Resolvent Compositions for Positive Linear Operators

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.

💡 Research Summary

This paper investigates the resolvent composition and cocomposition operators that combine a bounded linear operator L ∈ B(H,G) with a monotone set‑valued operator B : G → 2^G, focusing on the case where B is a positive (strongly monotone) linear operator, i.e., B ∈ S(G). The authors begin by recalling the necessary background on Hilbert spaces, the Löwner partial order on self‑adjoint operators, and the resolvent J_B = (Id + B)⁻¹. They define the resolvent composition L γ⋄B = L* ⊲ (B + γ⁻¹Id_G) − γ⁻¹Id_H and the resolvent cocomposition L γ˛B = (L 1/γ⋄B⁎)⁎, where “⊲” denotes the parallel composition L* ⊲ B = (L* ∘ B⁻¹ ∘ L)⁻¹.

The main contributions are as follows:

1. Löwner Order Relations – Proposition 3.3 establishes that for any γ > 0, \