A trace formula for functions of contractions and analytic operator Lipschitz functions

Mark Malamud; Hagen Neidhardt; Vladimir Peller

doi:10.1016/j.crma.2017.06.003

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 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

Comptes Rendus. Mathématique, Volume 355 (2017) no. 7, pp. 806-811.

Résumé
Abstract

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 $T - R \in S_{1}$ , alors $f (T) - f (R) \in S_{1}$ . 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 $L^{1} (T)$ pour laquelle la formule de trace suivante est vraie : $trace (f (T) - f (R)) = \int_{T} f^{'} (ζ) ξ (ζ) d ζ$ 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. $T - R \in S_{1}$ , 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 $f (T) - f (R) \in S_{1}$ . The main result of the note says that there exists a function ξ (a spectral shift function) on the unit circle $T$ of class $L^{1} (T)$ such that the following trace formula holds: $trace (f (T) - f (R)) = \int_{T} f^{'} (ζ) ξ (ζ) d ζ$ , whenever T and R are contractions with $T - R \in S_{1}$ , and f is an operator Lipschitz function analytic in $D$ .

Reçu le : 2017-04-06
Accepté le : 2017-06-07
Publié le : 2017-07-01

DOI : 10.1016/j.crma.2017.06.003

Affiliations des auteurs :

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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     title = {A trace formula for functions of contractions and analytic operator {Lipschitz} functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {806--811},
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     number = {7},
     year = {2017},
     doi = {10.1016/j.crma.2017.06.003},
     language = {en},
}

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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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