×g(`2,`3) −→g(`1,`3) ⊂ top([0,`1], [0,`3]) corresponds by the adjunction to the continuous map [0,`1] ×g(`1,`2) ×g(`2,`3) −→ [0,`3] which takes (t,x,y) to y(x(t)). the enriched category g is an example of a reparametrization category in the sense of [14, definition 4.3] which is different from the terminal category. it is introduced in [14, proposition 4.9]. another example is given in [14, proposition 4.11]. homotopy theory of moore flows (ii) 165 proposition 2.5. the topology of g(`1,`2) is the compact-open topology. in particular, it is metrizable. a sequence (φn)n>0 of g(`1,`2) converges to φ ∈g(`1,`2) if and only if it converges pointwise. it means that the topology of the pointwise convergence of g(`1,`2) is ∆generated. proposition 2.5 has an interesting generalization in corollary 6.12. proof. the compact-open topology on g(`1,`2) is metrizable by [21, proposition a.13]. the metric is given by the distance of the uniform convergence. consider a ball b(φ,�) for this metric. let ψ ∈ b(φ,�). then for all h ∈ [0, 1], | ( hψ(t) + (1 −h)φ(t) ) −φ(t)| = |h(ψ(t) −φ(t))| < h� 6 �. thus, the compact-open topology is locally path-connected. the compactopen topology is therefore equal to its ∆-kelleyfication. the last assertion is then a consequence of the second dini theorem. a multipointed space is a pair (|x|,x0) where: • |x| is a topological space called the underlying space of x; • x0 is a subset of |x| called the set of states of x. a morphism of multipointed spaces f : x = (|x|,x0) → y = (|y |,y 0) is a commutative square x0 f0 // �� y 0 �� |x| |f| // |y |. the corresponding category is denoted by mtop. notation 2.6. the maps f0 and |f| will be often denoted by f if there is no possible confusion. we have the well-known proposition: proposition 2.7. (the moore composition) let u be a topological space. let γi : [0,`i] −→ u be n continuous maps with 1 6 i 6 n with n > 1. suppose that γi(`i) = γi+1(0) for 1 6 i < n. then there exists a unique continuous map 166 p. gaucher γ1 ∗ · · · ∗γn : [ 0, ∑ i `i ] −→ u such that (γ1 ∗ · · · ∗γn)(t) = γi ( t− ∑ j 0. let µ` : [0,`] → [0, 1] be the homeomorphism defined by µ`(t) = t/`. definition 2.9. the map γ1 ∗ γ2 is called the moore composition of γ1 and γ2. the composite γ1 ∗n γ2 : [0, 1] (µ2) −1 −−−−−→ [0, 2] γ1∗γ2−−−−→ u is called the normalized composition. one has (γ1 ∗n γ2)(t) = { γ1(2t) if 0 6 t 6 1 2 , γ2(2t− 1) if 12 6 t 6 1. the normalized composition being not associative, a notation like γ1∗n · · ·∗n γn will mean, by convention, that ∗n is applied from the left to the right. a multipointed d-space x is a triple (|x|,x0,pgx) where: • the pair (|x|,x0) is a multipointed space. the space |x| is called the underlying space of x and the set x0 the set of states of x. • the set pgx is a set of continous maps from [0, 1] to |x| called the execution paths, satisfying the following axioms: – for any execution path γ, one has γ(0),γ(1) ∈ x0. – let γ be an execution path of x. then any composite γφ with φ ∈ g(1, 1) is an execution path of x. – let γ1 and γ2 be two composable execution paths of x; then the normalized composition γ1 ∗n γ2 is an execution path of x. a map f : x → y of multipointed d-spaces is a map of multipointed spaces from (|x|,x0) to (|y |,y 0) such that for any execution path γ of x, homotopy theory of moore flows (ii) 167 the map pgf : γ 7−→ f.γ is an execution path of y . notation 2.10. the mapping pgf will be often denoted by f if there is no ambiguity. the following examples play an important role in the sequel. (1) any set e will be identified with the multipointed d-space (e,e,∅). (2) the topological globe of z of length ` > 0, which is denoted by globg` (z), is the multipointed d-space defined as follows: • the underlying topological space is the quotient space 1 {0, 1}t (z × [0,`]) (z, 0) = (z′, 0) = 0, (z, 1) = (z′, 1) = 1 ; • the set of states is {0, 1}; • the set of execution paths is the set of continuous maps {δzφ : φ ∈g(1,`),z ∈ z} with δz(t) = (z,t). it is equal to the underlying set of g(1,`) ×z. in particular, globg` (∅) is the multipointed d-space {0, 1} = ( {0, 1},{0, 1}, ∅ ) . for ` = 1, we set globg(z) = globg1 (z). (3) the directed segment is the multipointed d-space −→ i g = globg({0}). the category of multipointed d-spaces is denoted by gdtop. the subset of execution paths from α to β is the set of γ ∈ pgx such that γ(0) = α and γ(1) = β; it is denoted by pgα,βx: α is called the initial state and β the final state of such a γ. an execution path having the same initial and final state is called a loop. the set pgα,βx is equipped with the ∆-kelleyfication of the relative topology induced by the inclusion pgα,βx ⊂ top([0, 1], |x|). in other terms, a set map u → pgα,βx is continuous if and only if the composite 1it is the suspension of z. 168 p. gaucher set map u → pgα,βx ⊂ top([0, 1], |x|) is continuous. the category gdtop is locally presentable by [11, theorem 3.5]. proposition 2.11. ([16, proposition 6.5]) the mapping ω : x 7−→( |x|,x0 ) induces a functor from gdtop to mtop which is topological and fibre-small. the ω-final structure is generated by the finite normalized composition of execution paths. we will come back on this point in theorem 3.9. note that proposition 2.11 holds both by working with ∆-generated spaces and with ∆-hausdorff ∆-generated spaces. the following proposition is implicitly assumed (for ` = 1) in all the previous papers about multipointed d-spaces: proposition 2.12. let z be a topological space. then there is the homeomorphism pg0,1glob g ` (z) ∼= g(1,`) ×z. proof. the set map ψ : g(1,`) ×z −→ pg0,1glob g ` (z) (φ,z) 7−→ δzφ is continuous because the mapping (t,φ,z) 7→ (z,φ(t)) from [0, 1]×g(1,`)×z to |globg` (z)| is continuous. it is a bijection since, by definition of glob g ` (z), the underlying set of pg0,1glob g ` (z) is equal to the underlying set of g(1,`)×z. the composite set map pg0,1glob g ` (z) −→ (p g 0,1glob g ` (z))co −→ z×]0, 1[ pr1−−−→ z γ 7−→ pr1 ( γ ( 1 2 )) where (pg0,1glob g ` (z))co is the set p g 0,1glob g ` (z) equipped with the compactopen topology is continuous. the continuous map z →{0} induces a continuous map pg0,1glob g ` (z) −→ p g 0,1glob g ` ({0}) ∼= g(1,`) γ 7−→ p.γ, where p : |globg` (z)|→ [0, 1] is the projection map. therefore the set map ψ−1 : pg0,1glob g ` (z) −→ g(1,`) ×z γ 7−→ ( p.γ, pr1 ( γ ( 1 2 ))) homotopy theory of moore flows (ii) 169 is continuous and ψ is a homeomorphism. definition 2.13. let x be a multipointed d-space x. denote again by pgx the topological space pgx = ⊔ (α,β)∈x0×x0 pgα,βx. a straightforward consequence of the definition of the topology of pgx is: proposition 2.14. let x be a multipointed d-space. let f : [0, 1] → pgx be a continuous map. then f factors as composite of continuous maps f : [0, 1] → pgα,βx → p gx for some (α,β) ∈ x0 ×x0. proof. it is due to the fact that [0, 1] is connected. proposition 2.15. let x be a multipointed d-space such that x0 is a totally disconnected subset of |x|. then the topology of pgx is the ∆-kelleyfication of the relative topology induced by the inclusion pgx ⊂ top([0, 1], |x|). proof. call for this proof (pgx)+ the set pgx equipped with the ∆-kelleyfication of the relative topology induced by the inclusion pgxλ ⊂ top([0, 1], |x|). there is a continuous bijection pgx → (pgx)+. using corollary 2.3, the proof is complete since x0 a totally disconnected subset of |x| and since [0, 1] is connected. theorem 2.16. the functor pg : mdtop → top is a right adjoint. in particular, it is limit preserving and accessible. proof. the left adjoint is constructed in [11, proposition 4.9] in the case of ∆-generated spaces. the proof still holds for ∆-hausdorff ∆-generated spaces. it relies on the fact that top is cartesian closed and that every ∆generated space is homeomorphic to the disjoint sum of its path-connected components which are also its connected components. the construction is similar to the construction of the left adjoint of the path p-space functor for p-flows [14, theorem 6.13] and to the construction of the left adjoint of the path functor for flows [15, theorem 5.9]. 170 p. gaucher the q-model structure of multipointed d-spaces is the unique combinatorial model structure such that{ globg(sn−1) ⊂ globg(dn) : n > 0 } ∪ { c : ∅ →{0},r : {0, 1}→{0} } is the set of generating cofibrations, the maps between globes being induced by the closed inclusion sn−1 ⊂ dn, and such that{ globg(dn ×{0}) ⊂ globg(dn+1) : n > 0 } is the set of generating trivial cofibrations, the maps between globes being induced by the closed inclusion (x1, . . . ,xn) 7→ (x1, . . . ,xn, 0) (e.g., [16, theorem 6.16]). the weak equivalences are the maps of multipointed d-spaces f : x → y inducing a bijection f0 : x0 ∼= y 0 and a weak homotopy equivalence pgf : pgx → pgy and the fibrations are the maps of multipointed d-spaces f : x → y inducing a q-fibration pgf : pgx → pgy of topological spaces. 3. moore composition and ω-final structure notation 3.1. let φi : [0,`i] ∼=+ [0,`′i] for n > 1 and 1 6 i 6 n. then the map φ1 ⊗···⊗φn : [ 0, ∑ i `i ] ∼=+ [ 0, ∑ i `′i ] denotes the homeomorphism defined by (φ1⊗. . .⊗φn)(t) =   φ1(t) if 0 6 t 6 `1, φ2(t− `1) + `′1 if `1 6 t 6 `1 + `2, ... φi ( t− ∑ j 1. consider `1, . . . ,`n > 0 with n > 1 such that ∑i=n i=1 `i = `. then there exists a unique decomposition of φ of the form φ = φ1 ⊗···⊗φn such that φi : [0,`i] ∼=+ [0,`′i] for 1 6 i 6 n. homotopy theory of moore flows (ii) 171 proof. by definition of φ1 ⊗···⊗φn, we necessarily have φ (∑ j6i `j ) = φi (∑ j6i `j − ∑ j 1 by the formula φ (∑ j6i `j ) − ∑ j 1. assume that∑i=n i=1 `i = ∑i=n i=1 `′i = 1 and that φ (∑ j6i `j ) = ∑ j6i `′j for 1 6 i 6 n. then there exist (unique) φi : [0,`i] ∼=+ [0,`′i] for 1 6 i 6 n such that φ = φ1 ⊗···⊗φn. proposition 3.4. let u be a topological space. let γi : [0, 1] → u be n continuous maps with 1 6 i 6 n and n > 1. let φi : [0,`i] ∼=+ [0,`′i] for 1 6 i 6 n. then we have( (γ1µ`′1 ) ∗ · · · ∗ (γnµ`′n) ) (φ1 ⊗···⊗φn) = (γ1µ`′1φ1) ∗ · · · ∗ (γnµ`′nφn). proof. for ∑ j 1. let `i > 0 with 1 6 i 6 n be nonzero real numbers with ∑ i `i = 1. then for all ` > 0, we have( (γ1µ`1 ) ∗ · · · ∗ (γnµ`n) ) µ` = (γ1µ`1`) ∗ · · · ∗ (γnµ`n`). proof. for all 1 6 j 6 n, we have by definition of the moore composition ( (γ1µ`1 ) ∗ · · · ∗ (γnµ`n) ) µ`(t) = γj ( 1 `j (t ` − ∑ i 2 and 1 6 i 6 n such that γ1 ∗n · · · ∗n γn exists. then there is the equality γ1 ∗n · · · ∗n γn = ( γ1µ 1 2n−1 ) ∗ ( γ2µ 1 2n−1 ) ∗ ( γ3µ 1 2n−2 ) ∗ · · · ∗ ( γnµ1 2 ) . in particular, for n = 2, we have γ1 ∗n γ2 = ( γ1µ1 2 ) ∗ ( γ2µ1 2 ) . proof. the proof is by induction on n > 2. the map µ1 2 : [0, 1 2 ] ∼=+ [0, 1] which takes t to 2t gives rise to a homeomorphism µ1 2 ⊗ µ1 2 : [0, 1] ∼=+ [0, 2] which is equal to µ−12 : [0, 1] ∼=+ [0, 2]. we then write γ1 ∗n γ2 = (γ1 ∗γ2)µ−12 (by definition of ∗n ) = (γ1 ∗γ2) ( µ1 2 ⊗µ1 2 ) (because µ−12 = µ1 2 ⊗µ1 2 ) = ( γ1µ1 2 ) ∗ ( γ2µ1 2 ) (by proposition 3.4). the statement is therefore proved for n = 2. assume that the statement is proved for some n > 2 and for n = 2. then we obtain γ1∗n · · · ∗n γn+1 = (( γ1µ 1 2n−1 ) ∗ ( γ2µ 1 2n−1 ) ∗ ( γ3µ 1 2n−2 ) ∗ · · · ∗ ( γnµ1 2 )) ∗n γn+1 = ((( γ1µ 1 2n−1 ) ∗ ( γ2µ 1 2n−1 ) ∗ ( γ3µ 1 2n−2 ) ∗ · · · ∗ ( γnµ1 2 )) µ1 2 ) ∗ ( γn+1µ1 2 ) = (( γ1µ 1 2n ) ∗ ( γ2µ 1 2n ) ∗ ( γ3µ 1 2n−1 ) ∗ · · · ∗ ( γnµ 1 22 )) ∗ ( γn+1µ1 2 ) = ( γ1µ 1 2n ) ∗ ( γ2µ 1 2n ) ∗ ( γ3µ 1 2n−1 ) ∗ · · · ∗ ( γnµ 1 22 ) ∗ ( γn+1µ1 2 ) , the first equality by induction hypothesis, the second equality by the case n = 2, the third equality by proposition 3.5, and the last equality by associativity of the moore composition. we have proved the statement for n + 1. 