Note on the spectral theorem for unbounded non-selfadjoint operators

Abstract

In this paper, we deal with non-selfadjoint operators with the compact resolvent. Having been 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 operator 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.

Keywords

• spectral theorem
• abel-lidskii basis property
• schatten-von neumann class
• operator function
• evolution equation
• базисное свойство абеля-лидского
• класс шаттена-фон неймана
• оператор функция
• эволюционное уравнение