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Comptes Rendus Mathématique
Volume 353, n° 8
pages 723-728 (août 2015)
Doi : 10.1016/j.crma.2015.05.005
Received : 6 April 2015 ;  accepted : 22 May 2015
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
 

Aleksei Aleksandrov a, Fedor Nazarov b, Vladimir Peller c
a Saint Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 Saint Petersburg, Russia 
b Department of Mathematics, Kent State University, Kent, OH 44242, USA 
c Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 

Abstract

We study perturbations of functions   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  , then we have the following Lipschitz-type estimate in the Schatten–von Neumann norm  ,  :  . However, the condition   does not imply the Lipschitz-type estimate in   with  . The main tool is Schatten–von Neumann norm estimates for triple operator integrals.

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Résumé

Nous examinons les perturbations de fonctions   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  , nous obtenons l'estimation lipschitzienne en norme de Schatten–von Neumann  ,   :  . Par ailleurs, la condition   n'implique pas l'estimation lipschitzienne en norme de   pour  . L'outil principal consiste en l'estimation d'intégrales triples opératorielles dans les normes de  .

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