On the Commutativity of a Certain Class of Toeplitz Operators

Issam Louhichi,Fanilo Randriamahaleo,Lova Zakariasy

doi:10.2478/conop-2014-0001

Abstract

AbstractOne of the major goals in the theory of Toeplitz operators on the Bergman space over the unit disk D in the complex place C is to completely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. Here we shall study the commutants of a certain class of quasihomogeneous Toeplitz operators defined on the harmonic Bergman space.

Highlights

LetD be the unit disk of the complex plane C, and dA = rdr dθ π, where (r, θ) are polar coordinates, be the normalized Lebesgue measure, so that the area of D is one
If f and g are two bounded harmonic functions in D, Tf Tg = TgTf if and only if (a) both f and g are analytic in D, or (b) both f and g are antianalytic in D, or (c) f = αg + β, where α, β are constant in C
Rao proved that analytic Toeplitz operators commute only with other such operators

Summary

Introduction

Where (r, θ) are polar coordinates, be the normalized Lebesgue measure, so that the area of D is one. If f and g are two bounded harmonic functions in D, Tf Tg = TgTf if and only if (a) both f and g are analytic in D, or (b) both f and g are antianalytic in D, or (c) f = αg + β, where α, β are constant in C Basically if both symbols are harmonic, the product is commutative only in the trivial case. Rao proved that analytic Toeplitz operators commute only with other such operators Their result can be stated as follows: Theorem 2 (Axler, Cu ̆ckovic & Rao). When dealing with the product of quasihomogeneous Toeplitz operators, we are often confronted with the Mellin convolution of the radial functions in their quasihomogeneous symbols.

Tf commutes with

Hence or