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. 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. 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. 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 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. 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 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 � 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. 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), 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. [8] hoffman, m. e., multiple harmonic series, pacific j. math. 152 (1992), 275-290. [9] kölbig, k. s. nielsen’s generalized polylogarithms. siam j. math. anal. 17 (1986), no. 5, 1232–1258. [10] lewin, r. polylogarithms and associated functions. north holland, new york, 1981. [11] markett, c. triple sums and the riemann zeta function. j. number theory 48 (1994), no. 2, 113–132. [12] sitaramachandra rao, r. a formula of s. ramanujan. j. number theory 25 (1987), no. 1, 1–19. [13] sofo, a. polylogarithmic connections with euler sums. sarajevo j. math. 12(24) (2016), no. 1, 17–32. [14] sofo, a. integrals of logarithmic and hypergeometric functions. commun. math. 24 (2016), no. 1, 7–22. [15] sofo, a. and cvijović, d. extensions of euler harmonic sums, appl. anal. discrete math. 6 (2012), 317–328. [16] sofo, a. integrals of inverse trigonometric and polylogarithmic functions. submitted, 2019. [17] sofo, a.; srivastava, h. m. a family of shifted harmonic sums. ramanujan j. 37 (2015), no. 1, 89–108. [18] sofo, a. new classes of harmonic number identities. j. integer seq. 15 (2012), no. 7, article 12.7.4, 12 pp. [19] sofo, a. families of integrals of polylogarithmic functions, special functions and applications. editors choi, j. and shilin, i. mathematics (2019), 7, 143; doi:10.3390/math7020143, published by mdpi ag, basel, switzerland. [20] xu, ce. yan, yuhuan. shi, zhijuan. euler sums and integrals of polylogarithm functions. j. number theory 165 (2016), 84–108. [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), 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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[17] a.e. livingston, the coefficients of multivalent close-to-convex functions, proc. amer. math. soc. 21 (1969), 545 û552. [18] k.i. noor, hankel determinant problem for the class of functions with bounded boundary rotation, rev. roumaine math. pures appl. 28 (8) (1983), 731 – 739. [19] h. orhan, p. zaprawa, third hankel determinants for starlike and convex functions of order alpha, bull. korean math. soc. 55 (1) (2018), 165 – 173. [20] ch. pommerenke, on the coefficients and hankel determinants of univalent functions, j. lond. math. soc. s1-41 (1) (1966), 111 – 122. [21] ch. pommerenke, g. jensen “univalent functions”, studia mathematica/mathematische lehrbücher 25, vandenhoeck und ruprecht, gottingen, 1975. [22] m. raza, s.n. malik, upper bound of third hankel determinant for a class of analytic functions related with lemniscate of bernoulli, j. inequal. appl. (2013), 2013:412, 1 – 8. 42 d. vamshee krishna, d. shalini [23] t.v. sudharsan, s.p. vijayalakshmi, b.a. stephen, third hankel determinant for a subclass of analytic functions, malaya j. math. 2 (4) (2014), 438 – 444. [24] d. vamshee krishna, t. ramreddy, coefficient inequality for certain p-valent analytic functions, rocky mountain j. math. 44 (6) (2014), 1941 – 1959. [25] p. zaprawa, third hankel determinants for subclasses of univalent functions, mediterr. j. math. 14 (1) (2017), article 19, 1 – 10. [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 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. 