Pictures of monotone operators

Abstract

Let E be a real Banach space with dual E*. We associate with any nonempty subset H of E×E* a certain compact convex subset of the first quadrant in ℝ2, which we call the picture of H, Π(H). In general, Π(H) may be empty, but Π(M) is nonempty if M is a nonempty monotone subset of E×E*. If E is reflexive and M is maximal monotone then Π(M) is a single point on the diagonal of the first quadrant of ℝ2. On the other hand, we give an example (for E the nonreflexive space L1[0,1]) of a maximal monotone subset M of E×E* such that (0,1)∈Π(M) and (1,1)∈Π(M) but (1,0)∉Π(M). We show that the results for reflexive spaces can be recovered for general Banach spaces by using monotone operator of type ‘(NI)’ — a class of multifunctions from E into E* which includes the subdifferentials of all proper, convex, lower semicontinuous functions on E, all surjective operators and, if E is reflexive, all maximal monotone operators. Our results lead to a simple proof of Rockafellar's result that if E is reflexive and S is maximal monotone on E then S+J is surjective. Our main tool is a classical minimax theorem.