Proximity between selfadjoint operators and between their associated spectral measures EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 4, 2018, 893-910 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Proximity between selfadjoint operators and between their associated spectral measures Alain Boudou1, Sylvie Viguier-Pla2,1,∗ 1 Equipe de Stat. et Proba., Institut de Mathématiques, UMR5219, Université Paul Sabatier, 118 Route de Narbonne, F-31062 Toulouse Cedex 9, France 2 LAMPS, Université de Perpignan, 56 avenue Paul Alduy, F-66860 Perpignan Cedex, France Abstract. We study how the proximity between two selfadjoint bounded operators can be ex- pressed as a proximity between the associated spectral measures. Between two operators, we use a classical distance. For projector-valued spectral measures, we introduce the notion of α−equivalence, which is based on a partial order relation on the set of projectors. Assuming an hypothesis of commutativity, we show that the proximity between operators is equivalent with the proximity between the associated spectral measures. We develop the particular case where the operators are compact, and give some illustrations. 2010 Mathematics Subject Classifications: 60G57, 60G10, 60B15, 60H05 Key Words and Phrases: Random measures, Stationary processes, Convolution, Spectral mea- sures 1. Introduction The question of the association of a spectral measure (s.m.) with an operator is a usefull technique, and often considered for the analysis of their properties ([4], [3]). Therefore, if P is an orthogonal projector, and λ a real, then A = λP is a selfadjoint bounded operator, which associated s.m. is E = δ0P⊥ + δλP . Besides, the null operator O is associated with the s.m. ER = δ0I. As ‖A−O‖ = |λ|, A and O are two selfadjoint operators as close as we want, as far as we can get |λ| as small as we want. Nevertheless, let us consider the proximity between their associated s.m.’s. For any B of a σ−field defined on R, we have E(B)− ER(B) = δ0(B)P⊥ + δλ(B)P − δ0(B)P⊥ − δ0(B)P = δλ(B)P − δ0(B)P . So ‖E(B)− ER(B)‖ = |δλ(B)− δ0(B)|, which maximum is obviously equal to 1. This shows that the proximity between two s.m.’s, evaluated by sup{‖E(B)−ER(B)‖;B ∈ BR}, is not appropriate to be linked with the proximity between their associated operators. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v11i4.2552 Email addresses: boudou@math.univ-toulouse.fr (A. Boudou), viguier@univ-perp.fr (S. Viguier-Pla) http://www.ejpam.com 893 c© 2018 EJPAM All rights reserved. A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 894 In [2], we have seen how the proximity between unitary operators could be equivalent with proximity between their associated s.m.’s. In this paper, we develop tools for the study of the association between selfadjoint bounded operators and s.m.’s. Our study of proximity is placed in this context. In order to fix notation, Section 2 is devoted to the recall of some notions, as random measure, spectral measure, space of the measurable applications of integrable square with respect to a random measure, and projector-valued spectral measure. We introduce in a third section the notion of α−equivalence, for α positive real, between two s.m.’s. This concept of proximity can be translated by proximity between two selfadjoint bounded operators, what we will develop in Section 4. The fift section will be dedicated to the case of the compact operators, which is frequently encountered in numerical applications. A numerical illustration is given in the last section. 2. Recalls 2.1. Orthogonal families of projectors In this text, H, L(H) and P(H) are respectively a C−Hilbert space, the set of bounded endomorphisms of H (which is a Banach space for the norm ‖A‖L = sup{‖Ax‖; ‖x‖ = 1}), and the set of the orthogonal projectors on H. Let P1 and P2 be two elements of P(H), P1 is said to be “less or equal” to P2, what we denote P1 � P2, if P1P2 = P1. This defines, on P(H), a partial order relation. A family {Pn;n ∈ N} of elements of P(H) is said to be orthogonal when Pn ◦ Pm = 0, for any pair (n,m) of distinct elements of N. Then we can show that a) for any X of H, the family {PnX;n ∈ N} is summable; b) the application P : X ∈ H 7→ ∑ n∈N PnX ∈ H is an element of P(H) which we name the sum of the orthogonal family of orthogonal projectors {Pn;n ∈ N} and which we denote ∑ n∈N Pn; c) for any n of N, we have Pn ◦ P = Pn. We emphasize the fact that an orthogonal family of orthogonal projectors can be not summable (as a family of elements of the Banach space L(H)). The following first result, which can be easily proved, will be used in Section 3. Lemma 2.1.1. If {Dn;n ∈ N} and {D′n;n ∈ N} are two orthogonal families of orthogonal projectors of respective sum D and D′ such that Dn � D′n, for any n of N, then D � D′. 