Maximal Monotone Operators with unique Representation

Abstract

For a monotone operator A : X ⇒ X∗, we define FA as the family of all representations of A through convex lower semicontinuous functions on X × X∗. These representations facilitate the application of convex analysis techniques in the study of monotone operators. Bartz et al. have shown that when the operator is the subdifferential of a sublinear function, FA is a singleton. In this work, we investigate the inverse problem: what conditions must an operator satisfy to have a unique representation? We establish specific properties and, under certain assumptions, provide a complete characterization of such operators.