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

Bernd Hofmann, Stefan Kindermann

TL;DR

This work analyzes how bounded linear operators between Banach spaces can be well- or ill-posed, highlighting that Banach spaces admit multiple, non-equivalent definitions linked to complemented versus uncomplemented null-spaces. It contrasts the Hilbert-space intuition—where ill-posedness type II is tied to compactness and type I to non-compactness—with Banach-space phenomena such as strictly singular but non-compact injective operators and uncomplemented null-spaces, which give rise to hybrid and NV-type ill-posedness. A central contribution is the refined Definition \ref{def:general} for general Banach-space operators, plus a detailed analysis of generalized inverses and their role in characterizing various notions of well-posedness (HaWP, HSWP, NVWP, WP, and $\mathcal{R}$-WP). The paper also surveys alternative ill-posedness classifications and provides concrete examples (e.g., embeddings $\ell^p \to \ell^q$ and Mazur maps) to illustrate the intricate landscape beyond the Hilbert-space setting, with implications for regularization and numerical analysis in Banach spaces.

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.

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

TL;DR

This work analyzes how bounded linear operators between Banach spaces can be well- or ill-posed, highlighting that Banach spaces admit multiple, non-equivalent definitions linked to complemented versus uncomplemented null-spaces. It contrasts the Hilbert-space intuition—where ill-posedness type II is tied to compactness and type I to non-compactness—with Banach-space phenomena such as strictly singular but non-compact injective operators and uncomplemented null-spaces, which give rise to hybrid and NV-type ill-posedness. A central contribution is the refined Definition \ref{def:general} for general Banach-space operators, plus a detailed analysis of generalized inverses and their role in characterizing various notions of well-posedness (HaWP, HSWP, NVWP, WP, and -WP). The paper also surveys alternative ill-posedness classifications and provides concrete examples (e.g., embeddings and Mazur maps) to illustrate the intricate landscape beyond the Hilbert-space setting, with implications for regularization and numerical analysis in Banach spaces.

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.
Paper Structure (7 sections, 15 theorems, 16 equations, 4 figures)

This paper contains 7 sections, 15 theorems, 16 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Hilbert space setting
  3. Banach space setting for injective operators
  4. Banach space setting for general operators
  5. Alternative concepts of well-posedness
  6. Characterizations of well-posedness by generalized inverses
  7. Alternatives concepts of types of ill-posedness

Key Result

Proposition 1

A bounded linear operator $A:X \to Y$ mapping between Banach spaces is strictly singular if the closed subspaces $Z$ of $X$, for which the restriction $A|_{Z}$ has a bounded inverse, are necessarily finite dimensional.

Figures (4)

  • Figure 1: Case distinction for bounded linear operators between infinite-dimensional Hilbert spaces. Here, strictly singular operators are always compact.
  • Figure 2: Case distinction for injective bounded linear operators mapping between infinite-dimensional Banach spaces (cf. FHV15).
  • Figure 3: Case distinction for bounded linear operators between infinite-dimensional Banach spaces with complemented and uncomplemented null-spaces.
  • Figure 4: Position of hybrid operators.

Theorems & Definitions (36)

  • Definition 1: Naive well-posedness and ill-posedness characterization
  • Definition 2: Strictly singular operator (cf. GoldThorp63)
  • Proposition 1: Kato58
  • Proposition 2: Kato58
  • Lemma 1
  • proof
  • Proposition 3: GoldThorp63
  • Proposition 4: Pitt's theorem
  • Proposition 5
  • Proposition 6
  • ...and 26 more