A New Old Class of Maximal Monotone Operators

In a recent paper ["Brøndsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces", J. Convex Analysis 15(4) (2008) 693--706] the authors studied 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 strict version of the Brøndsted-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 strict Brøndsted-Rockafellar property is equivalent to the assumption of the conjugate of the Fitzpatrick function to majorize the duality product.