Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators

Aleksandrov, Aleksei; Nazarov, Fedor; Peller, Vladimir

doi:10.1016/j.crma.2015.05.005

Harmonic analysis/Functional analysis

Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators
[Intégrales triples opératorielles en normes de Schatten–von Neumann et fonctions d'opérateurs perturbés ne commutant pas]

Présenté par : Gilles Pisier

Aleksandrov, Aleksei ¹ ; Nazarov, Fedor ² ; Peller, Vladimir ³

¹ Saint Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 Saint 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, Tome 353 (2015) no. 8, pp. 723-728.

Résumé
Abstract

Nous examinons les perturbations de 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 de Schatten–von Neumann $S_{p}$ , $1 \leq p \leq 2$ : ${‖ f (A_{1}, B_{1}) - f (A_{2}, B_{2}) ‖}_{S_{p}} \leq const ({‖ A_{1} - A_{2} ‖}_{S_{p}} + {‖ B_{1} - B_{2} ‖}_{S_{p}})$ . Par ailleurs, la condition $f \in B_{\infty, 1}^{1} (R^{2})$ n'implique pas l'estimation lipschitzienne en norme de $S_{p}$ pour $p > 2$ . L'outil principal consiste en l'estimation d'intégrales triples opératorielles dans les normes de $S_{p}$ .

We study perturbations of 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 Schatten–von Neumann norm $S_{p}$ , $1 \leq p \leq 2$ : ${‖ f (A_{1}, B_{1}) - f (A_{2}, B_{2}) ‖}_{S_{p}} \leq const ({‖ A_{1} - A_{2} ‖}_{S_{p}} + {‖ B_{1} - B_{2} ‖}_{S_{p}})$ . However, the condition $f \in B_{\infty, 1}^{1} (R^{2})$ does not imply the Lipschitz-type estimate in $S_{p}$ with $p > 2$ . The main tool is Schatten–von Neumann norm estimates for triple operator integrals.

Reçu le : 2015-04-06
Accepté le : 2015-05-22
Publié le : 2015-06-17

DOI : 10.1016/j.crma.2015.05.005

Affiliations des auteurs :

Aleksandrov, Aleksei ¹ ; Nazarov, Fedor ² ; Peller, Vladimir ³

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Aleksandrov, Aleksei; Nazarov, Fedor; Peller, Vladimir. Triple operator integrals in Schatten–von Neumann norms and functions of perturbed noncommuting operators. Comptes Rendus. Mathématique, Tome 353 (2015) no. 8, pp. 723-728. doi : 10.1016/j.crma.2015.05.005. http://www.numdam.org/articles/10.1016/j.crma.2015.05.005/

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