Abstract evolution equations with an operator function at the right-hand side

Maksim V. Kukushkin

Published 2022-07-10Version 1

In this paper, we deal with non-selfadjoint operators with the compact resolvent. Having inspired by the Lidskii idea involving a notion of convergence of a series on the root vectors of the operator in a weaker -- Abel-Lidskii sense, we proceed constructing theory in the direction. The main concept of the paper is a generalization of the spectral theorem for a non-selfadjoint operator. In this way, we come to the definition of the function of an unbounded non-selfadjoint operator. As an application, we notice some approaches allowing us to principally broaden conditions imposed on the right-hand side of the evolution equation in the abstract Hilbert space and produce some concrete examples significant in applied sciences. In conection with this such operator as a Riemann-Liouville fractional differential operator, Kipriyanov fractional differential operator, Riesz potential are involved.