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Faces of quasidensity

Stephen Simons

TL;DR

The paper develops a unified, operator-theoretic framework for maximal monotone and quasidense subsets of $B=E\times E^*$ by introducing and linking dual constructs $F_M$, $G_M$, $P_M$, $~q$, $~L$, and $~r$ with their primal counterparts. It proves that CMQd sets are maximally monotone and introduces the Gossez extension $M^\sharp$, establishing nine equivalent characterizations and extending Rockafellar-type surjectivity results to nonreflexive spaces, alongside a nonreflexive Brezis–Browder-type generalization. The work also develops limiting results via biconjugates and nets, clarifying the relationships between type (NI), type (D), and type (WD) across linear subspaces and general monotone sets. Overall, the study provides a comprehensive, interlocked set of criteria and constructions that systematize quasidensity, monotonicity, and dual extensions in both reflexive and nonreflexive settings, with concrete examples such as the tail operator illustrating the boundaries of these classes.

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).

Faces of quasidensity

TL;DR

The paper develops a unified, operator-theoretic framework for maximal monotone and quasidense subsets of by introducing and linking dual constructs , , , , , and with their primal counterparts. It proves that CMQd sets are maximally monotone and introduces the Gossez extension , establishing nine equivalent characterizations and extending Rockafellar-type surjectivity results to nonreflexive spaces, alongside a nonreflexive Brezis–Browder-type generalization. The work also develops limiting results via biconjugates and nets, clarifying the relationships between type (NI), type (D), and type (WD) across linear subspaces and general monotone sets. Overall, the study provides a comprehensive, interlocked set of criteria and constructions that systematize quasidensity, monotonicity, and dual extensions in both reflexive and nonreflexive settings, with concrete examples such as the tail operator illustrating the boundaries of these classes.

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).
Paper Structure (13 sections, 32 theorems, 110 equations)

This paper contains 13 sections, 32 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Banach space notation
  3. $q$, $L$ and $r$
  4. Monotone, Quasidense and CMQd sets
  5. The function $P_M\colon B \to [0,\infty]$
  6. $\widetilde{q}$, ${\widetilde{L}}$ and $\widetilde{r}$
  7. The function $F_M\colon B^* \to\ ]-\infty,\infty]$
  8. The function $G_M\colon B^* \to\ ]-\infty,\infty]$
  9. Type (NI)
  10. Linear subspaces and their adjoints
  11. Two sets of equivalences
  12. Limiting results for certain biconjugates
  13. Type CMQd, type (D) and type (WD)

Key Result

Theorem 2.2

Let $f \in {\cal PCLSC}(X)$ and $x \in X$. Then $f^{**}(\widehat{x}) = f(x)$, that is to say, $f(x) = \sup\nolimits_{x^* \in X^*}[\langle x,x^*\rangle - f^*(x^*)]$.

Theorems & Definitions (88)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Definition 4.1
  • Theorem 4.2: Quasidensity and maximality
  • proof
  • Remark 4.3
  • Lemma 5.1
  • proof
  • Theorem 5.2
  • ...and 78 more