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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.08002 |
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| _version_ | 1866909690770751488 |
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| author | Ghosh, Gargi Roy, Subrata Shyam |
| author_facet | Ghosh, Gargi Roy, Subrata Shyam |
| contents | Let $Ω\subseteq\mathbb C^n$ be a bounded symmetric domain and $f :Ω\to Ω^\prime\subseteq \mathbb C^n$ be a proper holomorphic mapping which is factored by a finite complex reflection group $G.$ We identify a family of reproducing kernel Hilbert spaces on $Ω^\prime$ arising naturally from the isotypic decomposition of the regular representation of $G$ on the Hardy space $H^2(Ω).$ Each element of this family can be realized as a closed subspace of some $L^2$-space on the Šilov boundary of $Ω^\prime$. The reproducing kernel Hilbert space associated to the sign representation of $G$ is the Hardy space $H^2(Ω^\prime).$ We establish a Brown-Halmos type characterization for the Toeplitz operators on $H^2(Ω^\prime),$ where $Ω^\prime$ is the image of the open unit polydisc $\mathbb D^n$ in $\mathbb C^n$ under a proper holomorphic mapping factored by the finite complex reflection group $G(m,p,n).$ Moreover, we prove various multiplicative properties of Toeplitz operators on $H^2(Ω^\prime)$, where $Ω^\prime$ is a proper holomorphic image of a bounded symmetric domain. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_08002 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Brown-Halmos type Theorems on the proper images of bounded symmetric domains Ghosh, Gargi Roy, Subrata Shyam Complex Variables Functional Analysis 30H10, 47B35, 32A10 Let $Ω\subseteq\mathbb C^n$ be a bounded symmetric domain and $f :Ω\to Ω^\prime\subseteq \mathbb C^n$ be a proper holomorphic mapping which is factored by a finite complex reflection group $G.$ We identify a family of reproducing kernel Hilbert spaces on $Ω^\prime$ arising naturally from the isotypic decomposition of the regular representation of $G$ on the Hardy space $H^2(Ω).$ Each element of this family can be realized as a closed subspace of some $L^2$-space on the Šilov boundary of $Ω^\prime$. The reproducing kernel Hilbert space associated to the sign representation of $G$ is the Hardy space $H^2(Ω^\prime).$ We establish a Brown-Halmos type characterization for the Toeplitz operators on $H^2(Ω^\prime),$ where $Ω^\prime$ is the image of the open unit polydisc $\mathbb D^n$ in $\mathbb C^n$ under a proper holomorphic mapping factored by the finite complex reflection group $G(m,p,n).$ Moreover, we prove various multiplicative properties of Toeplitz operators on $H^2(Ω^\prime)$, where $Ω^\prime$ is a proper holomorphic image of a bounded symmetric domain. |
| title | Brown-Halmos type Theorems on the proper images of bounded symmetric domains |
| topic | Complex Variables Functional Analysis 30H10, 47B35, 32A10 |
| url | https://arxiv.org/abs/2405.08002 |