# Vestnik КRAUNC. Fiz.-Mat. nauki. 2022. vol. 39. no. 2. P. 42-61. ISSN 2079-6641

**MSC 47B28, 47A10, 47B12, 47B10, 34K30, 58D25 **

**Research Article**

**Note on the spectral theorem for unbounded non-selfadjoint operators**

**M. V. Kukushkin**

Moscow State University of Civil Engineering, Yaroslavl highway, 26, 129337, Moscow, Russia

E-mail: kukushkinmv@rambler.ru

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.

*Key words: Spectral theorem; Abel-Lidskii basis property; Schatten-von Neumann class; operator function; evolution equation.*

**DOI: 10.26117/2079-6641-2022-39-2-42-61**

**Original article submitted: 04.07.2022**

**Revision submitted: 11.08.2022**

**For citation.** Kukushkin M. V. Note on the spectral theorem for unbounded non-selfadjoint operators. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 39: 2, 42-61. DOI: 10.26117/2079-6641-2022-39-2-42-61

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Kukushkin M. V., 2022

**Competing interests.** The author declares that there are no conflicts of interest with respect to authorship and publication.

**Contribution and responsibility.** The author contributed to the writing of the article and is solely responsible for submitting the final version of the article to the press. The final version of the manuscript was approved by the author.

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* Kukushkin Maksim Vladimirovich *– PhD (Phys. & Math.), Senior Lecturer, Department of Higher Mathematics, Moscow State University of Civil Engineering, Moscow, Russia, ORCID 0000-0003-0598-032X.