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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2008.10558 |
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| _version_ | 1866917993135472640 |
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| author | Sampat, Jeet |
| author_facet | Sampat, Jeet |
| contents | For spaces of analytic functions defined on an open set in $\mathbb{C}^n$ that satisfy certain nice properties, we show that operators that preserve shift-cyclic functions are necessarily weighted composition operators. Examples of spaces for which this result holds true consist of the Hardy space $H^p(\mathbb{D}^n) \, (0 < p < \infty)$, the Drury-Arveson space $\mathcal{H}^2_n$, and the Dirichlet-type space $\mathcal{D}_α \, (α\in \mathbb{R})$. We focus on the Hardy spaces and show that when $1 \leq p < \infty$, the converse is also true. The techniques used to prove the main result also enable us to prove a version of the Gleason-Kahane-Żelazko theorem for partially multiplicative linear functionals on spaces of analytic functions in more than one variable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_10558 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Cyclicity preserving operators on spaces of analytic functions in $\mathbb{C}^n$ Sampat, Jeet Complex Variables For spaces of analytic functions defined on an open set in $\mathbb{C}^n$ that satisfy certain nice properties, we show that operators that preserve shift-cyclic functions are necessarily weighted composition operators. Examples of spaces for which this result holds true consist of the Hardy space $H^p(\mathbb{D}^n) \, (0 < p < \infty)$, the Drury-Arveson space $\mathcal{H}^2_n$, and the Dirichlet-type space $\mathcal{D}_α \, (α\in \mathbb{R})$. We focus on the Hardy spaces and show that when $1 \leq p < \infty$, the converse is also true. The techniques used to prove the main result also enable us to prove a version of the Gleason-Kahane-Żelazko theorem for partially multiplicative linear functionals on spaces of analytic functions in more than one variable. |
| title | Cyclicity preserving operators on spaces of analytic functions in $\mathbb{C}^n$ |
| topic | Complex Variables |
| url | https://arxiv.org/abs/2008.10558 |