174 p. gaucher proposition 3.7. let u be a topological space. let γi : [0, 1] → u be n continuous maps with n > 1 and 1 6 i 6 n such that γ1∗n · · ·∗n γn exists. let φ ∈ g(1, 1). then there exist φ1 : [0,`1] ∼=+ [0, 12n−1 ], φ2 : [0,`2] ∼=+ [0, 12n−1 ], φ3 : [0,`3] ∼=+ [0, 12n−2 ], . . . , φn : [0,`n] ∼=+ [0, 12 ] such that φ = φ1 ⊗···⊗φn (which implies ∑ i `i = 1) and there is the equality( γ1∗n · · ·∗n γn ) φ = ( γ1µ 1 2n−1 φ1 ) ∗ ( γ2µ 1 2n−1 φ2 ) ∗ ( γ3µ 1 2n−2 φ3 ) ∗· · ·∗ ( γnµ1 2 φn ) . proof. let `1, . . . ,`n > 0 such that ∑ i `i = 1 and such that φ(`1) = 1 2n−1 , φ(`1 + `2) = 1 2n−1 + 1 2n−1 , φ(`1 + `2 + `3) = 1 2n−1 + 1 2n−1 + 1 2n−2 , ... φ(`1 + `2 + `3 + · · · + `n) = 1 2n−1 + 1 2n−1 + 1 2n−2 + · · · + 1 2 = 1. by proposition 3.2, there exist φ1 : [0,`1] ∼=+ [0, 12n−1 ], φ2 : [0,`2] ∼=+ [0, 12n−1 ], φ3 : [0,`3] ∼=+ [0, 12n−2 ], . . . , φn : [0,`n] ∼=+ [0, 12 ] such that φ = φ1 ⊗···⊗φn. we obtain( γ1∗n · · · ∗n γn+1 ) φ = (( γ1µ 1 2n ) ∗ ( γ2µ 1 2n ) ∗ ( γ3µ 1 2n−1 ) ∗ · · · ∗ ( γnµ 1 22 ) ∗ ( γn+1µ1 2 )) φ = ( γ1µ 1 2n−1 φ1 ) ∗ ( γ2µ 1 2n−1 φ2 ) ∗ ( γ3µ 1 2n−2 φ3 ) ∗ · · · ∗ ( γnµ1 2 φn ) , the first equality by proposition 3.6 and the second by proposition 3.4. proposition 3.8. let u be a topological space. let γi : [0, 1] → u be n continuous maps with n > 2 and 1 6 i 6 n such that γ1 ∗n · · · ∗n γn exists. let `1, . . . ,`n > 0 be nonzero real numbers such that ∑ i `i = 1. let φ1 : [0, 1 2n−1 ] ∼=+ [0,`1], φ2 : [0, 12n−1 ] ∼=+ [0,`2], φ3 : [0, 12n−2 ] ∼=+ [0,`3], . . . , φn : [0, 1 2 ] ∼=+ [0,`n] and let φ = φ1 ⊗···⊗φn. then φ ∈g(1, 1), and there is the equality(( γ1µ`1 ) ∗ ( γ2µ`2 ) ∗ ( γ3µ`3 ) ∗ · · · ∗ ( γnµ`n )) φ =( γ1µ`1φ1µ −1 1 2n−1 ) ∗n ( γ2µ`2φ2µ −1 1 2n−1 ) ∗n ( γ3µ`3φ3µ −1 1 2n−2 ) ∗n · · · ∗n ( γnµ`nφnµ −1 1 2 ) . homotopy theory of moore flows (ii) 175 proof. we have(( γ1µ`1 ) ∗ ( γ2µ`2 ) ∗ ( γ3µ`3 ) ∗ · · · ∗ ( γnµ`n )) φ = ( γ1µ`1φ1 ) ∗ ( γ2µ`2φ2 ) ∗ ( γ3µ`3φ3 ) ∗ · · · ∗ ( γnµ`nφn ) = ( γ1µ`1φ1µ −1 1 2n−1 µ 1 2n−1 ) ∗ ( γ2µ`2φ2µ −1 1 2n−1 µ 1 2n−1 ) ∗ ( γ3µ`3φ3µ −1 1 2n−2 µ 1 2n−2 ) ∗ · · · ∗ ( γnµ`nφnµ −1 1 2 µ1 2 ) = ( γ1µ`1φ1µ −1 1 2n−1 ) ∗n ( γ2µ`2φ2µ −1 1 2n−1 ) ∗n ( γ3µ`3φ3µ −1 1 2n−2 ) ∗n · · · ∗n ( γnµ`nφnµ −1 1 2 ) , where the first equality is due to proposition 3.4, the second equality is due to the fact that µ−1` µ` is the identity of [0,`] for all nonzero real numbers ` > 0, and the last equality is a consequence of proposition 3.6. theorem 3.9. consider a cocone (ω(xi)) •−−→ (|x|,x0) of mtop. let x be the ω-final lift. let fi : xi → x be the canonical maps. then the set of execution paths of x is the set of finite moore compositions of the form (f1γ1µ`1 )∗·· ·∗(fnγnµ`n) such that γi is an execution path of xi for all 1 6 i 6 n with ∑ i `i = 1. proof. let p(x) be the set of execution paths of x of the form (f1γ1µ`1 )∗ · · · ∗ (fnγnµ`n) such that γi is an execution path of xi for all 1 6 i 6 n with ∑ i `i = 1. the final structure is generated by the finite normalized composition of execution paths (f1γ1) ∗n · · · ∗n (fnγn) (with the convention that the ∗n are calculated from the left to the right) and all reparametrizations by φ running over g(1, 1). by proposition 3.7, there exist φ1 : [0,`1] ∼=+ [0, 1 2n−1 ], φ2 : [0,`2] ∼=+ [0, 12n−1 ], φ3 : [0,`3] ∼=+ [0, 12n−2 ], . . . , φn : [0,`n] ∼=+ [0, 1 2 ] such that φ = φ1 ⊗···⊗φn and we have( (f1γ1) ∗n · · · ∗n (fnγn) ) φ = ( f1γ1µ 1 2n−1 φ1 ) ∗ ( f2γ2µ 1 2n−1 φ2 ) ∗ ( f3γ3µ 1 2n−2 φ3 ) ∗ · · · ∗ ( fnγnµ1 2 φn ) = ( f1γ1µ 1 2n−1 φ1µ −1 `1 µ`1 ) ∗ ( f2γ2µ 1 2n−1 φ2µ −1 `2 µ`2 ) ∗ ( f3γ3µ 1 2n−2 φ3µ −1 `3 µ`3 ) ∗ · · · ∗ ( fnγnµ1 2 φnµ −1 `n µ`n ) = ( f1γ ′ 1µ`1 ) ∗ ( f2γ ′ 2µ`2 ) ∗ ( f3γ ′ 3µ`3 ) ∗ · · · ∗ ( fnγ ′ nµ`n ) , 176 p. gaucher the first equality by proposition 3.7, the second equality because µ−1` µ` is the identity of [0,`] for all ` > 0 and the third equality because of the following notations:   γ′1 = γ1 ( µ 1 2n−1 φ1µ −1 `1 ) , γ′2 = γ2 ( µ 1 2n−1 φ1µ −1 `2 ) , γ′3 = γ3 ( µ 1 2n−2 φ3µ −1 `3 ) , ... γ′n = γn ( µ1 2 φnµ −1 `n ) . it implies that the set p(x) contains the final structure. conversely, let (f1γ1µ`1 ) ∗ ·· · ∗ (fnγnµ`n) be an element of p(x). choose φ1 : [0, 1 2n−1 ] ∼=+ [0,`1], φ2 : [0, 1 2n−1 ] ∼=+ [0,`2], φ3 : [0, 12n−2 ] ∼=+ [0,`3], . . . , φn : [0, 12 ] ∼=+ [0,`n] and let φ = φ1 ⊗···⊗φn. using proposition 3.8, we obtain(( f1γ1µ`1 ) ∗ ( f2γ2µ`2 ) ∗ ( f3γ3µ`3 ) ∗ · · · ∗ ( fnγnµ`n )) φ = ( f1γ1µ`1φ1µ −1 1 2n−1 ) ∗n ( f2γ2µ`2φ2µ −1 1 2n−1 ) ∗n ( f3γ3µ`3φ3µ −1 1 2n−2 ) ∗n . . .∗n ( fnγnµ`nφnµ −1 1 2 ) . the continuous maps µ`1φ1µ −1 1 2n−1 ,µ`2φ2µ −1 1 2n−1 ,µ`3φ3µ −1 1 2n−2 , . . . ,µ`nφnµ −1 1 2 from [0, 1] to itself belong to g(1, 1). thus γ′1, . . . ,γ ′ n defined by the equalities  γ′1 = γ1 ( µ`1φ1µ −1 1 2n−1 ) , γ′2 = γ2 ( µ`2φ1µ −1 1 2n−1 ) , γ′3 = γ3 ( µ`3φ3µ −1 1 2n−2 ) , ... γ′n = γn ( µ`nφnµ −1 1 2 ) , are execution paths of x1, . . . ,xn respectively. we obtain(( f1γ1µ`1 ) ∗ ( f2γ2µ`2 ) ∗ ( f3γ3µ`3 ) ∗ · · · ∗ ( fnγnµ`n )) φ = ( f1γ ′ 1 ) ∗n ( f2γ ′ 2 ) ∗n ( f3γ ′ 3 ) ∗n · · · ∗n ( fnγ ′ n ) . we deduce that the set of paths p(x) is included in the ω-final structure. homotopy theory of moore flows (ii) 177 4. from multipointed d-spaces to moore flows definition and notation 4.1. the enriched category of enriched presheaves from g to top is denoted by [gop, top]. the underlying set-enriched category of enriched maps of enriched presheaves is denoted by [gop, top]0. the objects of [gop, top]0 are called the g-spaces. let fg op ` u = g(−,`) ×u ∈ [g op, top]0 where u is a topological space and where ` > 0. proposition 4.2. ([12, proposition 5.3 and proposition 5.5]) the category [gop, top]0 is a full reflective and coreflective subcategory of topg op 0 . for every g-space f : gop → top, every ` > 0 and every topological space x, we have the natural bijection of sets [gop, top]0(fg op ` x,f) ∼= top(x,f(`)). theorem 4.3. ([14, theorem 5.14]) let d and e be two g-spaces. let d ⊗e := ∫ (`1,`2) g(−,`1 + `2) ×d(`1) ×e(`2). the pair ([gop, top]0,⊗) has the structure of a closed symmetric semimonoidal category, i.e., a closed symmetric nonunital monoidal category. notation 4.4. let d be a g-space. let φ : ` → `′ be a map of g. let x ∈ d(`′). we will use the notation x.φ := d(φ)(x). intuitively, x is a path of length `′ and x.φ is a path of length ` which is the reparametrization by φ of x. proposition 4.5 sheds light on the meaning of the tensor product of gspaces. it is used in the proof of theorem 7.2. it is not in [14]. the proof is given in this section and not in section 7 to recall [14, corollary 5.13] which also helps to understand the geometric contents of the tensor product of g-spaces. 178 p. gaucher proposition 4.5. let d1, . . . ,dn be n g-spaces with n > 1. then the mapping (x1, . . . ,xn) 7→ (id,x1, . . . ,xn) yields a surjective continuous map φd1,...,dn : ⊔ (`1,...,`n) `1+···+`n=l d1(`1) ×···×dn(`n) −→ (d1 ⊗···⊗dn)(l). proof. by [14, corollary 5.13], the space (d1⊗···⊗dn)(l) is the quotient of the space ⊔ (`1,...,`n) g(l,`1 + · · · + `n) ×d1(`1) ×···×dn(`n). by the identifications (ψ,x1φ1, . . . ,xnφn) = ((φ1 ⊗···⊗φn)ψ,x1, . . . ,xn) for all `1,` ′ 1, . . . ,`n,` ′ n > 0, all ψ ∈g(l,`1 + · · · + `n), all xi ∈ di(`′i) and all φi ∈g(`i,`′i). let ` ′′ 1, . . . ,` ′′ n > 0 defined by induction on i by the equation `′′i = ψ −1 (∑ 16j6i `j ) − ∑ 16j 0 are called the execution paths of length `. the category of moore flows, denoted by gflow, is locally presentable by [14, theorem 6.11]. a map of moore flows f : x → y induces a set map f0 : x0 → y 0 and a map of g-spaces pα,βf : pα,βx → pf(α),f(β)y for each (α,β) ∈ x0 ×x0. let px = ⊔ (α,β)∈x0×x0 pα,βx, py = ⊔ (α,β)∈y 0×y 0 pα,βy, pf = ⊔ (α,β)∈x0×x0 pα,βf. notation 4.7. the map pf : px → py can be denoted by f : px → py is there is no ambiguity. the set map f0 : x0 → y 0 can be denoted by f : x0 → y 0 is there is no ambiguity. every set s can be viewed as a moore flow with an empty g-space of execution paths denoted in the same way. let d : gop → top be a g-space. we denote by glob(d) the moore flow defined as follows: glob(d)0 = {0, 1}, p0,0glob(d) = p1,1glob(d) = p1,0glob(d) = ∆g0∅, p0,1glob(d) = d. there is no composition law. this construction yields a functor glob : [gop, top]0 →gflow. 180 p. gaucher there exists a unique model structure on gflow such that{ glob(fg op ` s n−1) ⊂ glob(fg op ` d n) : n > 0, ` > 0 } ∪ { c,r } is the set of generating cofibrations and such that all objects are fibrant. the set of generating trivial cofibrations is{ glob(fg op ` d n) ⊂ glob(fg op ` d n+1) : n > 0, ` > 0 } where the maps dn ⊂ dn+1 are induced by the mappings (x1, . . . ,xn) 7→ (x1, . . . ,xn, 0). the weak equivalences are the map of moore flows f : x → y inducing a bijection x0 ∼= y 0 and such that for all (α,β) ∈ x0 × x0, the map of g-spaces pα,βx → pf(α),f(β)y is an objectwise weak homotopy equivalence. the fibrations are the map of moore flows f : x → y such that for all (α,β) ∈ x0 × x0, the map of g-spaces pα,βx → pf(α),f(β)y is an objectwise q-fibration of spaces. it is called the q-model structure and we use the terminology of q-cofibration and q-fibration for naming the cofibrations and the fibrations respectively. definition 4.8. let x be a multipointed d-space. let p`α,βx be the subspace of continuous maps from [0,`] to |x| defined by p`α,βx = { t 7→ γµ` : γ ∈ pgα,βx } . its elements are called the execution paths of length ` from α to β. let p`x = ⊔ (α,β)∈x0×x0 p`α,βx. a map of multipointed d-spaces f : x → y induces for each ` > 0 a continuous map p`f : p`x → p`y by composition by f (in fact by |f|). note that p1α,βx = p g α,βx, that there is a homeomorphism p ` α,βx ∼= pgα,βx for all ` > 0, and that for any topological space z, we have the homeomorphism p`0,1(glob g(z)) ∼= g(`, 1) ×z for any ` > 0 by proposition 2.12. the definition above of an execution path of length ` > 0 is not restrictive. indeed, we have: homotopy theory of moore flows (ii) 181 proposition 4.9. let x be a multipointed d-space. let φ : [0,`] ∼=+ [0,`]. let γ ∈ p`x. then γφ ∈ p`x. proof. by definition of p`x, there exists γ ∈ pgx such that γ = γµ`. we obtain γφ = γµ`φµ −1 ` µ`. since µ`φµ −1 ` ∈ g(1, 1), we deduce that γµ`φµ −1 ` ∈ p gx and that γφ ∈ p`x. proposition 4.10. let x be a multipointed d-space. let γ1 and γ2 be two execution paths of x with γ1(1) = γ2(0). let `1,`2 > 0. then( γ1µ`1 ∗γ2µ`1 ) µ−1`1+`2 is an execution path of x. proof. let φ1 : [0, 1 2 ] ∼=+ [0,`1] and φ2 : [0, 12 ] ∼=+ [0,`2]. then we have φ1 ⊗φ2 : [0, 1] ∼=+ [0,`1 + `2]. we obtain the sequence of equalities( (γ1µ`1 )∗(γ2µ`2 ) ) µ−1`1+`2 = ( (γ1µ`1 ) ∗ (γ2µ`2 ) )( φ1 ⊗φ2 )( φ1 ⊗φ2 )−1 µ−1`1+`2 = ( (γ1µ`1φ1) ∗ (γ2µ`1φ2) )( φ1 ⊗φ2 )−1 µ−1`1+`2 = ( (γ1µ`1φ1µ −1 1 2 µ1 2 ) ∗ (γ2µ`2φ2µ −1 1 2 µ1 2 ) )( φ1 ⊗φ2 )−1 µ−1`1+`2 = ( (γ1 µ`1φ1µ −1 1 2︸︷︷︸ ∈g(1,1) ) ∗n (γ2 µ`2φ2µ −1 1 2︸︷︷︸ ∈g(1,1) ) )( φ1 ⊗φ2 )−1 µ−1`1+`2︸︷︷︸ ∈g(1,1) , the first equality because φ1⊗φ2 is invertible, the second equality by proposition 3.4, the third equality because µ1 2 is invertible, and finally the last equality by proposition 3.7. the proof is complete because the set of execution paths of x is invariant by the action of g(1, 1). proposition 4.11. let x be a multipointed d-space. let `1,`2 > 0. the moore composition of continuous maps yields a continuous maps p`1x ×p`2x → p`1+`2x. proof. it is a consequence of definition 4.8 and proposition 4.10. theorem 4.12. let x be a multipointed d-space. then the following data 182 p. gaucher • the set of states x0 of x; • for all α,β ∈ x0 and all real numbers ` > 0, let p`α,βm g(x) := p`α,βx; • for all maps [0,`] ∼=+ [0,`′], a map f : [0,`′] → |x| of p` ′ α,βm g(x) is mapped to the map [0,`] ∼=+ [0,`′] f →|x| of p`α,βm g(x); • for all α,β,γ ∈ x0 and all real numbers `,`′ > 0, the composition maps ∗ : p`α,βm g(x) ×p` ′ β,γm g(x) −→ p`+` ′ α,γ m g(x) of proposition 4.11; assemble to a moore flow mg(x). this mapping induces a functor mg : gdtop −→gflow which is a right adjoint. note that the left adjoint mg! : gflow → gdtop preserves the set of states as well as the functor mg : gdtop →gflow. proof. these data give rise to a g-space pα,βmg(x) for each pair (α,β) of states of x0 and, thanks to proposition 4.11, to an associative composition law ∗ : p`1α,βm g(x)×p`2β,γm g(x) → p`1+`2α,γ mg(x) which is natural with respect to (`1,`2). by [14, section 6], these data assemble to a moore flow. since limits and colimits of g-spaces are calculated objectwise, the functor mg : gdtop → gflow is limit-preserving and accessible by theorem 2.16. therefore it is a right adjoint by [1, theorem 1.66]. proposition 4.13. let x be a multipointed d-space. let ` > 0 be a real number. let z be a topological space. then there is a bijection of sets gdtop(globg` (z),x) ∼= ⊔ (α,β)∈x0×x0 top(z,p`α,βx) which is natural with respect to z and x. proof. a map f of multipointed d-spaces from globg` (z) to x is determined by: homotopy theory of moore flows (ii) 183 • the image by f of 0 and 1 which will be denoted by α and β respectively; • a continuous map (still denoted by f) from |globg` (z)| to |x| such that for all x ∈ z and all φ : [0, 1] ∼=+ [0,`], the map t 7→ f(x,φ(t)) from [0, 1] to |x| belongs to pgα,βx. by definition of p`α,βx, for every x ∈ z, the continuous map f(x,−) from [0,`] to |x| belongs to p`α,βx since f(x,−) = f(x,φ(−)).φ −1 for any φ : [0, 1] ∼=+ [0,`]. since f is continuous and since top is cartesian closed, the mapping x 7→ f(x,−) actually yields a continuous map from z to p`α,βx. conversely, starting from a continuous map g : z → p`α,βx, one can define a map of multipointed d-spaces from globg` (z) to x by taking 0 and 1 to α and β respectively and by taking (x,t) ∈ |globg` (z)| to g(x)(t). we want to recall for the convenience of the reader: proposition 4.14. ([14, proposition 6.10]) let d : gop → top be a g-space. let x be a moore flow. then there is the natural bijection gflow(glob(d),x) ∼= ⊔ (α,β)∈x0×x0 [gop, top]0(d,pα,βx). proposition 4.15. for all topological spaces z and all ` > 0, there are the natural isomorphisms mg(globg` (z)) ∼= glob(f gop ` (z)) and m g ! (glob(f gop ` (z))) ∼= globg` (z). proof. by definition of mg and by proposition 2.12, the only nonempty path g-space of mg(globg` (z)) is p0,1mg(globg` (z)) = g(−,`) ×z and we obtain the first isomorphism. there is the sequence of natural bijections, for any multipointed d-space x, gdtop ( mg! (glob(f gop ` (z))),x ) ∼= gflow(glob(fgop` (z)),mgx) ∼= ⊔ (α,β)∈x0×x0 [gop, top]0 ( fg op ` (z),pα,βx ) ∼= ⊔ (α,β)∈x0×x0 top(z,p`α,βx) ∼= gdtop(globg` (z),x), 184 p. gaucher the first bijection by adjunction, the second bijection by proposition 4.14, the third bijection by proposition 4.2 and the last bijection by proposition 4.13. the proof of the second isomorphism is then complete thanks to the yoneda lemma. 