Functions of perturbed tuples of self-adjoint operators

Fedor Nazarov; Vladimir Peller

doi:10.1016/j.crma.2012.04.010

Mathematical Analysis/Functional Analysis

Functions of perturbed tuples of self-adjoint operators

Gilles Pisier

Fedor Nazarov ¹; Vladimir Peller ²

¹Department of Mathematics, Kent State University, Kent, OH 44242, USA
²Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 349-354

Abstract (Original language)
Résumé

We generalize earlier results of Aleksandrov and Peller (2010) [2,3], Aleksandrov et al. (2011) [6], Peller (1985) [13], Peller (1990) [14] to the case of functions of n-tuples of commuting self-adjoint operators. In particular, we prove that if a function f belongs to the Besov space $B_{\infty, 1}^{1} (R^{n})$ , then f is operator Lipschitz and we show that if f satisfies a Hölder condition of order α, then $‖ f (A_{1}, \dots, A_{n}) - f (B_{1}, \dots, B_{n}) ‖ ⩽ const \max_{1 ⩽ j ⩽ n} ‖ A_{j} - B_{j} ‖^{α}$ for all n-tuples of commuting self-adjoint operators $(A_{1}, \dots, A_{n})$ and $(B_{1}, \dots, B_{n})$ . We also consider the case of arbitrary moduli of continuity and the case when the operators $A_{j} - B_{j}$ belong to the Schatten–von Neumann class $S_{p}$ .

Dans cette Note nous généralisons des résultats de Aleksandrov et Peller (2010) [2,3], Aleksandrov et al. (2011) [6], Peller (1985) [13], Peller (1990) [14] en cas de fonctions dʼopérateurs auto-adjoints et dʼopérateurs normaux. Nous considérons le problème similaire pour les fonctions de n-uplets dʼopérateurs auto-adjoints qui commutent. En particulier, nous démontrons que si f est une fonction de la classe de Besov $B_{\infty, 1}^{1} (R^{n})$ , alors elle est lipschitzienne opératorielle. En outre, nous montrons que si f appartient à lʼespace de Hölder dʼordre α, alors $‖ f (A_{1}, \dots, A_{n}) - f (B_{1}, \dots, B_{n}) ‖ ⩽ const \max_{1 ⩽ j ⩽ n} ‖ A_{j} - B_{j} ‖^{α}$ por tous n-uplets $(A_{1}, \dots, A_{n})$ et $(B_{1}, \dots, B_{n})$ dʼopérateurs auto-adjoints qui commutent. Nous considérons aussi le cas de module de continuité arbitraire et le cas où les opérateurs $A_{j} - B_{j}$ appartiennent à lʼespace de Schatten–von Neumann $S_{p}$ .

Article information
Cite this article

Received: 2012-03-07
Accepted: 2012-04-10
Published online: 2012-04-01

DOI: 10.1016/j.crma.2012.04.010

Author's affiliations:

Fedor Nazarov ¹ ; Vladimir Peller ²

¹ Department of Mathematics, Kent State University, Kent, OH 44242, USA
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Fedor Nazarov; Vladimir Peller. Functions of perturbed tuples of self-adjoint operators. Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 349-354. doi: 10.1016/j.crma.2012.04.010

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