We consider functions $f(A,B)$ of noncommuting self-adjoint operators $A$ and
$B$ that can be defined in terms of double operator integrals. We prove that if
$f$ belongs to the Besov class $B_{\be,1}^1(\R^2)$, then we have the following
Lipschitz type estimate in the trace norm:
$\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_1}\le\const(\|A_1-A_2\|_{\bS_1}+\|B_1-B_2\|_{\bS_1})$.
However, the condition $f\in B_{\be,1}^1(\R^2)$ does not imply the Lipschitz
type estimate in the operator norm.Comment: 6 page

Aleksandrov, Aleksei

Nazarov, Fedor

Peller, Vladimir

English

arXiv

Aleksei Aleksandrov

Fedor Nazarov

Vladimir Peller

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 353 (2015) 209–214Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comMathematical analysis/Functional analysisFunctions of perturbed noncommuting self-adjoint operatorsFonctions d’opérateurs perturbés auto-adjoints ne commutant pasAleksei Aleksandrov a, Fedor Nazarov b, Vladimir Peller ca St-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St-Petersburg, Russiab Department of Mathematics, Kent State University, Kent, OH 44242, USAc Department of Mathematics, Michigan State University, East Lansing, MI 48824, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 7 November 2014Accepted after revision 19 December 2014Available online 29 January 2015Presented by Gilles PisierWe consider functions f (A, B) of noncommuting self-adjoint operators A and B that can be defined in terms of double operator integrals. We prove that if f belongs to the Besov class B1∞,1(R2), then we have the following Lipschitz-type estimate in the trace norm: ‖ f (A1, B1) − f (A2, B2)‖S1 ≤ const(‖A1 − A2‖S1 + ‖B1 − B2‖S1 ). However, the condition f ∈ B1∞,1(R2) does not imply the Lipschitz-type estimate in the operator norm.© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous considérons les fonctions f (A, B) d’opérateurs auto-adjoints A et B qui ne com-mutent pas. De telles fonctions peuvent être définies en termes d’intégrales doubles opératorielles. Pour f dans l’espace de Besov B1∞,1(R2), nous obtenons l’estimation lipschitzienne en norme trace : ‖ f (A1, B1) − f (A2, B2)‖S1 ≤ const(‖A1 − A2‖S1 +‖B1 − B2‖S1 ). Par ailleurs, la condition f ∈ B1∞,1(R2) n’implique pas l’estimation lip-schitzienne en norme opératorielle.© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Version française abrégéeIl est bien connu (voir [3]) q’une fonction lipschitzienne sur R n’est pas nécessairement lipschitzienne opératorielle, c’est-à-dire que l’inégalité ‖ f (A) − f (B)‖ ≤ const ‖A − B‖ pour les opérateurs auto-adjoints A et B peut être fausse. Dans [7]et [8], des conditions nécessaires et des conditions suffisantes sont données pour qu’une fonction f soit lipschitzienne opératorielle. En particulier, il est démontré dans [7] et [8] que si f appartient à l’espace de Besov B1∞1(R), alors f est lipschitzienne opératorielle. Il est aussi bien connu qu’une fonction f est lipschitzienne opératorielle si et seulement si la condition A − B appartient à la classe S1 (classe trace) implique que f (A) − f (B) ∈ S1 et ‖ f (A) − f (B)‖S1 ≤ const ‖A − B‖S1 .Par ailleurs, il est démontré dans [1] que, si f est une fonction hölderienne d’ordre α, 0 < α < 1, alors ‖ f (A) − f (B)‖≤ const ‖A − B‖α pour les opérateurs auto-adjoints A et B .E-mail address: peller@math.msu.edu (V. Peller).http://dx.doi.org/10.1016/j.crma.2014.12.0051631-073X/© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.210 A. Aleksandrov et al. / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 209–214Les résultats ci-dessus ont été géneralisés dans [2] au cas de fonctions d’opérateurs normaux est dans [5] au cas de fonctions de n-uplets d’opérateurs auto-adjoints commutants.Dans cette note, nous considérons les fonctions f (A, B) d’opérateurs auto-adjoints A et B qui ne commutent pas forcé-ment. Si une fonction f sur R2 est un multiplicateurs de Schur, on définit f (A, B) comme l’intégrale double opératoriellef (A, B) =¨f (x, y)dE A(x)dE B(y)où E A et E B sont les mesures spectrales de A et de B (voir [1] pour des informations sur les multiplicateurs de Schur et sur les intégrales doubles opératorielles).Nous démontrons que si f ∈ B1∞1(R2), alors l’inégalité suivante du type lipschitzenne en norme S 1 est vraie :∥∥ f (A1, B1) − f (A2, B2)∥∥S1 ≤ const ‖ f ‖B1∞,1(‖A1 − A2‖S1 + ‖B1 − B2‖S1),pour tous les opérateurs A1, A2, B1 et B2 auto-adjoints tels que A1 − A2 ∈ S1 et B1 − B2 ∈ S1. Pour démontrer l’inégalité ci-dessus, nous utilisons la représentation suivante en termes d’intégrale triple opératorielle :f (A1, B) − f (A2, B) =ˆRˆRˆRf (x1, y) − f (x2, y)x1 − x2 dE A1(x1)(A1 − A2)dE A2(x2)dE B(y).L’intégrale triple opératorielle est bien définie parce que la différence divisée appartient au produit tensorielL∞(R) ⊗h L∞(R) ⊗h L∞(R) (voir la définition dans la version anglaise). Plus précisément, si f est une fonction bornée sur R2 dont la transformée de Fourier a un support dans le disque unité, nous obtenons la représentation tensorielle suivante :f (x1, y) − f (x2, y)x1 − x2 =∑j,k∈Zsin(x1 − jπ)x1 − jπ ·sin(x2 − kπ)x2 − kπ ·f ( jπ, y) − f (kπ, y)jπ − kπ .En outre, on a :∑j∈Zsin2(x1 − jπ)(x1 − jπ)2 =∑k∈Zsin2(x2 − kπ)(x2 − kπ)2 = 1, x1 x2 ∈R,etsupy∈R∥∥∥∥{f ( jπ, y) − f (kπ, y)jπ − kπ}j,k∈Z∥∥∥∥B≤ const ‖ f ‖L∞(R),où B est l’espace d’opérateurs bornés sur 2. Si j = k, alors ( f ( jπ, y) − f (kπ, y))( jπ − kπ)−1 def= ∂ f∂x ( jπ, y).Par ailleurs, il se trouve que, pour la même classe de fonctions, il est impossible d’obtenir une inégalité du type lipschit-zienne pour la norme d’opérateurs. Plus précisément, nous pouvons démontrer qu’il n’y a pas de nombre positif M pour lequel ∥∥ f (A1, B1) − f (A2, B2)∥∥ ≤ M‖ f ‖L∞(R2)(‖A1 − A2‖ + ‖B1 − B2‖),chaque fois que f est une fonction bornée sur R dont la transformée de Fourier a un support dans le disque unité et A1, A2, B1, B2 sont des opérateurs auto-adjoints de rang fini.En outre, on peut trouver des opérateurs auto-adjoints de rang fini tels que ‖A1 − A2‖ ≤ 2π , ‖B1 − B2‖ ≤ 2π , et une fonction f sur R2 dont la transformée de Fourier a un support dans le disque unité et telle que ‖ f ‖L∞(R) ≤ 1, pour lesquels la norme ‖ f (A1, B1) − f (A2, B2)‖ est arbitrairement grande.1. IntroductionIt was shown