  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.