2.2. Random measure Let ξ be a σ−field of subsets of a set E. For any e of E, δe stands for the Dirac measure defined on ξ and concentrated on e. A random measure (r.m.) defined on ξ, taking values in H, is an application Z from ξ into H such that: A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 895 a) Z(A ∪ B) = Z(A) + Z(B) and < Z(A), Z(B) >= 0, for any pair (A,B) of disjoint elements of ξ; b) limn→∞Z(An) = 0 for any sequence (An)n∈N of elements of ξ which decreasingly con- verges to ∅. We then have the following properties. a) The application µZ : A ∈ ξ 7→ ‖Z(A)‖2 ∈ R+ is a bounded measure; b) there exists one, and only one, isometry from L2(E, ξ, µZ) onto HZ = vect{Z(A);A ∈ ξ} such that for any A of ξ, ZA is the image of 1A. The stochastic integral of an element ϕ of L2(E, ξ, µZ) with respect to the r.m. Z, is the image of ϕ by this isometry, and we note it ∫ ϕdZ. For example, if {Zn, n ∈ N} is a summable orthogonal family of elements of H, if (en)n∈N is a sequence of elements of E such that {en} ∈ ξ, then, for any B of ξ, the family {δen(B)Zn, n ∈ N}, of elements of H, is summable. The application Z : B ∈ ξ 7→∑ n∈N δen(B)Zn ∈ H is a r.m.. If f is an element of L2(µZ), then {f(en)Zn, n ∈ N} is a summable family of elements of H, of sum ∫ fdZ. In the all text, (E′, ξ′) denotes a second measurable space. Let f be a measurable application from E into E′. We can affirm the following. a) The application f(Z) : A′ ∈ ξ′ 7→ Z(f−1A′) ∈ H is a r.m., named r.m. image of Z by f ; b) f(µZ) = µf(Z); c) if ϕ′ is a element of L2(E′, ξ′, µf(Z)), then ϕ′◦f belongs to L2(E, ξ, µZ) and ∫ ϕ′df(Z) =∫ ϕ′ ◦ fdZ. A stationary continuous random function (c.r.f.) (Xt)t∈R is a family of elements of H such that the application t ∈ R 7→ Xt ∈ H is continuous and such that < Xt, Xt′ >=< Xt−t′ , X0 > for any pair (t, t′) of elements of R. There exists one, and only one r.m. Z, named r.m. associated with the stationary c.r.f. (Xt)t∈R, defined on BR, such that Xt = ∫ ei.tdZ, for any t of R. Then we will say that two c.r.f.’s (Xt)t∈R and (X ′t)t∈R are stationarily correlated when < Xt, X ′ t′ >=< Xt−t′ , X ′ 0 >, for any pair (t, t′) of elements of R. This property can be expressed in terms of associated r.m.’s: if Z and Z ′ are two r.m.’s associated with two stationary c.r.f.’s, stationarily correlated, then, for any pair (A,A′) of disjoint elements of BR, we have < Z(A), Z ′(A′) >= 0. 2.3. Spectral measure A spectral measure (s.m.) E on ξ for H is an application from ξ into P(H) such that a) E(E) = IH ; b) E(A ∪B) = E(A) + E(B), for any pair (A,B) of disjoint elements of ξ; c) limn→∞E(An)X = 0, for any X of H and for any sequence (An)n∈N of elements of ξ which decreasingly converges to ∅. Then we easily check the following properties. A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 896 a) For any X of H, the application ZXE : A ∈ ξ 7→ E(A)X ∈ H is a r.m.. b) If f is a measurable application from E into E′, the application f(E) : A′ ∈ ξ′ 7→ E(f−1A′) ∈ P(H) is a r.m. on ξ′ for H named s.m. image of E by f . For any X of H, we have f(ZXE ) = ZXf(E). Let us now examine more particularly the s.m.’s on BR, the Borel σ−field of R, for H. Two s.m.’s E1 and E2, on BR for H, commute when, for any pair (A1, A2) of elements of BR, E1(A1) and E2(A2) commute. If E1 and E2 are two s.m.’s on BR for H, which commute, then there exists one s.m., and only one, on BR⊗BR for H, denoted E1⊗E2, such that E1⊗E2(A1×A2) = E1(A1)◦E2(A2), for any (A1, A2) of BR × BR. We then name convolution product (or shortly convolution) of E1 and E2, and we denote it E1 ∗ E2, the s.m. image of E1⊗E2 by the measurable application S : (λ1, λ2) ∈ R×R 7→ λ1 + λ2 ∈ R (cf. [1]). This convolution has got an identity element. More precisely, the application ER : A ∈ BR 7→ δ0(A)IH ∈ P(H) is a s.m. which commutes with any s.m. E , on BR for H. Moreover, E ∗ ER = E . When one of the elements of the product E1 ∗ E2 is concentrated on a countable family of reals Λ = {λj ; j ∈ N}, we get a result which seems natural for a convolution. If E1 is a s.m., on BR for H, such that E1(Λ) = IH , and which commutes with a second s.m. E2, on BR for H, then, for any A of BR, the set {E1({λj}) ◦ E2(A− λj); j ∈ N}, is an orthogonal family of projectors which sum equals (E1 ∗ E2)(A). Let us end these recalls with algebraic properties of the convolution. If E is a s.m. on BR for H, if f1 and f2 are two measurable applications from R into itself, then the s.m.’s f1(E) and f2(E) commute and (f1 + f2)(E) = (f1(E)) ∗ (f2(E)). If we denote by w the measurable application x ∈ R 7→ −x ∈ R, then, taking into account the previous results, for any s.m. E on BR for H, we can write E ∗ (w(E)) = (IR + w)(E) = O (E) = ER. This means that any s.m. E has got its symmetric, the s.m. w(E), for the convolution. When it exists, the convolution is associative. Finally, if E1 and E2 commute, then the s.m.’s E1 and E1 ∗ E2 also commute. 