Functions of perturbed noncommuting self-adjoint operators

Aleksei Aleksandrov; Fedor Nazarov; Vladimir Peller

doi:10.1016/j.crma.2014.12.005

Mathematical analysis/Functional analysis

Functions of perturbed noncommuting self-adjoint operators

Presented by: Gilles Pisier

Aleksei Aleksandrov ¹; Fedor Nazarov ²; Vladimir Peller ³

¹ St-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St-Petersburg, Russia
² Department of Mathematics, Kent State University, Kent, OH 44242, USA
³ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 209-214.

Abstract
Résumé

We consider functions $f (A, B)$ of noncommuting self-adjoint operators A and B that can be defined in terms of double operator integrals. We prove that if f belongs to the Besov class $B_{\infty, 1}^{1} (R^{2})$ , then we have the following Lipschitz-type estimate in the trace norm: ${‖ f (A_{1}, B_{1}) - f (A_{2}, B_{2}) ‖}_{S_{1}} \leq const ({‖ A_{1} - A_{2} ‖}_{S_{1}} + {‖ B_{1} - B_{2} ‖}_{S_{1}})$ . However, the condition $f \in B_{\infty, 1}^{1} (R^{2})$ does not imply the Lipschitz-type estimate in the operator norm.

Nous considérons les fonctions $f (A, B)$ d'opérateurs auto-adjoints A et B qui ne commutent pas. De telles fonctions peuvent être définies en termes d'intégrales doubles opératorielles. Pour f dans l'espace de Besov $B_{\infty, 1}^{1} (R^{2})$ , nous obtenons l'estimation lipschitzienne en norme trace : ${‖ f (A_{1}, B_{1}) - f (A_{2}, B_{2}) ‖}_{S_{1}} \leq const ({‖ A_{1} - A_{2} ‖}_{S_{1}} + {‖ B_{1} - B_{2} ‖}_{S_{1}})$ . Par ailleurs, la condition $f \in B_{\infty, 1}^{1} (R^{2})$ n'implique pas l'estimation lip-schitzienne en norme opératorielle.

Received: 2014-11-07
Accepted: 2014-12-19
Published online: 2015-01-29

DOI: 10.1016/j.crma.2014.12.005

Author's affiliations:

Aleksei Aleksandrov ¹; Fedor Nazarov ²; Vladimir Peller ³

¹ St-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St-Petersburg, Russia
² Department of Mathematics, Kent State University, Kent, OH 44242, USA
³ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

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     author = {Aleksei Aleksandrov and Fedor Nazarov and Vladimir Peller},
     title = {Functions of perturbed noncommuting self-adjoint operators},
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     language = {en},
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Aleksei Aleksandrov; Fedor Nazarov; Vladimir Peller. Functions of perturbed noncommuting self-adjoint operators. Comptes Rendus. Mathématique, Volume 353 (2015) no. 3, pp. 209-214. doi : 10.1016/j.crma.2014.12.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.12.005/

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