5. cellular multipointed d-spaces let λ be an ordinal. in this section, we work with a colimit-preserving functor x : λ −→gdtop such that: • the multipointed d-space x0 is a set, in other terms x0 = (x0,x0,∅) for some set x0; • for all ν < λ, there is a pushout diagram of multipointed d-spaces globg(snν−1) �� gν // xν �� globg(dnν ) ĝν // xν+1 with nν > 0. let xλ = lim−→ν<λ xν. note that for all ν 6 λ, there is the equality x 0 ν = x0. denote by cν = ∣∣globg(dnν )∣∣\∣∣globg(snν−1)∣∣ the ν-th cell of xλ. it is called a globular cell. like in the usual setting of cw-complexes, ĝν induces a homeomorphism from cν to ĝν(cν) equipped with the relative topology which will be therefore denoted in the same way. it also means that ĝν(cν) equipped with the relative topology is ∆-generated. the closure of cν in |xλ| is denoted by ĉν = ĝν (∣∣globg(dnν )∣∣). the boundary of cν in |xλ| is denoted by ∂cν = ĝν (∣∣globg(snν−1)∣∣). the state ĝν(0) ∈ x0 (ĝν(1) ∈ x0 resp.) is called the initial (final resp.) state of cν. the integer nν + 1 is called the dimension of the globular cell cν. it homotopy theory of moore flows (ii) 185 is denoted by dim cν. the states of x 0 are also called the globular cells of dimension 0. definition 5.1. the cellular multipointed d-space xλ is finite if λ is a finite ordinal and x0 is finite. proposition 5.2. the space |xλ| is a cellular space. it contains x0 as a discrete closed subspace. the space |xλ| is weakly hausdorff. for every 0 6 ν1 6 ν2 6 λ, the continuous map |xν1| → |xν2| is a q-cofibration of spaces, and in particular a closed t1-inclusion. proof. by [9, theorem 8.2], the continuous map∣∣globg(snν−1)∣∣ −→ ∣∣globg(dnν )∣∣ is a q-cofibration of spaces for all ν > 0 between cellular spaces. since the functor x 7→ |x| is colimit-preserving, the space |xλ| is a cellular space. it is therefore weakly hausdorff. for every 0 6 ν1 6 ν2 6 λ, the continuous map |xν1| → |xν2| is a transfinite composition of q-cofibrations, and hence a qcofibration. the map x0 → xλ is a transfinite composition of q-cofibrations, and therefore a q-cofibration, and in particular a closed t1-inclusion. every subset of x0 is closed since x0 is equipped with the discrete topology. consequently, x0 is a discrete closed subspace of |xλ|. proposition 5.3. for all 0 6 ν1 6 ν2 6 λ, there is the equality pgxν1 = p gxν2 ∩ top([0, 1], |xν1|). proof. it is trivial for ν1 = ν2. for ν2 = ν1 + 1, there is a pushout diagram of multipointed d-spaces globg(snν1−1) �� gν1 // xν1 �� globg(dnν1 ) ĝν1 // xν2. the equality holds because the set of execution paths of xν2 is obtained as a ω-final structure. we conclude by a transfinite induction on ν2. proposition 5.4. for all 0 6 ν1 6 ν2 6 λ, the continuous map pgxν1 → pgxν2 is a ∆-inclusion. 186 p. gaucher proof. consider a set map [0, 1] → pgxν1 such that the composite set map [0, 1] −→ pgxν1 −→ p gxν2 is continuous. then by adjunction, we obtain a continuous map [0, 1] × [0, 1] −→|xν2|. by hypothesis, it factors as a composite of set maps [0, 1] × [0, 1] −→|xν1| −→ |xν2|. by proposition 5.2, the left-hand map is continuous since [0, 1] × [0, 1] is compact. the proof is complete by proposition 5.3 and proposition 2.2. proposition 5.5. let k be a compact subspace of |xλ|. then k intersects finitely many cν. proof. we mimick the proof of [21, proposition a.1] for the transfinite case. assume that there exists an infinite set s = {mj : j > 0} with mj ∈ k ∩ cνj , where (νj)j>0 is a sequence of mutually distinct ordinals. by transfinite induction on ν > 0, let us prove that s ∩|xν| is a closed subset of |xν|. the assertion is trivial for ν = 0. there is the pushout diagram of spaces for all ν < λ |globg(snν−1)| �� gν // |xν| �� |globg(dnν )| ĝν // |xν+1|. by induction hypothesis, g−1ν (s ∩ |xν|) is a closed subset of |glob g(snν−1)| and ĝν −1(s ∩ |xν+1|) is equal to g−1ν (s ∩ |xν|) union at most one point. therefore, s ∩|xν+1| is a closed subset of |xν+1| because the latter space is equipped with the final topology by proposition 5.2. suppose that we have proved that for all ν < ν′, s∩|xν| is a closed subset of |xν| where ν′ is a limit ordinal. then, since the topology of |xν′| is the final topology (it is a tower of one-to-one maps), s∩|xν′| is a closed subset of |xν′|. thus, by transfinite induction on ν > 0, we prove that s is closed in |xν| for all 0 6 ν 6 λ. the same argument proves that every subset of s is closed in |xλ|. thus s has the discrete topology. but it is compact, being a closed subset of the compact space k, and therefore finite. contradiction. homotopy theory of moore flows (ii) 187 colimits of multipointed d-spaces are calculated by taking the colimit of the underlying spaces and of the sets of states and by taking the ω-final structure which is generated by the free finite compositions of execution paths. consequently, the composite functor gdtop p g −−−→ top ⊂−−−→ set is finitely accessible. it is unlikely that the functor pg : gdtop → top, which is a right adjoint by theorem 2.16, is finitely accessible. however, we have: theorem 5.6. the composite functor λ x−−−→gdtop p g −−−→ top is colimit-preserving. in particular the continuous bijection lim−→(p g.x) −→ pg lim−→x is a homeomorphism. moreover the topology of pg lim−→x is the final topology. note that theorem 5.6 holds both for ∆-generated spaces and ∆-hausdorff ∆-generated spaces. proof. consider the set of ordinals{ ν 6 λ : ν limit ordinal and lim−→ ν′<ν (pgxν′) −→ pgxν not isomorphism } assume this set nonempty. let ν be its smallest element. the topology of lim−→ν′<ν p gxν′ is the final topology because the continuous maps pgxν′ → pgxν′+1 are one-to-one. let f : [0, 1] → pgxν be a continuous map. therefore the composite map [0, 1] f −−→ pgxν ⊂ top([0, 1], |xν|) is continuous. it gives rise by adjunction to a continuous map [0, 1]× [0, 1] → |xν|. since the functor x : λ → gdtop is colimit-preserving, there is the homeomorphism |xν| ∼= lim−→ν′<ν |xν′|. since [0, 1] × [0, 1] is compact, the latter continuous map then factors as a composite [0, 1] × [0, 1] → |xν′| → |xν| for some ordinal ν′ < ν by proposition 5.2. since pgxν′ = pgxν ∩ 188 p. gaucher top([0, 1], |xν′|) by proposition 5.3, f factors as a composite [0, 1] → pgxν′ → pgxν. using corollary 2.3. we obtain the homeomorphism lim−→ν′<ν p gxν′ → pgxν: contradiction. theorem 5.7. the composite functor λ x−−→gdtop m g −−−−→gflow is colimit-preserving. in particular the natural map lim−→ ν<λ mg(xν) −→ mgxλ is an isomorphism. proof. theorem 5.6 states that there is the homeomorphism lim−→ ν<λ pgxν −→ pgxλ. we have, by definition of the functor mg, the equality of functors pg = p1.mg. it means that there is the homeomorphism lim−→ ν<λ p1mg(xν) −→ p1mg(xλ). since all maps the reparametrization category g are isomorphisms, we obtain for all ` > 0 the homeomorphism lim−→ ν<λ p`mg(xν) −→ p`mg(xλ). since colimits of g-spaces are calculated objectwise, we obtain the isomorphism of g-spaces lim−→ ν<λ pmgxν −→ pmgxλ. the proof is complete thanks to the universal property of the colimits. definition 5.8. an execution path γ of a multipointed d-space x is minimal if γ(]0, 1[) ∩x0 = ∅. homotopy theory of moore flows (ii) 189 for any (q-cofibrant or not) topological space z, every execution path of the multipointed d-space globg(z) is minimal. the following theorem proves that execution paths of cellular multipointed d-spaces have a normal form. theorem 5.9. let γ be an execution path of xλ. then there exist minimal execution paths γ1, . . . ,γn and `1, . . . ,`n > 0 with ∑ i `i = 1 such that γ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n). moreover, if there is the equality γ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n) = (γ ′ 1µ`′1 ) ∗ · · · ∗ (γn′µ`′n′) such that all γ′j are also minimal and with ` ′ 1, . . . ,` ′ n′ > 0, then n = n ′ and γi = γ ′ i and `i = ` ′ i for all 1 6 i 6 n. proof. the set of execution paths of xλ is obtained as a ω-final structure. using theorem 3.9, we obtain γ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n) for some n > 1 with `1 + · · · + `n = 1 such that for all 1 6 i 6 n, there exists a globular cell cνi such that γi(]0, 1[) ⊂ cνi, γi(0) = ĝνi(0), γi(0) = ĝνi(1). therefore there exists a finite set {t0, . . . , tn} with t0 = 0 < t1 < · · · < tn = 1 and n > 1 such that γ([0, 1])∩x0 = {γ(ti) : 0 6 i 6 n}. we necessarily have `i = ti − ti−1 for 1 6 i 6 n. let `0 = 0. then we deduce that ∑ j ν0. it means that there exists a point ĝν(z,t) of ∂cν which belongs to cν0 with z ∈ snν−1 and, since cν0 ∩ x0 = ∅, with t ∈]0, 1[. therefore the carrier of the execution path ĝνδz contains the globular cell cν0 . we deduce that there exists φ ∈g(1, 1) such that ĝνδzφ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n) with γ = γi for at least one i ∈{1, . . . ,n}, γ1, . . . ,γn minimal and ∑ i `i = 1. in particular, we deduce that γ([0, 1]) ⊂ ∂cν: we have proved (1) ⇒ (2). definition 5.15. let cν be a globular cell of xλ. let 0 < h < 1. let ĉν[h] = { ĝν(z,h) : (z,h) ∈ |globg(dnν )| } . it is called an achronal slice of the globular cell cν. proposition 5.16. for any globular cell cν of xλ and any minimal execution path γ and any h ∈]0, 1[, the cardinal of the set{ t ∈]0, 1[ : γ(t) ∈ ĉν[h] } is at most one. in other terms, a minimal execution path of xλ intersects any achronal slice at most one time. remember that execution paths of xλ are locally injective, i.e., they do not contain zero speed points. proposition 5.16 does not hold in general for a non-minimal execution path because it could go back to the initial state of the globular cell after reaching its final state, which moreover could be equal to the initial state of the globular cell. proof. if the set γ(]0, 1[) ∩ ĉν[h] is nonempty, then the minimal execution path γ has at least one point of γ(]0, 1[) belonging to ĉν. if [cν] is the carrier of γ, then γ = δzφ with z ∈ dnν\snν−1 and φ ∈g(1, 1). we then have{ t ∈]0, 1[ : γ(t) ∈ ĉν[h] } = { φ−1(h) } . otherwise, by proposition 5.14, there is the inclusion γ([0, 1]) ⊂ ∂cν and there exists an execution path ĝνδzφ for some z ∈ snν−1 and φ ∈ g(1, 1) from the 192 p. gaucher initial state of cν to its final state with ĝνδzφ = (γ1µ`1 ) ∗ · · · ∗ (γnµ`n) with all γi minimal and γ ∈ {γ1, . . . ,γn}. since γ(]0, 1[) ∩ ĉν[h] is nonempty, we have h ∈ ] φ (∑ j 1. consider the set j1 = { h ∈]0, 1[ : ĉν[h] ∩ ( x0\ { ĝν(0) }) 6= ∅ } . if j1 is nonempty, then consider a sequence (h 1 n)n>0 of j1 converging to the greatest lower bound inf j1 of j1. for all n > 0, let z 1 n ∈ snν−1 such that ĝν ( z1n,h 1 n ) ∈ x0\ { ĝν(0) } . by extracting a subsequence, we can suppose that the sequence (z1n)n>0 converges to z1∞ ∈ snν−1. since the space |glob g(dnν )| is compact, the subspace ĉν is a compact subspace of the weakly hausdorff space |xλ|. the set ĉν ∩x0 is therefore finite because x0 is discrete in |xλ| by proposition 5.2. since( ĉν ∩x0 ) \ { ĝν(0) } ⊂ ĉν ∩x0 is discrete finite as well, the sequence (ĝν(z 1 n,h 1 n))n>0, which converges to ĝν(z 1 ∞, inf j1) by continuity of ĝν, eventually becomes constant. thus, ĝν ( z1∞, inf j1 ) 6= ĝν(0). it implies that inf j1 > 0. it means that whether j1 is empty or not, there exists a ∈]0, 1[ such that for 194 p. gaucher all h ∈]0,a], one has ĉν[h] ∩x0 ⊂{ĝν(0)}. consider the set j2 = { h ∈]0,a] : ĉν[h] ∩x0 = {ĝν(0)} } . if j2 is nonempty, then consider a sequence (h 2 n)n>0 of j2 converging to inf j2. for all n > 0, there exists z2n ∈ snν−1 such that ĝν ( z2n,h 2 n ) = ĝν(0). by extracting a subsequence, one can suppose that the sequence (z2n)n>0 of snν−1 converges to z2∞ ∈ snν−1. consider the sequence of globular cells (cνn)n>0 such that for all n > 0, the globular cell cνn is the first globular cell appearing in carrier(ĝνδz2n), i.e., carrier ( ĝνδz2n ) = [cνn,cn] where cn is a sequence of globular cells which is necessarily nonempty because h2n < 1. using proposition 5.5, we have that the compact subspace ĉν intersects finitely many globular cells of xλ. consequently, by extracting a subsequence again, we can suppose that the sequence of globular cells (cνn)n>0 is constant and equal to the globular cell cν′ for some ν ′ < ν. write ĝνδz2n = ( ĝν′δz′nφnµtn ) ∗ (γnµ1−tn) with, for all n > 0, 0 < tn 6 h 2 n < 1, z ′ n ∈ d nν′\snν′−1, φn ∈g(1, 1), ĝν′(0) = ĝν(0) (the globular cells cν and cν′ have the same initial state), ĝν′(1) = (ĝνδz2n)(tn), γn ∈ p g ĝν′(1),ĝν(1) x with carrier(γn) = [cn]. by extracting a subsequence, one can suppose that the sequence (z′n)n>0 of dnν′ converges to z′∞. since carrier(ĝν′δz′∞) exists (it is a sequence of globular cells intersecting ĉν′), the execution path ĝν′δz′∞ is not constant. thus, there exists t ∈]0, 1[ such that ĝν′ ( z′∞,t ) 6= ĝν′(0). by extracting again