Operator ordering by ill-posedness in Hilbert and Banach spaces
Stefan Kindermann, Bernd Hofmann
TL;DR
This paper develops an abstract ordering of ill-posedness among operators, defined via connecting operators so that $A' = T A S$, and proves its equivalence with Mathé–Hofmann's definition in Hilbert spaces for compact and non-compact operators. It shows that, in the compact Hilbert-space setting, several natural orderings (including those using isometries and singular-value comparisons) are equivalent and closely tied to the decay of singular values, with a stable interpretation under norm limits. The framework is extended to non-compact operators, Banach spaces, and nonlinear problems, where the ordering can be interpreted through polar decompositions, range inclusions, and linearization properties, and where nonlinearity conditions (e.g., tangential cone conditions and rotleft) can ensure that nonlinear ill-posedness mirrors linear ill-posedness. The results illuminate how regularization performance relates to operator orderings, provide explicit representations and counterexamples (e.g., Hausdorff–Cesàro compositions and non-compact multiplication operators), and suggest practical criteria for assessing and comparing ill-posedness in linear and nonlinear settings.
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 Mathé 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.