Composition operators on Hardy-Smirnov spaces
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2021
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| author | Fávaro, V. V. Hai, P. V. Pellegrino, D. M. Severiano, O. R. |
| author_facet | Fávaro, V. V. Hai, P. V. Pellegrino, D. M. Severiano, O. R. |
| contents | We investigate composition operators $C_Φ$ on the Hardy-Smirnov space $H^{2}(Ω)$ induced by analytic self-maps $Φ$ of an open simply connected proper subset $Ω$ of the complex plane. When the Riemann map $τ:\mathbb{U}\rightarrowΩ$ used to define the norm of $H^{2}(Ω)$ is a linear fractional transformation, we characterize the composition operators whose adjoints are composition operators. As applications of this fact, we provide a new proof for the adjoint formula discovered by Gallardo-Gutiérrez and Montes-Rodríguez and we give a new approach to describe all Hermitian and unitary composition operators on $H^{2}(Ω).$ Additionally, if the coefficients of $τ$ are real, we exhibit concrete examples of conjugations and describe the Hermitian and unitary composition operators which are complex symmetric with respect to specific conjugations on $H^{2}(Ω).$ We finish this paper showing that if $Ω$ is unbounded and $Φ$ is a non-automorphic self-map of $Ω$ with a fixed point, then $C_Φ$ is never complex symmetric on $H^{2}(Ω).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_10609 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Composition operators on Hardy-Smirnov spaces Fávaro, V. V. Hai, P. V. Pellegrino, D. M. Severiano, O. R. Functional Analysis 47B33, 47B32, 47B99 We investigate composition operators $C_Φ$ on the Hardy-Smirnov space $H^{2}(Ω)$ induced by analytic self-maps $Φ$ of an open simply connected proper subset $Ω$ of the complex plane. When the Riemann map $τ:\mathbb{U}\rightarrowΩ$ used to define the norm of $H^{2}(Ω)$ is a linear fractional transformation, we characterize the composition operators whose adjoints are composition operators. As applications of this fact, we provide a new proof for the adjoint formula discovered by Gallardo-Gutiérrez and Montes-Rodríguez and we give a new approach to describe all Hermitian and unitary composition operators on $H^{2}(Ω).$ Additionally, if the coefficients of $τ$ are real, we exhibit concrete examples of conjugations and describe the Hermitian and unitary composition operators which are complex symmetric with respect to specific conjugations on $H^{2}(Ω).$ We finish this paper showing that if $Ω$ is unbounded and $Φ$ is a non-automorphic self-map of $Ω$ with a fixed point, then $C_Φ$ is never complex symmetric on $H^{2}(Ω).$ |
| title | Composition operators on Hardy-Smirnov spaces |
| topic | Functional Analysis 47B33, 47B32, 47B99 |
| url | https://arxiv.org/abs/2111.10609 |