Algebraic Properties of Toeplitz Operators on the Pluriharmonic Bergman Space

Liu Liu,Yufeng Lu,Jingyu Yang

doi:10.1155/2013/578436

Abstract

We study some algebraic properties of Toeplitz operators with radial or quasihomogeneous symbols on the pluriharmonic Bergman space. We first give the necessary and sufficient conditions for the product of two Toeplitz operators with radial symbols to be a Toeplitz operator and discuss the zero-product problem for several Toeplitz operators with radial symbols. Next, we study the finite-rank product problem of several Toeplitz operators with quasihomogeneous symbols. Finally, we also investigate finite rank commutators and semicommutators of two Toeplitz operators with quasihomogeneous symbols.

Highlights

Let dν denote the normalized Lebesgue volume measure on the unit ball Bn of Cn
We study some algebraic properties of Toeplitz operators with radial or quasihomogeneous symbols on the pluriharmonic Bergman space
On the pluriharmonic Bergman space of the unit ball, Lee and Zhu [26] characterized commuting Toeplitz operators with holomorphic symbols and obtained the necessary and sufficient condition for the product of two Toeplitz operators with pluriharmonic symbols to be equal to a Teoplitz operator

Summary

Introduction

Let dν denote the normalized Lebesgue volume measure on the unit ball Bn of Cn. L2(Bn, dν) is the Hilbert space of Lebesgue square integrable functions on Bn with the inner product:. Guo et al [21] completely characterized the finite rank commutator and semicommutator of two Toeplitz operators with bounded harmonic symbols on the Bergman space of the unit disk. Cuckovicand Louhichi [14] studied the finite rank semi-commutators and commutators of Toeplitz operators with quasihomogeneous symbols on the Bergman space of the unit disk and obtained different results from the case of harmonic Toeplitz operators. On the pluriharmonic Bergman space of the unit ball, Lee and Zhu [26] characterized commuting Toeplitz operators with holomorphic symbols and obtained the necessary and sufficient condition for the product of two Toeplitz operators with pluriharmonic symbols to be equal to a Teoplitz operator They gave a complete description of holomorphic symbols for which the associated Toeplitz operators have zero semicommutator.

The Mellin Transform and Mellin Convolution

Products of Toeplitz Operators with Radial Symbols

Product of n Toeplitz Operator

Finite Rank Commutators

Finite Rank Semicommutators