in [3] that a Lipschitz function f on R does not have to be operator Lipschitz, i.e., it does not have to satisfy the inequality ‖ f (A) − f (B)‖ ≤ const ‖A − B‖ for self-adjoint operators A and B . In [7] and [8], it was shown that the condition f ∈ B1∞,1(R) is sufficient for f to be operator Lipschitz (see [6] and [9] for information about Besov spaces Bsp,q). It is also well known that f is operator Lipschitz if and only if f (A) − f (B) belongs to trace class S 1 whenever A − B ∈ S1and ‖ f (A) − f (B)‖S1 ≤ const ‖A − B‖S1 . It was shown in [1] that if f is a Hölder function of order α, 0 < α < 1, then it is operator Hölder of order α, i.e., ‖ f (A) − f (B)‖ ≤ const ‖A − B‖α for self-adjoint operators A and B .Later the above results were generalized in [2] to functions of normal operators and in [5] to functions of commuting n-tuples of self-adjoint operators.In this paper we are going to consider functions of noncommuting pairs of self-adjoint operators.Let A and B be self-adjoint operators on Hilbert space and let E A and E B be their spectral measures. Suppose that fis a function of two variables that is defined at least on σ(A) × σ(B). If f is a Schur multiplier with respect to the pair (E A, E B), we define the function f (A, B) of A and B byA. Aleksandrov et al. / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 209–214 211f (A, B)def=¨f (x, y)dE A(x)dE B(y) (1)(we refer the reader to [1] for definitions of Schur multipliers and double operator integrals). Note that this functional calculus f → f (A, B) is linear, but not multiplicative.If we consider functions of bounded operators, without loss of generality we may deal with periodic functions with a sufficiently large period. Clearly, we can rescale the problem and assume that our functions are 2π -periodic in each variable.If f is a trigonometric polynomial of degree N , we can represent f in the formf (x, y) =N∑j=−Nei jx(N∑k=−Nf̂ ( j,k)eiky).Thus f belongs to the projective tensor product L∞⊗̂L∞ and‖ f ‖L∞⊗̂L∞ ≤N∑j=−Nsupy∣∣∣∣∣N∑k=−Nf̂ ( j,k)eiky∣∣∣∣∣ ≤ (1 + 2N)‖ f ‖L∞ .It follows that every periodic function f of class B1∞1(R2) belongs to L∞⊗̂L∞ and the operator f (A, B) is well defined by (1).2. Lipschitz-type estimates in the trace normIn this paper we use triple operator integrals to estimate functions of perturbed noncommuting operators in trace norm. Let E1, E2, and E3 be spectral measures on measurable spaces (X1, M1), (X2, M2), and (X3, M3). We say that a function Ψ on X1 × X2 × X3 belongs to the Haagerup tensor product of the spaces L∞(E j), j = 1, 2, 3, (notationally,Ψ ∈ L∞(E1) ⊗h L∞(E2) ⊗h L∞(E3)) if Ψ admits a representationΨ (x1, x2, x3) =∑j,k≥0α j(x1)β jk(x2)γk(x3); (2)here {α j} j≥0, {γk}k≥0 ∈ L∞(2) and {β jk} j,k≥0 ∈ L∞(B), where B is the space of infinite matrices that induce bounded linear operators on 2. We refer the reader to [11] for Haagerup tensor products. It is well known (see [4]) that for a function Ψsatisfying (2) and for bounded linear operators T and R , one can define the triple operator integralW =ˆX1ˆX2ˆX3Ψ (x1, x2, x3)dE1(x1)T dE2(x2)R dE3(x3) (3)and‖W ‖ ≤ ‖Ψ ‖L∞⊗h L∞⊗h L∞‖T ‖ · ‖R‖, (4)where‖Ψ ‖L∞⊗h L∞⊗h L∞ def= inf{∥∥{α j} j≥0∥∥L∞(2)∥∥{β jk} j,k≥0∥∥L∞(B)∥∥{γk}k≥0∥∥L∞(2)},the infimum being taken over all representations of Ψ in the form (2). Note that this extends the definition of triple operator integrals given in [10] for projective tensor products of L∞ spaces.We would like to define triple operator integrals of the form (3) in the case when one of the operators T or R is of trace class and find conditions on Ψ under which the operator W must be in S 1.Definition. A function Ψ is said to belong to the tensor product L∞(E1) ⊗h L∞(E2) ⊗h L∞(E3) if it admits a representationΨ (x1, x2, x3) =∑j,k≥0α j(x1)βk(x2)γ jk(x3), x j ∈ X j, (5)with {α j} j≥0, {βk}k≥0 ∈ L∞(2) and {γ jk} j,k≥0 ∈ L∞(B). For a bounded linear operator R and for a trace class operator T , we define the triple operator integralW =ˆX1ˆX2ˆX3Ψ (x1, x2, x3)dE1(x1)T dE2(x2)R dE3(x3)as the following continuous linear functional on the class of compact operators:212 A. Aleksandrov et al. / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 209–214Q → trace((ˆX1ˆX2ˆX3Ψ (x1, x2, x3)dE2(x2)R dE3(x3)Q dE1(x1))T). (6)The fact that the linear functional (6) is continuous on the class of compact operators is a consequence of inequality (4), which also implies the following estimate:‖W ‖S1 ≤ ‖Ψ ‖L∞⊗h L∞⊗h L∞‖T ‖S1‖R‖,where ‖Ψ ‖L∞⊗h L∞⊗h L∞ is the infimum of ‖{α j} j≥0‖L∞(2)‖{βk}k≥0‖L∞(2)‖{γ jk} j,k≥0‖L∞(B) over all representations in (5).Similarly, suppose that Ψ ∈ L∞(E1) ⊗h L∞(E2) ⊗h L∞(E3), i.e., Ψ admits a representationΨ (x1, x2, x3) =∑j,k≥0α jk(x1)β j(x2)γk(x3)where {β j} j≥0, {γk}k≥0 ∈ L∞(2), {α jk} j,k≥0 ∈ L∞(B), T is a bounded linear operator, and R ∈ S1. Then the continuous linear functionalQ → trace((ˆX1ˆX2ˆX3Ψ (x1, x2, x3)dE3(x3)Q dE1(x1)T dE2(x2))R)on the class of compact operators determines a trace class operatorWdef=ˆX1ˆX2ˆX3Ψ (x1, x2, x3)dE1(x1)T dE2(x2)R dE3(x3).Moreover,‖W ‖S1 ≤ ‖Ψ ‖L∞⊗h L∞⊗h L∞‖T ‖ · ‖R‖S1 .Note that the above definitions of triple operator integrals extend the definition given in [10] in terms of the projective tensor product of the L∞ spaces.We would like to obtain a sufficient condition on a function f on R2 under which∥∥ f (A1, B1) − f (A2, B2)∥∥S1 ≤ const(‖A1 − A2‖S1 + ‖B1 − B2‖S1),whenever (A1, B1) and (A2, B2) are pairs of (not necessarily commuting) self-adjoint operators such that A1 − A2 ∈ S1 and B1 − B2 ∈ S1. Clearly, it suffices to verify the following inequalities:∥∥ f (A1, B) − f (A2, B)∥∥ ≤ const ‖A1 − A2‖S1 and ∥∥ f (A, B1) − f (A, B2)∥∥ ≤ const ‖B1 − B2‖S1 .Theorem 2.1. Let f be a bounded function on R2 whose Fourier transform is supported in the ball {ξ ∈ R2 : ‖ξ‖ ≤ 1}. Thenf (x1, y) − f (x2, y)x1 − x2 =∑j,k∈Zsin(x1 − jπ)x1 − jπ ·sin(x2 − kπ)x2 − kπ ·f ( jπ, y) − f (kπ, y)jπ − kπ .Moreover,∑j∈Zsin2(x1 − jπ)(x1 − jπ)2 =∑k∈Zsin2(x2 − kπ)(x2 − kπ)2 = 1, x1, x2 ∈ R,andsupy∈R∥∥∥∥{f ( jπ, y) − f (kπ, y)jπ − kπ}j,k∈Z∥∥∥∥B≤ const ‖ f ‖L∞(R).Note that if j = k, we assume that ( f ( jπ, y) − f (kπ, y))( jπ − kπ)−1 = ∂ f∂x ( jπ, y).Theorem 2.1 implies the following result:Theorem 2.2. Let f be a bounded function on R2 whose Fourier transform is supported in the