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In this paper we study non-selfadjoint operators using the methods of thespectral theory. The main challenge is to represent a complete description ofan operator belonging to the Schatten-von Neumann class having used the orderof the Hermitian real component. The latter fundamental result is advantageoussince many theoretical statements based upon it and one of them is a concept onthe root series expansion to say nothing on a wide spectrum of applications inthe theory of evolution equations. In this regard the evolution equations offractional order with the operator function in the term not containing the timevariable are involved. Constructing the operator class satisfying theasymptotic more subtle than one of the power type, we show convexly therelevance of the obtained fundamental results.

Schatten-von Neumann classes with more subtle asymptotics than one of the power type