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The first aim is to clarify the results [1] obtained by Lidsky V.B. devoted to the decomposition of the root vector system of the non-selfadjoint 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 in contrast to the results of Lidsky V.B., where a sequence of contours of the exponential type was used. As an application, we study the Cauchy problems for the evolution equation in the abstract Hilbert space [2]. The made approach allows us to obtain a solution analytically for the right-hand side belonging to a sufficiently wide class of operators. In this regard such operators as the Riemann-Liouville fractional differential operator, the Riesz potential, the difference operator have been involved. Moreover, we produce the artificially constructed normal operator for which the clarification of the Lidsky results relevantly works.

References:
[1] Lidskii V.B. Summability of series in terms of the principal vectors of non-selfadjoint operators. Tr. Mosk. Mat. Obs., 11 (1962), 3-35.
[2] Kukushkin M.V. Evolution Equations in Hilbert Spaces via the Lacunae Method. arXiv, arXiv:2202.07338 [math.FA], https://doi.org/10.48550/arXiv.2202.07338.