ball {ξ ∈ R2 : ‖ξ‖ ≤ σ }. Then the function Ψ defined byΨ (x1, x2, y) = f (x1, y) − f (x2, y) , x1, x2, y ∈R,x1 − x2A. Aleksandrov et al. / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 209–214 213belongs to the tensor product L∞(R) ⊗h L∞(R) ⊗h L∞(R),‖Ψ ‖L∞⊗h L∞⊗h L∞ ≤ constσ‖ f ‖L∞(R2)andf (A1, B) − f (A2, B) =ˆRˆRˆRf (x1, y) − f (x2, y)x1 − x2 dE A1(x1)(A1 − A2)dE A2(x2)dE B(y), (7)whenever A1 , A2 , and B are self-adjoint operators such that A1 − A2 ∈ S1 .In a similar way we can prove the following fact:Theorem 2.3. Under the same hypotheses on f , the function Ψ defined byΨ (x, y1, y2) = f (x, y1) − f (x, y2)y1 − y2 , x, y1, y2 ∈R,belongs to the tensor product L∞(R) ⊗h L∞(R) ⊗h L∞(R), ‖Ψ ‖L∞⊗h L∞⊗h L∞ ≤ constσ‖ f ‖L∞(R2) , andf (A, B1) − f (A, B2) =ˆRˆRˆRf (x, y1) − f (x, y2)y1 − y2 dE A(x)dE B1(x1)(B1 − B2)dE B2(y2), (8)whenever A, B1 , and B2 are self-adjoint operators such that B1 − B2 ∈ S1 .Theorems 2.2 and 2.3 imply the main result of this section:Theorem 2.4. Let f ∈ B1∞,1(R2). Then∥∥ f (A1, B1) − f (A2, B2)∥∥S1 ≤ const ‖ f ‖B1∞,1(‖A1 − A2‖S1 + ‖B1 − B2‖S1), (9)whenever (A1, B1) and (A2, B2) are pairs of self-adjoint operators, A1 − A2 ∈ S1 and B1 − B2 ∈ S1 .We have defined functions f (A, B) for f in B1∞,1(R2) only for bounded self-adjoint operators A and B . However, formu-lae (7) and (8) allow us to define the difference f (A1, B1) − f (A2, B2) in the case when f ∈ B1∞,1(R2) and the self-adjoint operators A1, A2, B1, B2 are possibly unbounded once we know that the pair (A2, B2) is a trace class perturbation of the pair (A1, B1). Moreover, inequality (9) also holds for such operators.3. Lipschitz-type estimates in the operator normThe main result of this section shows that unlike in the case of commuting pairs of self-adjoint operators, the condition f ∈ B1∞,1(R2) does not imply Lipschitz-type estimates in the operator norm.Theorem 3.1. There is no positive number M such that∥∥ f (A1, B1) − f (A2, B2)∥∥ ≤ M‖ f ‖L∞(R2)(‖A1 − A2‖ + ‖B1 − B2‖)for all bounded functions f on R2 with Fourier transform supported in [−1, 1]2 and for all finite rank self-adjoint operators A1, A2, B1, B2 .Construction. Let { f j}1≤ j≤N and {gk}1≤k≤N be orthonormal systems. Consider the rank one projections P j and Q j defined byP ju = (u, f j) f j and Q ju = (u, g j)g j, 1 ≤ j ≤ N.We define the self-adjoint operators A1, A2, and B = B1 = B2 byA1 =N∑j=14 jπ P j, A2 =N∑j=1(4 j + 2)π P j, and B =N∑k=14kπ Q k.Clearly, ‖A1 − A2‖ ≤ 2π . Putϕ(x, y) = 4 · 1 − cos x2· 1 − cos y2, x, y ∈R.x y214 A. Aleksandrov et al. / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 209–214Given a matrix {τ jk}1≤ j,k≤N , we define the function f byf (x, y) =∑1≤ j,k≤Nτ jkϕ(x − 4π j, y − 4πk).It is easy to see that‖ f ‖L∞(R2) ≤ const max1≤ j,k≤N|τ jk|, f (4 jπ,4kπ) = τ jk, f((4 j + 2)π,4kπ) = 0, 1 ≤ j,k ≤ N,and the Fourier transform of f is supported in [−1, 1] × [−1, 1]. One can easily verify thatf (A1, B) =∑1≤ j,k≤Nτ jk P j Q k, while f (A2, B) = 0.It can easily be shown that the supremum of ‖ f (A1, B)‖ over all P j and Q k is the Schur multiplier norm of the matrix {τ jk}1≤ j,k≤N . This norm can be made