Strongly-Representable Monotone Operators

Recently M. Marques Alves and B. F. Svaiter ["Brønsted-Rockafellar property and maximality of monotone operators representable by convex functions in non-reflexive Banach spaces", J. Convex Analysis 15(4) (2008) 693--706] introduced a new class of maximal monotone operators. In this note we study domain-range properties as well as connections with other classes and calculus rules for these operators we called strongly-representable. While not every maximal monotone operator is strongly-representable, every maximal monotone NI operator is strongly-representable, and every strongly-representable operator is locally maximal monotone, maximal monotone locally, strongly maximal monotone, and ANA. As a consequence the conjugate of the Fitzpatrick function of a maximal monotone operator is not necessarily a representative function