A note on Hausdorff-convergence of pseudospectra

TL;DR

This paper investigates how the pseudospectra of bounded linear operators on Banach spaces converge in the Hausdorff sense, providing conditions based on spectral quantity convergence.

Contribution

It offers new necessary and sufficient conditions for Hausdorff convergence of pseudospectra using pointwise spectral convergence criteria.

Findings

Established conditions for Hausdorff convergence of pseudospectra.

Linked spectral quantity convergence to pseudospectra approximation.

Provided a theoretical framework for spectral approximation analysis.

Abstract

For a bounded linear operator on a Banach space, we study approximation of the spectrum and pseudospectra in the Hausdorff distance. We give sufficient and necessary conditions in terms of pointwise convergence of appropriate spectral quantities.