Note on the spectral theorem for unbounded non-selfadjoint operators

M. V. Kukushkin

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 39 (2022) no. 2, pp. 42-61

Résumé

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.

Détail

Keywords: Spectral theorem, Abel-Lidskii basis property, Schatten-von Neumann class, operator function
Mots-clés : evolution equation.

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