On some questions about composition operators on weighted Hardy spaces
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911882893328384 |
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| author | Lefèvre, Pascal Li, Daniel Queffélec, Hervé Rodríguez-Piazza, Luis |
| author_facet | Lefèvre, Pascal Li, Daniel Queffélec, Hervé Rodríguez-Piazza, Luis |
| contents | We first consider some questions raised by N. Zorboska in her thesis. In particular she asked for which sequences $β$ every symbol $φ\colon \mathbb{D} \to \mathbb{D}$ with $φ\in H^2 (β)$ induces a bounded composition operator $C_ϕ$ on the weighted Hardy space $H^2 (β)$. We give partial answers and investigate when $H^2 (β)$ is an algebra. We answer negatively another question in showing that there are a sequence $β$ and $φ\in H^2 (β)$ such that $\| φ\|_\infty < 1$ and the composition operator $C_φ$ is not bounded on $H^2 (β)$. In a second part, we show that for $p \neq 2$, no automorphism of $\mathbb{D}$, except those that fix $0$, induces a bounded composition operator on the Beurling-Sobolev space $\ell^p_A$, and even on any weighted version of this space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_01062 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On some questions about composition operators on weighted Hardy spaces Lefèvre, Pascal Li, Daniel Queffélec, Hervé Rodríguez-Piazza, Luis Functional Analysis We first consider some questions raised by N. Zorboska in her thesis. In particular she asked for which sequences $β$ every symbol $φ\colon \mathbb{D} \to \mathbb{D}$ with $φ\in H^2 (β)$ induces a bounded composition operator $C_ϕ$ on the weighted Hardy space $H^2 (β)$. We give partial answers and investigate when $H^2 (β)$ is an algebra. We answer negatively another question in showing that there are a sequence $β$ and $φ\in H^2 (β)$ such that $\| φ\|_\infty < 1$ and the composition operator $C_φ$ is not bounded on $H^2 (β)$. In a second part, we show that for $p \neq 2$, no automorphism of $\mathbb{D}$, except those that fix $0$, induces a bounded composition operator on the Beurling-Sobolev space $\ell^p_A$, and even on any weighted version of this space. |
| title | On some questions about composition operators on weighted Hardy spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2311.01062 |