On the spectrum of operator families on discrete groups over minimal dynamical systems

TL;DR

This paper generalizes the spectral invariance results for operator families over minimal dynamical systems from selfadjoint and non-selfadjoint cases to a broad class of bounded linear operators on Banach-space valued -spaces over countable discrete groups, using limit operator techniques.

Contribution

It extends spectral and pseudospectral invariance results to a wider class of operator families on Banach-space valued -spaces over discrete groups, broadening previous findings.

Findings

Spectra of operator families coincide over minimal dynamical systems.

Pseudospectra are equal for operators within these families.

Techniques from limit operator theory are effective for analysis.

Abstract

It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in $\ell^p (\ZM)$. Here, we generalize this to a large class of bounded linear operator families on Banach-space valued $\ell^p$-spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are techniques from limit operator theory.