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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

Frank Werner, Bernd Hofmann

Abstract

Covering ill-posed problems with compact and non-compact operators regarding the degree of ill-posedness is a never ending story written by many authors in the inverse problems literature. This paper tries to add a new narrative and some new facets with respect to this story under the auspices of the spectral theorem. The latter states that any self-adjoint and bounded operator is unitarily equivalent to a multiplication operator on some (semi-finite) measure space. We will exploit this fact and derive a distribution function from the corresponding multiplier, the growth behavior of which at zero allows us to characterize the degree of ill-posedness. We prove that this new concept coincides with the well-known one for compact operators (by means of their singular values), and illustrate the implications along examples including the Hausdorff moment operator and convolutions.

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

Abstract

Covering ill-posed problems with compact and non-compact operators regarding the degree of ill-posedness is a never ending story written by many authors in the inverse problems literature. This paper tries to add a new narrative and some new facets with respect to this story under the auspices of the spectral theorem. The latter states that any self-adjoint and bounded operator is unitarily equivalent to a multiplication operator on some (semi-finite) measure space. We will exploit this fact and derive a distribution function from the corresponding multiplier, the growth behavior of which at zero allows us to characterize the degree of ill-posedness. We prove that this new concept coincides with the well-known one for compact operators (by means of their singular values), and illustrate the implications along examples including the Hausdorff moment operator and convolutions.
Paper Structure (12 sections, 9 theorems, 65 equations, 1 figure)

This paper contains 12 sections, 9 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The degree of ill-posedness for equations with compact operators
  3. The spectral theorem as a tool for defining the degree of ill-posedness also in the non-compact case
  4. The spectral theorem according to Halmos
  5. The distribution function measuring the strength of ill-posedness
  6. Ill-posedness of equations with multiplication operators on the real axis
  7. Chances and limitations of Lebesgue measure spaces
  8. Examples
  9. The Hausdorff moment operator
  10. Convolution operators
  11. The case of unbounded operators
  12. Conclusion and outlook

Key Result

Theorem 1

The interval of ill-posedness of eq:model is given by

Figures (1)

  • Figure 1: Illustrative preview of occurring case distinctions

Theorems & Definitions (37)

  • Definition 1
  • Example 1: Riemann–Liouville fractional integration of order $\alpha>0$ (cf. VuGo94)
  • Example 2: Multivariate integration in the $d$-dimensional case (cf. HF23)
  • Theorem 1
  • proof
  • Corollary 1
  • Example 3: Inverse of negative Laplace operator in the $d$-dimensional case
  • Remark 1
  • Lemma 1
  • proof
  • ...and 27 more