Localisation of pseudospectra on discrete groups

Abstract

In this paper we generalise two of the methods and corresponding results from our previous paper ``On spectral inclusion sets and computing the spectra and pseudospectra of bounded linear operators'' [J. Spectr. Theory 14 (2024), 719--804] from tridiagonal operators on ${\ell }^2(\mathbb{Z})$ to band operators $A$ on ${\ell }^2(G,Y)$ with a countable Abelian group $G$ and a Hilbert space $Y$. Again, we cover the pseudospectra of $A$, with error-control, via a union of pseudospectra of finite and moderately sized ``local patches'' of $A$. While a major application is to understand the case $G={\mathbb{Z}}^d$ that is immanent in many physical problems, our new approach to the so-called $\ddot{I}„$ and $\ddot{I}{„}_1$ methods immediately extends to countable Abelian groups $G$.