Analytic operator Lipschitz functions in the disk and a trace formula for functions of contractions

Abstract

In this paper we prove that for an arbitrary pair $\{T_1,T_0\}$ of contractions on Hilbert space with trace class difference, there exists a function $\mathbf{\xi }$ in $L^1(\mathbb{T})$ (called a spectral shift function for the pair $\{T_1,T_0\}$ ) such that the trace formula $\mathrm{trace}(f(T_1)-f(T_0))={\int }_{\mathbb{T}}{f}^{\prime }(\zeta )\mathbf{\xi }(\zeta )d\zeta$) holds for an arbitrary operator Lipschitz function $f$ analytic in the unit disk.

2 3pages