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

## TL;DR

This paper establishes a trace formula for differences of functions of contractions on Hilbert space with trace class difference, involving a spectral shift function on the unit circle for operator Lipschitz functions.

## Contribution

It proves the existence of a spectral shift function on the unit circle that satisfies a trace formula for functions of contractions with trace class difference.

## Key findings

- Existence of a spectral shift function in L^1 on the unit circle.
- Trace formula relating trace of function differences to integral involving the spectral shift.
- Applicable to operator Lipschitz functions analytic in the unit disk.

## Abstract

In this note we study the problem of evaluating the trace of $f(T)-f(R)$, where $T$ and $R$ are contractions on Hilbert space with trace class difference, i.e., $T-R\in\boldsymbol{S}_1$ and $f$ is a function analytic in the unit disk ${\Bbb D}$. It is well known that if $f$ is an operator Lipschitz function analytic in ${\Bbb D}$, then $f(T)-f(R)\in\boldsymbol{S}_1$. The main result of the note says that there exists a function $\boldsymbol{\xi}$ (a spectral shift function) on the unit circle ${\Bbb T}$ of class $L^1({\Bbb T})$ such that the following trace formula holds: $\operatorname{trace}(f(T)-f(R))=\int_{\Bbb T} f'(\zeta)\boldsymbol{\xi}(\zeta)\,d\zeta$, whenever $T$ and $R$ are contractions with $T-R\in\boldsymbol{S}_1$ and $f$ is an operator Lipschitz function analytic in ${\Bbb D}$.