Faces of quasidensity

TL;DR

This paper explores the structure and properties of maximally monotone and quasidense sets in Banach spaces, introducing new subclasses, characterizations, and generalizations of classical theorems to broader contexts.

Contribution

It introduces six subclasses of maximally monotone sets, defines the Gossez extension, and generalizes key theorems to nonreflexive Banach spaces, expanding the theoretical framework.

Findings

Identified six subclasses of maximally monotone sets equivalent to quasidense sets.

Provided nine characterizations of elements in the Gossez extension in the quasidense case.

Generalized Rockafellar's surjectivity theorem to nonreflexive Banach spaces.

Abstract

This paper is about the maximally monotone and quasidense subsets of the product of a real Banach space and its dual. We discuss six subclasses of the maximal monotone sets that are equivalent to the quasidense ones. We define the Gossez extension to the dual of a maximally monotone set, and give nine equivalent characterizations of an element of this set in the quasidense case. We discuss maximally monotone sets of "type (NI)'' (one of the six 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. We also discuss briefly maximally monotone subsets of "type (D)'' and "type (WD)'' (two more of the six classes referred to above).