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∗. In this talk we shall discuss the singular case, that is, the family collapses to a singleton. We show that every such operator is cyclically monotone (hence, A = ∂f for some convex function f) if and only if it is 3-monotone. In Radon–Nikod´ym spaces, under mild conditions (which become superfluous in finite dimensions), we prove that a subdifferential operator A = ∂f is uniquely representable if and only if f is the sum of a support and an indicator function of suitable convex sets. This is joint work with Aris Daniilidis.