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. [6] a. boua, commutativity of near-rings with certain constrains on jordan ideals, bol. soc. parana. mat. (3) 36 (4) (2018), 159 – 170. [7] m. brešar, j. vukman, on left derivations and related mappings, proc. amer. math. soc. 110 (1) (1990), 7 – 16. [8] a. enguady, a. boua, on lie ideals with left derivations in 3-prime nearrings, an. ştiinţ. univ. al. i. cuza iaşi. mat. (n.s.) 68 (1) (2022), 123 – 132. [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), 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 compactness theorem and risk measures, j. math. anal. appl. 485(1) (2020), article 123764. doi.org/10.1016/j.jmaa.2019.123764 [7] f. delbaen, k. owari, convex functions on dual orlicz spaces, positivity 23 (5) (2019), 1051 – 1064. doi.org/10.1007/s11117-019-00651-x [8] j. gil de lamadrid, topology of mappings and differentiation processes, illinois j. math. 3 (1959), 408 – 420. doi.org/10.1215/ijm/1255455262 [9] j. gómez gil, on local convexity of bounded weak topologies on banach spaces, pacific j. math. 110 (1) (1984), 71 – 76. doi.org/10.2140/pjm.1984.110.71 https://doi.org/10.1007/3-540-29587-9 https://doi.org/10.1090/surv/105 https://doi.org/10.1016/0022-0531(72)90136-6 https://doi.org/10.1090/s0002-9947-1955-0069386-1 https://doi.org/10.1016/s0304-0208(08)72315-6 https://doi.org/10.1016/j.jmaa.2019.123764 https://doi.org/10.1007/s11117-019-00651-x https://doi.org/10.1215/ijm/1255455262 https://doi.org/10.2140/pjm.1984.110.71 mackey continuity of convex functions 195 [10] m. 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(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. � 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 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 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). 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. 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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. � 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 anonymous referee for suggestions that improved the exposition in contextualising theorem 2.3 and corollary 2.4 in this manuscript. references [1] m. akian, s. gaubert, b. lemmens, r.d. nussbaum, iteration of order preserving subhomogeneous maps on a cone, math. proc. cambridge philos. soc. 140 (2006), 157 – 176. [2] m. akian, s. gaubert, b. lemmens, stability and convergence in discrete convex monotone dynamical systems, j. fixed point theory appl. 9 (2011), 295 – 325. 242 s. jayaraman, y.k. prajapaty, s. sridharan [3] j. bernik, r. drnovsek, t. kosir, t. laffey, g. macdonald, r. meshulam, m. omladic, h. radjavi, common fixed points and common eigenvectors for sets of matrices, linear multilinear algebra 53 (2005), 137 – 146. [4] m.p. drazin, j.w. dungey, k.w. grunberg, some theorems on commutative matrices, j. london math. soc. 26 (1951), 221 – 228. [5] r.a. horn, c.r. johnson, “ matrix analysis ”, second edition, cambridge university press, cambridge, 2013. [6] b.p. kitchens, “ symbolic dynamics: one-sided, two-sided and countable state markov shifts ”, universitext, springer-verlag, berlin, 1998. [7] b. lemmens, nonlinear perron-frobenius theory and dynamics of cone maps, in “ positive systems ”, lect. notes control inf. sci., 341, springer, berlin, 2006, 399 – 406. [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. 31, núm. 2, 227 – 233 (2016) a study on ricci solitons in generalized complex space form m.m. praveena, c.s. bagewadi department of mathematics, kuvempu university, shankaraghatta 577 451, shimoga, karnataka, india mmpraveenamaths@gmail.com prof bagewadi@yahoo.co.in presented by marcelo epstein received june 18, 2016 abstract: in this paper we obtain the condition for the existence of ricci solitons in non-flat generalized complex space form by using eisenhart problem. also it is proved that if (g, v, λ) is ricci soliton then v is solenoidal if and only if it is shrinking or steady or expanding depending upon the sign of scalar curvature. key words: kähler manifolds, generalized complex space form, parallel second order covariant tensor field, einstein space, ricci soliton. ams subject class. (2010): 53c15, 53c21, 53c35, 53c55, 53c56. 