Decomposition on the root vector series of the non-selfadjoint operators with the s-number asymptotics more subtle than one of the power type

The first our aim is to clarify the results obtained by Lidskii V.B. devoted to the decomposition on the root vector system of a non-selfadjoint compact operator. We use a technique of the entire function theory and introduce a so-called Schatten-von Neumann class of the convergence exponent. Considering strictly accretive operators satisfying special conditions formulated in terms of the norm, we construct a sequence of contours of the power type on the contrary to the results by Lidskii V.B., where a sequence of contours of the exponential type was used. This approach allows us to obtain a decomposition on the root vector series of the non-selfadjoint operators with the s-number asymptotics more subtle than one of the power type.
Finally, we produce applications to differential equations in the abstract Hilbert space. In particular, the existence and uniqueness theorems for fractional order evolution equations, with respect to the time variable, containing a differential operator with a fractional derivative in final terms are covered by the invented abstract method. In this regard such operators as the Riemann-Liouville fractional differential operator, the Kipriyanov operator, the Riesz potential, the difference operator, and other operators generated by strongly continuous semigroups of contractions can be involved.