On One Method of Studying Spectral Properties of Non-selfadjoint Operators

Maksim V Kukushkin

doi:10.1155/2020/1461647

Abstract

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

Highlights

It is remarkable that initially, the perturbation theory of selfadjoint operators was born in the works of Keldysh [1,2,3] and had been motivated by the works of famous scientists such as Carleman [4] and Tamarkin [5]
If we consider a case where in the representation the operator T is neither selfadjoint nor normal and we cannot approach the required representation in an obvious way, it is possible to use another technique based on the properties of the real component of the initial operator
Note that in this case, the made assumptions related to the initial operator W allow us to consider a m-accretive operator class which was thoroughly studied by mathematicians such as Kato [20] and Okazawa [21, 22]

Summary

Introduction

It is remarkable that initially, the perturbation theory of selfadjoint operators was born in the works of Keldysh [1,2,3] and had been motivated by the works of famous scientists such as Carleman [4] and Tamarkin [5]. If we consider a case where in the representation the operator T is neither selfadjoint nor normal and we cannot approach the required representation in an obvious way, it is possible to use another technique based on the properties of the real component of the initial operator. Note that in this case, the made assumptions related to the initial operator W allow us to consider a m-accretive operator class which was thoroughly studied by mathematicians such as Kato [20] and Okazawa [21, 22]. In the one-dimensional case, the Kipriaynov operator coincides with the Marchaud operator, in which relationship with the Weyl and Riemann-Liouville operators is well known [31, 32]

Preliminaries

Main Results

Applications