Algebraic Properties of Quasihomogeneous and Separately Quasihomogeneous Toeplitz Operators on the Pluriharmonic Bergman Space

Yufeng Lu,Liu Liu,Hongyan Guan

doi:10.1155/2013/252037

Abstract

We study some algebraic properties of Toeplitz operator with quasihomogeneous or separately quasihomogeneous symbol on the pluriharmonic Bergman space of the unit ball inℂn. We determine when the product of two Toeplitz operators with certain separately quasi-homogeneous symbols is a Toeplitz operator. Next, we discuss the zero-product problem for several Toeplitz operators, one of whose symbols is separately quasihomogeneous and the others are quasi-homogeneous functions, and show that the zero-product problem for two Toeplitz operators has only a trivial solution if one of the symbols is separately quasihomogeneous and the other is arbitrary. Finally, we also characterize the commutativity of certain quasihomogeneous or separately quasihomogeneous Toeplitz operators.

Highlights

For n ≥ 1, let Cn be the cartesian product of n copies of C.For any points z = (z1, z2, . . . , zn) and w = (w1, w2, . . . , wn) in Cn, we use the notions ⟨z, w⟩ = z1w1 + z2w2 + ⋅ ⋅ ⋅ + znwn and |z| = √|z1|2 + |z2|2 + ⋅ ⋅ ⋅ + |zn|2 for the inner product and the ball awsshoicchiatceodnEsiustcsliodfeapnoinnotsrmz .∈LeCt nBwn idthen|oz|te
Ξp s ξ which implies that φ is a separately quasihomogeneous function of degree (p, s)
If |p| > |s|, according to Lemma 3, (59) holds which implies that φ (z + 2 󵄨󵄨󵄨󵄨p󵄨󵄨󵄨󵄨 − 2 |s|) ψ (z) = ψ (z + 2 󵄨󵄨󵄨󵄨p󵄨󵄨󵄨󵄨 − 2 |s|) φ (z), (61) z ∈ {z : Re z > 0}

Summary

Introduction

For n ≥ 1, let Cn be the cartesian product of n copies of C. On the Bergman space of the unit ball, Dong and Zhou [10] investigated the zero-product problem of two Toeplitz operators, one of whose symbols is separately quasihomogeneous and the other is arbitrary bounded. Brown and Halmos [2] firstly considered the commutativity of two Toeplitz operators on the Hardy space They showed that two bounded Toeplitz operators Tφ and Tψ commute if and only if (1) both φ and ψ are analytic, (2) both φ and ψ are coanalytic, or (3) one is a linear function of the other. In 2012, Dong and Zhou [28] and Louhichi and Zakariasy [29] characterized the commuting Toeplitz operators with radial or quasihomogeneous symbols on the harmonic Bergman space of the unit disk. The commutativity of certain (separately) quasihomogeneous Toeplitz operators is discussed

Preliminaries

The Product of Toeplitz Operators with Separately Quasihomogeneous Symbols