Essential Spectra of Weighted Composition Operators Induces by Elliptic Automorphisms

TL;DR

This paper investigates the spectra and essential spectra of weighted composition operators induced by elliptic automorphisms on weighted Bergman spaces, removing the usual continuity assumption on the weight function.

Contribution

It provides new spectral analysis results for weighted composition operators with elliptic automorphisms without requiring boundary continuity of the weight.

Findings

Spectral properties are characterized without boundary continuity of the weight.

Essential spectra are explicitly described for elliptic automorphisms.

Results extend classical spectral theory for weighted composition operators.

Abstract

The spectrum of a weighted composition operator $C_{\psi, \varphi}$ who is induced by an automorphism has been investigated for over fifty years. However, many results are got only under the condition that the weight function $\psi$ is continuous up to the boundary. In this paper we study the spectra and essential spectra of $C_{\psi,\varphi}$ on weighted Bergman spaces when $\varphi$ is an elliptic automorphism, without the assumption that $\psi$ is continuous up to the boundary.