Spectral properties of the Cauchy transform on modified Bergman spaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910920460992512 |
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| author | Chbichib, Khaled Ghiloufi, Noureddine Snoun, Safa |
| author_facet | Chbichib, Khaled Ghiloufi, Noureddine Snoun, Safa |
| contents | In this paper, we determine the singular values $s_n(T_{α,β})$ and $s_n(R_{α,β})$ of the operators $T_{α,β}=\mathcal C\mathbb P_{α,β}$ and $R_{α,β}=\mathbb P_{α,β}\mathcal C\mathbb P_{α,β}$ where $\mathcal C$ is the integral Cauchy transform and $\mathbb P_{α,β}$ is the orthogonal projection from $L^2(\mathbb D,μ_{α,β})$ onto the modified Bergman space $\mathcal A^2(\mathbb D,μ_{α,β})$. These singular values will be expressed in terms of some series involving hypergeometric functions. We show that in both cases the sequence $n^{α+1}s_n(.)$ has a finite limit as $n\to+\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19286 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral properties of the Cauchy transform on modified Bergman spaces Chbichib, Khaled Ghiloufi, Noureddine Snoun, Safa Complex Variables 47G10, 47A75, 30H20 In this paper, we determine the singular values $s_n(T_{α,β})$ and $s_n(R_{α,β})$ of the operators $T_{α,β}=\mathcal C\mathbb P_{α,β}$ and $R_{α,β}=\mathbb P_{α,β}\mathcal C\mathbb P_{α,β}$ where $\mathcal C$ is the integral Cauchy transform and $\mathbb P_{α,β}$ is the orthogonal projection from $L^2(\mathbb D,μ_{α,β})$ onto the modified Bergman space $\mathcal A^2(\mathbb D,μ_{α,β})$. These singular values will be expressed in terms of some series involving hypergeometric functions. We show that in both cases the sequence $n^{α+1}s_n(.)$ has a finite limit as $n\to+\infty$. |
| title | Spectral properties of the Cauchy transform on modified Bergman spaces |
| topic | Complex Variables 47G10, 47A75, 30H20 |
| url | https://arxiv.org/abs/2504.19286 |