Toeplitz Operators on the Weighted Bergman Space over the Two-Dimensional Unit Ball

Alma García,Nikolai Vasilevski

doi:10.1155/2015/306168

Abstract

We extend the known results on commutative Banach algebras generated by Toeplitz operators with radial quasi-homogeneous symbols on the two-dimensional unit ball. Spherical coordinates previously used hid a possibility to detect an essentially wider class of symbols that can generate commutative Banach Toeplitz operator algebras. We characterize these new algebras describing their properties and, under a certain extra condition, construct the corresponding Gelfand theory.

Highlights

We extend the known results on commutative Banach algebras generated by Toeplitz operators with radial quasi-homogeneous symbols on the two-dimensional unit ball
After a detailed study of the commutative C∗-algebras generated by Toeplitz operators acting on the weighted Bergman spaces on the unit ball [1, 2] it was quite unexpectedly observed [3] that, contrary to the one-dimensional case of the unit disk, there exist many other Banach algebras generated by Toeplitz operators that are commutative on each weighted Bergman space
Note that the Introduction of [4] stated that the paper studies the unique commutative Toeplitz operator Banach algebra on the two-dimensional ball. This was completely true for the only known by that time generating radial quasi-homogeneous symbols. As it later turns out the spherical coordinates used in [4] hide the possibility to detect many other commutative Banach algebras on the two-dimensional ball generated by Toeplitz operators with symbols of a more general type

Summary

Introduction

After a detailed study of the commutative C∗-algebras generated by Toeplitz operators acting on the weighted Bergman spaces on the unit ball [1, 2] it was quite unexpectedly observed [3] that, contrary to the one-dimensional case of the unit disk, there exist many other Banach (not C∗!) algebras generated by Toeplitz operators that are commutative on each weighted Bergman space They were generated by Toeplitz operators with radial and the so-called quasi-homogeneous symbols, and the commutativity of the corresponding algebras was established just by an observation that the generating operators commute among themselves. Note that the Introduction of [4] stated that the paper studies the unique commutative Toeplitz operator Banach algebra on the two-dimensional ball This was completely true for the only known by that time generating radial quasi-homogeneous symbols.

Preliminaries

Radical