Resolvents of functions of operators with Hilbert-Schmidt hermitian components

Abstract

Let H be a separable Hilbert space with the unit operator I. We derive a sharp norm estimate for the operator function (?I-f(A))-1 (? ? C), where A is a bounded linear operator in H whose Hermitian component (A- A*)/2i is a Hilbert-Schmidt operator and f(z) is a function holomorphic on the convex hull of the spectrum of A. Here A* is the operator adjoint to A. Applications of the obtained estimate to perturbations of operator equations, whose coefficients are operator functions and localization of spectra are also discussed.