On Toeplitz operators on $H^1(\mathbb{C}^+)$
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913727989678080 |
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| author | Bellavita, Carlo Peloso, Marco M. |
| author_facet | Bellavita, Carlo Peloso, Marco M. |
| contents | In this paper we consider Toeplitz operators with anti-analytic symbols on $H^1(\mathbb{C}^+)$. It is well known that there are no bounded Toeplitz operators $T_{\overlineΘ}\colon H^1(\mathbb{C}^+) \to H^1(\mathbb{C}^+)$, where $Θ\in H^\infty(\mathbb{C}^+)$. We consider the subspace $H^1_Θ=\left\lbrace f \in H^1(\mathbb{C}^+)\colon \int_{\mathbb{R}}f \overlineΘ=0\right\rbrace$ and show that it is natural to study the boundedness of $T_{\overlineΘ}\colon H^1_Θ\to H^1(\mathbb{C}^+)$. We provide several different conditions equivalent to such boundedness. We prove that when $Θ=e^{iτ(\cdot)}$, with $τ>0$ $T_{\overlineΘ}\colon H^1_Θ\to H^1(\mathbb{C}^+)$ is bounded. Finally, we discuss a number of related open questions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07281 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Toeplitz operators on $H^1(\mathbb{C}^+)$ Bellavita, Carlo Peloso, Marco M. Functional Analysis Classical Analysis and ODEs Complex Variables 47B35, 30H10 In this paper we consider Toeplitz operators with anti-analytic symbols on $H^1(\mathbb{C}^+)$. It is well known that there are no bounded Toeplitz operators $T_{\overlineΘ}\colon H^1(\mathbb{C}^+) \to H^1(\mathbb{C}^+)$, where $Θ\in H^\infty(\mathbb{C}^+)$. We consider the subspace $H^1_Θ=\left\lbrace f \in H^1(\mathbb{C}^+)\colon \int_{\mathbb{R}}f \overlineΘ=0\right\rbrace$ and show that it is natural to study the boundedness of $T_{\overlineΘ}\colon H^1_Θ\to H^1(\mathbb{C}^+)$. We provide several different conditions equivalent to such boundedness. We prove that when $Θ=e^{iτ(\cdot)}$, with $τ>0$ $T_{\overlineΘ}\colon H^1_Θ\to H^1(\mathbb{C}^+)$ is bounded. Finally, we discuss a number of related open questions. |
| title | On Toeplitz operators on $H^1(\mathbb{C}^+)$ |
| topic | Functional Analysis Classical Analysis and ODEs Complex Variables 47B35, 30H10 |
| url | https://arxiv.org/abs/2503.07281 |