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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.18455 |
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| _version_ | 1866909269298774016 |
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| author | Wang, Penghui Zhao, Chong Zhu, Zeyou |
| author_facet | Wang, Penghui Zhao, Chong Zhu, Zeyou |
| contents | In the present paper, we prove that all the quotient modules in $H^2(\mathbb D^2)$, associated to the finitely generated submodules containing a distinguished homogenous polynomial, are essentially normal, which is the first result on the essential normality of non-algebraic quotient modules in $H^2(\mathbb D^2)$. Moreover, we obtain the equivalence of the essential normality of a quotient module and the Hilbert-Schmidtness of its associated submodule in $H^2(\mathbb D^2)$, in the case that the submodule contains a distinguished homogenous polynomial. As an application, we prove that each finitely generated submodule containing a polynomial is Hilbert-Schmidt, which partially gives an affirmative answer to the conjecture of Yang \cite{Ya3}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18455 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Essential normality of quotient modules vs. Hilbert-Schmidtness of submodules in $H^2(\mathbb D^2)$ Wang, Penghui Zhao, Chong Zhu, Zeyou Functional Analysis In the present paper, we prove that all the quotient modules in $H^2(\mathbb D^2)$, associated to the finitely generated submodules containing a distinguished homogenous polynomial, are essentially normal, which is the first result on the essential normality of non-algebraic quotient modules in $H^2(\mathbb D^2)$. Moreover, we obtain the equivalence of the essential normality of a quotient module and the Hilbert-Schmidtness of its associated submodule in $H^2(\mathbb D^2)$, in the case that the submodule contains a distinguished homogenous polynomial. As an application, we prove that each finitely generated submodule containing a polynomial is Hilbert-Schmidt, which partially gives an affirmative answer to the conjecture of Yang \cite{Ya3}. |
| title | Essential normality of quotient modules vs. Hilbert-Schmidtness of submodules in $H^2(\mathbb D^2)$ |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2407.18455 |