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Comptes Rendus Mathématique
Volume 355, n° 7
pages 806-811 (juillet 2017)
Doi : 10.1016/j.crma.2017.06.003
Received : 6 April 2017 ;  accepted : 7 June 2017
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
 

Mark Malamud a, b, Hagen Neidhardt c, Vladimir Peller d, b
a Institute of Applied Mathematics and Mechanics, NAS of Ukraine, Slavyansk, Ukraine 
b RUDN University, 6 Miklukho-Maklay St., Moscow, 117198, Russia 
c Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany 
d Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA 

Abstract

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

The full text of this article is available in PDF format.
Résumé

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

The full text of this article is available in PDF format.


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