The Douglas question on the Bergman and Fock spaces
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| Format: | Preprint |
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2024
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| author | Chen, Jian-hua Leng, Qianrui Zhao, Xianfeng |
| author_facet | Chen, Jian-hua Leng, Qianrui Zhao, Xianfeng |
| contents | Let $μ$ be a positive Borel measure and $T_μ$ be the bounded Toeplitz operator induced by $μ$ on the Bergman or Fock space. In this paper, we mainly investigate the invertibility of the Toeplitz operator $T_μ$ and the Douglas question on the Bergman and Fock spaces. In the Bergman-space setting, we obtain several necessary and sufficient conditions for the invertibility of $T_μ$ in terms of the Berezin transform of $μ$ and the reverse Carleson condition in two classical cases: (1) $μ$ is absolutely continuous with respect to the normalized area measure on the open unit disk $\mathbb D$; (2) $μ$ is the pull-back measure of the normalized area measure under an analytic self-mapping of $\mathbb D$. Nonetheless, we show that there exists a Carleson measure for the Bergman space such that its Berezin transform is bounded below but the corresponding Toeplitz operator is not invertible. On the Fock space, we show that $T_μ$ is invertible if and only if $μ$ is a reverse Carleson measure, but the invertibility of $T_μ$ is not completely determined by the invertibility of the Berezin transform of $μ$. These suggest that the answers to the Douglas question for Toeplitz operators induced by positive measures on the Bergman and Fock spaces are both negative in general cases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_05412 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Douglas question on the Bergman and Fock spaces Chen, Jian-hua Leng, Qianrui Zhao, Xianfeng Functional Analysis 47B35 Let $μ$ be a positive Borel measure and $T_μ$ be the bounded Toeplitz operator induced by $μ$ on the Bergman or Fock space. In this paper, we mainly investigate the invertibility of the Toeplitz operator $T_μ$ and the Douglas question on the Bergman and Fock spaces. In the Bergman-space setting, we obtain several necessary and sufficient conditions for the invertibility of $T_μ$ in terms of the Berezin transform of $μ$ and the reverse Carleson condition in two classical cases: (1) $μ$ is absolutely continuous with respect to the normalized area measure on the open unit disk $\mathbb D$; (2) $μ$ is the pull-back measure of the normalized area measure under an analytic self-mapping of $\mathbb D$. Nonetheless, we show that there exists a Carleson measure for the Bergman space such that its Berezin transform is bounded below but the corresponding Toeplitz operator is not invertible. On the Fock space, we show that $T_μ$ is invertible if and only if $μ$ is a reverse Carleson measure, but the invertibility of $T_μ$ is not completely determined by the invertibility of the Berezin transform of $μ$. These suggest that the answers to the Douglas question for Toeplitz operators induced by positive measures on the Bergman and Fock spaces are both negative in general cases. |
| title | The Douglas question on the Bergman and Fock spaces |
| topic | Functional Analysis 47B35 |
| url | https://arxiv.org/abs/2405.05412 |