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A note on the Goldberg-Thorp example in light of the classification of linear ill-posed problems in Banach spaces

Bernd Hofmann, Jens Flemming

Abstract

This note considers the strictly singular mapping, denoted by $B$, from $\ell^1$ onto $\ell^2$ of an example by Goldberg and Thorp from 1963 as a typical hybrid-type operator in the context of the classification of ill-posed linear operators in infinite-dimensional Banach spaces. The null-spaces of hybrid-type operators are not complemented and therefore need special attention. More generally, a given well-posedness definition for linear operators requiring both closed range and complemented null-space is motivated by the continuity of occurring pseudo-inverse operators as a stability criterion. With respect to the operator $B$, structure, representation and properties of the operator and its adjoint are summarized in a theorem. Moreover, limitations and opportunities of regularization approaches for the treatment of $B$ are outlined.

A note on the Goldberg-Thorp example in light of the classification of linear ill-posed problems in Banach spaces

Abstract

This note considers the strictly singular mapping, denoted by , from onto of an example by Goldberg and Thorp from 1963 as a typical hybrid-type operator in the context of the classification of ill-posed linear operators in infinite-dimensional Banach spaces. The null-spaces of hybrid-type operators are not complemented and therefore need special attention. More generally, a given well-posedness definition for linear operators requiring both closed range and complemented null-space is motivated by the continuity of occurring pseudo-inverse operators as a stability criterion. With respect to the operator , structure, representation and properties of the operator and its adjoint are summarized in a theorem. Moreover, limitations and opportunities of regularization approaches for the treatment of are outlined.
Paper Structure (4 sections, 9 theorems, 24 equations, 1 figure)

This paper contains 4 sections, 9 theorems, 24 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Characterization of well-posedness and ill-posedness with focus on the hybrid case
  3. The operator of the Goldberg-Thorp example
  4. Limitations and chances of regularization approaches for Goldberg-Thorp

Key Result

Proposition 2.4

For an operator equation eq:opeq of hybrid-type, the operator $A: X \to Y$ is not compact, and its null-space $\mathcal{N}(A)$ is always uncomplemented.

Figures (1)

  • Figure 1: Case distinction for bounded linear operators between infinite-dimensional Banach spaces with dense range

Theorems & Definitions (23)

  • Definition 2.1: Well- and ill-posedness characterization and classification
  • Definition 2.2: Strictly singular operators
  • Definition 2.3: Hybrid-type operators
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Definition 3.1: Mazur-type operators
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 13 more