Brown Halmos Operator Identity and Toeplitz Operators on the Dirichlet Space

TL;DR

This paper extends the classical Brown-Halmos operator identity characterization of Toeplitz operators from the Hardy space to the Dirichlet space, introducing a new class of symbols and establishing a similar operator identity.

Contribution

It introduces a new class of Toeplitz operators on the Dirichlet space and characterizes them via an operator identity analogous to the classical Hardy space case.

Findings

Characterization of Toeplitz operators on the Dirichlet space using an operator identity.

Introduction of a symbol class combining bounded analytic functions and multipliers.

Extension of the Brown-Halmos identity to the Dirichlet space setting.

Abstract

A well known result of Brown and Halmos shows that the Toeplitz operators induced by $L^{\infty}(\mathbb T)$ symbols on the Hardy space of the unit disc $\mathbb D$ are characterized by the operator identity $T_{\bar{z}}AT_z=A,$ where $T_z, T_{\bar{z}}$ are the Toeplitz operators induced by the function $z$ and $\bar{z}$ on the unit circle $\mathbb T$ respectively. In this paper we introduce and study a class of Toeplitz operators on the Dirichlet space $\mathcal{D} _0$ induced by a symbol class $\mathcal T(\mathcal D _0)= \overline{H^{\infty}_0(\mathbb D)} + \mathcal M(\mathcal D_0 ),$ where $H^{\infty}_0(\mathbb D)$ denotes the set of all bounded analytic function on $\mathbb D$ vanishing at $0$ and $\mathcal M(\mathcal D _0)$ denotes the multiplier algebra of the Dirichlet space $\mathcal D_0.$ We find that the Toeplitz operators on the Dirichlet space $\mathcal D$ induced by the symbol class $\mathcal T(\mathcal D _0)$ is completely characterized by the operator identity $T_{\bar{z}}AT_z=A.$