Sectorial perturbations of self-adjoint matrices and operators

TL;DR

This paper studies how the spectrum of matrices and operators of the form A + γB changes as γ varies, revealing new spectral behaviors in both finite and infinite-dimensional settings, especially for non-self-adjoint perturbations.

Contribution

It introduces monodromy-type results describing spectral behavior in different asymptotic regimes and extends many findings to infinite-dimensional operators.

Findings

Spectral behavior in the limit as |γ| approaches 0 or infinity.

Properties of spectra for intermediate γ values.

Extension of results to infinite-dimensional operators.

Abstract

This paper considers $N\times N$ matrices of the form $A_\gamma =A+ \gamma B$, where $A$ is self-adjoint, $\gamma \in C$ and $B$ is a non-self-adjoint perturbation of $A$. We obtain some monodromy-type results relating the spectral behaviour of such matrices in the two asymptotic regimes $|\gamma |\to\infty$ and $|\gamma |\to 0$ under certain assumptions on $B$. We also explain some properties of the spectrum of $A_\gamma$ for intermediate sized $\gamma$ by considering the limit $N\to\infty$, concentrating on properties that have no self-adjoint analogue. A substantial number of the results extend to operators on infinite-dimensional Hilbert spaces.