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

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.