Generalized Hilbert operators arising from Hausdorff matrices
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909646331052032 |
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| author | Bellavita, Carlo Chalmoukis, Nikolaos Daskalogiannis, Vassilis Stylogiannis, Georgios |
| author_facet | Bellavita, Carlo Chalmoukis, Nikolaos Daskalogiannis, Vassilis Stylogiannis, Georgios |
| contents | For a finite, positive, Borel measure $μ$ on $(0,1)$ we consider an infinite matrix $Γ_μ$, related to the classical Hausdorff matrix defined by the same measure $μ$, in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When $μ$ is the Lebesgue measure, $Γ_μ$ reduces to the classical Hilbert matrix. We prove that the matrices $Γ_μ$ are not Hankel, unless $μ$ is a constant multiple of the Lebesgue measure,
we give necessary and sufficient conditions for their boundedness on the scale of Hardy spaces $H^p, \, 1 \leq p < \infty$, and we study their compactness and complete continuity properties. In the case $2\leq p<\infty$, we are able to compute the exact value of the norm of the operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_15334 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Generalized Hilbert operators arising from Hausdorff matrices Bellavita, Carlo Chalmoukis, Nikolaos Daskalogiannis, Vassilis Stylogiannis, Georgios Functional Analysis Complex Variables 30H10, 47B91 For a finite, positive, Borel measure $μ$ on $(0,1)$ we consider an infinite matrix $Γ_μ$, related to the classical Hausdorff matrix defined by the same measure $μ$, in the same algebraic way that the Hilbert matrix is related to the Cesáro matrix. When $μ$ is the Lebesgue measure, $Γ_μ$ reduces to the classical Hilbert matrix. We prove that the matrices $Γ_μ$ are not Hankel, unless $μ$ is a constant multiple of the Lebesgue measure, we give necessary and sufficient conditions for their boundedness on the scale of Hardy spaces $H^p, \, 1 \leq p < \infty$, and we study their compactness and complete continuity properties. In the case $2\leq p<\infty$, we are able to compute the exact value of the norm of the operator. |
| title | Generalized Hilbert operators arising from Hausdorff matrices |
| topic | Functional Analysis Complex Variables 30H10, 47B91 |
| url | https://arxiv.org/abs/2307.15334 |