In Peller (1985, 1990) [10,11], Aleksandrov and Peller (2009, 2010, 2010) [1–3] 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 .
On a obtenu dans Peller (1985, 1990) [10,11], Aleksandrov et Peller (2009, 2010, 2010) [1–3] 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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Aleksei Aleksandrov 1; Vladimir Peller 2; Denis Potapov 3; Fedor Sukochev 3
@article{CRMATH_2010__348_9-10_553_0, author = {Aleksei Aleksandrov and Vladimir Peller and Denis Potapov and Fedor Sukochev}, title = {Functions of perturbed normal operators}, journal = {Comptes Rendus. Math\'ematique}, pages = {553--558}, publisher = {Elsevier}, volume = {348}, number = {9-10}, year = {2010}, doi = {10.1016/j.crma.2010.04.015}, language = {en}, }
TY - JOUR AU - Aleksei Aleksandrov AU - Vladimir Peller AU - Denis Potapov AU - Fedor Sukochev TI - Functions of perturbed normal operators JO - Comptes Rendus. Mathématique PY - 2010 SP - 553 EP - 558 VL - 348 IS - 9-10 PB - Elsevier DO - 10.1016/j.crma.2010.04.015 LA - en ID - CRMATH_2010__348_9-10_553_0 ER -
Aleksei Aleksandrov; Vladimir Peller; Denis Potapov; Fedor Sukochev. Functions of perturbed normal operators. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 553-558. doi : 10.1016/j.crma.2010.04.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.04.015/
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