Resolvent Compositions for Positive Linear Operators

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 L\"owner 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.