Spectral inclusions of perturbed normal operators and applications

TL;DR

This paper investigates how the spectrum of a normal operator on a Hilbert space is affected by perturbations, providing bounds and stability results under various spectral assumptions and perturbation conditions.

Contribution

It offers new bounds for the spectrum of perturbed normal operators without symmetry assumptions, including cases with infinite spectral gaps and stability of the essential spectrum.

Findings

Spectrum of T+A lies between hyperbolas or in rotated hyperbolic regions.

Provides bounds for spectrum when the imaginary part of T's spectrum is bounded.

Establishes stability of the essential spectrum under perturbations.

Abstract

We consider a normal operator $T$ on a Hilbert space $H$. Under various assumptions on the spectrum of $T$, we give bounds for the spectrum of $T+A$ where $A$ is $T$-bounded with relative bound less than 1 but we do not assume that $A$ is symmetric or normal. If the imaginary part of the spectrum of $T$ is bounded, then the spectrum of $T+A$ is contained in the region between certain hyperbolas whose asymptotic slope depends on the $T$-bound of $A$. If the spectrum of $T$ is contained in a bisector, then the spectrum of $T+A$ is contained in the area between certain rotated hyperbola. The case of infinite gaps in the spectrum of $T$ is studied. Moreover, we prove a stability result for the essential spectrum of $T+A$. If $A$ is even $p$-subordinate to $T$, then we obtain stronger results for the localisation of the spectrum of $T+A$.