a subsequence, one can suppose that the sequence homotopy theory of moore flows (ii) 195 (tnφ −1 n (t))n>0 of [0, 1] converges to t∞. we have ĝνδz2n ( tnφ −1 n (t) ) = ĝν′δz′n ( φnµtn ( tnφ −1 n (t) )) = ĝν′(z ′ n,t) for all n > 0. we obtain by passing to the limit ĝνδz2∞(t∞) = ĝν′ ( z′∞,t ) . we deduce that ĝνδz2∞(t∞) 6= ĝν(0) and therefore that 0 < t∞. from the inequalities tnφ −1 n (t) 6 tn 6 h 2 n for all n > 0, we obtain by passing to the limit the inequalities 0 < t∞ 6 inf j2. it means that whether j2 is empty or not, there exists b ∈]0, 1[ such that for all h ∈]0,b], one has ĉν[h] ∩x0 = ∅. theorem 5.18. let γ∞ be an execution path of xλ. let ν0 < λ. there exists an open neighborhood ω of γ∞ in pgxλ such that for all execution paths γ ∈ ω, the number of copies of cν0 in the carrier of γ cannot exceed the length of the carrier of γ∞. proof. let carrier(γ∞) = [cν1, . . . ,cνn]. consider the decomposition of theorem 5.9 γ∞ = (γ 1 ∞µ`1 ) ∗ · · · ∗ (γ n ∞µ`n) with ∑ i `i = 1 and all execution paths γ i ∞ minimal for i = 1, . . . ,n. for 1 6 i 6 n, let νi < λ, φi ∈g(1, 1) and zi ∈ dnνi\snνi−1 such that carrier(γi∞) = [cνi], γ i ∞(]0, 1[) ⊂ cνi, γ i ∞ = δziφi. using proposition 5.17, pick h ∈]0, 1[ such that ĉν0 [h] ∩ x0 = ∅. for all 1 6 i 6 n, the set { t ∈]0, 1[ : γi∞(t) ∈ ĉν0 [h] } contains at most one point ti by proposition 5.16; if the set above is empty, 196 p. gaucher let ti = 1 2 . for all 1 6 i 6 n, let li and l ′ i be two real numbers such that 0 < li < ti < l ′ i < 1. for 1 6 i 6 n, consider the covering of the segment [ ∑ j0 be a sequence of execution paths of xλ which converges in pgxλ. let cν0 be a globular cell of xλ. let ik be the number of times that cν0 appears in carrier(γk). then the sequence of integers (ik)k>0 is bounded. proof. write γ∞ for the limit of (γk)k>0 in pgxλ. by theorem 5.18, there exists an open ω containing γ∞ such that for all γ ∈ ω, the number of copies of cν0 in the carrier of γ does not exceed the length of the carrier of γ∞. since the sequence (γk)k>0 converges to γ∞, there exists n > 0 such that for all k > n, γk belongs to ω. the proof is complete. theorem 5.20. let 0 6 ν < λ. then every execution path of xν+1 can be written in a unique way as a finite moore composition (f1γ1µ`1 ) ∗ · · · ∗ (fnγnµ`n) with n > 1 such that (1) ∑ i `i = 1; (2) fi = f and γi is an execution path of xν or fi = ĝν and γi = δziφi with zi ∈ dnν\snν−1 and some φ ∈g(1, 1); (3) for all 1 6 i < n, either fiγi or fi+1γi+1 (or both) is (are) of the form ĝνδzφ for some z ∈ dnν\snν−1 and some φ ∈ g(1, 1): intuitively, there is no possible simplification using the moore composition inside xν. proof. we use the normal form of theorem 5.9 and we use proposition 4.10 to compose successive execution paths of xν. 6. chains of globes let z1, . . . ,zp be p nonempty topological spaces with p > 1. consider the multipointed d-space x = globg(z1) ∗ · · · ∗ globg(zp), homotopy theory of moore flows (ii) 199 with p > 1 where the ∗ means that the final state of a globe is identified with the initial state of the next one by reading from the left to the right. let {α0,α1, . . . ,αp} be the set of states such that the canonical map globg(zi) → x takes the initial state 0 of globg(zi) to αi−1 and the final state 1 of globg(zi) to αi. as a consequence of the associativity of the semimonoidal structure on g-spaces recalled in theorem 4.3 and of [14, proposition 5.16], we have proposition 6.1. let u1, . . . ,up be p topological spaces with p > 1. let `1, . . . ,`p > 0. there is the natural isomorphism of g-spaces fg op `1 u1 ⊗···⊗fg op `p up ∼= fg op `1+···+`p(u1 ×···×up). the case p = 1 of proposition 6.3 is treated in proposition 2.12 and already used in proposition 4.15. an additional argument is required for the case p > 1. at first, we prove a lemma which is an addition to proposition 2.5. lemma 6.2. the set map (−)−1 : g(1,p) → g(p, 1) which takes f : [0, 1] ∼=+ [0,p] to its inverse f−1 : [0,p] ∼=+ [0, 1] is continuous. proof. since all ∆-generated spaces are sequential, it suffices to prove that (−)−1 : g(1,p) →g(p, 1) is sequentially continuous. let (fn)n>0 be a sequence of g(1,p) which converges to f ∈ g(1,p). let t ∈ [0,p]. then the sequence (f−1n (t))n>0 of [0, 1] has at least one limit point denoted by l(t). by extracting a subsequence of the sequence (fn(f −1 n (t)))n>0, we obtain f(l(t)) = t, which implies l(t) = f−1(t). thus every subsequence of (f−1n (t))n>0 has a unique limit point f−1(t). suppose that the sequence (f−1n (t))n>0 does not converge to f−1(t). then there exists an open neighborhood v of f−1(t) such that for all n > 0, f−1n (t) ∈ v c which is compact: contradiction. therefore the sequence (f−1n )n>0 pointwise converges to f −1. by proposition 2.5, we deduce that the sequence (f−1n )n>0 converges to f −1. proposition 6.3. with the notations of this section. there is a homeomorphism pgα0,αpx ∼= g(1,p) ×z1 ×···×zp. proof. the moore composition of paths induced a map of g-spaces p0,1mgglobg(z1) ⊗···⊗p0,1mgglobg(zp) −→ pα0,αpm g(x). 200 p. gaucher by proposition 4.15, there is the isomorphism of g-spaces p0,1mgglobg(z) ∼= fg op 1 z for all topological spaces z. we obtain a map of g-spaces fg op 1 z1 ⊗···⊗f gop 1 zp −→ pα0,αpm g(x). by proposition 6.1, and since p1α0,αpm g(x) = pgα0,αpx by definition of the functor mg, we obtain a continuous map ψ : g(1,p) ×z1 ×···×zp −→ pgα0,αpx (φ,z1, . . . ,zp) 7−→ (δz1φ1) ∗ · · · ∗ (δzpφp), where φi ∈ g(`i, 1) with ∑ i `i = 1 and φ = φ1 ⊗ ··· ⊗ φp being the decomposition of proposition 3.2. since all executions paths of globes are oneto-one, the map ψ above is a continuous bijection. the continuous maps zi → {0} for 1 6 i 6 p induce by functoriality a map of multipointed dspaces x → −→ i g ∗ · · · ∗ −→ i g (p times) and then a continuous map k : pgα0,αpx −→ p g α0,αp (−→ i g ∗ · · · ∗ −→ i g ) = g(1,p) (δz1φ1) ∗ · · · ∗ (δzpφp) 7−→ (δ0φ1) ∗ · · · ∗ (δ0φp) = φ1 ⊗···⊗φp. let i ∈{1, . . . ,p}. then we have, with γ = (δz1φ1) ∗ · · · ∗ (δzpφp), γ ( k(γ)−1 ( i− 1 2 )) = γ ( φ−1 ( i− 1 2 )) = δziφiφ −1 ( i− 1 2 ) = ( zi, 1 2 ) , the first equality by definition of k : pgα0,αpx → g(1,p), the second equality since i−1 < i− 1 2 < i and by definition of γ, and the last equality by definition of the φi’s. the set map pgα0,αpx −→ |glob g(zi)| γ 7−→ γ ( k(γ)−1 ( i− 1 2 )) is continuous since k : pgα0,αpx → g(1,p) and (−) −1 : g(1,p) → g(p, 1) are both continuous (see lemma 6.2 for the latter map). consequently, the set map k : pgα0,αpx −→ z1 ×···×zp (δz1φ1) ∗ · · · ∗ (δzpφp) 7−→ (z1, . . . ,zp) homotopy theory of moore flows (ii) 201 is continuous. it implies that the set map ψ−1 = (k,k) : (δz1φ1) ∗ · · · ∗ (δzpφp) 7−→ (φ1 ⊗···⊗φp,z1, . . . ,zp) is continuous as well and that ψ is a homeomorphism. until the end of this section, we work like in section 5 with a cellular multipointed d-space xλ, with the attaching map of the globular cell cν for ν < λ denoted by ĝν : glob g(dnν ) −→ xλ. each carrier c = [cν1, . . . ,cνn] gives rise to a map of multipointed d-spaces from a chain of globes to xλ ĝc : glob g(dnν1 ) ∗ · · · ∗ globg(dnνn ) −→ xλ by “concatenating” the attaching maps of the globular cells cν1, . . . ,cνn. let αi−1 (αi resp.) be the initial state (the final state resp.) of glob g(dnνi ) for 1 6 i 6 n in globg(dnν1 ) ∗ · · · ∗ globg(dnνn ). it induces a continuous map pgĝc : xc := pgα0,αn ( globg(dnν1 ) ∗ · · · ∗ globg(dnνn ) ) −→ pgxλ. proposition 6.4. let γ be an execution path of xλ. consider a nondecreasing set map φ : [0, 1] → [0, 1] preserving the extremities such that γφ = γ. then φ is the identity of [0, 1]. proof. note that it is not assumed that φ is continuous. suppose that there exist t < t′ such that φ(t) = φ(t′). then for t′′ ∈ [t,t′], γ(t′′) = γ(φ(t′′)) = γ(φ(t)) because φ(t) 6 φ(t′′) 6 φ(t′), which contradicts the fact that γ is locally injective by proposition 5.13. thus the set map φ is strictly increasing. let carrier(γ) = [cν1, . . . ,cνn]. let γ = (γ1µ`1 )∗· · ·∗(γnµ`n) with `1+· · ·+`n = 1 such that for all 1 6 i 6 n, there exist zi ∈ dnνi\snνi−1 and φi ∈g(1, 1) such that for all t ∈]0, 1[, γi(t) = (zi,φi(t)) ∈ cνi, γi(0) = ĝνi(0) and γi(1) = ĝνi(1). then { t ∈ [0, 1] : γ(t) ∈ x0 } = {0 = t0 < t1 < · · · < tn = 1} with ti = ∑ 16j6i `j for 0 6 i 6 n. we deduce that 0 = φ(t0) < φ(t1) < · · · < φ(tn) = 1 because the set map φ is strictly increasing. since γ(φ(ti)) = γ(ti) ∈ x0 for 0 6 i 6 n, one obtains φ(ti) = ti for 0 6 i 6 n and φ(]ti−1, ti[) ⊂ 202 p. gaucher ]ti−1, ti[ for all 1 6 i 6 n. then, observe that (zi,φi(φ(t))) = (zi,φi(t)), ∀1 6 i 6 n, ∀t ∈]ti−1, ti[ . since φi is bijective, it means that the restriction φ� ]ti−1,ti[ is the identity of ]ti−1, ti[ for all 1 6 i 6 n. notation 6.5. let φ be a set map from a segment [a,b] to a segment [c,d]. let φ(x−) = sup{φ(t) : t < x}, φ(x+) = inf{φ(t) : x < t}. theorem 6.6. let γ1 and γ2 be two execution paths of xλ such that there exist two nondecreasing set maps φ1,φ2 : [0, 1] → [0, 1] preserving the extremities such that γ1(φ1(t)) = γ2(t) and γ1(t) = γ2(φ2(t)) for all t ∈ [0, 1]. then φ1,φ2 ∈g(1, 1) and φ2 = φ−11 . proof. note that it is not assumed that φ1 and φ2 are continuous. for all t ∈ [0, 1], we have γ1(φ1(φ2(t))) = γ2(φ2(t)) = γ1(t). using proposition 6.4, we deduce that φ1φ2 = id[0,1]. in the same way, we have φ2φ1 = id[0,1]. this proves that φ1 and φ2 are two bijective set maps preserving the extremities which are inverse to each other. suppose e.g., that there exists t ∈ [0, 1] such that φ1(t −) < φ1(t). then φ1 cannot be surjective: contradiction. by using similar arguments, we deduce that for all t ∈ [0, 1], φ1(t−) = φ1(t) = φ1(t+) and φ2(t −) = φ2(t) = φ2(t +). consequently, the set maps φ1 and φ2 are continuous. proposition 6.7. let c be the carrier of some execution path of xλ. every execution path of the image of pgĝc is of the form( ĝν1δz1 ∗ · · · ∗ ĝνnδzn ) φ with φ ∈g(1,n) and zi ∈ dnνi for 1 6 i 6 n. proof. the first assertion is a consequence of the definition of ĝc and of proposition 6.3. homotopy theory of moore flows (ii) 203 notation 6.8. let c be the carrier of some execution path of xλ. using the identification provided by the homeomorphism of proposition 6.3, we can use the notation (pgĝc)(φ,z1, . . . ,zn) = (ĝν1δz1 ∗ · · · ∗ ĝνnδzn)φ. before proving the main theorem of this section, we need the following topological lemmas: lemma 6.9. let u1, . . . ,up be p first-countable ∆-hausdorff ∆-generated spaces with p > 1. then the product u1 ×···× up in the category top of general topological spaces and continuous maps coincides with the product in top. proof. consider u1 ×···×up equipped with the product topology in the category top of general topological spaces and continuous maps. this topology is first-countable as a finite product of first-countable topologies. each space ui is locally path-connected, being ∆-generated. thus, the finite product u1 ×···×up equipped with the product topology in top is locally pathconnected. we deduce that u1 ×···×up equipped with the product topology in top is ∆-generated: the ∆-kelleyfication functor is not required. moreover since each ui is ∆-hausdorff, the product in top is ∆-hausdorff as well. it means that u1×···×up equipped with the product topology in top coincides with the product in top. lemma 6.10. let u1, . . . ,up be p first-countable ∆-hausdorff ∆-generated spaces with p > 1. let (uin)n>0 be a sequence of ui for 1 6 i 6 p which converges to ui∞ ∈ ui. then the sequence ((u1n, . . . ,u p n))n>0 converges to (u1∞, . . . ,u p ∞) ∈ u1 ×···×up for the product calculated in top. note that the converse is obvious: if the sequence ((u1n, . . . ,u p n))n>0 converges to (u1∞, . . . ,u p ∞) ∈ u1 ×···×up, then the sequences (uin)n>0 converge to ui∞ ∈ ui for all 1 6 i 6 p because of the existence of the projection maps u1 ×···×up → ui for all 1 6 i 6 p. a sequence converges to some point in a ∆-generated space if and only if the corresponding application from the onepoint compactification n = n∪{∞} of the discrete space n to the ∆-generated space is continuous. the point is that the one-point compactification of n is not ∆-generated: its ∆-kelleyfication is a discrete space. therefore it does not seem possible to use the universal property of the finite product in top to prove lemma 6.10. 