The resolvent average of monotone operators: dominant and recessive properties

TL;DR

This paper extends the concept of proximal averages from convex functions to monotone operators in Hilbert spaces, introducing the resolvent average and analyzing its properties, including dominance and recessiveness.

Contribution

It introduces the resolvent average for monotone operators, classifies properties as dominant or recessive, and enhances understanding of proximal averages in convex analysis.

Findings

The resolvent average has many desirable properties.

Classification of properties as dominant or recessive.

New results on the proximal average's properties.

Abstract

Within convex analysis, a rich theory with various applications has been evolving since the proximal average of convex functions was first introduced over a decade ago. When one considers the subdifferential of the proximal average, a natural averaging operation of the subdifferentials of the averaged functions emerges. In the present paper we extend the reach of this averaging operation to the framework of monotone operator theory in Hilbert spaces, transforming it into the resolvent average. The theory of resolvent averages contains many desirable properties. In particular, we study a detailed list of properties of monotone operators and classify them as dominant or recessive with respect to the resolvent average. As a consequence, we recover a significant part of the theory of proximal averages. Furthermore, we shed new light on the proximal average and present novel results and desirable properties the proximal average possesses which have not been previously available.