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Comptes Rendus Mathématique
Volume 348, n° 9-10
pages 553-558 (mai 2010)
Doi : 10.1016/j.crma.2010.04.015
Received : 23 Mars 2010 ;  accepted : 6 April 2010
Functions of perturbed normal operators
Fonctions d’opérateurs perturbés normaux
 

Aleksei Aleksandrov a, Vladimir Peller b , Denis Potapov c, Fedor Sukochev c
a St-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St-Petersburg, Russia 
b Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 
c School of Mathematics & Statistics, University of NSW, Kensington NSW 2052, Australia 

Abstract

In Peller (1985, 1990) [10, 11], Aleksandrov and Peller (2009, 2010, 2010) [1, 2, 3] sharp estimates for   were obtained for self-adjoint operators A and B and for various classes of functions f on the real line  . 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  ,  , of functions of two variables, and   and   are normal operators, then  . We obtain a more general result for functions in the space   for an arbitrary modulus of continuity ω . We prove that if f belongs to the Besov class  , then it is operator Lipschitz, i.e.,  . We also study properties of   in the case when   and   belongs to the Schatten–von Neumann class  .

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Résumé

On a obtenu dans Peller (1985, 1990) [10, 11], Aleksandrov et Peller (2009, 2010, 2010) [1, 2, 3] des estimations précises de  , où A et B sont des opérateurs autoadjoints et f est une fonction sur la droite réelle  . 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  ,  , alors   pour tous opérateurs normaux   et  . Nous obtenons aussi un résultat plus général pour les fonctions de la classe  . Nous montrons que si f appartient à l’espace de Besov  , alors f est une fonction lipschitzienne opératorielle, c’est-à-dire   pour tous opérateurs normaux   et  . Nous étudions aussi les propriétés de   quand   et   et   sont des opérateurs normaux tells que   appartient à l’espace   de Schatten–von Neumann.

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