Comparing the ill-posedness for linear operators in Hilbert spaces

TL;DR

The paper introduces a partial ordering framework to compare the ill-posedness strength of linear operator equations in Hilbert spaces, defining $A' \prec_{R,S} A$ via $A' = R A S$ to account for non-closed ranges and domain/target differences. For compact operators, the ordering aligns with singular-value decay, $s_n(A') = O(s_n(A))$, and extends to non-compact operators through the spectral theorem and range-inclusion concepts (Douglas' theorem). It analyzes how the ordering interacts with regularization, including the dichotomy of convergent vs. unbounded behavior depending on the right-hand side's range, and provides a variety of illustrative examples, including compositions with non-compact operators and multiplication/Hausdorff moment operators. Overall, the framework offers a principled tool to compare ill-posedness strength across operator equations and to inform regularization design in inverse problems within Hilbert spaces.

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.

Figures (2)

• Figure 1: Comparison of operators $A$ and $A^\prime$ via factorization, where using the orthogonal mapping $R$.
• Figure 2: Comparison of self-adjoint operators $H$ and $H^\prime$

Theorems & Definitions (34)

• Definition 1: partial ordering
• Remark 1
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Corollary 1
• Remark 2