Sub-Bergman Hilbert spaces on the unit disk III
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913580595544064 |
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| author | Luo, Shuaibing Zhu, Kehe |
| author_facet | Luo, Shuaibing Zhu, Kehe |
| contents | For a bounded analytic function $φ$ on the unit disk $\D$ with $\|φ\|_\infty\le1$ we consider the defect operators $D_φ$ and $D_{\overlineφ}$ of the Toeplitz operators $T_φ$ and $T_{\overlineφ}$, respectively, on the weighted Bergman space $A^2_α$. The ranges of $D_φ$ and $D_{\overlineφ}$, written as $H(φ)$ and $H(\overlineφ)$ and equipped with appropriate inner products, are called sub-Bergman spaces.
We prove the following three results in the paper: for $-1<α\le0$ the space $H(φ)$ has a complete Nevanlinna-Pick kernel if and only if $φ$ is a Möbius map; for $α>-1$ we have $H(φ)=H(\overlineφ)=A^2_{α-1}$ if and only if the defect operators $D_φ$ and $D_{\overlineφ}$ are compact; and for $α>-1$ we have $D^2_φ(A^2_α)= D^2_{\overlineφ}(A^2_α)=A^2_{α-2}$ if and only if $φ$ is a finite Blaschke product. In some sense our restrictions on $α$ here are best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_01980 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sub-Bergman Hilbert spaces on the unit disk III Luo, Shuaibing Zhu, Kehe Complex Variables Functional Analysis 30H15, 30H10, 30H05, 47B35 For a bounded analytic function $φ$ on the unit disk $\D$ with $\|φ\|_\infty\le1$ we consider the defect operators $D_φ$ and $D_{\overlineφ}$ of the Toeplitz operators $T_φ$ and $T_{\overlineφ}$, respectively, on the weighted Bergman space $A^2_α$. The ranges of $D_φ$ and $D_{\overlineφ}$, written as $H(φ)$ and $H(\overlineφ)$ and equipped with appropriate inner products, are called sub-Bergman spaces. We prove the following three results in the paper: for $-1<α\le0$ the space $H(φ)$ has a complete Nevanlinna-Pick kernel if and only if $φ$ is a Möbius map; for $α>-1$ we have $H(φ)=H(\overlineφ)=A^2_{α-1}$ if and only if the defect operators $D_φ$ and $D_{\overlineφ}$ are compact; and for $α>-1$ we have $D^2_φ(A^2_α)= D^2_{\overlineφ}(A^2_α)=A^2_{α-2}$ if and only if $φ$ is a finite Blaschke product. In some sense our restrictions on $α$ here are best possible. |
| title | Sub-Bergman Hilbert spaces on the unit disk III |
| topic | Complex Variables Functional Analysis 30H15, 30H10, 30H05, 47B35 |
| url | https://arxiv.org/abs/2302.01980 |