Note on spectra of non-selfadjoint operators over dynamical systems

TL;DR

This paper studies spectra of non-selfadjoint operators over dynamical systems, introducing pseudo-ergodic elements, and shows that spectra are constant and match the essential spectrum for these operators, especially under minimality.

Contribution

It introduces the concept of pseudo-ergodic elements and proves spectral constancy for operators over dynamical systems, extending understanding of spectral properties in this context.

Findings

Operators over pseudo-ergodic elements share the same spectrum

The spectrum equals the essential spectrum for these operators

Spectral constancy holds under minimality of the dynamical system

Abstract

We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.