Resolvents of Functions of Operators with Hilbert-Schmid Hermitian Components

Abstract

Let  $\cH$ be  a separable  Hilbertspace  with  the   unit operator $I$.
We derive a sharp norm estimate for the operator function 
$(\la I-f(A))\mi$   $(\la\in\bc)$,
where $A$ is  a bounded linear operator in  $\cH$ 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.