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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.08514 |
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| _version_ | 1866915383905091584 |
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| author | Gómez-Cabello, Carlos González-Doña, F. Javier |
| author_facet | Gómez-Cabello, Carlos González-Doña, F. Javier |
| contents | Let $(φ_t)_{t\geq 0}$ be a parabolic semigroup of analytic functions on $\mathbb{D}$, with Koenigs function $h$ and Koenigs domain $Ω= h(\mathbb{D})$. We study the point spectrum $σ_p(Δ\mid_{H^p})$ of $Δ$, the infinitesimal generator of the $C_0$-semigroup $(C_{φ_t})_{t\geq 0}$ of composition operators on $H^p$. This reduces to characterizing the frequencies of $Ω$. That is, those $λ\in \mathbb{C}$ such that $e^{λh} \in H^p$. We first derive containment relations for $σ_p(Δ\mid_{H^p})$ and provide sufficient conditions for its complete characterization. Our approach relies heavily on the geometric properties of $Ω$ and on careful estimates of the harmonic measure of some boundary subsets of $Ω$. Furthermore, assuming that $Ω$ is convex, we also obtain necessary conditions for $λ$ to be a frequency of $Ω$. Using these, we are able to completely describe $σ_p(Δ\mid_{H^p})$ in a broad range of situations e.g. when $Ω$ contains an angular sector. We conclude with some consequences regarding the spectrum of the composition operators $(C_{φ_t})_{t\geq 0}$. These results extend a previous work of Betsakos on hyperbolic semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08514 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On frequencies of parabolic Koenigs domains Gómez-Cabello, Carlos González-Doña, F. Javier Complex Variables Functional Analysis Primary 47B33, 47D06, 30C45, 30C85 Let $(φ_t)_{t\geq 0}$ be a parabolic semigroup of analytic functions on $\mathbb{D}$, with Koenigs function $h$ and Koenigs domain $Ω= h(\mathbb{D})$. We study the point spectrum $σ_p(Δ\mid_{H^p})$ of $Δ$, the infinitesimal generator of the $C_0$-semigroup $(C_{φ_t})_{t\geq 0}$ of composition operators on $H^p$. This reduces to characterizing the frequencies of $Ω$. That is, those $λ\in \mathbb{C}$ such that $e^{λh} \in H^p$. We first derive containment relations for $σ_p(Δ\mid_{H^p})$ and provide sufficient conditions for its complete characterization. Our approach relies heavily on the geometric properties of $Ω$ and on careful estimates of the harmonic measure of some boundary subsets of $Ω$. Furthermore, assuming that $Ω$ is convex, we also obtain necessary conditions for $λ$ to be a frequency of $Ω$. Using these, we are able to completely describe $σ_p(Δ\mid_{H^p})$ in a broad range of situations e.g. when $Ω$ contains an angular sector. We conclude with some consequences regarding the spectrum of the composition operators $(C_{φ_t})_{t\geq 0}$. These results extend a previous work of Betsakos on hyperbolic semigroups. |
| title | On frequencies of parabolic Koenigs domains |
| topic | Complex Variables Functional Analysis Primary 47B33, 47D06, 30C45, 30C85 |
| url | https://arxiv.org/abs/2507.08514 |