Spectrum of Weighted Composition Operators. Part IX. The spectrum and essential spectra of some weighted composition operators on uniform algebras

TL;DR

This paper investigates the spectral properties of weighted composition operators on uniform algebras, showing that their spectra are connected, rotation-invariant subsets of the complex plane under certain boundary conditions.

Contribution

It provides new results on the spectrum and upper semi-Fredholm spectrum of weighted composition operators on uniform algebras, especially for analytic cases.

Findings

Spectrum is a connected rotation-invariant subset of the complex plane.

Upper semi-Fredholm spectrum is rotation invariant.

Results depend on the map mapping the Shilov boundary onto itself.

Abstract

We obtain some results about the spectrum and the upper semi-Fredholm spectrum of weighted composition operators on uniform algebras, assuming that the corresponding map maps the Shilov boundary onto itself. In particular, it follows from our results that in the case of analytic uniform algebras the spectrum is a connected rotation invariant subset of the complex plane, and that the upper semi-Fredholm spectrum is rotation invariant as well.