Quasi-parabolic Composition Operators on Weighted Bergman Spaces

TL;DR

This paper investigates the essential spectra of a class of quasi-parabolic composition operators on weighted Bergman spaces, introducing a novel method to analyze their spectral properties and essential normality.

Contribution

It develops a new approach to show these operators belong to a C*-algebra of Toeplitz operators and Fourier multipliers, enabling spectral calculations and examples of essential normality.

Findings

Identified the essential spectra of quasi-parabolic composition operators.

Established that these operators belong to a specific C*-algebra.

Provided new examples of essentially normal composition operators.

Abstract

In this work we study the essential spectra of composition operators on weighted Bergman spaces of analytic functions which might be termed as "quasi-parabolic." This is the class of composition operators on $A_{\alpha}^{2}$ with symbols whose conjugate with the Cayley transform on the upper half-plane are of the form $\varphi(z)=$ $z+\psi(z)$, where $\psi\in$ $H^{\infty}(\mathbb{H})$ and $\Im(\psi(z)) > \epsilon > 0$. We especially examine the case where $\psi$ is discontinuous at infinity. A new method is devised to show that this type of composition operators fall in a C*-algebra of Toeplitz operators and Fourier multipliers. This method enables us to provide new examples of essentially normal composition operators and to calculate their essential spectra.