Maximal monotonicity, conjugation and the duality product

TL;DR

This paper establishes a one-to-one correspondence between convex functions satisfying specific inequalities and maximal monotone operators, deepening the understanding of their duality and conjugation relationships.

Contribution

It proves that every convex function meeting certain inequalities uniquely corresponds to a maximal monotone operator, completing the characterization.

Findings

Convex functions satisfying the inequalities are associated with maximal monotone operators.

The paper establishes a bijective relationship between these functions and operators.

It advances the theoretical framework linking convex analysis and monotone operator theory.

Abstract

Recently, the authors studied the connection between each maximal monotone operator T and a family H(T) of convex functions. Each member of this family characterizes the operator and satisfies two particular inequalities. The aim of this paper is to establish the converse of the latter fact. Namely, that every convex function satisfying those two particular inequalities is associated to a unique maximal monotone operator.