Commutative Banach Algebras Generated by Toeplitz Operators on the Bergman Space

TL;DR

This paper investigates the structure of commutative Banach algebras generated by Toeplitz operators with generalized symbols on the Bergman space, extending previous results and analyzing their Gelfand theory and radicals.

Contribution

It introduces a generalized framework for Toeplitz operator algebras on the Bergman space and develops their Gelfand theory and radical description, advancing the understanding of their structure.

Findings

Describes the Gelfand spectrum of the algebra.

Provides a characterization of the radical of the algebra.

Extends previous results to more general symbols.

Abstract

We present and study commutative Banach algebras generated by Toeplitz operators with generalized quasi-radial pseudo-homogeneous symbols acting on the Bergman space over the unit ball. We develop the Gelfand theory of these algebras and give some structural information about them. In particular, we provide a description of the radical of these algebras. This paper generalizes and completes the results from previous works related to Toeplitz operators with quasi-radial quasi-homogeneous symbols.