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

Abstract

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.