arbitrarily large as N → ∞, while assumption |τ jk| ≤ 1, 1 ≤ j, k ≤ N , can still hold.Remark. The above construction shows that there exist a function f on R2 whose Fourier transform is supported in [−1, 1]2such that ‖ f ‖L∞(R) ≤ 1 and self-adjoint operators of finite rank A1, A2, B1, B2 such that ‖A1 − A2‖ ≤ 2π and ‖B1 − B2‖ ≤2π , but ‖ f (A1, B1) − f (A2, B2)‖ is greater than any given positive number. In particular, the fact that f is a Hölder function of order α ∈ (0, 1) does not imply the estimate ‖ f (A1, B1) − f (A2, B2)‖ ≤ const(‖A1 − A2‖α + ‖B1 − B2‖α).We conclude the paper with a theorem that can be deduced from the results of this section.Theorem 3.2. There are spectral measure E1, E2 and E3 on Borel subsets of R, a function Ψ in the Haagerup tensor productL∞(E1) ⊗h L∞(E2) ⊗h L∞(E3) and an operator Q in S1 such that˚Ψ (x1, x2, x2)dE1(x1)dE2(x2)Q dE3(x3) /∈ S1.AcknowledgementsThe research of the first author is partially supported by RFBR grant 14-01-00198, the research of the second author is partially supported by NSF grant DMS 126562, the research of the third author is partially supported by NSF grant DMS 1300924.References[1] A.B. Aleksandrov, V.V. Peller, Operator Hölder–Zygmund functions, Adv. Math. 224 (2010) 910–966.[2] A.B. Aleksandrov, V.V. Peller, D. Potapov, F. Sukochev, Functions of normal operators under perturbations, Adv. Math. 226 (2011) 5216–5251.[3] Yu.B. Farforovskaya, The connection of the Kantorovich–Rubinshtein metric for spectral resolutions of selfadjoint operators with functions of operators, Vestn. Leningr. Univ. 19 (1968) 94–97 (in Russian).[4] K. Juschenko, I.G. Todorov, L. Turowska, Multidimensional operator multipliers, Trans. Amer. Math. Soc. 361 (2009) 4683–4720.[5] F.L. Nazarov, V.V. Peller, Functions of n-tuples of commuting self-adjoint operators, J. Funct. Anal. 266 (2014) 5398–5428.[6] J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Press, Durham, NC, USA, 1976.[7] V.V. Peller, Hankel operators in the theory of perturbations of unitary and self-adjoint operators, Funkc. Anal. Prilozh. 19 (2) (1985) 37–51 (in Russian). English transl.: Funct. Anal. Appl. 19 (1985) 111–123.[8] V.V. Peller, Hankel operators in the perturbation theory of unbounded self-adjoint operators, in: Analysis and Partial Differential Equations, in: Lect. Notes Pure Appl. Math., vol. 122, Marcel Dekker, New York, 1990, pp. 529–544.[9] V.V. Peller, Hankel Operators and Their Applications, Springer-Verlag, New York, 2003.[10] V.V. Peller, Multiple operator integrals and higher operator derivatives, J. Funct. Anal. 233 (2006) 515–544.[11] G. Pisier, Introduction to Operator Space Theory, Lond. Math. Soc. Lect. Note Ser., vol. 294, Cambridge University Press, Cambridge, UK, 2003.

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Functions of perturbed noncommuting self-adjoint operators

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FUNCTIONS OF PERTURBED NONCOMMUTING SELF-ADJOINT OPERATORS

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Functions of perturbed noncommuting self-adjoint operators

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