New aspects of ill-posedness classification in Banach spaces
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914146283421696 |
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| author | Flemming, Jens Hofmann, Bernd |
| author_facet | Flemming, Jens Hofmann, Bernd |
| contents | Motivated by a seminal paper of professor M. Z. Nashed published in 1987 on classification of ill-posed linear operator equations and distinguishing two types of ill-posedness in Banach and Hilbert spaces, we present, illustrate and justify a new classification scheme in this context. This scheme classifies bounded linear operators mapping between infinite-dimensional Banach spaces with respect to ill-posedness types, including non-injective operators that may have uncomplemented null-spaces. The hybrid case of strictly singular operators the range of which contains a closed infinite-dimensional subspace plays a prominent role there. By a series of new theorems we complement moreover the theory of $\ell^1$-regularization with respect to ill-posedness phenomena and shed some light on the role of weak*-to-weak continuity in the context of $\ell^1$-regularization for operators with uncomplemented null-space. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_06690 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New aspects of ill-posedness classification in Banach spaces Flemming, Jens Hofmann, Bernd Functional Analysis 47A52, 47B01 Motivated by a seminal paper of professor M. Z. Nashed published in 1987 on classification of ill-posed linear operator equations and distinguishing two types of ill-posedness in Banach and Hilbert spaces, we present, illustrate and justify a new classification scheme in this context. This scheme classifies bounded linear operators mapping between infinite-dimensional Banach spaces with respect to ill-posedness types, including non-injective operators that may have uncomplemented null-spaces. The hybrid case of strictly singular operators the range of which contains a closed infinite-dimensional subspace plays a prominent role there. By a series of new theorems we complement moreover the theory of $\ell^1$-regularization with respect to ill-posedness phenomena and shed some light on the role of weak*-to-weak continuity in the context of $\ell^1$-regularization for operators with uncomplemented null-space. |
| title | New aspects of ill-posedness classification in Banach spaces |
| topic | Functional Analysis 47A52, 47B01 |
| url | https://arxiv.org/abs/2511.06690 |