1. introduction ricci flow is an excellent tool in simplifying the structure of the manifolds. it is defined for riemannain manifolds of any dimension. it is a process which deforms the metric of a riemannian manifold analogous to the diffusion of heat there by smoothing out the irregularity in the metric. it is given by ∂g(t) ∂t = −2 ric(g(t)), where g is riemannian metric dependent on time t and ric(g(t)) is ricci tensor. let 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 xp given by xpf = df(ϕt(p)) dt , f ∈ c∞(m). if y is a vector field then lxy = limt→0 ϕ∗t y −y t is known as lie derivative of y with respect to x. ricci solitons move under the ricci flow under ϕt : m −→ m of the initial metric i.e., they are stationary points of the ricci 227 228 m.m. praveena, c.s. bagewadi flow in space of metrics. if g0 is a metric on the codomain then g(t) = ϕ ∗ t g0 is the pullback of g0, is a metric on the domain. hence if g0 is a solution of the ricci flow on the codomain subject to condition lv g0 + 2ricg0 + 2λg0 = 0 on the codomain then g(t) is the solution of the ricci flow on the domain subject to the condition lv g + 2ricg + 2λg = 0 on the domain by [12] under suitable conditions. here g0 and g(t) are metrics which satisfy ricci flow. thus the equation in general 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. 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. [6] l.p. eisenhart, symmetric tensors of the second order whose first covariant derivatives are zero, trans. amer. math, soc. 25 (2) (1923), 297 – 306. [7] g. ingalahalli, c. s. bagewadi, ricci solitons in α-sasakian manifolds, isrn geometry, volume 2012 (2012), article id 421384, 13 pages. [8] h. levy, symmetric tensors of the second order whose covariant derivatives vanish, ann. of math. (2) 27 (2) (1925), 91 – 98. [9] z. olszak, on the existence of generalized complex space forms, isrel j. math. 65 (2) (1989), 214 – 218. [10] m.m. praveena, c.s. bagewadi, on almost pseudo bochner symmetric generalized complex space forms, acta mathematica. academiae paedagogicae ny¡regyháziensis 32 (1) (2016), 149 – 159. [11] r. sharma, second order parallel tensor in real and complex space forms, internat. j. math. and math. sci. 12 (4) (1989), 787 – 790. [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. 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. 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 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. 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. [2] d. gromoll, w. klingenberg, w. meyer, “ riemannsche geometrie im grossen ”, lecture notes in mathematics, no. 55, springer-verlag, berlinnew york, 1968. [3] a. haefliger, sur les feuilletages analytiques, c. r. acad. sci. paris 242 (1956), 2908 – 2910. [4] r. hermann, a sufficient condition that a mapping of riemannian manifolds be a fibre bundle, proc. amer. math. soc. 11 (1960), 236 – 242. [5] r. hermann, the differential geometry of foliations, ii, j. math. mech. 11 (1962), 305 – 315. [6] sh. kobayashi, k. nomizu, “ foundations of differential geometry, vol. ii ”, interscience publishers john wiley & sons, inc., new york-londonsydney, 1969. [7] c. lamoureux, feuilletages de codimension 1. transversales fermées, c. r. acad. sci. paris sér. a-b 270 (1970), a1659 – a1662. [8] r. langevin, a list of questions about foliations, in “ differential topology, foliations, and group actions ”, contemporary math., 161, amer. math. soc., providence, ri, 1994, 59 – 80. [9] p. molino, “ riemannian foliations ”, progress in mathematics, 73, birkhäuser boston, inc., boston, ma, 1988. [10] a.ya. narmanov, “ geometry of orbits of vector fields and singular foliations ”, monograph, tashkent university, 2015. [11] a.ya. narmanov, a.m. bayturayev, on geometry of riemannian manifolds, nuuz bulletin. -tashkent, 2010, no. 3, 143 – 147. [12] a. narmanov, g. kaipnazarova, metric functions on riemannian manifolds, uzbek. math. j. -tashkent, 2010, no. 1, 11 – 20. [13] a.ya. narmanov, a.s. sharipov, on the group of foliation isometries, methods funct. anal. topology 15 (2009), 195 – 200. [14] a.ya. narmanov, a.s. sharipov, on the geometry of submersions, international journal of geometry 3 (2014), 51 – 56. [15] g. reeb, sur certaines propriétés topologiques des variétés feuilletée, in actualités sci. ind., no. 1183, hermann & cie., paris, 1952, 5 – 89, 155 – 156. [16] b.l. reinhart, foliated manifolds withbundle-like metrics, ann. of math. (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. 