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 Hilbert space with trace class difference, i.e., $T-R\in {\mathbf{S}}_1$ and $f$ is a function analytic in the unit disk $\mathbb{D}$. It is well known that if $f$ is an operator Lipschitz function analytic in $\mathbb{D}$, then $f(T)-f(R)\in {\mathbf{S}}_1$. The main result of the note says that there exists a function $\mathbf{\xi }$ (a spectral shift function) on the unit circle $\mathbb{T}$ of class $L^1(\mathbb{T})$ such that the following trace formula holds: $\mathrm{trace}(f(T)-f(R))={\int }_{\mathbb{T}}{f}^{\prime }(\zeta )\mathbf{\xi }(\zeta )d\zeta$, whenever $T$ and $R$ are contractions with $T-R\in {\mathbf{S}}_1$ and $f$ is an operator Lipschitz function analytic in $\mathbb{D}$.

6 pages