Spectral theory for structured perturbations of linear operators

TL;DR

This paper develops a spectral theory framework for operators expressed as a sum of a simpler operator and a structured perturbation, with applications in control theory and domain perturbations.

Contribution

It provides a characterization of the spectrum of such operators, often reducing the problem to finding zeros of a characteristic equation, advancing understanding of structured perturbations.

Findings

Spectral values characterized as zeros of a characteristic equation.

Applicable to perturbations of operator domains.

Relevant to closed-loop control system analysis.

Abstract

We characterize the spectrum (and its parts) of operators which can be represented as G=A+BC for a simpler operator A and a structured perturbation BC. The interest in this kind of perturbations is motivated, e.g., by perturbations of the domain of an operator A but also arises in the theory of closed-loop systems in control theory. In many cases our results yield the spectral values of G as zeros of a "characteristic equation".