Curious ill-posedness phenomena in the composition of non-compact linear operators in Hilbert spaces
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914797776273408 |
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| author | Kindermann, Stefan Hofmann, Bernd |
| author_facet | Kindermann, Stefan Hofmann, Bernd |
| contents | We consider the composition of operators with non-closed range in Hilbert spaces and how the nature of ill-posedness is affected by their composition. Specifically, we study the \mbox{Hausdorff-,} Cesàro-, integration operator, and their adjoints, as well as some combinations of those. For the composition of the Hausdorff- and the Cesàro-operator, we give estimates of the decay of the corresponding singular values. As a curiosity, this provides also an example of two practically relevant non-compact operators, for which their composition is compact. Furthermore, we characterize those operators for which a composition with a non-compact operator gives a compact one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14701 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Curious ill-posedness phenomena in the composition of non-compact linear operators in Hilbert spaces Kindermann, Stefan Hofmann, Bernd Functional Analysis We consider the composition of operators with non-closed range in Hilbert spaces and how the nature of ill-posedness is affected by their composition. Specifically, we study the \mbox{Hausdorff-,} Cesàro-, integration operator, and their adjoints, as well as some combinations of those. For the composition of the Hausdorff- and the Cesàro-operator, we give estimates of the decay of the corresponding singular values. As a curiosity, this provides also an example of two practically relevant non-compact operators, for which their composition is compact. Furthermore, we characterize those operators for which a composition with a non-compact operator gives a compact one. |
| title | Curious ill-posedness phenomena in the composition of non-compact linear operators in Hilbert spaces |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2401.14701 |