Functions of perturbed normal operators

Aleksandrov, Aleksei; Peller, Vladimir; Potapov, Denis; Sukochev, Fedor

doi:10.1016/j.crma.2010.04.015

Harmonic Analysis/Functional Analysis

Functions of perturbed normal operators
[Fonctions d'opérateurs perturbés normaux]

Présenté par : Gilles Pisier

Aleksandrov, Aleksei ¹ ; Peller, Vladimir ² ; Potapov, Denis ³ ; Sukochev, Fedor ³

¹ St-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St-Petersburg, Russia
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
³ School of Mathematics & Statistics, University of NSW, Kensington NSW 2052, Australia

Comptes Rendus. Mathématique, Tome 348 (2010) no. 9-10, pp. 553-558.

Résumé
Abstract

On a obtenu dans Peller (1985, 1990) [10,11], Aleksandrov et Peller (2009, 2010, 2010) [1–3] des estimations précises de $f (A) - f (B)$ , où A et B sont des opérateurs autoadjoints et f est une fonction sur la droite réelle $R$ . Dans cette note nous obtenons des généralisations de ces résultats pour les opérateurs normaux et pour les fonctions f de deux variables. Nous démontrons que si f appartient à l'espace de Hölder $Λ_{α} (R^{2})$ , $0 < α < 1$ , alors $‖ f (N_{1}) - f (N_{2}) ‖ ⩽ const ‖ f ‖_{Λ_{α}} ‖ N_{1} - N_{2} ‖^{α}$ pour tous opérateurs normaux $N_{1}$ et $N_{2}$ . Nous obtenons aussi un résultat plus général pour les fonctions de la classe $Λ_{ω} (R^{2}) = {f : | f (ζ_{1}) - f (ζ_{2}) | ⩽ const ω (| ζ_{1} - ζ_{2} |)}$ . Nous montrons que si f appartient à l'espace de Besov $B_{\infty 1}^{1} (R^{2})$ , alors f est une fonction lipschitzienne opératorielle, c'est-à-dire $‖ f (N_{1}) - f (N_{2}) ‖ ⩽ const ‖ f ‖_{B_{\infty 1}^{1}} ‖ N_{1} - N_{2} ‖$ pour tous opérateurs normaux $N_{1}$ et $N_{2}$ . Nous étudions aussi les propriétés de $f (N_{1}) - f (N_{2})$ quand $f \in Λ_{α} (R^{2})$ et $N_{1}$ et $N_{2}$ sont des opérateurs normaux tells que $N_{1} - N_{2}$ appartient à l'espace $S_{p}$ de Schatten–von Neumann.

In Peller (1985, 1990) [10,11], Aleksandrov and Peller (2009, 2010, 2010) [1–3] sharp estimates for $f (A) - f (B)$ were obtained for self-adjoint operators A and B and for various classes of functions f on the real line $R$ . In this Note we extend those results to the case of functions of normal operators. We show that if f belongs to the Hölder class $Λ_{α} (R^{2})$ , $0 < α < 1$ , of functions of two variables, and $N_{1}$ and $N_{2}$ are normal operators, then $‖ f (N_{1}) - f (N_{2}) ‖ ⩽ const ‖ f ‖_{Λ_{α}} ‖ N_{1} - N_{2} ‖^{α}$ . We obtain a more general result for functions in the space $Λ_{ω} (R^{2}) = {f : | f (ζ_{1}) - f (ζ_{2}) | ⩽ const ω (| ζ_{1} - ζ_{2} |)}$ for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class $B_{\infty 1}^{1} (R^{2})$ , then it is operator Lipschitz, i.e., $‖ f (N_{1}) - f (N_{2}) ‖ ⩽ const ‖ f ‖_{B_{\infty 1}^{1}} ‖ N_{1} - N_{2} ‖$ . We also study properties of $f (N_{1}) - f (N_{2})$ in the case when $f \in Λ_{α} (R^{2})$ and $N_{1} - N_{2}$ belongs to the Schatten–von Neumann class $S_{p}$ .

Reçu le : 2010-03-23
Accepté le : 2010-04-06
Publié le : 2010-04-24

DOI : 10.1016/j.crma.2010.04.015

Affiliations des auteurs :

Aleksandrov, Aleksei ¹ ; Peller, Vladimir ² ; Potapov, Denis ³ ; Sukochev, Fedor ³

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Aleksandrov, Aleksei; Peller, Vladimir; Potapov, Denis; Sukochev, Fedor. Functions of perturbed normal operators. Comptes Rendus. Mathématique, Tome 348 (2010) no. 9-10, pp. 553-558. doi : 10.1016/j.crma.2010.04.015. http://www.numdam.org/articles/10.1016/j.crma.2010.04.015/

Bibliographie
Cité par

[1] Aleksandrov, A.B.; Peller, V.V. Functions of perturbed operators, C. R. Acad. Sci. Paris, Sér I, Volume 347 (2009), pp. 483-488

[2] Aleksandrov, A.B.; Peller, V.V. Operator Hölder–Zygmund functions, Advances in Math., Volume 224 (2010), pp. 910-966

[3] Aleksandrov, A.B.; Peller, V.V. Functions of operators under perturbations of class $S_{p}$ , J. Funct. Anal., Volume 258 (2010), pp. 3675-3724

[4] Birman, M.S.; Solomyak, M.Z. Double Stieltjes operator integrals, Problems of Math. Phys. Leningrad. Univ. (Topics Math. Physics), Volume 1 (1966), pp. 33-67 (in Russian). English transl.:, vol. 1, 1967 pp. 25–54

[5] Birman, M.S.; Solomyak, M.Z. Double Stieltjes operator integrals. II, Problems of Math. Phys. Leningrad. Univ. (Topics Math. Physics), Volume 2 (1967), pp. 26-60 (in Russian). English transl.:, vol. 2, 1968 pp. 19–46

[6] Birman, M.S.; Solomyak, M.Z. Double Stieltjes operator integrals. III, Problems of Math. Phys. Leningrad. Univ., Volume 6 (1973), pp. 27-53 (in Russian)

[7] Farforovskaya, Yu.B. The connection of the Kantorovich–Rubinshtein metric for spectral resolutions of selfadjoint operators with functions of operators, Vestnik Leningrad. Univ., Volume 19 (1968), pp. 94-97 (in Russian)

[8] Yu.B. Farforovskaya, L. Nikolskaya, Operator Hölderness of Hölder functions, Algebra i Analiz. (2010), in press (in Russian)

[9] Peetre, J. New Thoughts on Besov Spaces, Duke Univ. Press, Durham, NC, 1976

[10] Peller, V.V. Hankel operators in the theory of perturbations of unitary and self-adjoint operators, Funktsional. Anal. i Prilozhen., Volume 19 (1985) no. 2, pp. 37-51 (in Russian). English transl.: Funct. Anal. Appl., 19, 1985, pp. 111-123

[11] Peller, V.V. Hankel operators in the perturbation theory of unbounded self-adjoint operators, Analysis and Partial Differential Equations, Lecture Notes in Pure and Appl. Math., vol. 122, Dekker, New York, 1990, pp. 529-544

[12] Peller, V.V. Hankel Operators and Their Applications, Springer-Verlag, New York, 2003

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