Characterizations of maximal monotone operators

Abstract

LET X BE A Banach space, A be a subset of X, and T: A + 2x* be a monotone operator. It is known that in the case of an open A, the following assertions are equivalent (see [l]): (1) T is maximal monotone; (2) T is convex and w*-compact valued, and w*-upper semicontinuous; (3) T is minimal (with respect to graph inclusion) among all convex and w*-compact valued, w *-upper semicontinuous multivalued maps defined on A. In [2, corollary I] we proved that (1) is equivalent to (2) for monotone operators defined on much more general sets (e.g. for relatively open or dense subsets of the set of nonsupport points of a convex set whose affine hull is X). When A has “support points”, T may be unbounded and the above statements are no longer equivalent. In [2] we were able to characterize maximal monotone operators in terms of their behavior at the support points and a certain upper semicontinuity property. This note contains improvements of some of our results in [2] and also extensions of some results obtained in [3]. Among others, we give, in a very general context, characterizations of the maximality of monotone operators, which, for open sets, reduce to those stated at the beginning of this introduction.