An inequality for similarity condition numbers of unbounded operators with schatten - von neumann hermitian components

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.