2.4. The space M(E, E) Let E be a s.m. on ξ, σ−field of subsets of a set E, for H. M(E, E) is the set of the measurable applications ϕ, from E into C, such that a) ∫ |ϕ|2dµZXE < +∞, for any X of H; b) the set of the reals { ∫ |ϕ|2dµZXE ; ‖X‖ = 1} is bounded. Then, when ϕ is an element of M(E, E), we can consider the application Eϕ : X ∈ H 7→ ∫ ϕdZXE ∈ H, and we have the following property. Proposition 2.4.1. For any ϕ ofM(E, E), the application Eϕ is linear and bounded. A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 897 Proof. If ϕ is an element of M(E, E), then, for any (λ, λ′, X,X ′) of C× C×H ×H, it belongs to L2(µ ZλX+λ′X′ E + µZXE + µ ZX ′ E ) (because ∫ |ϕ|2d(µ ZλX+λ′X′ E + µZXE + µ ZX ′ E ) =∫ |ϕ|2dµ ZλX+λ′X′ E + ∫ |ϕ|2dµZXE + ∫ |ϕ|2dµ ZX ′ E ). Taking into account the properties of density of the indicator functions, ϕ can be writen as follows. ϕ = limn→∞ ∑ j∈Jn αn,j1Bn,j , |Jn| < +∞, Bn,j ∈ BR, in L2(µ ZλX+λ′X′ E + µZXE + µ ZX ′ E ). (2.4.1) As ‖.‖2L2(µ ZλX+λ′X′ E +µ ZXE +µ ZX ′ E ) = ‖.‖2L2(µ ZλX+λ′X′ E ) +‖.‖2L2(µ ZXE ) +‖.‖2L2(µ ZX ′ E ), the equal- ity (2.4.1) is exact in L2(µ ZλX+λ′X′ E ), L2(µZXE ), and L2(µ ZX ′ E ). Integrating it successively with respect to the r.m.’s ZλX+λ′X′ E , ZXE and ZX ′ E , we have: Eϕ(λX + λ′X ′) = limn→∞ ∑ j∈Jn αn,jZ λX+λ′X′ E (Bn,j), EϕX = limn→∞ ∑ j∈Jn αn,jZ X E (Bn,j), EϕX ′ = limn→∞ ∑ j∈Jn αn,jZ X′ E (Bn,j). Then the linearity comes from the fact that: ZλX+λ′X′ E (Bn,j) = E(Bn,j)(λX + λ′X ′) = λE(Bn,j)X + λ′E(Bn,j)X ′ = λZXE (Bn,j) + λ′ZX ′ E (Bn,j). Finally, the continuity comes from the fact that, for any normed element X of H: ‖EϕX‖2 = ‖ ∫ ϕdZXE ‖2 = ∫ |ϕ|2dµZXE 6 sup{ ∫ |ϕ|2dµZXE ; ‖X‖ = 1}. � Now it is clear that M(E, E) has got a vector space structure, as a subspace of the vector space of the measurable applications from E into C. From the linearity of the stochastic integral, we deduce the following proposition. Proposition 2.4.2. The application ϕ ∈M(E, E) 7→ Eϕ ∈ L(H) is linear. Let us now approach a result close to the transfert theorem. Proposition 2.4.3. When E is a s.m. on ξ, σ−field of subsets of a set E for H, and when f is a measurable application from E into E′, then, for any ϕ of M(E′, f(E)), we can affirm that ϕ ◦ f belongs to M(E, E). Moreover, we have (f(E))ϕ = Eϕ◦f . Proof. The properties of a r.m. image allow us, for any X of H, to obtain:∫ |ϕ ◦ f |2dµZXE = ∫ |ϕ|2 ◦ fdµZXE = ∫ |ϕ|2df(µZXE ) = ∫ |ϕ|2dµZX f(E) , so ϕ ◦ f belongs to M(E, E), because ϕ is an element of M(E′, f(E)). Moreover, (f(E))ϕX = ∫ ϕdZXf(E) = ∫ ϕdf(ZXE ) = ∫ ϕ ◦ fdZXE = (Eϕ◦f )(X), for any X of H, what ends the proof. � Let us now introduce a new notion. Definition 2.4.1. We say that a s.m. E, on BR for H, is bounded when there exists a real a > 0 such that E([−a, a[) = IH . This characteristic is stable by convolution as follows. Lemma 2.4.1. If E1 and E2 are two bounded r.m.’s, on BR for H, which commute, then E1 ∗ E2 is also bounded. A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 898 Proof. Because E1 and E2 are bounded, there exists two strictly positive reals a1 and a2 such that IH = E1([−a1, a1[) = E2([−a2, a2[). If we denote a = a1 + a2, as [−a1, a1[×[−a2, a2[⊂ S−1[−a, a[, we can write IH = E1 ⊗ E2([−a1, a1[×[−a2, a2[)� E1 ⊗ E2(S−1[−a, a[) = E1 ∗ E2([−a, a[) 6 IH , then IH = E1 ∗ E2([−a, a[), what ends the proof. � Let us end this section by results which we will use later. For any n of N, it is clear that the application jn : λ ∈ R 7→ λn ∈ C is measurable and, when µ is a bounded measure defined on BR having a compact support, then vect{jn, n ∈ N} is dense in L2(R,BR, µ). So, when E is a bounded s.m. on BR for H, then vect{jn, n ∈ N} = L2(R,BR, µZXE ), this for any X of H. Moreover, jn belongs to M(R, E), so we can consider the applications Ejn . Lemma 2.4.2. If {Dp; p ∈ N} is an orthogonal family of projectors of sum I, if (µp)p∈N∗ is a real sequence which decreasingly strictly converges to 0 and if we set µ0 = 0, then we can affirm that a) for any B of BR, {δµp(B)Dp; p ∈ N} is an orthogonal family of projectors; b) the application E : B ∈ BR 7→ ∑ p∈N δµp(B)Dp ∈ P(H) is a bounded s.m.; c) {µpDp; p ∈ N} is a family of elements of L(H), summable of sum Ej. Proof. For any B of BR, it is clear that {δµp(B)Dp; p ∈ N} is an orthogonal family of projectors. If we denote by E(B) its sum, for any X of H, {δµp(B)DpX; p ∈ N} is a summable family of sum (E(B))X. As the set of the indexes is N, we can write (E(B))X = limn ∑n p=0 δµp(B)DpX, and then ‖(E(B))X‖2 = limn‖ ∑n p=0 δµp(B)DpX‖2 = limn ∑n p=0 δµp(B)‖DpX‖2 = ∑ p∈N δµp(B)‖DpX‖2. (2.4.2) Of course, E(R) = I (because ∑ p∈NDp = I). Moreover, if (B1, B2) is a pair of disjoint elements of BR, we have E(B1 ∪B2) = E(B1) + E(B2), because for any p of N we have: δµp(B1 ∪B2)DpX = δµp(B1)DpX + δµp(B2)DpX. Let (Bn)n∈N be a sequence of elements of BR which decreasingly converges to ∅ and X an element of H. In order to prove that E is a s.m., it remains to be proved that limn(E(Bn))X = 0. For this, let us first