Operator ordering by ill-posedness in Hilbert and Banach spaces

TL;DR

This paper introduces a new operator ordering based on ill-posedness that applies to both Hilbert and Banach spaces, providing a unified framework and extending to nonlinear problems.

Contribution

It proposes a novel ill-posedness ordering for operators, unifying existing definitions and extending applicability to Banach spaces and nonlinear problems.

Findings

The new ordering is equivalent to existing definitions for compact and non-compact operators in Hilbert spaces.

It compares favorably with traditional measures like singular value decay and norm estimates.

The approach extends to nonlinear operators via linearization.

Abstract

For operators representing ill-posed problems, an ordering by ill-posedness is proposed, where one operator is considered more ill-posed than another one if the former can be expressed as a cocatenation of bounded operators involving the latter. This definition is motivated by a recent one introduced by Math\'e and Hofmann [Adv. Oper. Theory, 2025] that utilizes bounded and orthogonal operators, and we show the equivalence of our new definition with this one for the case of compact and non-compact linear operators in Hilbert spaces. We compare our ordering with other measures of ill-posedness such as the decay of the singular values, norm estimates, and range inclusions. Furthermore, as the new definition does not depend on the notion of orthogonal operators, it can be extended to the case of linear operators in Banach spaces, and it also provides ideas for applications to nonlinear problems in Hilbert spaces. In the latter context, certain nonlinearity conditions can be interpreted as ordering relations between a nonlinear operator and its linearization.