204 p. gaucher proof. each convergent sequence gives rise to a continuous map n → ui for 1 6 i 6 p. we obtain a continuous map n → u1 ×···×up by using the universal property of the finite product in top thanks to lemma 6.9 and the proof is complete. the sequence (φk)k>1 of g(1, 1) depicted in figure 2 has no limit point because the only possibility is the set map which takes 0 to 0 and the other points of [0, 1] to 1: it does not belong to g(1, 1). thus, the topological space g(1,n), which is homeomorphic to g(1, 1) for all n > 1, is not sequentially compact. however, theorem 6.11 holds anyway. (0 , 0) ϕk (1 , 1) 1 − 1 k 1 k figure 2: a sequence (φk)k>1 of g(1, 1) without limit point theorem 6.11. let c be the carrier of some execution path of xλ. (1) consider a sequence (γk)k>0 of the image of pgĝc which converges pointwise to γ∞ in pgxλ. let γk = ( pgĝc )( φk,z 1 k, . . . ,z n k ) with φk ∈g(1,n) and zik ∈ d nνi for 1 6 i 6 n and k > 0. then there exist φ∞ ∈g(1,n) and zi∞ ∈ d nνi for 1 6 i 6 n such that γ∞ = ( pgĝc )( φ∞,z 1 ∞, . . . ,z n ∞ ) and ( φ∞,z 1 ∞, . . . ,z n ∞ ) is a limit point of the sequence (( φk,z 1 k, . . . ,z n k )) k>0 . (2) the image of pgĝc is closed in pgxλ. homotopy theory of moore flows (ii) 205 proof. (1) by a cantor diagonalization argument, we can suppose that: • the sequence (zik)k>0 converges to z i ∞ ∈ d nνi for each 1 6 i 6 n; • the sequence (φk(r))k>0 converges to a real number denoted by φ∞(r) ∈ [0,m] for each r ∈ [0, 1] ∩q; • the sequence (φ−1k (r))k>0 converges to a real number denoted by φ −1 ∞ (r) ∈ [0, 1] for each r ∈ [0,n] ∩q. since the sequence of execution paths (γk)k>0 converges pointwise to γ∞, we obtain γ∞(r) = ( ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞ ) (φ∞(r)) for all r ∈ [0, 1] ∩q, γ∞ ( φ−1∞ (r) ) = ( ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞ ) (r) for all r ∈ [0,n] ∩q. for r1 < r2 ∈ [0, 1]∩q, φk(r1) < φk(r2) for all k > 0. therefore by passing to the limit, we obtain φ∞(r1) 6 φ∞(r2). note that φ∞(0) = 0 and φ∞(1) = n since 0, 1 ∈ q. in the same way, we see that φ−1∞ : [0,n] ∩ q → [0, 1] is nondecreasing and that φ−1∞ (0) = 0 and φ −1 ∞ (n) = 1. for t ∈]0, 1[, let us extend the definition of φ∞ as follows: φ∞(t) = sup { φ∞(r) : r ∈ ]0, t] ∩q } . the upper bound exists since {φ∞(r) : r ∈ ]0, t] ∩ q} ⊂ [0,n]. for each t ∈ [0, 1]\q, there exists a nondecreasing sequence (rk)k>0 of rational numbers converging to t. then lim k→∞ φ∞(rk) = φ∞(t). by continuity, we deduce that γ∞(t) = ( ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞ ) (φ∞(t)) for all t ∈ [0, 1]. it is easy to see that the set map φ∞ : [0, 1] → [0,n] is nondecreasing and that it preserves extremities. for t ∈]0, 1[, extend the definition of φ−1∞ as well as follows: φ−1∞ (t) = sup { φ−1∞ (r) : r ∈ ]0, t] ∩q } . the upper bound exists since {φ−1∞ (r) : r ∈ ]0, t] ∩ q} ⊂ [0, 1]. for each t ∈ [0,n]\q, there exists a nondecreasing sequence (rk)k>0 of rational numbers converging to t. then lim k→∞ φ−1∞ (rk) = φ −1 ∞ (t). 206 p. gaucher by continuity, we deduce that γ∞ ( φ−1∞ (t) ) = ( ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞ ) (t) for all t ∈ [0,n]. it is easy to see that the set map φ−1∞ : [0,n] → [0, 1] is nondecreasing and that it preserves extremities. we obtain for all t ∈ [0, 1] γ∞(t) = ( ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞ )( µ−1n µnφ∞(t) ) , γ∞ ( φ−1∞ µ −1 n (t) ) = ( ĝν1δz1∞ ∗ · · · ∗ ĝνnδzn∞ )( µ−1n (t) ) . using theorem 6.6, we obtain that µnφ∞ : [0, 1] → [0, 1] and φ−1∞ µ−1n : [0, 1] → [0, 1] are homeomorphisms which are inverse to each other. we deduce that φ∞ : [0, 1] → [0,n] and φ−1∞ : [0,n] → [0, 1] are homeomorphisms which are inverse to each other. let t ∈ [0, 1]\q. since the sequence (φk(t))k>0 belongs to the sequential compact [0,n], it has at least one limit point `. there exists a subsequence of (φk(t))k>0 which converges to `. we obtain φ∞(r) 6 ` 6 φ∞(r ′), ∀r ∈ [0, t] ∩q, ∀r′ ∈ [t, 1] ∩q. since φ∞ ∈g(1,n) and by density of q, we deduce that ` = φ∞(t) necessarily. now suppose that the sequence (φk(t))k>0 does not converge to φ∞(t). then there exists an open neighborhood v of φ∞(t) in [0,n] such that for all k > 0, φk(t) /∈ v . we deduce that the sequence (φk(t))k>0 of [0,n] has no limit point: contradiction. we have proved that the sequence (φk)k>0 converges pointwise to φ∞. using proposition 2.5, we deduce that (φk)k>0 converges uniformly to φ∞. we deduce that (φ∞,z 1 ∞, . . . ,z n ∞) is a limit point of the sequence( (φk,z 1 k, . . . ,z n k ) ) k>0 in g(1,n) × dnν1 × ···× dnνn by proposition 2.5 and lemma 6.10. (2) let ( pgĝc(γn) ) n>0 be a sequence of ( pgĝc )( xc ) which converges in pgxλ. the limit γ∞ ∈ pgxλ of the sequence of execution paths ( pgĝc(γn) ) n>0 is also a pointwise limit. we can suppose by extracting a subsequence that the sequence (γn)n>0 of xc is convergent in xc. thus, by continuity of pgĝc, we obtain γ∞ = ( pgĝc ) (γ∞) for some γ∞ ∈ xc. we deduce that pgĝc ( xc ) is sequentially closed in pgxλ. since pgxλ is sequential, being a ∆-generated space, the proof is complete. corollary 6.12. suppose that xλ is a finite cellular multipointed dspace without loops. then the topology of pgxλ is the topology of the pointwise convergence which is therefore ∆-generated. homotopy theory of moore flows (ii) 207 we do not know whether the “without loops” hypothesis can be removed and whether finite can be replaced by locally finite. proof. let (γn)n>0 be a sequence of execution paths of xλ which pointwise converges to γ∞. since xλ is finite and without loop, the set t = { carrier(γ) : γ ∈ pgxλ } is finite. we obtain a finite covering by (closed) subsets pgxλ = ⋃ c∈t ( pĝc ) (xc) because each execution path has a carrier by theorem 5.9. suppose that (γn)n>0 does not converge to γ∞ in pgxλ. then there exists an open neighborhood v of γ∞ in pgxλ such that for n > 0 and all γn /∈ v . since t is finite, one can suppose by extracting a subsequence that ∃c ∈t such that γn ∈ ( pĝc ) (xc) for all n > 0. by theorem 6.11, the sequence (γn)n>0 has a limit point which belongs to the complement of v which is closed. this limit point is necessarily the pointwise limit γ∞. we obtain γ∞ /∈ v : contradiction. corollary 6.12 can be viewed as a second dini theorem for the space of execution paths of a finite cellular multipointed d-space without loops. indeed, if xλ = −→ i g (the directed segment), then pgxλ = g(1, 1) and we recover the fact that the topology of g(1, 1) coincides with the pointwise convergence topology by proposition 2.5. 7. the unit and the counit of the adjunction on q-cofibrant objects consider in this section the following situation: a pushout diagram of multipointed d-spaces globg(sn−1) �� g // a f �� globg(dn) ĝ // x 208 p. gaucher with n > 0 and a cellular. note that a0 = x0. let d = fg op 1 s n−1 and e = fg op 1 d n. consider the moore flow x defined by the pushout diagram of figure 3 where the two equalities mg(globg(sn−1)) = glob(d), mg(globg(dn)) = glob(e) come from proposition 4.15 and where the map ψ is induced by the universal property of the pushout. mg(globg(sn−1)) = glob(d) �� mg(g) // mg(a) mg(f) �� f �� mg(globg(dn)) = glob(e) g // mg(ĝ) // x ψ o o o ''o o o mg(x). figure 3: definition of x the g-space of execution paths of the moore flow x can be calculated by introducing a diagram of g-spaces df over a reedy category pg(0),g(1)(a0) whose definition is recalled in appendix a. let t be the g-space defined by the pushout diagram of [gop, top]0 d �� pmg(g) // pg(0),g(1)mg(a) pf �� e pg // t. consider the diagram of spaces df : pg(0),g(1)(a0) → [gop, top]0 defined as follows: df ((u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un)) = zu0,u1⊗zu1,u2⊗···⊗zun−1,un with zui−1,ui = { pui−1,uim g(a) if �i = 0, t if �i = 1. homotopy theory of moore flows (ii) 209 in the case �i = 1, (ui−1,ui) = (g(0),g(1)) by definition of pg(0),g(1)(a0). the inclusion maps i′is are induced by the map pf : pg(0),g(1)m g(a) → t. the composition maps c′is are induced by the compositions of paths of the moore flow mg(a). theorem 7.1. ([14, theorem 9.7]) we obtain a well-defined diagram of g-spaces df : pg(0),g(1)(a0) −→ [gop, top]0. there is the isomorphism of g-spaces lim−→d f ∼= px. by the universal property of the pushout, we obtain a canonical map of g-spaces pψ : lim−→d f −→ pmgx. the goal of theorem 7.2 and of theorem 7.3 is to prove that the canonical map of g-spaces pψ is an isomorphism of g-spaces. the proof is twofold: at first, it is proved that it is an objectwise continous bijection, and then that it is an objectwise homeomorphism. theorem 7.2. under the hypotheses and the notations of this section. the map of g-spaces pψ : lim−→d f −→ pmg(x) is an objectwive bijection. proof. throughout the proof, the reader must keep in mind that for any map of multipointed d-spaces q : x1 → x2 and for any execution path γ ∈ p`x1 = p`mg(x1) of length ` of the multipointed d-space x1, or equivalently of the moore flow mg(x1), there is by definition 4.8 and theorem 4.12 the tautological equality( pmg(q) ) (γ) = |q|.γ, the right-hand term meaning the composite of the underlying continuous map |q| : |x1| → |x2| between the underlying spaces of x1 and x2 with the execution path γ : [0, 1] →|x1|. it will be denoted qγ or q.γ, as it was always done so far. in other terms, we will be using the abuse of notation pmg(q) = q 210 p. gaucher for any map of multipointed d-spaces q. the reader must also keep in mind that if γ ∈ p`x1 and γ′ ∈ p` ′ x1 are two composable execution paths of x1 of length ` and `′ respectively, then the moore composition of execution paths (cf. proposition 2.7) γ ∗γ′ ∈ p`+` ′ x1 is also by theorem 4.12 the composition of paths in mg(x1) for tautological reasons. the proof of this theorem is divided in several parts. • objectwise calculation. it suffices to prove that the continuous map p1ψ : lim−→d f (1) −→ p1mg(x) = pgx is a bijection to complete the proof since all objects of the reparametrization category g are isomorphic and since colimits of g-spaces are calculated objectwise. • the final topology . if we can prove that the continuous p1ψ : lim−→d f (1) −→ pgx is a bijection with the colimit lim−→d f (1) equipped with the final topology, then the proof will be complete even in the category of ∆-hausdorff ∆-generated topological spaces because of the following facts: let i : a → b be a continuous one-to-one map between ∆-generated spaces such that b is also ∆-hausdorff, then a is ∆-hausdorff: let f : [0, 1] → a be a continuous map; then f being one-to-one, f([0, 1]) = i−1((i.f)([0, 1])) is closed. the space pgx is, by definition, equipped with the ∆-kelleyfication of the relative topology induced by the inclusion of set pgx ⊂ top([0, 1], |x|). if we work in the category of ∆-hausdorff ∆-generated topological spaces, then the space top([0, 1], |x|) will be ∆-hausdorff, hence the space pgx will be ∆-hausdorff, and therefore lim−→d f equipped with the final topology will be ∆-hausdorff as well. • surjectivity of p1ψ. the map ψ of figure 3 is induced by the universal property of the pushout. it is bijective on states. the multipointed d-space x is equipped with the ω-final structure because it is defined as a colimit in the category of multipointed d-spaces. by theorem 3.9, every execution path of x is therefore a moore composition of the form homotopy theory of moore flows (ii) 211 (f1γ1µ`1 ) ∗ · · · ∗ (fnγnµ`n) such that fi ∈{f, ĝ} for all 1 6 i 6 n, where{ γi ∈ pgglobg(dn) if fi = ĝ, γi ∈ pga if fi = f, with ∑ i `i = 1. then for all 1 6 i 6 n γiµ`i ∈ p `iglobg(dn) = p`img(globg(dn)) or γiµ`i ∈ p `ia = p`img(a). it gives rise to the execution path pf1(γ1µ`1 ) ∗ · · · ∗pfn(γnµ`n) with { fi = g if fi = ĝ, fi = f if fi = f, of length 1 of the moore flow x. by the commutativity of figure 3, we obtain the equality (f1γ1µ`1 ) ∗ · · · ∗ (fnγnµ`n) = ( p1ψ )( pf1(γ1µ`1 ) ∗ · · · ∗pfn(γnµ`n) ) . it means that the map of moore flows ψ : x → mg(x) induces a surjective continuous map from p1x to p1mg(x) = pgx. in other terms, the map p1ψ is a surjection. • the map ĉ. consider the diagram of topological spaces ef : pg(0),g(1)(a0) −→ top defined as follows: ef ( (u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un) ) = ⊔ (`1,...,`n) `1+···+`n=1 zu0,u1 (`1) ×···×zun−1,un(`n), with 212 p. gaucher zui−1,ui(`i) = { p`iui−1,uim g(a) = p`iui−1,uia if �i = 0, t(`i) if �i = 1 [ ⇒(ui−1, ui) = (g(0), g(1)) ]. the composition maps c′is are induced by the moore composition of execution paths of a. the inclusion maps i′is are induced by the continuous maps p`f : p` g(0),g(1) mg(a) → t(`) for ` > 0. we obtain a well-defined diagram of topological spaces 2 ef : pg(0),g(1)(a0) −→ top and, by proposition 4.5, there is an objectwise continuous surjective map k : ef −→df (1). we deduce that lim−→k is surjective. we want to prove that the composite map ĉ : lim−→e f lim−→k // // ( lim−→df)(1) p1ψ // p1mg(x) = pgx is a continuous bijection. we already know that the map p1ψ is surjective, and therefore that the map ĉ : lim−→e f → pgx is surjective as well. to prove that ĉ : lim−→e f → pgx is one-to-one, we must first introduce the notion of simplified element. let x be an element of some vertex of the diagram of spaces ef . we say that x ∈ef (n) is simplified if 3 d(n) = min { d(m) : ∃m ∈ obj(pg(0),g(1)(a0)) and ∃y ∈ef (m) such that y = x ∈ lim−→e f } . let x be a simplified element belonging to some vertex ef (n) of the diagram ef with n = ( (u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un) ) . • case 1 . it is impossible to have �i = �i+1 = 0 for some 1 6 i < n. indeed, otherwise x would be of the form (. . . ,γiµ`i,γi+1µ`i+1, . . . ) 2 this point is left as an exercice; verifying the commutativity relations is easy. 3 d is the degree function of the reedy category, see appendix a. homotopy theory of moore flows (ii) 213 where γi and γi+1 would be two execution paths of a. using the equality ci ( (. . . ,γiµ`i,γi+1µ`i+1, . . . ) ) = (. . . ,γiµ`i ∗γi+1µ`i+1, . . . ), the tuple x can then be identified in the colimit with the tuple( . . . , ( γiµ`i ∗γi+1µ`i+1 ) µ−1`i+`i+1︸︷︷︸ ∈pga by proposition 4.10 µ`i+`i+1, . . . ) ∈ef (n′) with d(n′) = n− 1 + ∑ i �i < d(n). it is a contradiction because x is simplified by hypothesis. • case 2 . suppose that �i = 1 for some 1 6 i 6 n and that x is of the form (. . . ,pg(δziφiµ`i), . . . ). if zi ∈ sn−1, then using the equality ii ( (. . . ,gδziφiµ`i, . . . ) ) = ( . . . ,pg(δziφiµ`i), . . . ) , the tuple x can then be identified in the colimit with the tuple (. . . ,gδziφiµ`i, . . . ) ∈e f (n′) with d(n′) = n + (∑ i �i ) − 1 < d(n). it is a contradiction because x is simplified by hypothesis. we deduce that in this case, zi ∈ dn\sn−1. • partial conclusion. consequently, for all simplified elements x = (x1, . . . , xn) of ef , we have ĉ(x) = (f1x1) ∗ · · · ∗ (fnxn) with for all 1 6 i 6 n,{ fi = f and xi ∈ p`ia, fi = pψ and xi = pg(δziφiµ`i) with zi ∈ d n\sn−1, and there are no two consecutive terms of the first form (i.e., fi = fi+1 = f for some i). it means that it is the finite moore composition of ĉ(x) of theorem 5.20. 