Some integral operators acting on $H^{\infty}$
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916121282609152 |
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| author | Anderson, Austin Jovovic, Mirjana Smith, Wayne |
| author_facet | Anderson, Austin Jovovic, Mirjana Smith, Wayne |
| contents | Let $f$ and $g$ be analytic on the unit disc $\mathbb{D}$. The integral operator $T_g$ is defined by $ T_g f(z) = \int_0^z f(t)g'(t)\,dt$, $z \in \mathbb{D}$. The problem considered is characterizing those symbols $g$ for which $T_g$ acting on $H^\infty$, the space of bounded analytic functions on $\mathbb{D}$, is bounded or compact. When the symbol is univalent, these become questions in univalent function theory. The corresponding problems for the companion operator, $ S_g f(z)= \int_0^z f'(t)g(t)\, dt$, acting on $H^\infty$ are also studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06774 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some integral operators acting on $H^{\infty}$ Anderson, Austin Jovovic, Mirjana Smith, Wayne Complex Variables Functional Analysis 47B38 (Primary) 47G10, 30H05 (Secondary) Let $f$ and $g$ be analytic on the unit disc $\mathbb{D}$. The integral operator $T_g$ is defined by $ T_g f(z) = \int_0^z f(t)g'(t)\,dt$, $z \in \mathbb{D}$. The problem considered is characterizing those symbols $g$ for which $T_g$ acting on $H^\infty$, the space of bounded analytic functions on $\mathbb{D}$, is bounded or compact. When the symbol is univalent, these become questions in univalent function theory. The corresponding problems for the companion operator, $ S_g f(z)= \int_0^z f'(t)g(t)\, dt$, acting on $H^\infty$ are also studied. |
| title | Some integral operators acting on $H^{\infty}$ |
| topic | Complex Variables Functional Analysis 47B38 (Primary) 47G10, 30H05 (Secondary) |
| url | https://arxiv.org/abs/2402.06774 |