Lipschitz Estimates and an application to trace formulae (2312.08706v3)

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 $|T_0|<1$, 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 $\cir$. Using our Lipschitz estimates, we establish a modified Krein trace formula applicable to a specific category of pairs of contractions featuring Hilbert-Schmidt perturbations.