Faces and Support Functions for the Values of Maximal Monotone Operators

TL;DR

This paper develops new representation formulas for faces and support functions of maximal monotone operators in specific Banach space settings, enhancing understanding of their structure and selection properties.

Contribution

It introduces novel formulas for faces and support functions of maximal monotone operators in Banach spaces with convex duals or nonempty interior domains, advancing their structural analysis.

Findings

Formulas for faces and support functions established

Operators are characterized by limit values of selections

Structural properties like decomposition are derived

Abstract

Representation formulas for faces and support functions of the values of maximal monotone operators are established in two cases: either the operators are defined on uniformly Banach spaces with uniformly convex duals, or their domains have nonempty interiors on reflexive real Banach spaces. Faces and support functions are characterized by the limit values of the minimal-norm selections of maximal monotone operators in the first case while in the second case they are represented by the limit values of any selection of maximal monotone operators. These obtained formulas are applied to study the structure of maximal monotone operators: the local unique determination from their minimal-norm selections, the local and global decompositions, and the unique determination on dense subsets of their domains.