Faces of quasidensity

Abstract: This paper is about the maximally monotone and quasidense subsets of the product of a real Banach space and its dual. We discuss eight subclasses of the maximally monotone sets that are equivalent to the quasidense ones. We define the Gossez extension to the dual of a maximally monotone set, and give seven equivalent characterizations of an element of this set in the quasidense case. We discuss maximally monotone sets of "type (NI)" (one of the eight classes referred to above) and we show that the "tail operator" is not of type (NI), but it is the Gossez extension of a maximally monotone set that is of type (NI). We generalize Rockafellar's surjectivity theorem for maximally monotone subsets of reflexive Banach spaces to maximally monotone subsets of type (NI) of general Banach spaces. We discuss a generalization of the Brezis-Browder theorem on monotone linear subspaces of reflexive spaces to the nonreflexive situation.