Lipschitz Estimates and an application to trace formulae

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 $\text{cir}$ and for a pair of unitary operators $(U,V)$ with $U-V\in {\mathcal{S}}_2(\text{hilh})$ (the Hilbert-Schmidt class). As a consequence, we obtain the Schatten $2$-Lipschitz estimate $\mathrm{\parallel }f(U)-f(V){\mathrm{\parallel }}_2\le \mathrm{\parallel }f{\mathrm{\parallel }}_{\text{lip}(\text{cir})}\mathrm{\parallel }U-V{\mathrm{\parallel }}_2$ for all Lipschitz functions $f:\text{cir}\to \text{C}$. 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 $p\in (1,\mathrm{\infty })$ and for every pair of contractions $(T_0,T_1)$ with $\mathrm{\parallel }{T}_0\mathrm{\parallel }<1$, there exists a constant $d_{f,p,T_0}>0$ such that $\mathrm{\parallel }f(T_1)-f(T_0){\mathrm{\parallel }}_p\le d_{f,p,T_0}\mathrm{\parallel }{T}_1-T_0{\mathrm{\parallel }}_p$ for all Lipschitz functions on $\text{cir}$. Using our Lipschitz estimates, we establish a modified Krein trace formula applicable to a specific category of pairs of contractions featuring Hilbert-Schmidt perturbations.