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. � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 1 (2022), 139 – 151 doi:10.17398/2605-5686.37.1.139 available online march 7, 2022 order isomorphisms between bases of topologies∗ j. cabello sánchez departamento de matemáticas & imuex, universidad de extremadura avda. de elvas s/n, 06006 badajoz, spain coco@unex.es received august 30, 2021 presented by p. kozsmider accepted february 21, 2022 abstract: in this paper we will study the representations of isomorphisms between bases of topological spaces. it turns out that the perfect setting for this study is that of regular open subsets of complete metric spaces, but we have been able to show some results about arbitrary bases in complete metric spaces and also about regular open subsets in hausdorff regular topological spaces. key words: lattices; 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 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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[10] p.r. halmos, invariant subspaces of polynomially compact operators, pacific j. math. 16 (1966), 433 – 437. [11] p.r. halmos, quasitriangular operators, acta sci. math. (szeged) 29 (1968), 283 – 293. [12] v. lomonosov, invariant subspaces of the family of operators that commute with a completely continuous operator (russian), funkcional. anal. i priloz̆en 7 (1973), 55 – 56. [13] a.j. michael, hilden’s simple proof of lomonosov’s invariant subspace theorem, adv. math. 25 (1977), 56 – 58. [14] c. pearcy, n. salinas, an invariant-subspace theorem, michigan math. j. 20 (1973), 21 – 31. [15] c. perez-garcia, w.h. schikhof, “ locally convex spaces over nonarchimedean valued fields ”, cambridge studies in advanced mathematics 119, cambridge university press, cambridge, 2010. [16] h. radjavi, p. rosenthal, “ invariant subspaces ”, springer-verlag, new york-heidelberg, 1973. [17] w.h. schikhof, on p-adic compact operators, report 8911, department of mathematics, catholic university, nijmegen, the netherlands, 1989, 1 – 28. [18] p. schneider, “ nonarchimedean functional analysis ”, springer-verlag, berlin, 2002. [19] j.p. serre, endomorphismes complètement continus des espaces de banach p-adiques, inst. hautes études sci. publ. math. 12 (1962), 69 – 85. [20] w. śliwa, the invariant subspace problem for non-archimedean banach spaces, canad. math. bull. 51 (2008), 604 – 617. [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 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. [6] a. goshabulaghi, h. vaezi, adjoints of rationally induced composition operators on bergman and dirichlet spaces, turkish j. math. 38 (5) (2014), 862 – 871. [7] c. hammond, j. moorhouse, m.e. robbins, adjoints of composition operators with rational symbol, j. math. anal. appl. 341 (1) (2008), 626 – 639. [8] p.r. hurst, relating composition operators on different weighted hardy spaces, arch. math. 68 (6) (1997), 503 – 513. [9] b.d. maccluer, j.h. shapiro, angular derivatives and compact composition operators on the hardy and bergman spaces, canad. j. math. 38 (4) (1986), 878 – 906. [10] m.j. martín, d. vukotić, adjoints of composition operators on hilbert spaces of analytic functions, j. funct. anal. 238 (1) (2006), 298 – 312. [11] j.n. mcdonald, adjoints of a class of composition operators, proc. amer. math. soc. 131 (2) (2003), 601 – 606. [12] a. salaryan, h. vaezi, notes on generalized weighted composition operators on weighted hardy spaces, submitted. [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. � 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 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. � 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 � extracta mathematicae volumen 33, número 2, 2018 instituto de investigación de matemáticas de la universidad de extremadura extracta mathematicae vol. 37, num. 1 (2022), 111 – 138 