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In this paper we study spectral properties of non-selfadjoint operators withthe discrete spectrum. The main challenge is to represent a completedescription of belonging to the Schatten class through the properties of theHermitian real component. The method of estimating the singular values iselaborated by virtue of the established asymptotic formulas. The latterfundamental result is advantageous since many theoretical statements based uponit, one of them is a concept on the root vectors series expansion which leadsto a wide spectrum of applications in the theory of evolution equations. Inthis regard the evolution equations of fractional order with the operator inthe term not containing the time variable are involved. The concrete well-knownoperators are considered and the advantage of the represented method isconvexly shown.

Schatten index of the operator via the real component of its inverse