Abstract evolution equations with an operator function in the second term

TL;DR

This paper introduces a new approach to evolution equations in Hilbert spaces by defining functions of unbounded non-selfadjoint operators, broadening conditions on the right-hand side, and generalizing the spectral theorem.

Contribution

It presents a novel method for defining operator functions in evolution equations, extending spectral theory to non-selfadjoint operators in Hilbert spaces.

Findings

Established convergence of series on root vectors in Abel-Lidskii sense

Broadened conditions for the right-hand side of evolution equations

Generalized spectral theorem for non-selfadjoint operators

Abstract

In this paper, having introduced a convergence of a series on the root vectors in the Abel-Lidskii sense, we present a valuable application to the evolution equations. The main issue of the paper is an approach allowing us to principally broaden conditions imposed upon the right-hand side of the evolution equation in the abstract Hilbert space. In this way, we come to the definition of the function of an unbounded non-selfadjoint operator. Meanwhile, considering the main issue we involve an additional concept that is a generalization of the spectral theorem for a non-selfadjoint operator.