Almost commuting functions of almost commuting self-adjoint operators

Aleksei Aleksandrov; Vladimir Peller

doi:10.1016/j.crma.2015.04.012

Mathematical analysis/Functional analysis

Almost commuting functions of almost commuting self-adjoint operators
[Fonctions presque commutantes d'opérateurs auto-adjoints presque commutants]

Présenté par : Gilles Pisier

Aleksei Aleksandrov ^1,² ; Vladimir Peller ³

¹ St.-Petersburg Branch, Steklov Institute of Mathematics, Fontanka 27, 191023 St. Petersburg, Russia
² Department of Mathematics and Mechanics, Saint Petersburg State University, 28, Universitetski pr., St. Petersburg, 198504, Russia
³ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 583-588.

Résumé
Abstract

On dit que des opérateurs A et B sont presque commutants si leur commutateur $[A, B]$ appartient à la classe trace. Pour des opérateurs A et B auto-adjoints qui presque commutent, nous construisons un calcul fonctionnel $φ \mapsto φ (A, B)$ , $φ \in B_{\infty, 1}^{1} (R^{2})$ , où $B_{\infty, 1}^{1} (R^{2})$ est la classe de Besov. Ce calcul a les propriétés suivantes : il est linéaire, les opérateurs $φ (A, B)$ et $ψ (A, B)$ presque commutent pour toutes les fonctions φ et ψ dans $B_{\infty, 1}^{1} (R^{2})$ , $φ (A, B) = u (A) v (B)$ si $φ (s, t) = u (s) v (t)$ , et la formule des traces de Helton et Howe est vraie. L'outil principal est la notion d'intégrales triples opératorielles.

Let A and B be almost commuting (i.e, $A B - B A \in S_{1}$ ) self-adjoint operators. We construct a functional calculus $φ \mapsto φ (A, B)$ for φ in the Besov class $B_{\infty, 1}^{1} (R^{2})$ . This functional calculus is linear, the operators $φ (A, B)$ and $ψ (A, B)$ almost commute for $φ, ψ \in B_{\infty, 1}^{1} (R^{2})$ , $φ (A, B) = u (A) v (B)$ whenever $φ (s, t) = u (s) v (t)$ , and the Helton–Howe trace formula holds. The main tool is triple operator integrals.

Reçu le : 2014-12-11
Accepté le : 2015-04-22
Publié le : 2015-05-11

DOI : 10.1016/j.crma.2015.04.012

Affiliations des auteurs :

Aleksei Aleksandrov ^{1, 2} ; Vladimir Peller ³

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     title = {Almost commuting functions of almost commuting self-adjoint operators},
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     pages = {583--588},
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Aleksei Aleksandrov; Vladimir Peller. Almost commuting functions of almost commuting self-adjoint operators. Comptes Rendus. Mathématique, Volume 353 (2015) no. 7, pp. 583-588. doi : 10.1016/j.crma.2015.04.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.04.012/

Bibliographie
Cité par

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[3] M.S. Birman; M.Z. Solomyak Double Stieltjes operator integrals, Problems of Math. Phys., vol. 1, Leningrad University, 1966, pp. 33-67 (Russian). English translation: Topics in Mathematical Physics, vol. 1, Consultants Bureau Plenum Publishing Corporation, New York, 1967, pp. 25–54

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[17] G. Pisier Introduction to Operator Space Theory, London Math. Society Lect. Notes Series, vol. 294, Cambridge University Press, Cambridge, UK, 2003

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