Local convergence of spectra and pseudospectra

TL;DR

This paper investigates how the spectra and pseudospectra of sequences of linear operators converge locally in Hilbert spaces, especially when the limit operator has a non-empty essential spectrum, providing new insights into spectral approximation.

Contribution

It establishes local convergence results for spectra and pseudospectra of operator sequences, including spectral and pseudospectral exactness outside the essential spectrum, with perturbation analysis.

Findings

Spectral convergence outside the essential spectrum

Pseudospectral convergence outside the near spectrum

Perturbation properties of spectra and pseudospectra

Abstract

We prove local convergence results for the spectra and pseudospectra of sequences of linear operators acting in different Hilbert spaces and converging in generalised strong resolvent sense to an operator with possibly non-empty essential spectrum. We establish local spectral exactness outside the limiting essential spectrum, local $\varepsilon$-pseudospectral exactness outside the limiting essential $\varepsilon$-near spectrum, and discuss properties of these two notions including perturbation results.