recall that, if (ap)p∈N is a sequence of elements of R+ such that∑ p∈N ap < +∞, if {fn,p; (n, p) ∈ N × N} is a family of positive reals such that, for any p of N, limnfn,p = 0, and if fn,p < ap for any (n, p) of N× N, then on one side, for any n of N, the family {fn,p; p ∈ N} is summable, and on another side, limn ∑ p∈N fn,p = 0. Now let us apply this result to ap = ‖DpX‖2 and fn,p = δµp(Bn)‖DpX‖2, for any (n, p) of N× N. It comes limn ∑ p∈N δµp(Bn)‖DpX‖2 = 0, or, taking into account the result of (2.4.2), limn‖(E(Bn))X‖2 = 0, so A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 899 limn(E(Bn))X = 0. So E is a s.m., and is clearly bounded (because, if we choose a such that {µp; p ∈ N} ⊂ [−a, a[, then E({[−a, a[) = 0). When X is an element of H, from recalls of Sec- tion 2.2, as {DpX; p ∈ N} is a summable orthogonal family of elements of H, the family {j(µp)DpX; p ∈ N}, and then the family {µpDpX; p ∈ N}, is summable of sum ∫ jdZXE , and hence of sum Ej(X). As the set of indices is N, we can write Ej(X) = limp→+∞ ∑p k=0 µkDkX. Let us consider ε an element of R∗+. There exists an integer Nε such that µNε 6 ε (because limnµn = 0). For any finite part J of N, disjoint of {0, 1, . . . , Nε− 1, Nε}, we can write ‖ ∑ p∈J µpDp‖ 6 max{µp; p ∈ J} < µNε 6 ε. This allows us to affirm that {µpDp; p ∈ N} is a summable family of elements of L(H). As the set of indices of this summable family is N, it comes ( ∑ p∈N µpDp)X = (limp ∑p k=0 µpDp)X = limp ∑p k=0 µpDpX = Ej(X), taking into account what precedes. This ends the proof of point c). � 2.5. Spectral measure associated with an operator Many monographs (see, for instance, Dunford and Schwartz, 1963, Riesz and Nagy, 1991) evoque an association between a s.m. E and a selfadjoint operator A which allows to express this last one as an integral: A = ∫ λdE(λ). More precisely, we check the following. Let A be a bounded selfadjoint operator. There exists one, and only one, bounded s.m. E , named s.m. associated with A, such that, for any X of H, AX = ∫ jdZXE . This s.m. is such that ‖A‖ = inf{a ∈ R+; E([−a, a]) = IH}. Without pretention of giving exhaustive explanations, we will give some indications on the way the s.m. E is defined. If A is a bounded selfadjoint operator, it is easy to verify that (eitA(X))t∈R is a stationary c.r.f., and we denote by ZX its associated r.m.. So we obtain a family of stationary c.r.f.’s, which are pairwise stationarily correlated. From this fact, we deduce, on one hand, that for any B of BR, the application E(B) : X ∈ H 7→ ZX(B) ∈ H is a projector, and on another hand, that the application E : B ∈ BR 7→ E(B) ∈ P(H) is a bounded s.m. such that A = Ej , or, in other words, such that AX = ∫ jdZXE , for any X of H. For any a > ‖A‖, we can write A = limm→∞ ∑k=m−1 k=0 (−a+ k 2a m )E([−a+ k 2a m ,−a+ (k + 1)2am [), in L(H), expression which evoques a Riemann sum associated with an integral of the type ∫ λdE(λ). Let us now examine the following preliminary result. Lemma 2.5.1. If E is the s.m. associated with the bounded selfadjoint operator A, then, for any n of N, we have An = Ejn. Proof. The proof is obtained by induction. In fact, if n is an integer such that An = Ejn , then, for any X of H, we have: < An+1X,X >=< AnX,AX >=< Ejn(X), Ej(X) >=< ∫ jndZXE , ∫ jdZXE > =< jn, j >L2(µ ZXE )= ∫ jn+1dµZXE =< ∫ jn+1dZXE , ∫ R dZXE >=< Ejn+1X,X >. A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 900 Then An+1 = Ejn+1 , what means that the property is true for n+ 1. � So the proposition follows. Proposition 2.5.1. If A is a bounded selfadjoint operator of associated s.m. E, if T is an element of L(H) such that T ◦ A = A ◦ T , then, for any B of BR, T and E(B) commute. Proof. As µZTXE + µZXE has a compact support, for any B of BR, 1B can be writen as 1B = limm→∞ ∑ j∈Jm αj,mj nj,m , |Jm| < +∞, in L2(µZTXE + µZXE ). (2.5.1) Equality (2.5.1) is exact in L2(µZTXE ) and in L2(µZXE ), its integrations successively with respect to the r.m.’s ZTXE and ZXE give: (E(B))TX = limm→∞ ∑ j∈Jm αj,mA nj,mTX, (2.5.2) and (E(B))X = limm→∞ ∑ j∈Jm αj,mA nj,mX, hence T (E(B))X = limm→∞ ∑ j∈Jm αj,mTA nj,mX, what, taking into account result (2.5.2), and as Anj,m ◦ T = T ◦ Anj,m , allows us to write E(B) ◦ T = T ◦ E(B). � This property has got its converse. Proposition 2.5.2. If A is a bounded selfadjoint operator of associated s.m. E, if T is an element of L(H) such that T ◦ E(B) = E(B) ◦ T , then, for any B of BR, T and A commute. Proof. Taking into account the property of density of the indicator functions, the element j of L2(µZTXE + µZXE ) can be writen as: j = limm→∞ ∑ j∈Jm αj,m1Bj,m , Bj,m ∈ BR, |Jm| < +∞, in L2(µZTXE + µZXE ). (2.5.3) The equality (2.5.3) is true in L2(µZTXE ) and in L2(µZXE ), by integration with respect to the r.m.’s ZTXE and ZXE , we have: ATX = limm→∞ ∑ j∈Jm αj,m(E(Bj,m))TX, (2.5.4) and AX = limm→∞ ∑ j∈Jm αj,m(E(Bj,m))X, hence TAX = limm→∞ ∑ j∈Jm αj,mT (E(Bj,m))X, what