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

TL;DR

This paper introduces a unified spectral-theoretic framework to quantify the degree of ill-posedness in linear operator equations, encompassing both compact and non-compact cases, and demonstrates its consistency with classical measures through examples.

Contribution

It proposes a novel spectral-based measure for ill-posedness that applies to all bounded self-adjoint operators, extending existing concepts for compact operators.

Findings

The new measure aligns with classical singular value-based measures for compact operators.

It characterizes ill-posedness via the growth behavior of a distribution function at zero.

Illustrative examples include the Hausdorff moment operator and convolution operators.

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.