214 p. gaucher • injectivity of ĉ. let x and y be two elements of lim−→e f such that ĉ(x) = ĉ(y). we can suppose that both x and y are simplified. let x = (x1, . . . ,xm) and y = (y1, . . . ,yn). then (f1x1) ∗ · · · ∗ (fmxm) = (g1y1) ∗ · · · ∗ (gnyn). since both members of the equality are the finite moore composition of theorem 5.20, we deduce that m = n and that for all 1 6 i 6 m, we have fixi = giyi. for a given i ∈ [1,m], there are two mutually exclusive possibilities: 1. fi = gi = f and xi and yi are two paths of length `i of a. since f is one-to-one because sn−1 is a subset of dn, we deduce that xi = yi. 2. fi = gi = pψ, xi = gδziφiµ`i and yi = gδtiψiµ`i, with zi, ti ∈ d n\sn−1 and φi,ψi ∈ g(1, 1). we also have pψ(xi) = ĝδziφiµ`i and pψ(yi) = ĝδtiψiµ`i. the restriction of ĝ to globg(dn)\globg(sn−1) being one-to-one, we deduce that zi = ti, φi = ψi and therefore once again that xi = yi. we conclude that x = y and that the map ĉ : lim−→e f → pgx is one-to-one. • informal summary . the arrows of the small category pg(0),g(1)(a0) and the relations satisfied by them prove that each element of the colimit lim−→e f has a simplified rewriting and this simplified rewriting coincides with the normal form of theorem 5.20. the latter theorem relies on the fact that all execution paths of globes are one-to-one, and more generally that all execution paths of cellular multipointed d-spaces are locally injective. • injectivity of p1ψ. at this point of the proof, we have a composite continuous map lim−→e f continuous bijection ĉ 77 lim−→k // // lim−→d f (1) p1ψ // // pgx. let a,b ∈ lim−→d f (1) such that p1ψ(a) = p1ψ(b). let a,b ∈ lim−→e f such that( lim−→k ) (a) = a and ( lim−→k ) (b) = b. then a = b and therefore a = b. we have proved that the continuous map p1ψ : lim−→d f (1) → pgx is one-to-one. theorem 7.3. under the hypotheses and the notations of this section. the map of g-spaces pψ : lim−→d f −→ pmg(x) is an isomorphism of g-spaces. homotopy theory of moore flows (ii) 215 proof. we already know by theorem 7.2 that the map of g-spaces pψ : lim−→d f −→ pmg(x) is an objectwise continuous bijection. we want to prove that it is an objectwise homeomorphism. since all objects of the reparametrization category g are isomorphic, it suffices to prove that p1ψ : lim−→d f (1) −→ pgx is a homeomorphism. consider a set map ξ : [0, 1] → lim−→d f (1) such that the composite map ξ : [0, 1] ξ −−→ lim−→d f (1) p1ψ −−−−−→ pgx is continuous. by corollary 2.3, it suffices to prove that the set map ξ : [0, 1] −→ lim−→d f (1) is continuous as well. • first reduction. the composite continuous map ξ gives rise by adjunction to a continuous map ξ̂ : [0, 1] × [0, 1] −→|x|. since [0, 1] × [0, 1] is compact and since |x| is weakly hausdorff by proposition 5.2, the subset ξ̂ ( [0, 1] × [0, 1] ) is a compact subset of |x|. by proposition 5.5, ξ̂ ( [0, 1] × [0, 1] ) intersects a finite number of globular cells of the cellular multipointed d-space x. therefore we can suppose that the multipointed d-space x is finite by proposition 5.4. write {cj : j ∈ j} for its finite set of globular cells. • second reduction. it suffices to prove that there exists a finite covering {f1, . . . ,fn} of [0, 1] by closed subsets such that each restriction ξ�fi factors through the colimit lim−→d f (1). let t be the set defined as follows: t = { carrier ( ξ(u) ) : u ∈ [0, 1] } . suppose that t is infinite. since j is finite, there exist j0 ∈ j and a sequence ( ξ(un) ) n>0 of execution paths of x such that the numbers in of times 216 p. gaucher that cj0 appears in the carrier of ξ(un) for n > 0 give rise to a strictly increasing sequence of integers (in)n>0. since [0, 1] is sequentially compact, the sequence (un)n>0 has a convergent subsequence. by continuity, the sequence( ξ(un) ) n>0 has therefore a convergent subsequence in pgx. this contradicts theorem 5.19. consequently, the set t is finite. for each carrier c ∈t , let uc = { u ∈ [0, 1] : carrier(ξ(u)) = c } . consider the closure ûc of uc in [0, 1]. we obtain the finite covering of [0, 1] by closed subsets [0, 1] = ⋃ c∈t ûc. we replace [0, 1] by ûc which is compact, metrizable and therefore sequential. the carrier c = [cj1, . . . ,cjm] is fixed until the very end of the proof. • third reduction. the attaching maps ĝjk : glob g(dnjk ) −→ x for 1 6 k 6 m of the cells cj1, . . . ,cjm yield a map of multipointed d-spaces ĝc : glob g(dnj1 ) ∗ · · · ∗ globg(dnjm ) −→ x. let αi−1 (αi resp.) be the initial state (the final state resp.) of glob g(dnji ) for 1 6 i 6 m in globg(dnj1 ) ∗ · · · ∗ globg(dnjm ). we obtain a map of g-spaces fg op 1 (d nj1 ) ⊗···⊗fg op 1 (d njm ) −→df (m) for some m belonging to pg(0),g(1)(a0) such that df (m) = zĝc(α0),ĝc(α1) ⊗···⊗zĝc(αm−1),ĝc(αm). using proposition 6.1, we obtain a continuous map yc : g(1,m) × dnj1 ×···× dnjm −→ zc ⊂df (m)(1) where zc is, by definition, the image of yc. at this point, we have obtained that the continuous map ξ�uc : uc −→ p gx homotopy theory of moore flows (ii) 217 factors as a composite of maps ξ�uc : uc −→ zc ⊂d f (m)(1) pm−−−→ lim−→d f (1) −→ pgx. consider a sequence (un)n>0 of uc converging to u∞ ∈ ûc. then for each n > 0, ξ(un) belongs to the image of pgĝc which is a closed subset of the sequential space pgx by theorem 6.11. thus ξ(u∞) belongs to the image of pgĝc as well. since pgĝc = p1ψ.pm.yc, we have obtained that the continuous map ξ� ûc : ûc −→ pgx factors as a composite of maps ξ� ûc : ûc −→ zc ⊂df (m)(1) pm−−−→ lim−→d f (1) p1ψ −−−−→ pgx. they are all of them continuous except maybe the left-hand one from ûc to zc (cf. the remark after this proof). since ûc is sequential, it remains to prove that the map ξ� ûc : ûc −→ zc ⊂df (m)(1) pm−−−→ lim−→d f (1) is sequentially continuous to complete the proof. • sequential continuity . consider a sequence (un)n>0 of ûc which converges to u∞ ∈ ûc. write ξ(un) = pm ( yc(φn,z 1 n, . . . ,z m n ) ) for all n > 0. we obtain ξ(un) = ( pgĝc )( φn,z 1 n, . . . ,z m n ) for all n > 0. by theorem 6.11, the sequence ( (φn,z 1 n, . . . ,z m n ) ) n>0 has a limit point (φ∞,z 1 ∞, . . . ,z m ∞). we deduce that the sequence (ξ(un))n>0 has a limit point because both yc and pm are continuous. it is necessarily equal to ξ(u∞) because the map p1ψ : lim−→d f (1) −→ pgx 218 p. gaucher is continuous bijective by theorem 7.2 and because ξ = p1ψ.ξ. the same argument shows that every subsequence of ( ξ(un) ) n>0 has a limit point which is necessarily ξ(u∞). suppose that the sequence ( ξ(un) ) n>0 does not converge to ξ(u∞). then there exists an open neighborhood v of ξ(u∞) such that for all n > 0, ξ(un) /∈ v . since v c is closed in lim−→d f (1), it means that ξ(u∞) cannot be a limit point of the sequence ( ξ(un) ) n>0 . contradiction. it implies that the sequence ( ξ(un) ) n>0 converges to ξ(u∞). before expounding the consequences of theorem 7.2 and of theorem 7.3, let us add an additional remark about the proof of theorem 7.3. it can be proved that the inverse image p−1m (γ) for each γ ∈ lim−→d f (1) is always finite. when the multipointed d-space x does not contain any loop, i.e., when pgα,αx is empty for all α ∈ x0, the map pm is even one-to-one and it is then possible to prove that the set map ûc → zc is always continuous. on the contrary, when x contains loops, the set map ûc → zc is not necessarily continuous mainly because an inverse image p−1m (γ) may contain several points. corollary 7.4. suppose that a is a cellular multipointed d-space. consider a pushout diagram of multipointed d-spaces globg(sn−1) �� // a �� globg(dn) // x with n > 0. then there is the pushout diagram of moore flows mg(globg(sn−1)) = glob(fg op 1 s n−1) �� // mg(a) �� mg(globg(dn)) = glob(fg op 1 d n) // mg(x). corollary 7.5. let x be a q-cofibrant multipointed d-space. then mg(x) is a q-cofibrant moore flow. proof. for every q-cofibrant moore flow x, the canonical map ∅ → x is a retract of a transfinite composition of the q-cofibrations c : ∅ → {0}, homotopy theory of moore flows (ii) 219 r : {0, 1} → {0} and of the q-cofibrations glob(fg op ` s n−1) ⊂ glob(fg op ` d n) for ` > 0 and n > 0. the cofibration r : {0, 1} → {0} is not necessary to reach all q-cofibrant objects. therefore, this theorem is a consequence of theorem 5.7 and of corollary 7.4. theorem 7.6. let x be a q-cofibrant moore flow. then the unit of the adjunction x → mg ( mg! (x) ) is an isomorphism. proof. by proposition 4.15, the theorem holds when x is a globe. it also clearly holds for x = {0}. for every q-cofibrant moore flow x, the canonical map ∅ → x is a retract of a transfinite composition of the q-cofibrations c : ∅ → {0}, r : {0, 1} → {0} and of the q-cofibrations glob(fg op ` s n−1) ⊂ glob(fg op ` d n) for ` > 0 and n > 0. the cofibration r : {0, 1} → {0} is not necessary to reach all q-cofibrant objects. therefore, this theorem is also a consequence of theorem 5.7 and of corollary 7.4. corollary 7.7. let x be a q-cofibrant moore flow. then there is the homeomorphism p1x ∼= pg(mg! (x)). proof. apply the functor p1(−) to the isomorphism x ∼= mg(mg! (x)). theorem 7.8. let λ be a limit ordinal. let x : λ −→gdtop be a colimit preserving functor such that: • the multipointed d-space x is a set, in other terms x = (x0,x0,∅); • for all ν < λ, there is a pushout diagram of multipointed d-spaces globg(snν−1) �� gν // xν �� globg(dnν ) ĝν // xν+1 with nν > 0. let xλ = lim−→ν<λ xν. for all ν 6 λ, the counit map κν : mg! (m g(xν)) −→ xν is an isomorphism. 220 p. gaucher proof. the map κ0 is an isomorphism because x0 is a set. by theorem 5.7, and since mg! is a left adjoint, it suffices to prove that if κν is an isomorphism, then κν+1 is an isomorphism. assume that κν is an isomorphism. by corollary 7.4, there is the pushout diagram of moore flows mg ( globg(snν−1) ) = glob ( fg op 1 s nν−1 ) �� gν // mg(xν) �� mg ( globg(dnν ) ) = glob ( fg op 1 d nν ) ĝν // mg(xν+1). apply again the left adjoint mg! to this diagram, we obtain by using the induction hypothesis that κν+1 is an isomorphism. corollary 7.9. for every q-cofibrant multipointed d-space x, the counit of the adjunction mg! (m g(x)) → x is an isomorphism of multipointed dspaces. proof. it is due to the fact that every q-cofibrant multipointed d-space x is a retract of a cellular multipointed d-space (note that the cofibration r : {0, 1} → {0} is not required to reach all cellular multipointed d-spaces) and that a retract of an isomorphism is an isomorphism. 