Toeplitz Operators on the Weighted Pluriharmonic Bergman Space with Radial Symbols

Zhi Ling Sun,Yu Feng Lu

doi:10.1155/2011/210596

Abstract

We construct an operatorRwhose restriction onto weighted pluriharmonic Bergman Spacebμ2(Bn)is an isometric isomorphism betweenbμ2(Bn)andl2#. Furthermore, using the operatorRwe prove that each Toeplitz operatorTawith radial symbols is unitary to the multication operatorγa,μI. Meanwhile, the Wick function of a Toeplitz operator with radial symbol gives complete information about the operator, providing its spectral decomposition.

Highlights

Let ξBn be the ξ1, . . . , ξn open unit ball in Cn, let z·ξ in n j the complex vector space Cn
We will be concerned with the question of Toeplitz operators with radial symbols on the weighted pluriharmonic Bergman space
Using the operator R we prove that each Toeplitz operator Ta with radial symbols is unitary to the multication operator γa,μI acting on l2#

Summary

Introduction

Bn be the ξ1, . . . , ξn open unit ball in Cn, let z·ξ in n j the complex vector space Cn. Let QBμn be the Hilbert space orthogonal projection from L2μ Bn onto bμ[2] Bn. For a function u ∈ L2μ Bn , the Toeplitz operator Tu : bμ[2] Bn → bμ[2] Bn with symbol u is the linear operator defined by. The authors in 4 analyze the influence of the radial component of a symbol to spectral, compactness and Fredholm properties of Toeplitz operators on Bergman space on unit disk D. In 5 , they are devoted to study Toeplitz operators with radial symbols on the weighted Bergman spaces on the unit ball in Cn. In this paper, we will be concerned with the question of Toeplitz operators with radial symbols on the weighted pluriharmonic Bergman space. It turns out that in our particular radial symbols case the Wick symbols of a Toeplitz operator give complete information about the operator, providing its spectral decomposition

Pluriharmonic Bergman Space and Orthogonal Projection

Toeplitz Operator with Radial Symbols on bμ2 Bn