On condition numbers of spectral operators in a hilbert space

Abstract

We consider a linear unbounded operator \(A\) in a separable Hilbert space. with the following property: there is a normal operator \(D\) with a discrete spectrum, such \(\Vert A-D\Vert <\infty \). Besides, all the Eigen values of \(D\) are different. Under certain assumptions it is shown that \(A\) is similar to a normal operator and a sharp bound for the condition number is suggested. Applications of that bound to spectrum perturbations and operator functions are also discussed. As an illustrative example we consider a non-selfadjoint differential operator.