8. from multipointed d-spaces to flows the goals of this final section are to complete the proof of the quillen equivalence between multipointed d-spaces and moore flows in theorem 8.1, which together with the results of [14] establish that multipointed d-spaces and flows have quillen equivalent q-model structures, and to give a new and conceptual proof of [11, theorem 7.5] in theorem 8.14 independent of [8]. we also give a new presentation of the underlying homotopy type of flow in proposition 8.16. theorem 8.1. the adjunction mg! a m g : gflow � gdtop induces a quillen equivalence between the q-model structure of moore flows and the q-model structure of multipointed d-spaces. proof. since the q-fibrations of moore flows are the maps of moore flows inducing an objectwise q-fibration on the g-spaces of execution paths, the functor mg takes q-fibrations of multipointed d-spaces to q-fibrations of moore homotopy theory of moore flows (ii) 221 flows. since mg preserves the set of states and since trivial q-fibrations of moore flows are maps inducing a bijection on states and an an objectwise trivial q-fibration on the g-spaces of execution paths, the functor mg takes trivial q-fibrations of multipointed d-spaces to trivial q-fibrations of moore flows. therefore, the functor mg : gdtop →gflow is a right quillen adjoint. by theorem 7.6, the map x → mg(mg! (x)) is a weak equivalence of moore flows for every q-cofibrant moore flow x. let y be a (q-fibrant) multipointed d-space. then the composite map of multipointed d-spaces mg! (m g(y cof )) ∼=−−→ y cof '−−→ y where y cof is a q-cofibrant replacement of y , is a weak equivalence of multipointed d-spaces because: 1) the left-hand map is an isomorphism by corollary 7.9; 2) the right-hand map is a weak equivalence by definition of a cofibrant replacement. let us give now some reminders about flows and the categorization functor cat from multipointed d-spaces to flows. definition 8.2. ([7, definition 4.11]) a flow is a small semicategory enriched over the closed monoidal category (top,×). the corresponding category is denoted by flow. let us expand the definition above. a flow x consists of a topological space px of execution paths, a discrete space x0 of states, two continuous maps s and t from px to x0 called the source and target map respectively, and a continuous and associative map ∗ : {(x,y) ∈ px ×px : t(x) = s(y)}−→ px such that s(x∗y) = s(x) and t(x∗y) = t(y). a morphism of flows f : x → y consists of a set map f0 : x0 → y 0 together with a continuous map pf : px → py such that f0(s(x)) = s(pf(x)), f0(t(x)) = t(pf(x)), pf(x∗y) = pf(x) ∗pf(y). let pα,βx = {x ∈ px : s(x) = α and t(x) = β}. notation 8.3. the map pf : px → py can be denoted by f : px → py 222 p. gaucher is there is no ambiguity. the set map f0 : x0 → y 0 can be denoted by f : x0 → y 0 is there is no ambiguity. the category flow is locally presentable. every set can be viewed as a flow with an empty path space. the obvious functor set ⊂ flow is limitpreserving and colimit-preserving. one another example of flow is important for the sequel: example 8.4. for a topological space z, let glob(z) be the flow defined by glob(z)0 = {0, 1}, pglob(z) = p0,1glob(z) = z, s = 0, t = 1. this flow has no composition law. notation 8.5. c : ∅ →{0}, r : {0, 1}→{0}, −→ i = glob({0}). the q-model structure of flows is the unique combinatorial model structure such that { glob ( sn−1 ) ⊂ glob ( dn ) : n > 0 } ∪{c,r} is the set of generating cofibrations and such that{ glob ( dn ×{0} ) ⊂ glob ( dn+1 ) : n > 0 } is the set of generating trivial cofibrations (e.g., [16, theorem 7.6]) where the maps dn ⊂ dn+1 are induced by the mappings (x1, . . . ,xn) 7→ (x1, . . . ,xn, 0). the weak equivalences are the maps of flows f : x → y inducing a bijection f0 : x0 ∼= y 0 and a weak homotopy equivalence pf : px → py and the fibrations are the maps of flows f : x → y inducing a q-fibration pf : px → py of topological spaces. let x be a multipointed d-space. consider for every (α,β) ∈ x0×x0 the coequalizer of spaces pα,βx = lim−→ ( pgα,βx ×g(1, 1) ⇒ p g α,βx ) where the two maps are (c,φ) 7→ c and (c,φ) 7→ c.φ. let [−]α,β : pgα,βx → pα,βx be the canonical map. homotopy theory of moore flows (ii) 223 theorem 8.6. ([11, theorem 7.2]) let x be a multipointed d-space. then there exists a flow cat(x) with cat(x)0 = x0, pα,βcat(x) = pα,βx and the composition law ∗ : pα,βx × pβ,γx → pα,γx is for every triple (α,β,γ) ∈ x0 × x0 × x0 the unique map making the following diagram commutative: pgα,βx ×p g β,γx ∗n // [−]α,β×[−]β,γ �� pgα,γx [−]α,γ �� pα,βx ×pβ,γx // pα,γx where ∗n is the normalized composition (cf. definition 2.9). the mapping x 7→ cat(x) induces a functor from gdtop to flow. definition 8.7. the functor cat : gdtop → flow is called the categorization functor. the motivation for the constructions of this paper and of [14] comes from the following theorem which is added for completeness. theorem 8.8. the categorization functor cat : gdtop → flow is neither a left adjoint nor a right adjoint. in particular, it cannot be a left or a right quillen equivalence. proof. this functor is not a left adjoint by [11, proposition 7.3]. suppose that it is a right adjoint. let l : flow →gdtop be the left adjoint. pick a nonempty topological space z. the adjunction yields the bijection of sets gdtop ( l(glob(z)), −→ i g ) ∼= flow(glob(z),−→i ). since z is nonempty, a map of flows from glob(z) to −→ i is determined by the choice of a map from z to {0}. we deduce that there is exactly one map f of multipointed d-spaces from l(glob(z)) to −→ i g. suppose that l(glob(z)) contains at least one execution path φ : [0, 1] →|l(glob(z))|. then f.φ is an execution path of −→ i g. every map g ∈gdtop( −→ i g, −→ i g) ∼= {[0, 1] ∼=+ [0, 1]} gives rise to and execution path g.f.φ of −→ i g. since g.f ∈gdtop ( l(glob(z)), −→ i g ) , we deduce that g.f = f. contradiction. we deduce that the multipointed d-space l(glob(z)) does not contain any execution path. therefore this multipointed d-space is of the form (uz,u 0 z,∅). we obtain the bijection mtop ( (uz,u 0 z), ([0, 1],{0, 1}) ) ∼= {f}. suppose that uz is nonempty. then for all g ∈ mtop ( ([0, 1],{0, 1}), ([0, 1],{0, 1}) ) , we have g.f = f. the only 224 p. gaucher possibilities are that f = 0 or f = 1. since f is the unique element, we deduce that uz = ∅. there are also the natural bijections of sets gdtop(l({0}),x) ∼= flow({0},cat(x)) ∼= cat(x)0 ∼= x0 ∼= gdtop({0},x). by the yoneda lemma, we obtain l({0}) = {0}. to summarize, if l : flow →gdtop is a left adjoint to the functor cat : gdtop → flow, then one has l({0}) = {0} and for all nonempty topological spaces z, there is the equalities l(glob(z)) = ∅. by [7, theorem 6.1], any flow is a colimit of globes and points. since l is colimit-preserving, we deduce that for all flows y , the multipointed d-space l(y ) is a set. we go back to the natural bijection given by the adjunction: gdtop(l(y ),x) ∼= flow(y,cat(x)). since l(y ) is a set, we obtain the natural bijection set(l(y ),x0) ∼= flow(y,cat(x)). we obtain the natural bijection gdtop(l(y ),x0) ∼= flow(y,cat(x)) and by adjunction the natural bijection flow(y,x0) ∼= flow(y,cat(x)) since cat(x0) = x0. by yoneda, we conclude that cat(x) = x0 for all multipointed d-spaces x, which is a contradiction. proposition 8.9. ([14, proposition 5.17]) let u and u ′ be two topological spaces. there is the natural isomorphism of g-spaces ∆gopu ⊗ ∆gopu ′ ∼= ∆gop(u ×u ′). let x be a flow. the moore flow m(x) is the enriched semicategory defined as follows: • the set of states is x0; • the g-space pα,βm(x) is the g-space ∆gop(pα,βx); • the composition law is defined, using proposition 8.9 as the composite map ∆gop(pα,βx)⊗∆gop(pβ,γx) ∼= ∆gop(pα,βx×pβ,γx) ∆gop(∗)−−−−−→ ∆gop(pα,γ)x. the construction above yields a well-defined functor m : flow →gflow. homotopy theory of moore flows (ii) 225 consider a g-flow y . for all α,β ∈ y 0, let yα,β = lim−→pα,βy . let (α,β,γ) be a triple of states of y . the composition law of the g-flow y induces a continuous map yα,β ×yβ,γ ∼= lim−→(pα,βy ⊗pβ,γy ) −→ lim−→pα,γy ∼= yα,γ which is associative. we obtain the proposition 8.10. ([14, proposition 10.6 and proposition 10.7]) for any g-flow y , the data • the set of states is y 0; • for all α,β ∈ y 0, let yα,β = lim−→pα,βy ; • for all α,β,γ ∈ y 0, the composition law yα,β ×yβ,γ → yα,γ; assemble to a flow denoted by m!(y ). it yields a well-defined functor m! : gflow → flow. there is an adjunction m! a m. theorem 8.11. there is the isomorphism of functors cat ∼= m!.mg. proof. first, let us notice that the functors cat : gdtop → flow (theorem 8.6), mg : gdtop → gflow (theorem 4.12) and m! : gflow → flow (proposition 8.10) preserve the set of states by definition of these functors. therefore, for every multipointed d-space x, the flows cat(x) and m!.mg(x) have the same set of states x0. let g1 be the full subcategory of g generated by 1: the set of objects is the singleton {1} and g1(1, 1) = g(1, 1). for (α,β) ∈ x0 ×x0, the inclusion functor ι : g1 ⊂g induces a continuous map lim−→g1 (( pα,βmgx ) .ι ) → lim−→g pα,βm gx. it turns out that there is the natural homeomorphisms lim−→g1 (( pα,βmgx ) .ι ) ∼= lim−→g1 p 1 α,βm gx ∼= lim−→g1 p g α,βx ∼= pα,βcat(x), the first one by definition of ι, the second one by definition of mg and the last one by definition of cat. we obtain a natural map of flows 226 p. gaucher cat(x) → (m!.mg)(x) which is bijective on states. let ` > 0 be an object of g. then the comma category (`↓ι) is characterized as follows: • the set of objects is g(`, 1) which is always nonempty for every ` > 0; • the set of maps from an arrow u : ` → 1 to an arrow v : ` → 1 is an element of mor(g)(u,v). the comma category (`↓ι) is connected since in any diagram of g of the form [0,`] u // [0, 1] k �� [0,`] v // [0, 1], there exists a map k ∈ g([0, 1], [0, 1]) making the square commute: take k = v.u−1. by [29, theorem 1 p. 213], we deduce that the natural map of flows cat(x) → (m!.mg)(x) induces a homeomorphism between the spaces of paths. corollary 8.12. suppose that a is a cellular multipointed d-space. consider a pushout diagram of multipointed d-spaces globg(sn−1) �� // a �� globg(dn) // x with n > 0. then there is the pushout diagram of flows glob(sn−1) �� // cat(a) �� glob(dn) // cat(x). proof. it is a consequence of corollary 7.4, theorem 8.11 and of the fact that m! : gflow → flow is a left adjoint. homotopy theory of moore flows (ii) 227 definition 8.13. we consider the composite functors (lcat) : gdtop (−)cof −−−−−→gdtop cat−−−−→ flow (lcat)−1 : flow m−−−→gflow (−)cof −−−−−→gflow mg !−−−−→gdtop where (−)cof is a q-cofibrant replacement functor. we can now write down the new proof of [11, theorem 7.5]. theorem 8.14. the categorization functor from multipointed d-spaces to flows cat : gdtop −→ flow takes q-cofibrant multipointed d-spaces to q-cofibrant flows. its total left derived functor in the sense of [4] induces an equivalence of categories between the homotopy categories of the q-model structures. proof. the functor cat ∼= m!.mg takes q-cofibrant multipointed d-spaces to q-cofibrant flows by corollary 7.5 and because m! is a left quillen adjoint. the rest of the proof is divided in four parts. • x ' y ⇒ (lcat)(x) ' (lcat)(y ). let x ' y be two weakly equivalent multipointed d-spaces in the q-model structure. then there is the weak equivalence xcof ' y cof . since all multipointed d-spaces are q-fibrant, the right quillen functor mg takes weak equivalences of multipointed d-spaces to weak equivalences of moore flows. since m! is a left quillen functor and since mg preserves q-cofibrancy by corollary 7.5, we deduce using theorem 8.11 the sequence of isomorphisms and weak equivalences (lcat)(x) ∼= m!mg ( xcof ) ' m!mg ( y cof ) ∼= (lcat)(y ). •x ' y ⇒ (lcat)−1(x) ' (lcat)−1(y ). let x ' y be two weakly equivalent flows in the q-model structure. since m is a right quillen functor and since all flows are q-fibrant, we obtain the weak equivalence of moore flows m(x) ' m(y ). by definition of (lcat)−1 and since mg! is a left quillen adjoint, we deduce the sequence of isomorphisms and weak equivalences (lcat)−1(x) ∼= mg! (m(x)) cof ' mg! (m(y )) cof ∼= (lcat)−1(y ). the functors (lcat) and (lcat)−1 therefore induce functors between the homotopy categories. 228 p. gaucher • (lcat)−1(lcat)(x) ' x. let x be a multipointed d-space. then we have the sequence of isomorphisms and of weak equivalences (lcat)−1(lcat)(x) ∼= mg! ( mm! q-cofibrant by corollary 7.5 mg ( xcof ) )cof ' mg! ( mg ( xcof ))cof ' mg! m g(xcof ) ∼= xcof ' x, the first isomorphism by definition of (lcat) and (lcat)−1 and by theorem 8.11, the first weak equivalence since the adjunction m! a m is a quillen equivalence by [14, theorem 10.9] and since mg! is a left quillen adjoint, the second weak equivalence by corollary 7.5 and again since mg! is a left quillen adjoint, the second isomorphism by corollary 7.9, and the last weak equivalence by definition of a q-cofibrant replacement. • (lcat)(lcat)−1(y ) ' y . let y be a flow. then we have the sequence of isomorphisms and of weak equivalences (lcat)(lcat)−1(y ) ∼= ( m!mg )( mg! (my ) cof )cof ' ( m!mg )( mg! (my ) cof ) ∼= m!(my )cof ' y, the first isomorphism by definition of (lcat) and (lcat)−1 and by theorem 8.11, the first weak equivalence because mg is a right quillen adjoint, because mg! (my ) cof is q-cofibrant, because mg preserves q-cofibrancy by corollary 7.5 and finally because m! is a left quillen adjoint, the second isomorphism by theorem 7.6, and finally the last weak equivalence since the adjunction m! a m is a quillen equivalence by [14, theorem 10.9]. the proof is complete. the underlying homotopy type of a flow is, morally speaking, the underlying space of states of a flow after removing the execution paths. it is defined only up to homotopy, not up to homeomorphism. we conclude the section and the paper by recovering it in a very intuitive way by using (lcat)−1. it is proved in [9, theorem 6.1] that for every cellular flow x, there exists a cellular multipointed d-space xtop such that there is an isomorphism cat(xtop) ∼= xcof . definition 8.15. ([9, section 6]) the underlying homotopy type of a flow x is the topological space ||x|| := ∣∣xtop∣∣, where |xtop| is the underlying homotopy theory of moore flows (ii) 229 topological space of the cellular multipointed d-space xtop. this yields a well defined functor ||− || : flow −→ ho(top) from the category of flows to the homotopy category of the q-model structure of top. proposition 8.16. for any flow x, there is the homotopy equivalence of topological spaces ||x|| ' ∣∣(lcat)−1(x)∣∣. proof. one has cat ( mg! ( m(x)cof )) = ( m!mg )( mg! ( m(x)cof )) ∼= m!