allows, thanks to (2.5.4) and to the fact that T ◦E(Bj,m) = E(Bj,m)◦T , to write: A ◦ T = T ◦A. � The following property is obtained combining these two last results. Proposition 2.5.3. Two bounded selfadjoint operators A and A′ commute if, and only if, their associated bounded s.m.’s commute. We are now able to examine the main result of this section. Proposition 2.5.4. If A and A′ are two bounded selfadjoint operators which com- mute, of respective associated s.m.’s E and E ′, then E ∗ E ′ is the s.m. associated with the operator A+A′. A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 901 Proof. Let us first notice that, because A and A′ commute, the bounded s.m.’s E and E ′ also commute. So we can consider the s.m.’s E ⊗ E ′ and E ∗ E ′, this last one being bounded. Let us denote by P (resp. P ′) the measurable application (λ, λ′) ∈ R2 7→ λ ∈ R (resp. (λ, λ′) ∈ R2 7→ λ′ ∈ R). We easily verify that P (E ⊗ E ′) = E (resp. P ′(E ⊗ E ′) = E ′). As E is a bounded s.m., j belongs toM(R, E), so toM(R, P (E⊗E ′)). Proposition 2.4.3 allows us to affirm that j ◦P belongs toM(R×R, E ⊗E ′) and that Ej = (E ⊗E ′)j◦P . In a same way, we get E ′j = (E⊗E ′)j◦P ′ . The linearity of the application f ∈M(R×R, E⊗E ′) 7→ (E ⊗ E ′)f ∈ L(H) allows us to write: Ej + E ′j = (E ⊗ E ′)j◦P + (E ⊗ E ′)j◦P ′ = (E ⊗ E ′)j◦P+j◦P ′ = (E ⊗ E ′)j◦S (2.5.5) Besides, as j is an element ofM(R, E∗E ′), so ofM(R, S(E⊗E ′)), from Proposition 2.4.3, j ◦ S belongs to M(R× R, E ⊗ E ′) and (S(E ⊗ E ′))j = (E ⊗ E ′)j◦S . So, taking into account (2.5.5), we have Ej + E ′j = (S(E ⊗ E ′))j = (E ∗ E ′)j , or also A+A′ = (E ∗ E ′)j , what ends the proof. � 3. The α−equivalence In this section, we will examine a proximity relation between s.m.’s on BR for H. We will frequently use the fact that, if B is a compact subset of R and α an element of R∗+, then B + [−α, α] is compact. Definition 3.1. Let α be an element of R∗+, we say that two s.m.’s E1 and E2 are α−equivalent, what we denote E1 α∼ E2, when, for any compact B, we have i) E1(B)� E2(B + [−α, α]); ii) E2(B)� E1(B + [−α, α]). Remark. When two s.m.’s are α−equivalent, for any compact B, we have: E1(B) � E2(B + [−α, α]) � E1(B + [−2α, 2α]). As far as α is small, the compact sets B, B + [−α, α] and B + [−2α, 2α] are close together. It is also the case for the projectors E1(B) and E1(B + [−2α, 2α]). So the projector E2(B + [−α, α]), which is between E1(B) and E1(B + [−2α, 2α]), is close to E1(B). This induces the proximity between E1(B) and E2(B). Apparently, the relation of α−equivalence is symmetric, so the following property is close to a property of transitivity. Proposition 3.1. If three s.m.’s E1, E2 and E3 are such that E1 α∼ E2 and E2 α′∼ E3, where α and α′ are elements of R∗+, then E1 α+α′∼ E3. Proof. It is the result of the relations: E1(B)� E2(B + [−α, α])� E3((B + [−α, α]) + [−α′, α′]) = E3(B + [−(α+ α′), α+ α′]), A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 902 and, in a symmetric way: E3(B)� E2(B + [−α′, α′])� E1((B + [−α′, α′]) + [−α, α]) = E1(B + [−(α+ α′), α+ α′]), for any compact B. � The α−equivalence provides a kind of continuity as follows. Proposition 3.2. If (αn)n∈N is a sequence of elements of R∗+ which decreasingly converges to α, element of R∗+, if E1 and E2 are two s.m.’s such that E1 αn∼ E2, for any n of N, then E1 α∼ E2. Proof. Let B be a compact. It is known that B + [−α, α] = ∩n∈N(B + [−αn, αn]). As (B+ [−αn, αn])n∈N is a decreasing sequence of elements of BR, for any X of H, the properties of continuity of the r.m.’s allow us to write: limn→∞E1(B + [−αn, αn])X = limn→∞Z X E1(B + [−αn, αn]) = ZXE1(∩n∈N(B + [−αn, αn])) = ZXE1(B + [−α, α]) = E1(B + [−α, α])X, then limn→∞(E2(B))(E1(B + [−αn, αn]))X = (E2(B))(E1(B + [−α, α]))X. As (E2(B))(E1(B + [−αn, αn]))X = E2(B), because E2(B)� E1(B + [−αn, αn]), what precedes lets us write (E2(B))X = (E2(B))(E1(B + [−α, α]))X. It is then clear that E2(B)� E1(B + [−α, α]). The relation E2(B) � E1(B + [−α, α]) can be proved in a similar way, what ends the proof. � When a s.m. is concentrated on the neigbourhood of 0, it is close to ER, this is what is expressed in the following result. Proposition 3.3. A s.m. E is α−equivalent to ER if and only if E([−α, α]) = IH . Proof. Let E be a s.m. such that E([−α, α]) = IH . Let us consider a compact B. If 0 ∈ B+[−α, α], then E(B)� IH = ER(B+[−α, α]). If 0 6∈ B+[−α, α], thenB∩[−α, α] = ∅ and then 0 = E(B ∩ [−α, α]) = (E(B))(E([−α, α])) = E(B)� ER(B + [−α, α]). In both cases we have E(B)� ER(B + [−α, α]). In order to prove that ER(B)� E(B + [−α, α]), we also have two possibilities. Either ER(B) = IH , either ER(B) = 0. In the first case, 0 ∈ B and then [−α, α] ⊂ B+[−α, α], so ER(B) = IH = E([−α, α])� E(B + [−α, α]). In the second case, ER(B) = 0� E(B + [−α, α]). So we can conclude to the α−equivalence between the s.m.’s E and ER. As for the converse, it comes from the relations IH = ER({0})� E({0}+ [−α, α]) = E([−α, α]) = IH . � Let us now start the study of the transmission of the α−equivalence through convolu- tion. First of all, let us introduce a preliminary result. Lemma 3.1. Let E and E ′ be two s.m.’s which commute. If E is α−equivalent with ER, then E ′ ∗ E α∼ E ′. Proof. For any compact B of R, A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 903 (R× [−α, α]) ∩ S−1B ⊂ (B + [−α, α])× [−α, α], and we can write, from the fact that E([−α, α]) = IH : (E ′ ∗ E)(B) = (E ′ ⊗ E)(S−1B) = (E ′ ⊗ E)(R× [−α, α])(E ′ ⊗ E)(S−1B) = (E ′ ⊗ E)((R× [−α, α]) ∩ S−1B) � (E ′ ⊗ E)((B + [−α, α])× [−α, α]) = E ′(B + [−α, α]). Moreover, as B × [−α, α] ⊂ S−1(B + [−α, α]), we also have E ′(B) = E ′ ⊗ E(B × [−α, α]) � E ′ ⊗ E(S−1(B + [−α, α])) = E ′ ∗ E(B + [−α, α]), what allows to conclude. � Let us now examine the case where a s.m. is concentrated on a countable family. Lemma 3.2. Let Λ = {λn;n ∈ N} be a countable family of reals, and E be a s.m. such that E(Λ) = IH . If E1 and E2 are two α−equivalent s.m.’s which commute with E, then E ∗ E1 α∼ E ∗ E2. Proof. Let B be a compact of R. From Section 2.3, we can affirm that i) {E({λn})E1(B − λn);n ∈ N} is an orthogonal family of projectors which sum is E ∗ E1B; ii) {E({λn})E2(B + [−α, α] − λn);n ∈ N} is an orthogonal family of projectors which sum is E ∗ E2(B + [−α, α]). As the s.m.’s E and E1 commute and from E1(B − λn) ◦ E2(B + [−α, α]− λn) = E1(B − λn), (because E1 α∼ E2) we can write E({λn})E1(B − λn)E({λn})E2(B + [−α, α]− λn) = E({λn})E1(B − λn)E2(B + [−α, α]− λn) = E({λn})E1(B − λn), and so E({λn})E1(B − λn)� E({λn})E2(B + [−α, α]− λn), this for any n ∈ N. From the recalls of Section 2.1, we can affirm that E ∗ E1(B)� E ∗ E2(B + [−α, α]). Conversely, we could prove that E ∗ E2(B)� E ∗ E1(B + [−α, α]), what allows to conclude. � A discretization of R allows to approximate any s.m. by a s.m. concentrated on a countable family. For this, let us denote by Ln the measurable application x ∈ R 7→ [nx] n ∈ R, where n is an element of N∗ and [nx] the integer part of nx. Then the proposition follows. Proposition 3.4. For any n of N∗ and for any s.m. E, we can affirm that a) LnE ∗ wE 1 n∼ ER; b) LnE 1 n∼ E. Proof. It is easy to verify that, for any x of R, − 1 n < (Ln + w)(x) 6 0, A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 904 we deduce from this that (Ln + w)−1[− 1 n , 1 n ] = R. So it comes (Ln + w)E([− 1 n , 1 n ]) = E((Ln + w)−1[− 1 n , 1 n ]) = E(R) = IH . Then, from Proposition 3.3, (Ln + w)E 1 n∼ ER, or in other words (LnE) ∗ (wE) 1 n∼ ER, and Point a) is proved. As for Point b), it is a consequence of Point a) and of Lemma 3.1. � This last property lets us generalize both Lemma 3.1 and Lemma 3.2. Proposition 3.5. If E1 and E2 are two α−equivalent s.m.’s, which commute with a third s.m. E, then we have E ∗ E1 α∼ E ∗ E2. Proof. For any n of N∗, we can affirm that LnE ∗ wE 1 n∼ ER. As the s.m.’s LnE ∗ wE and E ∗ E1 commute, Lemma 3.1 allows us to write (LnE ∗ wE) ∗ (E ∗ E1) 1 n∼ E ∗ E1, or, taking into account the associativity of the convolution, LnE ∗ E1 1 n∼ E ∗ E1. (3.1) In a similar way, we can prove that LnE ∗ E2 1 n∼ E ∗ E2. (3.2) As LnE is a s.m. concentrated on a countable family, and as E1 α∼ E2, Lemma 3.2 lets us write: LnE ∗ E1 α∼ LnE ∗ E2. (3.3) From relations (3.1), (3.2) and (3.3), we deduce E ∗ E1 α+ 2 n∼ E ∗ E2. As the sequence (α+ 2 n)n∈N∗ decreasingly converges to α, Proposition 3.2 allows us to conclude. � 4. Proximity between operators and α−equivalence If E is the s.m. associated with a bounded selfadjoint operatorA, then E([−‖A‖, ‖A‖]) = IH , and Proposition 3.3 induces the following result. Proposition 4.1. If E is the s.m. associated with a bounded selfadjoint operator A, then E ‖A‖∼ ER. This means that if A is close to the null operator O, then the s.m. E is close to ER, the s.m. associated with O. Indeed, as ZXER = δ0(.)X, it comes (ER)jX = ∫ jdδ0(.)X = j(0)X = 0, for any X of H. So it seams that the proximity between operators can be transposed to s.m.’s. The following result lets us approach this aspect. Lemma 4.1. If A and A′ are two bounded selfadjoint operators which commute, of respective associated s.m.’s E and E ′, then E ∗(wE ′) is the s.m. associated with the bounded selfadjoint operator A−A′. A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 905 Proof. It is clear that if E ′ is the s.m. associated with A′, then wE ′ is the s.m. associated with −A′. Indeed, from one side, wE ′ is bounded, and from another side, we have (wE ′)jX = ∫ jdZXwE ′ = ∫ jdw(ZXE ′ ) = ∫ j ◦ wdZXE ′ = ∫ −jdZXE ′ = −A′X, for any X of H. So, from Proposition 2.5.4, the s.m. associated with A−A′ is E ∗ (wE ′). � We are now able to enunciate a property which generalizes the previous result. Proposition 4.2. If A and A′ are two bounded selfadjoint operators which commute, of respective associated s.m.’s E and E ′, then E ‖A−A ′‖∼ E ′. Proof. As E ∗wE ′ is