Classification of ill-posedness for bounded linear operators in Banach spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913846651781120 |
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| author | Hofmann, Bernd Kindermann, Stefan |
| author_facet | Hofmann, Bernd Kindermann, Stefan |
| contents | In this article, concepts of well- and ill-posedness for linear operators in Hilbert and Banach spaces are discussed. While these concepts are well understood in Hilbert spaces, this is not the case in Banach spaces, as there are several competing definitions, related to the occurrence of uncomplemented subspaces. We provide an overview of the various definitions and, based on this, discuss the classification of type I and type II ill-posedness in Banach spaces. Furthermore, a discussion of borderline (hybrid) cases in this classification is given together with several example instances of operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12931 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Classification of ill-posedness for bounded linear operators in Banach spaces Hofmann, Bernd Kindermann, Stefan Functional Analysis 47A52, 47B01, 47B02, 65J20 In this article, concepts of well- and ill-posedness for linear operators in Hilbert and Banach spaces are discussed. While these concepts are well understood in Hilbert spaces, this is not the case in Banach spaces, as there are several competing definitions, related to the occurrence of uncomplemented subspaces. We provide an overview of the various definitions and, based on this, discuss the classification of type I and type II ill-posedness in Banach spaces. Furthermore, a discussion of borderline (hybrid) cases in this classification is given together with several example instances of operators. |
| title | Classification of ill-posedness for bounded linear operators in Banach spaces |
| topic | Functional Analysis 47A52, 47B01, 47B02, 65J20 |
| url | https://arxiv.org/abs/2505.12931 |