An Inequality for Similarity Condition Numbers of Unbounded Operators with Schatten - von Neumann Hermitian Components

Abstract

Let $H$ be a linear unbounded  operator in a separable Hilbert space. It is assumed the resolvent of $H$ is a compact operator and $H-H^*$ is a Schatten - von Neumann operator. Various   integro-differential operators satisfy these conditions. Under certain assumptions it is shown  that $H$ is similar to a normal operator and a sharp bound for the condition number is suggested.

We also discuss applications of that bound to spectrum perturbations and operator functions.