2025/11/10 by Jens Flemming, Flemming, Jens, Bernd Hofmann +1
Mathematics · Computer Science · #Numerical methods in inverse problems #Advanced Banach Space Theory #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2511.06690
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 ℓ1-regularization with respect to ill-posedness phenomena and shed some light on the role of weak*-to-weak continuity in the context of ℓ1-regularization for operators with uncomplemented null-space.