For each \(p\in (1,\infty )\) and for every pair of contractions \((T_0,T_1)\) with \(\Vert T_0\Vert <1\) , we show that 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 f on \(\mathbb {T}\) . The functional calculus f(T) for a contraction T and for a Lipschitz function f is formed through a unitary dilation of T and is shown to be independent of the dilation being used. Using the estimate in the display above, we establish a modified Krein trace formula applicable to a specific category of pairs of contractions featuring Hilbert-Schmidt perturbations. An old result of Birman and Solomyak gives an expression of \(f(U)-f(V)\) in the form of a double operator integral where f is a Lipschitz function on the unit circle \(\mathbb {T}\) with for \(\alpha >0\) and (U, V) is a pair of unitary operators with \(U-V\in \mathcal {S}_{2}(\mathcal {H})\) (the Hilbert-Schmidt class). We give an elementary proof of this formula for every Lipschitz function f on the unit circle \(\mathbb {T}\) , thereby enlarging the class of functions. As a consequence, we obtain the Schatten 2-Lipschitz estimate for all Lipschitz functions \(f:\mathbb {T}\rightarrow \mathbb {C}\) .
