Toeplitz Operators with Quasihomogeneous Symbols on the Bergman Space of the Unit Ball

Bo Zhang,Yufeng Lu

doi:10.1155/2012/414201

Abstract

We consider when the product of two Toeplitz operators with some quasihomogeneous symbols on the Bergman space of the unit ball equals a Toeplitz operator with quasihomogeneous symbols. We also characterize finite-rank semicommutators or commutators of two Toeplitz operators with quasihomogeneous symbols.

Highlights

Let dA z denote the Lebesgue volume measure on the unit ball Bn of Cn normalized so that the measure of Bn equals 1
Let P be the orthogonal projection from L2 Bn, dA onto L2a Bn
Given a function φ ∈ L∞ Bn, dA, the Toeplitz operator Tφ: L2a Bn → L2a Bn is defined by the formula

Summary

Introduction

Let dA z denote the Lebesgue volume measure on the unit ball Bn of Cn normalized so that the measure of Bn equals 1. On the Bergman space of the unit ball, Grudsky et al 3 gave necessary and sufficient conditions for boundedness of Toeplitz operators with radial symbols. These conditions give a characterization of the radial functions in L1 Bn, dA which correspond to bounded operators and show that the T-functions form a proper subset of L1 Bn, dA which contains all bounded and “nearly bounded” functions. On the Bergman space of the unit disk, Ahern and Cuckovic 5 and Ahern 6 obtained a similar characterization for Toeplitz operators with bounded harmonic symbols. Dong see 1, 12, 13 , we discuss the finite-rank commutator semicommutator of Toeplitz operators with more general symbols on the unit ball in this paper.

The Mellin Transform and Mellin Convolution

Products of Toeplitz Operators with Quasihomogeneous Symbols

F M s1p1M2

Finite-Rank Semicommutator

Finite-Rank Commutators