Invariance of Abel universality under composition and applications
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914630265208832 |
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| author | Charpentier, Stéphane Manolaki, Myrto Maronikolakis, Konstantinos |
| author_facet | Charpentier, Stéphane Manolaki, Myrto Maronikolakis, Konstantinos |
| contents | A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class $\mathcal{U}_A (\mathbb{D})$ of Abel universal functions if the family $\{f_r: 0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the Banach space of all continuous functions on $K$, endowed with the supremum norm, for any proper compact subset $K$ of the unit circle. We prove that this property is invariant under composition from the left with any non-constant entire function. As an application, we show that $\mathcal{U}_A (\mathbb{D})$ is strongly-algebrable. Furthermore, we prove that Abel universality is invariant under composition from the right with an automorphism $Φ$ of $\mathbb{D}$ if and only if $Φ$ a rotation. On the other hand, we establish the existence of a subset of $\mathcal{U}_A (\mathbb{D})$ which is residual in the space of holomorphic functions on $\mathbb{D}$ and is invariant under composition from the right with any automorphism of $\mathbb{D}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02367 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Invariance of Abel universality under composition and applications Charpentier, Stéphane Manolaki, Myrto Maronikolakis, Konstantinos Complex Variables 30K15, 30B30, 30E10 A holomorphic function $f$ on the unit disc $\mathbb{D}$ belongs to the class $\mathcal{U}_A (\mathbb{D})$ of Abel universal functions if the family $\{f_r: 0\leq r<1\}$ of its dilates $f_r(z):=f(rz)$ is dense in the Banach space of all continuous functions on $K$, endowed with the supremum norm, for any proper compact subset $K$ of the unit circle. We prove that this property is invariant under composition from the left with any non-constant entire function. As an application, we show that $\mathcal{U}_A (\mathbb{D})$ is strongly-algebrable. Furthermore, we prove that Abel universality is invariant under composition from the right with an automorphism $Φ$ of $\mathbb{D}$ if and only if $Φ$ a rotation. On the other hand, we establish the existence of a subset of $\mathcal{U}_A (\mathbb{D})$ which is residual in the space of holomorphic functions on $\mathbb{D}$ and is invariant under composition from the right with any automorphism of $\mathbb{D}$. |
| title | Invariance of Abel universality under composition and applications |
| topic | Complex Variables 30K15, 30B30, 30E10 |
| url | https://arxiv.org/abs/2401.02367 |