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arXiv · 2411.02962

Brown Halmos Operator Identity and Toeplitz Operators on the Dirichlet Space

Ashish Kujur·Md Ramiz Reza
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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 $T_{\bar{z}}AT_z=A.$

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Ashish Kujur, Md Ramiz Reza. 2024-11-05. Brown Halmos Operator Identity and Toeplitz Operators on the Dirichlet Space. https://arxiv.org/abs/2411.02962

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