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Comptes Rendus Mathématique
Volume 353, n° 7
pages 583-588 (juillet 2015)
Doi : 10.1016/j.crma.2015.04.012
Received : 11 December 2014 ;  accepted : 22 April 2015
Almost commuting functions of almost commuting self-adjoint operators
Fonctions presque commutantes d'opérateurs auto-adjoints presque commutants
 

Aleksei Aleksandrov a, b, Vladimir Peller c
a St.-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St. Petersburg, Russia 
b Department of Mathematics and Mechanics, Saint Petersburg State University, 28, Universitetski pr., St. Petersburg, 198504, Russia 
c Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 

Abstract

Let A and B be almost commuting (i.e,  ) self-adjoint operators. We construct a functional calculus   for φ in the Besov class  . This functional calculus is linear, the operators   and   almost commute for  ,   whenever  , and the Helton–Howe trace formula holds. The main tool is triple operator integrals.

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Résumé

On dit que des opérateurs A et B sont presque commutants si leur commutateur   appartient à la classe trace. Pour des opérateurs A et B auto-adjoints qui presque commutent, nous construisons un calcul fonctionnel  ,  , où   est la classe de Besov. Ce calcul a les propriétés suivantes : il est linéaire, les opérateurs   et   presque commutent pour toutes les fonctions φ et ψ dans  ,   si  , et la formule des traces de Helton et Howe est vraie. L'outil principal est la notion d'intégrales triples opératorielles.

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