A note on composition operators on the bidisc
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910878347034624 |
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| author | Beslikas, Athanasios |
| author_facet | Beslikas, Athanasios |
| contents | In this note we give a new sufficient condition for the boundedness of the composition operator on the Dirichlet-type space on the disc, via a two dimensional change of variables formula. With the same formula, we characterise the bounded composition operators on the anisotropic Dirichlet-type spaces $\mathfrak{D}_{\vec{a}}(\mathbb{D}^2)$ induced by holomorphic self maps of the bidisc $\mathbb{D}^2$ of the form $Φ(z_1,z_2)=(ϕ_1(z_1),ϕ_2(z_2))$. We also consider the problem of boundedness of composition operators $C_Φ:\mathfrak{D}(\mathbb{D}^2)\to A^2(\mathbb{D}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the the Dirichlet space of the bidisc. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_14423 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on composition operators on the bidisc Beslikas, Athanasios Complex Variables 32A37 In this note we give a new sufficient condition for the boundedness of the composition operator on the Dirichlet-type space on the disc, via a two dimensional change of variables formula. With the same formula, we characterise the bounded composition operators on the anisotropic Dirichlet-type spaces $\mathfrak{D}_{\vec{a}}(\mathbb{D}^2)$ induced by holomorphic self maps of the bidisc $\mathbb{D}^2$ of the form $Φ(z_1,z_2)=(ϕ_1(z_1),ϕ_2(z_2))$. We also consider the problem of boundedness of composition operators $C_Φ:\mathfrak{D}(\mathbb{D}^2)\to A^2(\mathbb{D}^2)$ for general self maps of the bidisc, applying some recent results about Carleson measures on the the Dirichlet space of the bidisc. |
| title | A note on composition operators on the bidisc |
| topic | Complex Variables 32A37 |
| url | https://arxiv.org/abs/2405.14423 |