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

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