Complexity Issues in Computing Spectra, Pseudospectra and Resolvents

TL;DR

This paper explores computational methods for spectra, pseudospectra, and resolvents of linear operators in Hilbert spaces and Banach algebras, comparing pseudospectral techniques and polynomial numerical hull theory.

Contribution

It introduces a unified framework combining pseudospectral methods and polynomial numerical hull theory for computing spectra and resolvents.

Findings

Pseudospectral techniques are effective for Hilbert space operators.

Polynomial numerical hull theory extends to elements in Banach algebras.

Discussion of new pseudospectra types and the Solvability Complexity Index.

Abstract

We display methods that allow for computations of spectra, pseudospectra and resolvents of linear operators on Hilbert spaces and also elements in unital Banach algebras. The paper considers two different approaches, namely, pseudospectral techniques and polynomial numerical hull theory. The former is used for Hilbert space operators whereas the latter can handle the general case of elements in a Banach algebra. This approach leads to multicentric holomorphic calculus. We also discuss some new types of pseudospectra and the recently defined Solvability Complexity Index