doi:10.17398/2605-5686.37.1.111 available online february 15, 2022 prolongations of g-structures related to weil bundles and some applications p.m. kouotchop wamba 1, g.f. wankap nono 2, a. ntyam 1 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 , achillentyam@yahoo.fr received november 1, 2021 presented by manuel de león accepted december 5, 2021 abstract: let m be a smooth manifold of dimension m ≥ 1 and p be a g-structure on m, where g is a lie subgroup of linear group gl(m). in [8], it has been defined the prolongations of g-structures related to tangent functor of higher order and some properties have been established. the aim of this paper is to generalize these prolongations to a weil bundles. more precisely, we study the prolongations of g-structures on a manifold m, to its weil bundle tam (a is a weil algebra) and we establish some properties. in particular, we characterize the canonical tensor fields induced by the a-prolongation of some classical g-structures. key words: g-structures, weil-frobenius algebras, weil functors, gauge functors and natural transformations. msc (2020): 58a32, 53c15 secondary 58a20, 58a10. introduction we recall that, a weil algebra a is a real commutative algebra with unit which is of the form a = r · 1a ⊕na, where na is a finite dimensional ideal of nilpotent elements of a (see [4] or [8]). it exists several examples of weil algebra, for instance the algebra generated by 1 and ε with ε2 = 0 denoted by d (sometimes it is called the algebra of dual numbers, it is also the truncated polynomial algebra of degree 1). another weil algebra is given by the spaces of all r-jets of rk into r with source 0 ∈ rk and denoted by jr0 (r k,r). the ideal of nilpotent elements is the finite vector space jr0 (r k,r)0. let a = r·1a⊕na be a weil algebra, we adopt the covariant approach of weil functor described by i. kolàr in [6]. we denote by nka the ideal generated by the product of k elements of na, there is one and only one natural number h such that n h a 6= 0 and nh+1a = 0. the integer h is called the order of a and the dimension k of 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/10.17398/2605-5686.37.1.111 mailto:wambapm@yahoo.fr mailto:georgywan@yahoo.fr mailto:achillentyam@yahoo.fr https://publicaciones.unex.es/index.php/em https://creativecommons.org/licenses/by-nc/3.0/ 112 p.m.k. wamba, g.f. wankap nono, a. ntyam the vector space na/n2a is said to the width of a. in this case, the weil algebra a is called (k,h)-algebra. if %,%1 : j h 0 (r k,r) → a are two surjective algebra homomomorphisms, then there is an isomorphism σ : jh0 (r k,r) → jh0 (r k,r) such that: %1 ◦σ = %. we say that, two maps ϕ,ψ : rk → m determine the same a-velocity if for every smooth map f : m → r % ( jh0 (f ◦ϕ) ) = % ( jh0 (f ◦ψ) ) . the equivalence class of the map ϕ : rk → m is denoted by jaϕ and will called a-velocity at 0 (see [6], [7] or [8]). we denote by tam the space of all a-velocities on m. more precisely, tam = { jaϕ, ϕ : rk → m } . tam is a smooth manifold of dimension m × dim a. for a local chart( u,u1, . . . ,um ) of m, the local chart of tam is ( tau,ui0, . . . ,u i k ) such that:{ ui0 ( jaϕ ) = ui(ϕ(0)) uiα ( jaϕ ) = a∗α ( ja(ui ◦ϕ) ) 1 ≤ α ≤ k where (a0, . . . ,ak) is basis of a and (a ∗ 0, . . . ,a ∗ k) is a dual basis. we denote by πam : t am → m the natural projection such that πam (j aϕ) = ϕ(0), so( tam,m,πam ) is a fibered manifold. for every smooth map f : m → m, induces a smooth map taf : tam → tam such that: for any jaϕ ∈ tam, taf(jaϕ) = ja(f ◦ϕ). in particular we have that ( f,taf ) is a fibered morphism from ( tam,m,πam ) to ( tam,m,πa m ) . this defines a bundle functor ta : mf → fm called weil functor induced by a. the bundle functor ta preserves product in the sense, that for any manifolds m and m, the map( ta(prm ),t a(prm ) ) : ta(m ×m) −→ tam ×tam where prm : m ×m → m and prm : m ×m → m are the projections, is an fm−isomorphism. hence we can identify ta(m ×m) with tam ×tam. let b be another (s,r) weil algebra and µ : a → b be an algebra homomorphism, %′ : jr0 (r s,r) → b the