Comptes Rendus
no. 7
Harmonic analysis/Functional analysis
A trace formula for functions of contractions and analytic operator Lipschitz functions
[Une formule de trace pour les fonctions de contraction et les fonctions analytiques opérateurs-lipschitziennes]

Présenté par : Gilles Pisier

Mark Malamud1, 2 ; Hagen Neidhardt3 ; Vladimir Peller2, 4
1 Institute of Applied Mathematics and Mechanics, NAS of Ukraine, Slavyansk, Ukraine
2 RUDN University, 6 Miklukho-Maklay St., Moscow, 117198, Russia
3 Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany
4 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 806-811.

Nous considérons dans cette note le problème qui consiste à trouver le trace de f(T)−f(R), où T et R sont des contractions dans un espace hilbertien et f est une fonction analytique dans le disque unité D. Il est bien connu que, si f est une fonction analytique dans D qui est opérateurs-lipschitzienne, la différence T−R est de classe trace, c'est-à-dire que si T−R∈S1, alors f(T)−f(R)∈S1. Le résultat principal de cette note établit qu'il existe une fonction ξ (une fonction de décalage spectral) sur le cercle unité T dans l'espace L1(T) pour laquelle la formule de trace suivante est vraie : trace(f(T)−f(R))=∫Tf′(ζ)ξ(ζ)dζ pour n'importe quelle fonction f opérateurs-lipschitzienne et analytique dans D.

In this note, we study the problem of evaluating the trace of f(T)−f(R), where T and R are contractions on a Hilbert space with trace class difference, i.e. T−R∈S1, and f is a function analytic in the unit disk D. It is well known that if f is an operator Lipschitz function analytic in D, then f(T)−f(R)∈S1. The main result of the note says that there exists a function ξ (a spectral shift function) on the unit circle T of class L1(T) such that the following trace formula holds: trace(f(T)−f(R))=∫Tf′(ζ)ξ(ζ)dζ, whenever T and R are contractions with T−R∈S1, and f is an operator Lipschitz function analytic in D.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2017.06.003
Mark Malamud 1, 2 ; Hagen Neidhardt 3 ; Vladimir Peller 2, 4

1 Institute of Applied Mathematics and Mechanics, NAS of Ukraine, Slavyansk, Ukraine
2 RUDN University, 6 Miklukho-Maklay St., Moscow, 117198, Russia
3 Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany
4 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 
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Mark Malamud; Hagen Neidhardt; Vladimir Peller. A trace formula for functions of contractions and analytic operator Lipschitz functions. Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 806-811. doi : 10.1016/j.crma.2017.06.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.06.003/

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