The Cesàro operator on local Dirichlet spaces
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909331763494912 |
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| author | Dellepiane, Eugenio Mashreghi, Javad Nasri, Mostafa Verreault, William |
| author_facet | Dellepiane, Eugenio Mashreghi, Javad Nasri, Mostafa Verreault, William |
| contents | The family of Cesàro operators $σ_n^α$, $n \geq 0$ and $α\in [0,1]$, consists of finite rank operators on Banach spaces of analytic functions on the open unit disc. In this work, we investigate these operators as they act on the local Dirichlet spaces $\mathcal{D}_ζ$. It is well-established that they provide a linear approximation scheme when $α> \frac{1}{2}$, with the threshold value $α= \frac{1}{2}$ being optimal. We strengthen this result by deriving precise asymptotic values for the norm of these operators when $α\leq \frac{1}{2}$, corresponding to the breakdown of approximation schemes. Additionally, we establish upper and lower estimates for the norm when $α> \frac{1}{2}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00828 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Cesàro operator on local Dirichlet spaces Dellepiane, Eugenio Mashreghi, Javad Nasri, Mostafa Verreault, William Functional Analysis Complex Variables 47B38 (Primary) 42B35, 26D15 (Secondary) The family of Cesàro operators $σ_n^α$, $n \geq 0$ and $α\in [0,1]$, consists of finite rank operators on Banach spaces of analytic functions on the open unit disc. In this work, we investigate these operators as they act on the local Dirichlet spaces $\mathcal{D}_ζ$. It is well-established that they provide a linear approximation scheme when $α> \frac{1}{2}$, with the threshold value $α= \frac{1}{2}$ being optimal. We strengthen this result by deriving precise asymptotic values for the norm of these operators when $α\leq \frac{1}{2}$, corresponding to the breakdown of approximation schemes. Additionally, we establish upper and lower estimates for the norm when $α> \frac{1}{2}$. |
| title | The Cesàro operator on local Dirichlet spaces |
| topic | Functional Analysis Complex Variables 47B38 (Primary) 42B35, 26D15 (Secondary) |
| url | https://arxiv.org/abs/2410.00828 |