the s.m. associated with the bounded selfadjoint operator A−A′, Proposition 4.1 allows us to write E ∗ wE ′ ‖A−A ′‖∼ ER. As E ′ commute with E ∗ wE ′, Lemma 3.1 allows us to affirm that E ′ ∗ (E ∗ wE ′) ‖A−A ′‖∼ E ′, so that E ‖A−A ′‖∼ E ′, and the property is proved. � The following property is, in some way, the converse of the previous one. Proposition 4.3. Let E and E ′ be respectively the s.m.’s associated with the bounded selfadjoint operators A and A′, which commute. If E and E ′ are α−equivalent, then ‖A− A′‖ 6 α. Proof. The hypothesis of the Proposition can be writen E α∼ E ′. As wE ′ commutes with E (because E commutes with E ′), Proposition 3.5 allows us to write wE ′ ∗ E α∼ wE ′ ∗ E ′, that is E ∗ wE ′ α∼ ER. From Proposition 3.3, we have then E ∗ wE ′([−α, α]) = IH . But, as ‖A − A′‖ = inf{a ∈ R∗+, E ∗ wE ′([−a, a]) = IH}, because E ∗ wE ′ is the s.m. associated with A−A′. We have then ‖A−A′‖ 6 α, what allows to conclude. � The notion of α−equivalence is a good translation of the proximity of s.m.’s associ- ated with two operators, because of the equivalence with the closeness of the associated operators. Remark. From Proposition 4.2, when two self-adjoint bounded operators which com- mute are close together, the same happens for their respectively associated s.m.’s, accord- ing to the α−equivalence. We examine here an example where, when the commutativity is not satisfied, the proximity of the operators does not imply the proximity of the associated s.m.’s. It illustrates the necessity of this hypothesis for this proposition. Let us consider two elements y and h of H such that < y, h >= 0, ‖y‖2 + ‖h‖2 = 1, ‖y‖.‖h‖ 6= 0, and ‖h‖ < 1 4 . Let x = y + h and x′ = y − h. A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 906 It is clear that P = x ⊗ x and P ′ = x′ ⊗ x′ are orthogonal projectors such that ‖P − P ′‖ 6 2‖h‖ < 1 2 . We can easily verify that < x, x′ >6= 0 and that {x, x′} is a free family. From these last two points we can deduce that P and P ′ do not commute. So the self- adjoint bounded operators A = λP and A′ = λP ′, λ being an element of R∗, also do not commute. The s.m.’s respectively associated with A and A′ are, with obvious notation: E = δ0(.)P⊥ + δλ(.)P and E ′ = δ0(.)P ′ ⊥ + δλ(.)P ′. If we assume that E ′ ‖A−A ′‖∼ E , then we have: P ′⊥ = E ′{0} � E({0} + [−‖A − A′‖, ‖A − A′‖]) = E([−‖A − A′‖, ‖A − A′‖]) = P⊥ + δλ([−‖A−A′‖, ‖A−A′‖])P = P⊥ (because λ 6∈ [−‖A−A′‖, ‖A−A′‖]). So P � P ′ and then PP ′ = P ′P = P , what is the opposite of the hypothesis made at the beginning. So the relation E ′ ‖A−A ′‖∼ E is false. 5. The case of compact operators In this section we will examine the particular case of the compact selfadjoint positive operators. This particular case plays an important role in various fields of mathematics, and in particular, in statistics, the covariance operators belong to this family. It is well known that any compact operator is the limit, in L(H), of a sequence of finite rank operators (An)n∈N. Moreover, when A is positive and selfadjoint, the operators An are linear combinations of projectors. More precisely, we have the following. If A is a positive selfadjoint compact operator, which image is of infinite dimension, we can say that − there exists a real sequence (λp)p∈N which strictly decreasingly converges to 0, − there exists a family {Pp; p ∈ N} of orthogonal projectors such that, for any p of N, Pp 6= O and dim ImPp < +∞. This real sequence and this family of projectors are such that a) the family {λpPp; p ∈ N} of elements of L(H), is summable of sum A; b) {λp; p ∈ N} is the family of the eigenvalues of A, different from 0; c) for any p of N, ImPp is the eigenspace of A associated with the eigenvalue λp; d) if we denote by D the projector, sum of the orthogonal family of projectors {Pp; p ∈ N}, then KerA = Im(I −D). Let us set µ0 = 0, D0 = I −D and, for any p of N∗, µp = λp−1 and Dp = Pp−1. Then we can affirm that a) for any p of N∗, µp is the pth largest eigenvalue of A; b) ImDp is the eigenspace associated with the eigenvalue µp; c) the family {µpDp; p ∈ N∗}, of elements of L(H), is summable of sum A; d) ImD0 = KerA; e) {Dp; p ∈ N} is an orthogonal family of projectors of sum I. With these notation, we can write the following result. Proposition 5.1. If, for any p of N∗, we denote by µp and Dp respectively the p−th greatest eigenvalue and the associated eigenprojector of A, positive selfadjoint operator, if A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 907 we set µ0 = 0 and D0 = I − ∑ p∈N∗ Dp, then, i) for any B of BR, {δµp(B)Dp; p ∈ N} is an orthogonal family of projectors; ii) the application E : B ∈ BR 7→ ∑ p∈N δµp(B)Dp ∈ P(H) is the bounded s.m. associated with A. Proof. It is clear that {Dp; p ∈ N} is a family of orthogonal projectors of sum D0 +∑ p≥1Dp = I −D + ∑ n∈N Pn = I. As (µp)p∈N∗ is a real sequence which strictly converges, Lemma 2.4.2 lets us