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

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.

ABSTRACT FRACTIONAL CALCULUS FOR m-ACCRETIVE OPERATORS

“…proves that R H is compact (see proof of Theorem 4, [20]) and as a result of the application of Theorem 3.3, [12, p.337], we get RW is compact. Thus the claim of Theorem 4, [20] remains true regarding the operators R H , RW .…”

“…Thus, the claim of Lemma 1 [20] is true regarding the operatorW . Using this fact, we conclude that the claim of Lemma 2, [20] is true regarding the operatorW i.e.W is m-accretive.…”

“…This formula is a crucial point of the matter, we can repeat the rest part of the proof of Theorem 5, [20] in terms H := ReW . By virtue of these facts, Theorems 7-9, [20], can be reformulated in terms H := ReW , since they are based on Lemmas 1, 3, Theorems 4, 5, [20].…”

“…Generally, to apply the last paper results for a concrete operator L we must be able to represent it by a sum L = T + A, where the senior term T must be either a selfadjoint or normal operator. In other cases we can use methods of the papers [21], [20], which are relevant if we deal with non-selfadjoint operators and allow us to study spectral properties of operators whether we have the mentioned above representation or not. These methods are applicable to some type of sectorial operators as opposed to similar results of [23], which can be applied to study a non-selfadjoint operator (see a detailed remark in [34]) if its numerical range of values is a subset of a parabolic domain of the complex plain.…”

“…Continuing this line of reasonings we generalize a differential operator with a fractional integro-differential composition in final terms to some transform of the corresponding infinitesimal generator and introduce a class of transforms of m-accretive operators. Further, we use methods obtained in the papers [20], [21] to study spectral properties of non-selfadjoint operators acting in a complex separable Hilbert space, these methods alow us to obtain an asymptotic equivalence between the real component of the resolvent and the resolvent of the real component of an operator. Due to such an approach we obtain relevant results since an asymptotic formula for the operator real component can be established in many cases (see [2], [32]).…”