Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866910541503528960 |
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| author | Beltrán-Meneu, María J. Bonet, José Jordá, Enrique |
| author_facet | Beltrán-Meneu, María J. Bonet, José Jordá, Enrique |
| contents | The behaviour of the generalized Hilbert operator associated with a positive finite Borel measure $μ$ on $[0,1)$ is investigated when it acts on weighted Banach spaces of holomorphic functions on the unit disc defined by sup-norms and on Korenblum type growth Banach spaces. It is studied when the operator is well defined, bounded and compact. To this aim, we study when it can be represented as an integral operator. We observe important differences with the behaviour of the Cesàro-type operator acting on these spaces, getting that boundedness and compactness are equivalent concepts for some standard weights. For the space of bounded holomorphic functions on the disc and for the Wiener algebra, we get also this equivalence, which is characterized in turn by the summability of the moments of the measure $μ.$ In the latter case, it is also equivalent to nuclearity. Nuclearity of the generalized Hilbert operator acting on related spaces, such as the classical Hardy space, is also analyzed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17646 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms Beltrán-Meneu, María J. Bonet, José Jordá, Enrique Functional Analysis Primary 47B38, Secondary 46E10, 46E15, 47A10, 47A16, 47A35, 47B35 The behaviour of the generalized Hilbert operator associated with a positive finite Borel measure $μ$ on $[0,1)$ is investigated when it acts on weighted Banach spaces of holomorphic functions on the unit disc defined by sup-norms and on Korenblum type growth Banach spaces. It is studied when the operator is well defined, bounded and compact. To this aim, we study when it can be represented as an integral operator. We observe important differences with the behaviour of the Cesàro-type operator acting on these spaces, getting that boundedness and compactness are equivalent concepts for some standard weights. For the space of bounded holomorphic functions on the disc and for the Wiener algebra, we get also this equivalence, which is characterized in turn by the summability of the moments of the measure $μ.$ In the latter case, it is also equivalent to nuclearity. Nuclearity of the generalized Hilbert operator acting on related spaces, such as the classical Hardy space, is also analyzed. |
| title | Generalized Hilbert operators acting on weighted spaces of holomorphic functions with sup-norms |
| topic | Functional Analysis Primary 47B38, Secondary 46E10, 46E15, 47A10, 47A16, 47A35, 47B35 |
| url | https://arxiv.org/abs/2407.17646 |