affirm that a) for any B of BR, {δµp(B)Dp; p ∈ N} is an orthogonal family of projectors; b) the application E : B ∈ BR 7→ ∑ p∈N δµp(B)Dp ∈ P(H) is a bounded s.m.; c) {µpDp; p ∈ N} is a family of elements of L(H), of sum Ej . To complete the proof, we need to verify that Ej = A, that is that the summable families {λpPp; p ∈ N} and {µpDp; p ∈ N} are of same sum. Therefore, as the set of indices of these families is N, we can write Ej = limn ∑n p=0 µpDp = µ0D0 + limn ∑n p=1 µpDp = limn ∑n p=1 λn−1Pn−1 = limn ∑n−1 k=0 λkPk = limn ∑n k=0 λkPk = A, what ends the proof. � Remark. From properties of a partition of the set of the indexes of a summable family, we can affirm that, for any B of BR, the family {Dn;n ∈ N such that µn ∈ B} is summable of sum E(B). Let us now examine how the results of the previous section will express in terms of eigenspaces of two compact operators. Proposition 5.2. Let A1 and A2 be two selfadjoint compact operators which commute. For any (i, p) of {1, 2} × N∗, let us denote respectively by µip and Di p the p−th largest eigenvalue of Ai and the associated eigenprojector. Then for any p of N∗, we have D1 p � ∑ k:µ1p−α6µ2k6µ1p+α D2 k � ∑ k:µ1p−2α6µ1k6µ1p+2α D1 k, with α = ‖A1 −A2‖. Proof. This proposition is a direct consequence of Proposition 4.2. Indeed, let us denote by E1 and E2 the respective bounded s.m.’s associated with the operators A1 and A2. As ‖A1 − A2‖ 6 α, and as A1 and A2 commute, we have E1 α∼ E2. Then, for any p of N∗, E1({µ1p})� E2([µ1p − α;µ1p + α])� E1([µ1p − 2α;µ1p + 2α]). In other terms, taking into account the previous remark: D1 p � ∑ k:µ1p−α6µ2k6µ1p+α D2 k � ∑ k:µ1p−2α6µ1k6µ1p+2αD 1 k. � Of course, if A1 and A2 are close enough such that in the interval [µ1p − 2α;µ1p + 2α] there is no other eigenvalue of A1 than µ1p, then we have the following. Corollary 5.1. Let A1 and A2 be two selfadjoint compact operators which commute. Let α = ‖A1 −A2‖ and, for any (i, p) of {1, 2} × N∗, let us denote respectively by µip and Di p the p−th largest eigenvalue of Ai and the associated eigenprojector. We can affirm A. Boudou, S. Viguier-Pla / Eur. J. Pure Appl. Math, 11 (4) (2018), 893-910 908 that if µ11 − µ12 > 2α, then D1 1 = ∑ k:µ1p−α6µ2k6µ1p+α D2 k. Moreover, for any integer p ≥ 2, if µ1p−1 − µ1p > 2α and if µ1p − µ1p+1 > 2α, we have D1 p = ∑ k:µ1p−α6µ2k<µ1p6α D2 k. Proof. It is enough to notice that if µ11 − µ12 > 2α, then {k ∈ N;µ11 − 2α 6 µ1k 6 µ11 + 2α} = {1} and then ∑ k:µ1p−α6µ1k<µ1p6α D1 k = D1 1, so the first point is proved, thanks to Proposition 5.2. The proof of the second point is analog if we remark that, from hypotheses, we have {k ∈ N;µ1p − 2α 6 µ1k 6 µ 1 p + 2α} = {p}. � Remark. There exists at least one eigenvalue, of A2, which belongs to [µ1p−α, µ1p +α]. 6. Numerical illustration Let {Pj , j = 1, . . . , k} be a set of orthogonal projectors from Rp into Rp, {λj , j = 1, . . . , k} be a set of real values, and, for any j of {1, . . . , k}, (λjn)n∈N be a sequence of real values converging to λj . Then, the sequence of the selfadjoint operators (An)n∈N, where An = ∑k j=1 λ j nPj , converges to the selfadjoint operator A = ∑k j=1 λjPj . Each An obviously commutes with A. So ( (eitAn)t∈R ) n∈N is a sequence of sets of unitary operators which converges to (eitA)t∈R. If X is a random vector which takes values in Rp, Yn = Re( ( (eitAn)X ) t∈R), where Re stands for the real part, is a continuous random function, and the sequence (Yn)n∈N converges to the continuous random function Y = Re( ( (eitA)X ) t∈R). The associated s.m. of this last c.r.f. is E = ∑k j=1 δλj (.)PjX, and the r.m. is µZX =∑k j=1 δλj (.)‖PjX‖2. In order to give a simple graph illustration, we consider here the case where p = 2, and compute simulated sequences of Yn, with X randomized from the normalized Gaussian distribution, (λ1, λ2) = (0.4, 2), and (λ1n, λ 2 n) = (0.4 + 1/2n, 2− 1/n). REFERENCES 909 Figure 1: Variations of Yn(t) for three values of n, t varying, in regard with the limit process Y (t) Figure 2: Variations of Yn(t) for two values of t, n varying, in regard with the limit value Y (t) In Figure 1, we see that the most n is high, the nearest the curve Yn is close to the curve Y . As for Figure 2, it shows how, for t small (equal to 1) and then t higher (equal to 10), convergence is reached. We can notice that the more t is high, the fastest convergence is reached. References [1] A. Boudou and Y. Romain. On product measures associated with stationary processes. In Oxford Univ. Press, editor, The Oxford handbook of functional data analysis., pages REFERENCES 910 423–451, Oxford, 2011. [2] A. Boudou and S. Viguier-Pla. Relation between unit operators proximity and their associated spectral measures. Statist. Proba. Letters, 80:1724–1732, 2010. [3] N. Dunford and J. Schwartz. Linear Operators. Interscience Publishers, New-York, 1963. [4] F. Riesz and B. Nagy. Functional Analysis. Dover Publications, 1991.