Brown Halmos Operator Identity and Toeplitz Operators on the Dirichlet Space

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 $T_{\bar{z}}A{T}_z=A,$ 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 $\mathcal{T}({\mathcal{D}}_0)=\overline{H_0^{\infty }(\mathbb{D})}+\mathcal{M}({\mathcal{D}}_0),$ where $H_0^{\infty }(\mathbb{D})$ 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 $T_{\bar{z}}A{T}_z=A.$

17 pages