Lipschitz Estimates and an application to trace formulae

Tirthankar Bhattacharyya, Arup Chattopadhyay, Saikat Giri, Chandan Pradhan

Published 2023-12-14Version 1

In this short 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 $\cir$ and for a pair of unitaries $(U,V)$ with $U-V\in\mathcal{S}_{2}(\hilh)$ (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 strict, which significantly extends the function class from results known earlier. More specifically, for each $p\in(1,\infty)$ and for every pair of contractions $(T_0,T_1)$ with $\|T_0\|<1$, there exists a constant $d_{f, p,T_0}>0$ such that $\|f(T_1)-f(T_0)\|_p\leq d_{f,p, T_0}\|T_1-T_0\|_p$ for all Lipschitz functions on $\cir$. Utilizing our Lipschitz estimates, we establish a modified Krein trace formula applicable to a specific category of contraction pairs featuring Hilbert-Schmidt perturbations.