Completeness and spectral synthesis of nonselfadjoint one-dimensional perturbations of selfadjoint operators

TL;DR

This paper investigates the spectral properties of nonselfadjoint rank one perturbations of selfadjoint operators, providing new criteria for completeness and spectral synthesis, and revealing complex spectral structures through a functional model approach.

Contribution

It introduces new criteria for completeness and spectral synthesis in rank one perturbations and develops a functional model translating operator properties into analytic function systems.

Findings

New criteria for spectral completeness and synthesis.

Counterexamples showing complex spectral structures.

Development of a functional model for spectral analysis.

Abstract

We study spectral properties of nonselfadjoint rank one perturbations of compact selfadjoint operators. The problems under consideration include completeness of eigenvectors, relations between completeness of the perturbed operator and its adjoint, and the spectral synthesis problem. We obtain new criteria for completeness and spectral synthesis in this class as well as a series of counterexamples which show that the spectral structure of rank one perturbations is, in general, unexpectedly rich and complicated. A parallel spectral theory is developed for one-dimensional singular perturbations of unbounded selfadjoint operators. Our approach is based on a functional model for this class which translates the properties of operators to completeness problems for systems of reproducing kernels and their biorthogonals in some spaces of analytic (entire) functions.