Comparing the ill-posedness for linear operators in Hilbert spaces

TL;DR

This paper introduces a partial ordering framework to compare the ill-posedness of linear operators in Hilbert spaces, based on properties like singular value decay and operator composition, enhancing understanding of operator difficulty.

Contribution

It proposes a novel partial ordering for classifying ill-posedness of bounded linear operators, including compact and non-compact cases, with theoretical characterizations and illustrative examples.

Findings

Range inclusions induce partial ordering

Singular value decay characterizes compact operators' ill-posedness

Operator compositions affect the degree of ill-posedness

Abstract

The difficulty for solving ill-posed linear operator equations in Hilbert space is reflected by the strength of ill-posedness of the governing operator, and the inherent solution smoothness. In this study we focus on the ill-posedness of the operator, and we propose a partial ordering for the class of all bounded linear operators which lead to ill-posed operator equations. For compact linear operators, there is a simple characterization in terms of the decay rates of the singular values. In the context of the validity of the spectral theorem the partial ordering can also be understood. We highlight that range inclusions yield partial ordering, and we discuss cases when compositions of compact and non-compact operators occur. Several examples complement the theoretical results.