The resolvent average of monotone operators: dominant and recessive properties (Q2792407)

In this very well-written paper, the authors introduce and investigate the notion of resolvent average of \(n\) monotone operators defined on a Hilbert space, by extending the idea of proximal averages of convex functions, which has several nice applications in convex analysis and beyond. Given the close connections between the theory of monotone operators and convex analysis, such a generalization makes complete sense not only per se, but also because of the possible positive implications in the original field, i.e., convex analysis. The basic idea behind the extension provided in this paper lies in the fact that starting from the subdifferential of the proximal average of \(n\) functions, a natural averaging operation of the subdifferentials of the averaged functions emerges. Different properties of the resolvent average (invertibility, connections to solutions of monotone inclusions, Brezis-Haraux type property, Fitzpatrick functions associated to it) are investigated in the second part of the paper. In the next ones, these and others (strong monotonicity, cocoercivity, paramonotonicity, single-valuedness etc.) are classified as dominant or recessive with respect to the resolvent average. Different other related results and remarks close the paper. Worth noticing is also the fact that a significant part of the theory of proximal averages can be recovered as a special case of the newly constructed theory, but also new results and properties belonging to it are discovered. Some illustrative examples are presented as well.