Classification of ill-posedness for bounded linear operators in Banach spaces

2025/05/19 by Bernd Hofmann, Hofmann, Bernd, Stefan Kindermann +1 · 2 citations
Computer Science · Mathematics · #47A52 #47B01 #47B02 #65J20 #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2505.12931

openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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.

Citations

• A unified concept of the degree of ill-posedness for compact and non-compact linear operator equations in Hilbert spaces under the auspices of the spectral theorem
• Curious ill-posedness phenomena in the composition of non-compact linear operators in Hilbert spaces
• The Hausdorff Moment Problem in the light of ill-posedness of type I
• Regularization of linear ill-posed problems involving multiplication\n operators
• On ill-posedness concepts, stable solvability and saturation

Cited by

• New aspects of ill-posedness classification in Banach spaces
• Sparse variational regularization with oversmoothing penalty term in the scale of sequence spaces

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• Comparing the ill-posedness for linear operators in Hilbert spaces
• On ill-posedness concepts, stable solvability and saturation
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