surjective algebra homomorphism. then prolongations of g-structures related to weil bundles 113 there is an algebra homomorphism µ̃ : jh0 (r k,r) → jr0 (r s,r) such that the following diagram jh0 (r k,r) µ̃ −−−−→ jr0 (r s,r) % y y %′ a −−−−→ µ b commutes. in particular, there is map fµ : rs → rk such that, µ̃(jh0 g) = jr0 (g◦fµ), where g ∈ c ∞(rk). for any manifold m of dimension m ≥ 1, it is proved in [7] that there is smooth map µm : t am → tbm defined by: µm (j aϕ) = jb(ϕ◦fµ). more precisely, µm : t am → tbm is a natural transformations and denoted by µ : ta → tb. the fundamental result, which reads that every product preserving bundle functor on mf is a weil functor. more precisely, if f is a product preserving bundle functor on mf, a : r × r → r and λ : r×r → r is the addition and the multiplication of reals, then fa : fr × fr → fr and fλ : fr ×fr → fr is the vector addition and the algebra multiplication in the weil algebra fr and f coincides with the weil functor tfr. every natural transformation µ : ta → tb are in bijection with the algebra homomorphism µr : a → b (see [8]). since % : jh0 (r k,r) → a is determined up to an isomorphism jh0 (r k,r) → jh0 (r k,r) it follows that this construction is independent of the choice of %. the weil functor generalizes the tangent functor, more precisely, when a is the space of all r-jets of rk into r with source 0 ∈ rk denoted by jr0 (r k,r), the corresponding weil functor is the functor of k-dimensional velocities of order r and denoted by trk . for k = 1, it is called tangent functor of order r and denoted by t r, this functor plays an essential role in the reduction of some hamiltonian systems of higher order. it has been clarified that, the theory of weil functor represents a unified technique for studying a large class of geometric problems related with product preserving functor. let a = r · 1a ⊕na be a weil algebra. for any multiindex 0 < |α| ≤ h, we put eα = j a(xα) is an element of na. for any ϕ ∈ c∞(rk,r) , we have: jaϕ = ϕ(0) · 1a + ∑ 1≤|α|≤h 1 α! dαϕ(0)eα. in particular the family {eα} generates the ideal na. we denote by ba the set of all multiindex such that (eα)α∈ba is a basis of na and ba her 114 p.m.k. wamba, g.f. wankap nono, a. ntyam complementary with respect to the set of all multiindex γ ∈ nn such that 1 ≤| γ |≤ h. for β ∈ ba, we have eβ = λαβeα. in particular, eα ·eβ = { eα+β if α + β ∈ ba , λ γ α+βeγ if α + β ∈ ba . it follows that, for any ϕ ∈ c∞(rk,r) , we have: jaϕ = ϕ(0) · 1a + ∑ α∈ba ( 1 α! dαϕ(0) + ∑ β∈ba λα β β! dβϕ(0) ) eα . let ( u,xi ) be a local coordinate system of m, a coordinate system induced by ( u,xi ) over the open tau of tam denoted by ( xi,xiα ) is given by{ xi = xi ◦πam = x i 0 , xiα = x i α + ∑ β∈ba λ α βx i β , where xiβ(j ag) = 1 β! · dβ(xi ◦ g)(0) and jag ∈ tau. in the particular case where a = d, the local coordinate system of tm induced by ( u,xi ) is denoted by ( xi, ẋi ) . let m be a smooth manifold of dimension m ≥ 1, with (tm,m,πm ) we denote its tangent bundle, and with (f(m),m,pm ) we denote the frame bundles of m. let g be a lie subgroup of gl(m), a g-structure on a manifold m is a g-subbundle (p,m,π) of the frame bundle f(m) of m. for the general theory of g-structures see, for instance [1]. the prolongations of g-structures from a manifold m to its tangent bundles of higher order trm has been studied by a. morimoto in [12]. in particular, it proves that if a manifold m has an integrable structure (resp. almost complex structure, symplectic structure, pseudo-riemannian structure), then trm has canonically the same kind of structure. since the tangent functor of higher order tr on the manifolds, considers all derivatives of higher order (up to order r), all the proofs are obtained by calculation in local coordinate. the situation should be much complicated for the weil functor ta. thus, the aim of this paper is to define the prolongations of g-structures from a manifold m to its weil bundle tam. in particular, we construct a canonical embedding ja,e of t a(fe) into f(tae), where f(e) denote the frame bundle of the