Parametrized Families of Resolvent Compositions

TL;DR

This paper introduces a parametrized resolvent composition, exploring its properties, monotonicity, and convergence behavior, with implications for operator theory and optimization.

Contribution

It provides new properties, examples, and monotonicity results for parametrized resolvent compositions, including cases with non-monotone operators.

Findings

Resolvent compositions can be viewed as parallel compositions of perturbed operators.

New monotonicity results are established even for non-monotone initial operators.

Asymptotic convergence results are derived for graph and Hausdorff distances.

Abstract

This paper presents an in-depth analysis of a parametrized version of the resolvent composition, an operation that combines a set-valued operator and a linear operator. We provide new properties and examples, and show that resolvent compositions can be interpreted as parallel compositions of perturbed operators. Additionally, we establish new monotonicity results, even in cases when the initial operator is not monotone. Finally, we derive asymptotic results regarding operator convergence, specifically focusing on graph-convergence and the $\rho$-Hausdorff distance.