A new old class of maximal monotone operators

TL;DR

This paper characterizes a class of maximal monotone operators in Banach spaces, showing that a property of one Fitzpatrick function extends to all, linking it to the NI class and clarifying conditions for the Brondsted-Rockafellar property.

Contribution

It proves that if one Fitzpatrick function's conjugate exceeds the duality product, then all do, identifying the class NI as exactly those operators, and clarifies conditions for the Brondsted-Rockafellar property.

Findings

The property extends from one Fitzpatrick function to all for a maximal monotone operator.

The class NI is characterized by the conjugate of Fitzpatrick functions exceeding the duality product.

An auxiliary condition is shown to be equivalent to the conjugate property for Fitzpatrick functions.

Abstract

In a recent paper in Journal of Convex Analysis the authors studied, in non-reflexive Banach spaces, a class of maximal monotone operators, characterized by the existence of a function in Fitzpatrick's family of the operator which conjugate is above the duality product. This property was used to prove that such operators satisfies a restricted version of Brondsted-Rockafellar property. In this work we will prove that if a single Fitzpatrick function of a maximal monotone operator has a conjugate above the duality product, then all Fitzpatrick function of the operator have a conjugate above the duality product. As a consequence, the family of maximal monotone operators with this property is just the class NI, previously defined and studied by Simons. We will also prove that an auxiliary condition used by the authors to prove the restricted Brondsted-Rockafellar property is equivalent to the assumption of the conjugate of the Fitzpatrick function to majorize the duality product.