On Spectral Properties of some Class of Non-selfadjoint Operators

TL;DR

This paper investigates spectral properties of a class of non-selfadjoint operators in Hilbert spaces, establishing compactness of the resolvent, asymptotic relations, classification, and eigenvalue asymptotics.

Contribution

It introduces new spectral analysis techniques for non-selfadjoint operators, including resolvent asymptotics and eigenvalue asymptotics, based on sesquilinear form theory.

Findings

Proved compactness of the resolvent for the class of operators.

Established asymptotic equivalence between real parts of resolvents.

Derived an eigenvalue asymptotic formula.

Abstract

In this paper we explore a certain class of non-selfadjoint operators acting in a complex separable Hilbert space. We consider a perturbation of a non-selfadjoint operator by an operator that is also non-selfadjoint. Our consideration is based on known spectral properties of the real component of a non-selfadjoint compact operator. Using a technic of the sesquilinear form theory we establish the compactness property of the resolvent, obtain the asymptotic equivalence between the real component of the resolvent and the resolvent of the real component for some class of non-selfadjoint operators. We obtain a classification of non-selfadjoint operators in accordance with belonging their resolvent to the Schatten-von Neumann class and formulate a sufficient condition of completeness of the root vectors system. Finally we obtain an asymptotic formula for eigenvalues of the considered class of non-selfadjoint operators.