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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.17181 |
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| _version_ | 1866912853078835200 |
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| author | Tian, Xiance Wang, Penghui Zhu, Zeyou |
| author_facet | Tian, Xiance Wang, Penghui Zhu, Zeyou |
| contents | In the present paper, we study the norms for symmetric and antisymmetric tensor products of weighted shift operators. By proving that for $n\geq 2$,
$$\|S_α^{l_1}\odot\cdots \odot S_α^{l_k}\odot S_α^{*l_{k+1}}\odot\cdots \odot S_α^{*l_{n}}\| =\mathop{\prod}_{i=1}^n\left \| S_α^{l_{i}}\right\|, \text{ for any} \ (l_1,l_2\cdots l_n)\in\mathbb N^n$$ if and only if the weight satisfies the regularity condition, we partially solve \cite[Problem 6 and Problem 7]{GA}. It will be seen that most weighted shift operators on function spaces, including weighted Bergman shift, Hardy shift, Dirichlet shift, etc, satisfy the regularity condition. Moreover, at the end of the paper, we solve \cite[Problem 1 and Problem 2]{GA}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17181 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The norms for symmetric and antisymmetric tensor products of the weighted shift operators Tian, Xiance Wang, Penghui Zhu, Zeyou Functional Analysis 47A13, 46H25 In the present paper, we study the norms for symmetric and antisymmetric tensor products of weighted shift operators. By proving that for $n\geq 2$, $$\|S_α^{l_1}\odot\cdots \odot S_α^{l_k}\odot S_α^{*l_{k+1}}\odot\cdots \odot S_α^{*l_{n}}\| =\mathop{\prod}_{i=1}^n\left \| S_α^{l_{i}}\right\|, \text{ for any} \ (l_1,l_2\cdots l_n)\in\mathbb N^n$$ if and only if the weight satisfies the regularity condition, we partially solve \cite[Problem 6 and Problem 7]{GA}. It will be seen that most weighted shift operators on function spaces, including weighted Bergman shift, Hardy shift, Dirichlet shift, etc, satisfy the regularity condition. Moreover, at the end of the paper, we solve \cite[Problem 1 and Problem 2]{GA}. |
| title | The norms for symmetric and antisymmetric tensor products of the weighted shift operators |
| topic | Functional Analysis 47A13, 46H25 |
| url | https://arxiv.org/abs/2507.17181 |