  In this paper we study spectral properties of non-selfadjoint operators with
the discrete spectrum. The main challenge is to represent a complete
description of belonging to the Schatten class through the properties of the
Hermitian real component. The method of estimating the singular values is
elaborated by virtue of the established asymptotic formulas. The latter
fundamental result is advantageous since many theoretical statements based upon
it, one of them is a concept on the root vectors series expansion which leads
to a wide spectrum of applications in the theory of evolution equations. In
this regard the evolution equations of fractional order with the operator in
the term not containing the time variable are involved. The concrete well-known
operators are considered and the advantage of the represented method is
convexly shown.
