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

TL;DR

This paper establishes a trace formula for functions of contraction operators with trace class differences, using spectral shift functions for operator Lipschitz functions analytic in the unit disk.

Contribution

It proves the existence of spectral shift functions for pairs of contractions with trace class difference, extending trace formulas to a broad class of operator Lipschitz functions.

Findings

Existence of spectral shift functions in L^1 for contraction pairs

Trace formula holds for operator Lipschitz functions

Extension of trace formulas to analytic functions in the disk

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 $\boldsymbol\xi$ in $L^1({\Bbb T})$ (called a spectral shift function for the pair $\{T_1,T_0\}$ ) such that the trace formula $\operatorname{trace}(f(T_1)-f(T_0))=\int_{\Bbb T} f'(\zeta)\boldsymbol{\xi}(\zeta)\,d\zeta$) holds for an arbitrary operator Lipschitz function $f$ analytic in the unit disk.