vector bundle (e → m). using the natural isomorphism κa,m : t a(tm) → t(tam) (see [5]) and the embedding ja,tm , we define this a-prolongation t ap of a g-structure p of prolongations of g-structures related to weil bundles 115 a manifold m, to its weil bundle tam. in particular, we prove that t ap is integrable if and only if p is integrable. in the last section, we use the theory of lifting of tensor fields defined in [3] and [6], to characterized the canonical tensor fields induced by the a-prolongation of some classical g-structures. in this paper, all manifolds and mappings are assumed to be differentiable of class c∞. in the sequel a will be a weil algebra of order h ≥ 2 and of width k ≥ 1. 1. preliminaries 1.1. lifts of functions and vector fields. let ` : a → r be a smooth function, for any smooth function f : m → r, we define the `-lift of f to tam by: f(`) = `◦ta(f) ; f(`) is a smooth function on tam. remark 1. let (eβ)β∈ba a basis of na, we denote by ( e0,eβ ) β∈ba the dual basis of a. for ` = eα, the smooth function f(`) is denoted by f(α). in particular, for any jaϕ ∈ tam, f(α)(jaϕ) = 1 α! dα(f ◦ϕ)(z) ∣∣ z=0 + ∑ β∈ba λα β β! dβ(f ◦ϕ)(z) ∣∣∣ z=0 and f(0) = f◦πam . for a coordinate system ( u,x1, . . . ,xm ) in m, the induced coordinate system { xi0,x i α } on tam is such that, xiα = ( xi )(α) . remark 2. for any smooth map ` : a → r, the map c∞(m) −→ c∞(tam) f 7−→ f(`) is r-linear. for all multiindex α such that |α| ≤ h, we denote by χ(α) : ta → ta the natural transformation defined for any vector bundle (e → m) and ϕ ∈ c∞(rk,e) by: χ (α) e (j aϕ) = ja(zαϕ) where zαϕ is a smooth map defined for any z ∈ rk by (zαϕ)(z) = zαϕ(z). 116 p.m.k. wamba, g.f. wankap nono, a. ntyam proposition 1. let a be a weil algebra. there exists one and only one family κa,m : t a(tm) → t(tam) of vector bundle isomorphisms such that πtam ◦κa,m = ta(πm ) and the following conditions hold: 1. for every smooth mapping f : m → n the following diagram ta(tm) ta(tf) −−−−−−→ ta(tn) κa,m y yκa,n t(tam) −−−−−−→ t(taf) t(tan) commutes. 2. for two manifolds m, n we have κa,m×n = κa,m ×κa,n . proof. see [5]. let x : m → tm be a vector field on a manifold m, then we put x(α) = κa,m ◦χ (α) tm ◦t a(x) . it is a vector bundle field on ta(m) called α-lift of x to tam. in the particular case where α = 0, the vector field x(0) is denoted by x(c) and it is called complete lift of x to tam. we put x(α) = 0, for |α| > 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. 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, valuable comments and suggestions to improving this paper. references [1] j. alonso, h. martini, s. wu, on birkhoff orthogonality and isosceles orthogonality in normed linear 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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. 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 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[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, 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. 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. 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. [10] y. katsov, toward homological characterization of semirings: serre’s conjecture and bass’s perfectness in a semiring context, algebra universalis 52 (2) (2004), 197 – 214. [11] y. katsov, t.g. nam, n.x. tuyen, on subtractive semisimple semirings, algebra colloq. 16 (3) (2009), 415 – 426. [12] p. nasehpour, on the content of polynomials over semirings and its applications, j. algebra appl. 15 (5) (2016), 1650088. [13] p. nasehpour, valuation semirings, j. algebra appl. 16 (11) (2018), 1850073. [14] j. ohm, d. rush, content modules and algebras, math. scand. 31 (1972), 49 – 68. [15] b. pareigis, h. rohrl, remarks on semimodules, arxiv: 1305.5531, 2013. [16] r. razavi nazari, sh. ghalandarzadeh, multiplication semimodules, submitted. [17] h. subramanian, von neumann regularity in semirings, math. nachr. 45 (1970), 73 – 79. [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. 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 = (