"Densities" and maximal monotonicity I

TL;DR

This paper introduces the concepts of L-positive sets, quasidensity, and their properties in Banach SN spaces, expanding the understanding of maximal monotonicity and providing calculus rules for quasidense multifunctions.

Contribution

It generalizes monotone operator theory by defining quasidensity, establishing calculus rules, and extending classical theorems to non-reflexive Banach spaces.

Findings

Quasidense monotone multifunctions obey a calculus rule.

Subdifferentials of convex functions are quasidense.

Generalizations of Brezis-Browder theorem to non-reflexive spaces.

Abstract

We discuss "Banach SN spaces", which include Hilbert spaces, negative Hilbert spaces, and the product of any real Banach space with its dual. We introduce "L-positive" sets, which generalize monotone multifunctions from a Banach space into its dual. We introduce the concepts of "$r_L$-density" and its specialization "quasidensity": the closed quasidense monotone multifunctions from a Banach space into its dual form a (generally) strict subset of the maximally monotone ones, though all surjective maximally monotone and all maximally monotone multifunctions on a reflexive space are quasidense. We give a sum theorem and a parallel sum theorem for closed monotone quasidense multifunctions under very general constraint conditions. That is to say, quasidensity obeys a very nice calculus rule. We give a short proof that the subdifferential of a proper convex lower semicontinuous function on a Banach space is quasidense, and deduce generalizations of the Brezis-Browder theorem on linear relations to non reflexive Banach spaces. We also prove that any closed monotone quasidense multifunction has a number of other very desirable properties. This version differs from the previous version in the removal of Sections 13-16, part of Section 17, and Section 18.