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New aspects of ill-posedness classification in Banach spaces

Jens Flemming, Bernd Hofmann

TL;DR

The paper addresses ill-posed linear operator equations in Banach spaces beyond Nashed's classic dichotomy by developing a switched, hybrid-type classification that accommodates non-injective operators with uncomplemented null-spaces and operators whose ranges contain (or fail to contain) infinite-dimensional closed subspaces. It introduces Mazur-type and hybrid-type operators, analyzes pseudoinverse behavior under uncomplemented null-spaces, and analyzes $\ell^1$-regularization under weak$^*$-to-weak continuity, including explicit counterexamples showing its limitations. Key contributions include a refined ill-posedness framework, a detailed examination of the interplay between $\mathcal{N}(A)$ and $\mathcal{R}(A)$, and demonstrations that regularization may fail for certain non-injective, non-compact constructions. The results clarify stability criteria for inverse problems in Banach spaces and guide the applicability of $\ell^1$-regularization in non-reflexive settings, with implications for practical regularization strategy design.

Abstract

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.

New aspects of ill-posedness classification in Banach spaces

TL;DR

The paper addresses ill-posed linear operator equations in Banach spaces beyond Nashed's classic dichotomy by developing a switched, hybrid-type classification that accommodates non-injective operators with uncomplemented null-spaces and operators whose ranges contain (or fail to contain) infinite-dimensional closed subspaces. It introduces Mazur-type and hybrid-type operators, analyzes pseudoinverse behavior under uncomplemented null-spaces, and analyzes -regularization under weak-to-weak continuity, including explicit counterexamples showing its limitations. Key contributions include a refined ill-posedness framework, a detailed examination of the interplay between and , and demonstrations that regularization may fail for certain non-injective, non-compact constructions. The results clarify stability criteria for inverse problems in Banach spaces and guide the applicability of -regularization in non-reflexive settings, with implications for practical regularization strategy design.

Abstract

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 -regularization with respect to ill-posedness phenomena and shed some light on the role of weak*-to-weak continuity in the context of -regularization for operators with uncomplemented null-space.

Paper Structure

This paper contains 6 sections, 7 theorems, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A new facet for ill-posedness classification in Banach spaces
  3. Preliminaries
  4. Updated classification with switched hybrid case and its illustration
  5. Discussion of Figure \ref{['fig:update']}
  6. Phenomena of $\ell^1$-regularization and ill-posedness

Key Result

Proposition 2.1

If for the bounded linear operator $A:X \to Y$ from equation eq:opeq the null-space $\mathcal{N}(A)$ is complemented in $X$ with the closed infinite-dimensional subspace $U$ of the Banach space $X$ such that the direct sum $X=\mathcal{N}(A)\oplus U$ takes place, then the pseudoinverse $A^\dagger_U:

Figures (1)

  • Figure 1: Case distinction for bounded linear operators between infinite-dimensional Banach spaces with complemented and uncomplemented null-spaces and shifted hybrid case

Theorems & Definitions (21)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Definition 2.3: Hybrid-type
  • Proposition 2.4
  • proof
  • Example 2.5: Mazur-type operators in sequence spaces
  • Definition 2.6: Well- and ill-posedness characterization and classification
  • Example 2.7: Non-compact compositions to Mazur-type operator $B$
  • Example 2.8: Compact compositions to Mazur-type operator $B$
  • ...and 11 more