Brown Halmos Operator Identity and Toeplitz Operators on the Dirichlet Space
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arXiv
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| Formato: | Preprint |
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2024
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| _version_ | 1866915005969989632 |
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| author | Kujur, Ashish Reza, Md Ramiz |
| author_facet | Kujur, Ashish Reza, Md Ramiz |
| contents | 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 $T_{\bar{z}}AT_z=A.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02962 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Brown Halmos Operator Identity and Toeplitz Operators on the Dirichlet Space Kujur, Ashish Reza, Md Ramiz Functional Analysis 47B35, 46E22, 47A62 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 $T_{\bar{z}}AT_z=A.$ |
| title | Brown Halmos Operator Identity and Toeplitz Operators on the Dirichlet Space |
| topic | Functional Analysis 47B35, 46E22, 47A62 |
| url | https://arxiv.org/abs/2411.02962 |