Comptes Rendus
no. 9-10
Mathematical Analysis
Functions of perturbed operators
[Fonctions d'opérateurs perturbés]

Présenté par : Gilles Pisier

Aleksei Aleksandrov1 ; Vladimir Peller2
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
Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 483-488.

Nous démontrons que si 0<α<1 et f appartient à la classe de Hölder Λα(R), alors pour tous les opérateurs A et B autoadjoints dont la différence est bornée on a : ‖f(A)−f(B)‖⩽const‖A−B‖α. Nous obtenons un résultat analogue pour les fonctions de la classe de Zygmund Λ1(R) : ‖f(A+K)−2f(A)+f(A−K)‖⩽const‖K‖, où A et K sont des opérateurs autoadjoints. Un résultat analogue est aussi vrai pour toutes les classes de Hölder–Zygmund Λα(R), α>0. Nous étudions aussi les propriétés des opérateurs f(A)−f(B) si f∈Λα(R) et A et B sont des opérateurs autoadjoints dont la différence appartient à la classe de Schatten–von Neumann Sp. Nous considérons le même problème pour les différences d'ordre arbitraire. On peut obtenir des résultats analogues pour les opérateurs unitaires et pour les contractions.

We prove that if 0<α<1 and f is in the Hölder class Λα(R), then for arbitrary selfadjoint operators A and B with bounded A−B, the operator f(A)−f(B) is bounded and ‖f(A)−f(B)‖⩽const‖A−B‖α. We prove a similar result for functions f of the Zygmund class Λ1(R): ‖f(A+K)−2f(A)+f(A−K)‖⩽const‖K‖, where A and K are selfadjoint operators. Similar results also hold for all Hölder–Zygmund classes Λα(R), α>0. We also study properties of the operators f(A)−f(B) for f∈Λα(R) and selfadjoint operators A and B such that A−B belongs to the Schatten–von Neumann class Sp. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.03.004
Aleksei Aleksandrov 1 ; Vladimir Peller 2

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 
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Aleksei Aleksandrov; Vladimir Peller. Functions of perturbed operators. Comptes Rendus. Mathématique, Volume 347 (2009) no. 9-10, pp. 483-488. doi : 10.1016/j.crma.2009.03.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.03.004/

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[6] Yu.B. Farforovskaya; L. Nikolskaya Modulus of continuity of operator functions, Algebra i Analiz, Volume 20 (2008) no. 3, pp. 224-242

[7] T. Kato Continuity of the map S↦|S| for linear operators, Proc. Japan Acad., Volume 49 (1973), pp. 157-160

[8] V.V. Peller 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 (Russian). English transl.: Funct. Anal. Appl., 19, 1985, pp. 111-123

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