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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.14401 |
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| _version_ | 1866914807714676736 |
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| author | Xia, Jingbo Yan, Congquan Zhao, Danjun Zhu, Jingming |
| author_facet | Xia, Jingbo Yan, Congquan Zhao, Danjun Zhu, Jingming |
| contents | By now it is a well-known fact that if $f$ is a multiplier for the Drury-Arveson space $H^2_n$, and if there is a $c>0$ such that $|f(z)|\geq c$ for every $z\in B$, then the reciprocal function 1/f is also a multiplier for $H^2_n$. We show that for such an $f$ and for every $t\in \mathbb{R}$, $f^t$ is also a multiplier for $H^2_n$. We do so by deriving a differentiation formula for $R^m(f^th)$.Moreover, by this formula the same result holds for spaces $H_{m,s}$ of the Besov-Dirichlet type. The same technique also gives us the result that for a non-vanishing multiplier $f$ of $H^2_n$, $log f$ is a multiplier of $H^2_n$ if and only if log $f$ is bounded on $B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14401 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Roots and Logarithms of Multipliers Xia, Jingbo Yan, Congquan Zhao, Danjun Zhu, Jingming Functional Analysis By now it is a well-known fact that if $f$ is a multiplier for the Drury-Arveson space $H^2_n$, and if there is a $c>0$ such that $|f(z)|\geq c$ for every $z\in B$, then the reciprocal function 1/f is also a multiplier for $H^2_n$. We show that for such an $f$ and for every $t\in \mathbb{R}$, $f^t$ is also a multiplier for $H^2_n$. We do so by deriving a differentiation formula for $R^m(f^th)$.Moreover, by this formula the same result holds for spaces $H_{m,s}$ of the Besov-Dirichlet type. The same technique also gives us the result that for a non-vanishing multiplier $f$ of $H^2_n$, $log f$ is a multiplier of $H^2_n$ if and only if log $f$ is bounded on $B$. |
| title | Roots and Logarithms of Multipliers |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2405.14401 |