(m(x)cof), the equality by theorem 8.11 and the isomorphism by theorem 7.6. using the quillen equivalence m! a m of [14, theorem 10.9], we obtain the weak equivalences of flows cat ( mg! ( m(x)cof )) ' x ' cat ( xtop ) . thanks to theorem 8.14, we obtain the weak equivalence of q-cofibrant multipointed d-spaces mg! ( m(x)cof ) ' xtop. we deduce the homotopy equivalence between the underlying q-cofibrant spaces∣∣mg! (m(x)cof)∣∣ ' ∣∣xtop∣∣ because the underlying topological space functor |−| is a left quillen functor by [11, proposition 8.1]. the composite functor gflow mg !−−−−→gdtop |−| −−−→ top is the composite of two left quillen functors. therefore the mapping x 7−→ |mg! (x cof )| induces a functor from gflow to ho(top). for all g-flows x, there is the isomorphism xcof ∼= mg ( mg! ( xcof )) by theorem 7.6. consequently, the functor∣∣mg! ((−)cof)∣∣ can be regarded as the underlying homotopy type functor for g-flows. 230 p. gaucher a. the reedy category pu,v(s): reminder let s be a nonempty set. let pu,v(s) be the small category defined by generators and relations as follows (see [15, section 3]): • u,v ∈ s (u and v may be equal). • the objects are the tuples of the form m = ( (u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un) ) with n > 1, u0, . . . ,un ∈ s, �1, . . . ,�n ∈{0, 1} and for i such that 1 6 i 6 n, �i = 1 ⇒ (ui−1,ui) = (u,v). • there is an arrow cn+1 : ( m, (x, 0,y), (y, 0,z),n ) −→ ( m, (x, 0,z),n ) for every tuple m = ( (u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un) ) with n > 1 and every tuple n = ( (u′0,� ′ 1,u ′ 1), (u ′ 1,� ′ 2,u ′ 2), . . . , (u ′ n′−1,� ′ n′,u ′ n′) ) with n′ > 1. it is called a composition map. • there is an arrow in+1 : ( m, (u, 0,v),n ) −→ ( m, (u, 1,v),n ) for every tuple m = ( (u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un) ) with n > 1 and every tuple n = ( (u′0,� ′ 1,u ′ 1), (u ′ 1,� ′ 2,u ′ 2), . . . , (u ′ n′−1,� ′ n′,u ′ n′) ) with n′ > 1. it is called an inclusion map. • there are the relations (group a) ci.cj = cj−1.ci if i < j (which means since ci and cj may correspond to several maps that if ci and cj are composable, then there exist cj−1 and ci composable satisfying the equality). • there are the relations (group b) ii.ij = ij.ii if i 6= j. by definition of these maps, ii is never composable with itself. • there are the relations (group c) ci.ij = { ij−1.ci if j > i + 2, ij.ci if j 6 i− 1. by definition of these maps, ci and ii are never composable as well as ci and ii+1. homotopy theory of moore flows (ii) 231 by [15, proposition 3.7], there exists a structure of reedy category on pu,v(s) with the n-valued degree map defined by d((u0,�1,u1), (u1,�2,u2), . . . , (un−1,�n,un)) = n + ∑ i �i. the maps raising the degree are the inclusion maps. the maps decreasing the degree are the composition maps. b. an explicit construction of the left adjoint mg! the proof of theorem 4.12 uses a well-known characterization of right adjoint functors between locally presentable categories. it is possible to describe more explicitly the functor mg! : gflow →gdtop as follows. notation b.1. the composite of two natural transformations µ : f ⇒ g and ν : g ⇒ h is denoted by ν � µ to make the distinction with the composition of maps and of functors. the category d(gflow) of all small diagrams of moore flows over all small categories is defined as follows. an object is a functor f : i → gflow from a small category i to gflow. a morphism from f : i1 → gflow to g : i2 → gflow is a pair (f : i1 → i2,µ : f ⇒ g.f) where f is a functor and µ is a natural transformation. if (g,ν) is a map from g : i2 →gflow to h : i3 →gflow, then the composite (g,ν).(f,µ) is defined by (g.f, (ν.f)�µ). the identity of f : i1 → gflow is the pair (idi1, idf ). if (h,ξ) : (h : i3 → gflow) → (i : i4 →gflow) is another map of d(gflow), then we have ((h,ξ).(g,ν)) .(f,µ) = (h.g,ξ.g �ν).(f,µ) = (h.g.f,ξ.g.f �ν.f �µ) = (h,ξ).(g.f,ν.f �µ) = (h,ξ). ((g,ν).(f,µ)) . thus the composition law is associative and the category d(gflow) is welldefined. it is well-known that the colimit of small diagrams induces a functor lim−→ : d(gflow) →gflow (see e.g., [15, proposition a.2]). theorem b.2. there exists a functor d : gflow →d(gflow) such that the composite functor gflow d−−−→d(gflow) lim−→−−−→gflow 232 p. gaucher is the identity functor and such that every vertex of d(x) for any moore flow x is one of the following kind: (1) the moore flow {0}, (2) the globe glob(d) of some g-space d, (3) the moore flow glob(d)∗glob(e) for two g-spaces d and e where the final state of glob(d) is identified with the initial state of glob(e) (it is the “composition” of the two globes, hence the notation). moreover, each g-space d and e used by the diagram is of the form pα,γx or pα,βx ⊗pβ,γx. proof. this theorem is proved in [7, theorem 6.1] for the category of flows which are semicategories enriched over the closed (semi)monoidal bicomplete category (top,×) (see definition 8.2). the diagram is depicted in figure 4. we refer to the proof of [7, theorem 6.1] for the definitions of the maps. since a moore flow is a semicategory enriched over the closed semimonoidal bicomplete category ([gop, top]0,⊗), the proof is complete. glob (pα,γx) glob (pα,βx ⊗pβ,γx) pα,β,γ oo rα,β,γ �� glob (pα,βx) ∗ glob (pβ,γx) {α} i α,β 1 �� k α,β,γ 1 44iiiiiiiiiiiiiiiiiii j α,β,γ 1 33 i α,γ 1 44 {β} k α,β,γ 2 oo i α,β 2ttiii iiii iiii iiii iiii i i β,γ 1 ** uuuu uuuu uuuu uuuu uuu {γ} k α,β,γ 3 jjuuuuuuuuuuuuuuuuuuuu i β,γ 2 �� j α,β,γ 3 kk i α,γ 2 jj glob (pα,βx) h α,β,γ 1sssssssssss 99sssssssssss glob (pβ,γx) h α,β,γ 3 kkkkkkkkkkk eekkkkkkkkkkk figure 4: the moore flow x as a colimit of globes and points (the definition of the maps are easily understandable, cf. the proof of [7, theorem 6.1] for further explanations) homotopy theory of moore flows (ii) 233 notation b.3. denote by b(gflow) 4 the full subcategory of gflow generated by {0} and glob(d) where d runs over the class of all g-spaces. theorem b.4. let k be a bicomplete category. a functor f : k−→gflow has a left adjoint f! : gflow −→k if and only if there exists a functor m : b(gflow) → k such that there are the natural bijections k ( m({0}),y ) ∼= f(y )0, k ( m(d),y ) ∼= gflow(glob(d),f(y )) for all objects y of k and all g-spaces d. proof. the “only if” direction comes from the fact that there is a natural bijection f(x)0 ∼= gflow({0},f(x)). let d and e be two g-spaces. let m(glob(d)∗glob(e)) be the object of k defined by the pushout diagram of k m({0}) m(0 7→1) // m(07→0) �� m(d) �� m(e) // m(glob(d) ∗ glob(e)). by taking the image by the functor k(−,y ) : k→ set, we obtain the pullback diagram of sets k(m(glob(d) ∗ glob(e)),y ) // �� k(m(d),y ) �� k(m(e),y ) // k(m({0}),y ) 4 b like “brick”: the globes and the point are the elementary bricks to build flows. 234 p. gaucher for all objects y of k. we therefore obtain the natural bijection of sets k(m(glob(d) ∗ glob(e)),y ) ∼= gflow(glob(d) ∗ glob(e),f(y )) for all objects y of k. let f!(x) := lim−→m(d(x)). this defines a functor from gflow to k. for all objects y of k, there is the sequence of natural bijections (note that in the calculation below, the colimits are taken over a same small category which depends only on x) k(f!(x),y ) ∼= k(lim−→m(d(x)),y ) ∼= lim←−k(m(d(x)),y ) ∼= lim←−gflow(d(x),f(y )) ∼= gflow(lim−→d(x),f(y )) ∼= gflow(x,f(y )), the first one by definition of f!, the second one and the fourth one by the universal property of the (co)limit, the third one by hypothesis and by the calculation above, and finally the last one by theorem b.2. after theorem b.4, it suffices now to find a multipointed d-space denoted by mg! ({0}) such that there is a natural bijection with respect to x gdtop ( mg! ({0}),x ) ∼= mg(x)0 and a multipointed d-space denoted by mg! (glob(d)) natural with respect to the g-space d such that there is a natural bijection with respect to d and x gdtop ( mg! (glob(d)),x ) ∼= gflow(glob(d),mg(x)). we have the natural bijections gdtop ( {0},x ) ∼= x0 ∼= mg(x)0, and therefore mg! ({0}) = {0}. homotopy theory of moore flows (ii) 235 we have the sequence of natural bijections gdtop (∫ ` globg` (d(`)),x ) ∼= ∫ ` gdtop ( globg` (d(`)),x ) ∼= ∫ ` ⊔ (α,β)∈x0×x0 top ( d(`),p`α,βm g(x) ) ∼= ⊔ (α,β)∈x0×x0 ∫ ` top ( d(`),p`α,βm g(x) ) ∼= ⊔ (α,β)∈x0×x0 topg op( d,pα,βmg(x) ) ∼= ⊔ (α,β)∈x0×x0 [gop, top]0 ( d,pα,βmg(x) ) ∼= gflow ( glob(d),mg(x) ) , the first bijection by definition of a (co)limit, the second bijection by proposition 4.13, the third bijection because the underlying diagram of this end is connected, the fourth bijection by [29, page 219 (2)], the fifth bijection since [gop, top]0 is a full subcategory of topg op , and finally the last bijection by proposition 4.14. we obtain mg! (glob(d)) = ∫ ` globg` (d(`)). c. the setting of k-spaces in this appendix, the category of k-spaces is denoted by topk and the category of ∆-generated spaces by top∆. the proofs are just sketched. we must at first prove the existence of the projective q-model structure of [gop, topk]0: [12] is written in the locally presentable setting. we do not know whether the arguments of [31] are valid here since they are written in the category of hausdorff k-spaces. anyway, it is possible to give a much simpler argument. the inclusion top∆ ⊂ topk has a right adjoint k∆ : topk → top∆, which gives rise to a quillen equivalence. in fact, the q-model structure of topk is right-induced by k∆ : topk → top∆ in the sense of [6, 23]. from the functor k∆, we obtain a right adjoint u : [gop, topk]0 → [gop, top∆]0. for an object x of [gop, topk]0, let path(x) = ` 7→ topk([0, 1],x(`)) where 236 p. gaucher topk is the internal hom of topk. since the composite functor k∆.topk is the internal hom of top∆, the quillen path object argument can be used to obtain that u : [gop, topk]0 → [gop, top∆]0 right-induces the projective q-model structure of [gop, topk]0. this technique still works in the cofibrantly generated non-combinatorial setting: it dates back to [32] (see also [24, theorem 11.3.2]). the q-model category of multipointed d-spaces is constructed in [16, theorem 6.14] by right-inducing it and by using the quillen path object argument again. the q-model category of g-flows is constructed in [14, theorem 8.8] by mimicking the method used in [13, theorem 3.11] which works for any convenient category of topological spaces. indeed, it uses isaev’s work [26] about model categories of fibrant objects which does not require to work in a locally presentable setting. theorem 2.16 is not valid anymore. see [15, theorem 5.10] for a treatment of the similar problem for flows. the reason is that a k-space is not necessarily homeomorphic to the disjoint sum of its path-connected components (e.g., the cantor space). it is used in theorem 4.12 together with the local presentability of the category of ∆-generated spaces to prove the existence of the left adjoint mg! : gflow → gdtop. in the framework of k-spaces, the existence of mg! can be proved using appendix b. we have therefore a quillen adjunction mg! a m g : gflow(topk) �gdtop(topk) between the q-model structures, where the notation (topk) is to specify the category of topological spaces which is used. a k-space, unlike a ∆-generated space, is not necessarily sequential. it is not clear how to adapt the proof of theorem 6.11 by replacing sequences by nets since there is a cantor diagonalization argument which does not seem to be adaptable at least with uncountable nets. it is not clear either how to modify accordingly the last part of the proof of theorem 7.3 about the sequential continuity because ûc is an arbitrary compact space now, and not necessarily a closed subset of [0, 1] anymore. to obtain theorem 1.1 for kspaces, another method must be used. for all k-spaces z, the canonical map k∆(z) → z is a weak homotopy equivalence which induces a bijection topk([0, 1],k∆(z)) ∼= top∆([0, 1],z) because [0, 1] ∈ top∆ ⊂ topk. from these observations, we obtain two left quillen equivalences gflow(top∆) → gflow(topk) and gdtop(top∆) → gdtop(topk). we obtain that is homotopy theory of moore flows (ii) 237 commutative the diagram of left quillen adjunctions gflow(top∆) ⊂ // �� gflow(topk) �� gdtop(top∆) ⊂ // gdtop(topk) because the g-flows {0} and glob(d) of gflow(top∆) are taken to the same multipointed d-space of gdtop(topk) by appendix b. using the two-out-ofthree property, we obtain that the quillen adjunction mg! a m g : gflow(topk) �gdtop(topk) is a quillen equivalence. let x be a q-cofibrant object of gdtop(topk). then x ∈ gdtop(top∆). by corollary 7.9, there is the isomorphism mg! 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