Spectral theory of rank one perturbations of normal compact operators

TL;DR

This paper develops a functional model for rank one perturbations of compact normal operators in Hilbert spaces of entire functions, extending spectral theory and addressing completeness and spectral synthesis problems.

Contribution

It introduces a generalized functional model for such perturbations, extending previous selfadjoint results and proving a new Ordering Theorem for invariant subspaces.

Findings

Extended spectral theory to non-selfadjoint cases

Simplified existing results in the area

Proved a new Ordering Theorem for invariant subspaces

Abstract

We construct a functional model for rank one perturbations of compact normal operators acting in a certain Hilbert spaces of entire functions generalizing de Branges spaces. Using this model we study completeness and spectral synthesis problems for such perturbations. Previously, we developed the spectral theory of rank one perturbations in the selfadjoint case. In the present paper we extend and significantly simplify most of known results in the area. We also prove an Ordering Theorem for invariant subspaces with common spectral part. This result is essentially new even for rank one perturbations of compact selfadjoint operators.