Fixed points in the family of convex representations of a maximal monotone operator

TL;DR

This paper investigates convex representations of maximal monotone operators, demonstrating the existence of a fixed point under Fenchel-Legendre conjugation, which enhances understanding of their structural properties.

Contribution

It proves the existence of a convex representation of a maximal monotone operator that remains invariant under Fenchel-Legendre conjugation, revealing a fixed point property.

Findings

Existence of a fixed point convex representation

Invariance of convex functions under conjugation

Enhanced understanding of operator structure

Abstract

Any maximal monotone operator can be characterized by a convex function. The family of such convex functions is invariant under a transformation connected with the Fenchel-Legendre conjugation. We prove that there exist a convex representation of the operator which is a fixed point of this conjugation.