Lipschitz Estimates and an application to trace formulae

Abstract

In this note, we provide an elementary proof for the expression of $f(U)-f(V)$ in the form of a double operator integral for every Lipschitz function $f$ on the unit circle $\cir$ and for a pair of unitary operators $(U,V)$ with $U-V\in\mathcal{S}_{2}(\hilh)$ (the Hilbert-Schmidt class). As a consequence, we obtain the Schatten $2$-Lipschitz estimate $\|f(U)-f(V)\|_2\leq \|f\|_{\lip(\cir)}\|U-V\|_2$ for all Lipschitz functions $f:\cir\to\C$. Moreover, we develop an approach to the operator Lipschitz estimate for a pair of contractions with the assumption that one of them is a strict contraction, which significantly extends the class of functions from results known earlier. More specifically, for each $p\in (1,\infty )$ and for every pair of contractions $(T_0,T_1)$ with $\Vert T_0\Vert <1$, there exists a constant $d_{f,p,T_0}>0$ such that $\Vert f(T_1)-f(T_0){\Vert }_p\le d_{f,p,T_0}\Vert T_1-T_0{\Vert }_p$ for all Lipschitz functions on $\cir$. Using our Lipschitz estimates, we establish a modified Krein trace formula applicable to a specific category of pairs of contractions featuring Hilbert-Schmidt perturbations.

Minor revision. To appear in the Banach Journal of Mathematical Analysis