New results on q-positivity

TL;DR

This paper explores the concept of q-positivity in symmetrically self-dual spaces, generalizing classical monotonicity, and provides new characterizations and examples of maximal q-positive sets.

Contribution

It introduces the notion of q-positivity, extends the theory of maximal monotonicity, and offers new characterizations and examples in symmetrically self-dual Banach spaces.

Findings

Generalization of monotonicity via q-positivity

New characterizations of maximal q-positivity

Two novel examples of q-positive sets

Abstract

In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q-positive, where q is the quadratic form induced by the original bilinear form. The notion of q-positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss concepts generalizing the representations of monotone sets by convex functions, as well as the number of maximally q-positive extensions of a q-positive set. We also discuss symmetrically self-dual Banach spaces, in which we add a Banach space structure, giving new characterizations of maximal q-positivity. The paper finishes with two new examples.