Algebraic Properties of Toeplitz Operators on the Pluri-harmonic Fock Space

Dieudonne Agbor

doi:10.5539/jmr.v9n6p67

Abstract

We study some algebraic properties of Toeplitz operators with radial and quasi homogeneous symbols on the pluriharmonic Fock space over $\mathbb{C}^{n}$. We determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator, the zero-product problem for the product of two Toeplitz operators. Next we characterize the commutativity of Toeplitz operators with quasi homogeneous symbols and finally we study finite rank of the product of Toeplitz operators with quasi homogeneous symbols.

Highlights

Let n ∈ N and consider a family {μs}s>0 of normalized Gaussian measures on Cn defined by dμs(z) :=−n exp { − |z|2 s } dv(z). (1)Here dv denotes the usual Lebesgue volume form on Cn R2n and we write | · | for the Euclidean norm on Cn
We study some algebraic properties of Toeplitz operators with radial and quasi homogeneous symbols on the pluriharmonic Fock space over Cn
We characterize the commutativity of Toeplitz operators with quasi homogeneous symbols and we study finite rank of the product of Toeplitz operators with quasi homogeneous symbols

Summary

Introduction

Let n ∈ N and consider a family {μs}s>0 of normalized Gaussian measures on Cn defined by dμs(z). On the Hardy space, it was shown in (Brown and Halmos, 1964), that if f and g are bounded functions on the unit circle, T f Tg is a Toeplitz operator if and only if either f or g is analytic. (Guan et al, 2013), studied and characterized commuting Toeplitz operator with quasihomogeneous and separately quasihomogeneous symbols on the pluriharmonic Bergman space over the unit ball of Cn. For the case of the Fock space, (Bauer and Lee, 2011), showed that if f and g have atmost exponential growth and f is radial and non-constant T f and Tg commute implies that g is radial.

Preliminaries

Product of Toeplitz Operators with Radial Symbols

The Zero Product Problem

Commuting Toeplitz Operators