A new old class of maximal monotone operators

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.

Submited for publication on April 2008

References in corpus (1)

• Maximal monotonicity, conjugation and the duality product

Cited by in corpus (9)

• On Gossez type (D) maximal monotone operators
• Maximal monotonicity, conjugation and the duality product in non-reflexive Banach spaces
• On the surjectivity properties of perturbations of maximal monotone operators in non-reflexive Banach spaces
• For maximally monotone linear relations, dense type, negative-infimum type, and Fitzpatrick-Phelps type all coincide with monotonicity of the adjoint
• Nonreflexive Banach SSD spaces
• Regular Maximal Monotone Multifunctions and Enlargements
• Geometric properties of maximal monotone operators and convex functions which may represent them
• Characterizations of ultramaximally monotone operators
• On the generalized parallel sum of two maximal monotone operators of Gossez type (D)