Automorphisms of subalgebras of bounded analytic functions
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913900506644480 |
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| author | Behera, Kanha Maurya, Rahul Muthukumar, P. |
| author_facet | Behera, Kanha Maurya, Rahul Muthukumar, P. |
| contents | Let $H^\infty$ denote the algebra of all bounded analytic functions on the unit disk. It is well-known that every (algebra) automorphism of $H^\infty$ is a composition operator induced by disc automorphism. Maurya et al., (J. Math. Anal. Appl. 530 : Paper No: 127698, 2024) proved that every automorphism of the subalgebras $\{f\in H^\infty : f(0) = 0\}$ or $\{f\in H^\infty : f'(0) = 0\}$ is a composition operator induced by a rotation. In this article, we give very simple proof of their results. As an interesting generalization, for any $ψ\in H^\infty$, we show that every automorphism of $ψH^\infty$ must be a composition operator and characterize all such composition operators. Using this characterization, we find all automorphism of $ψH^\infty$ for few choices of $ψ$ with various nature depending on its zeros. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_03245 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Automorphisms of subalgebras of bounded analytic functions Behera, Kanha Maurya, Rahul Muthukumar, P. Complex Variables Functional Analysis Primary: 30H05, Secondary: 47B33, 30H50, 30J10, 08A35 Let $H^\infty$ denote the algebra of all bounded analytic functions on the unit disk. It is well-known that every (algebra) automorphism of $H^\infty$ is a composition operator induced by disc automorphism. Maurya et al., (J. Math. Anal. Appl. 530 : Paper No: 127698, 2024) proved that every automorphism of the subalgebras $\{f\in H^\infty : f(0) = 0\}$ or $\{f\in H^\infty : f'(0) = 0\}$ is a composition operator induced by a rotation. In this article, we give very simple proof of their results. As an interesting generalization, for any $ψ\in H^\infty$, we show that every automorphism of $ψH^\infty$ must be a composition operator and characterize all such composition operators. Using this characterization, we find all automorphism of $ψH^\infty$ for few choices of $ψ$ with various nature depending on its zeros. |
| title | Automorphisms of subalgebras of bounded analytic functions |
| topic | Complex Variables Functional Analysis Primary: 30H05, Secondary: 47B33, 30H50, 30J10, 08A35 |
| url | https://arxiv.org/abs/2412.03245 |