Pictures of monotone operators

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 ², which we call the picture of H, (H). In general, (H) may be empty, but (M) is nonempty if M is a nonempty monotone subset of E×E^*. If E is reflexive and M is maximal monotone then (M) is a single point on the diagonal of the first quadrant of ². 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)(M) and (1,1)(M) but (1,0)(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.