Spectrum of Weighted Composition Operators Part VI Essential spectra of $d$-endomorphisms of Banach $C(K)$-modules

TL;DR

This paper studies the essential spectra of disjointness preserving operators on Banach $C(K)$-modules, providing conditions for rotation invariance and a full spectral description for certain operators.

Contribution

It offers new insights into the spectra of $d$-endomorphisms on Banach $C(K)$-modules, including rotation invariance and explicit spectral characterizations.

Findings

Upper semi-Fredholm spectrum is rotation invariant under mild conditions

Full description of spectra for operators on Kaplansky modules of the form $T=wU$

Spectrum of $U$ is contained in the unit circle

Abstract

We investigate properties of essential spectra of disjointness preserving operators acting on Banach $C(K)$-modules. In particular, we prove that under some very mild conditions the upper semi-Fredholm spectrum of such an operator is rotation invariant. In the last part of the paper we provide a full description of the spectrum and the essential spectra of operators acting on Kaplansky modules of the form $T = wU$, where $w \in C(K)$, $U$ is a $d$-isomorphism, and the spectrum of $U$ is a subset of the unit circle.