Resolvents of functions of operators with Hilbert-Schmidt Hermitian components

Let H be a separable Hilbert space with the unit operator I. We derive a sharp norm estimate for the operator function (λI − f (A))⁻¹ (λ ∈ 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.