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

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 ℓp(Z). Here, we generalize this to a large class of bounded linear operator families on Banach-space valued ℓp-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.