Composition operators between Beurling subspaces of Hardy space
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929463538745344 |
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| author | Anjali, V. A. Muthukumar, P. Shankar, P. |
| author_facet | Anjali, V. A. Muthukumar, P. Shankar, P. |
| contents | V. Matache (J. Operator Theory 73(1):243--264, 2015) raised an open problem about characterizing composition operators $C_ϕ$ on the Hardy space $H^2$ and nonzero singular measures $μ_1$, $μ_2$ on the unit circle such that $C_ϕ({S_{μ_1}} H^2)\subseteq {S_{μ_2}} H^2,$ where $S_{μ_i}$ denotes the singular inner function corresponding to the measure $μ_i,i=1,2$. In this article, we consider this problem in a more general setting. We characterize holomorphic self maps $ϕ$ of the unit disk $\mathbb{D}$ and inner functions $θ_1, θ_2$ such that $C_ϕ(θ_1 H^p)\subseteq θ_2 H^p,$ for $p>0$. Emphasis is given to Blaschke products and singular inner functions as a special case.
We also give an another measure-theoretic characterization to above question when $ϕ$ is an elliptic automorphism. For a given Blaschke product $θ$, we discuss about finding all self maps $ϕ$ such that $θH^p$ is invariant under $C_ϕ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_09759 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Composition operators between Beurling subspaces of Hardy space Anjali, V. A. Muthukumar, P. Shankar, P. Functional Analysis Complex Variables Primary 47B33, Secondary 47A15, 47B38, 30H10, 46E15, 46E22 V. Matache (J. Operator Theory 73(1):243--264, 2015) raised an open problem about characterizing composition operators $C_ϕ$ on the Hardy space $H^2$ and nonzero singular measures $μ_1$, $μ_2$ on the unit circle such that $C_ϕ({S_{μ_1}} H^2)\subseteq {S_{μ_2}} H^2,$ where $S_{μ_i}$ denotes the singular inner function corresponding to the measure $μ_i,i=1,2$. In this article, we consider this problem in a more general setting. We characterize holomorphic self maps $ϕ$ of the unit disk $\mathbb{D}$ and inner functions $θ_1, θ_2$ such that $C_ϕ(θ_1 H^p)\subseteq θ_2 H^p,$ for $p>0$. Emphasis is given to Blaschke products and singular inner functions as a special case. We also give an another measure-theoretic characterization to above question when $ϕ$ is an elliptic automorphism. For a given Blaschke product $θ$, we discuss about finding all self maps $ϕ$ such that $θH^p$ is invariant under $C_ϕ$. |
| title | Composition operators between Beurling subspaces of Hardy space |
| topic | Functional Analysis Complex Variables Primary 47B33, Secondary 47A15, 47B38, 30H10, 46E15, 46E22 |
| url | https://arxiv.org/abs/2408.09759 |