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

Michael Gil

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.