Resolvent Compositions for Positive Linear Operators
Diego J. Cornejo
TL;DR
This work addresses how to preserve monotonicity when coupling a monotone operator with a bounded linear operator through resolvent compositions. Focusing on the case where the monotone operator is positive, it derives Löwner-order relations, concavity of the associated operator maps, and precise asymptotic limits, while establishing nonexpansiveness with respect to the Thompson metric. A geometric interpolation between the parallel composition and the full sandwich form is introduced, and two nonlinear fixed-point problems based on resolvent compositions are shown to admit unique solutions via contraction arguments. Overall, the results extend the resolvent-average framework, yielding stable, monotonicity-preserving tools for monotone inclusions and convex optimization in coupled linear-monotone settings.
Abstract
Resolvent compositions were recently introduced as monotonicity-preserving operations that combine a set-valued monotone operator and a bounded linear operator. They generalize in particular the notion of a resolvent average. We analyze the resolvent compositions when the monotone operator is a positive linear operator. We establish several new properties, including Löwner partial order relations, concavity, and asymptotic behavior. In addition, we show that the resolvent composition operations are nonexpansive with respect to the Thompson metric. We also introduce a new form of geometric interpolation and explore its connections to resolvent compositions. Finally, we study two nonlinear equations based on resolvent compositions.