Gossez's skew linear map and its pathological maximally monotone multifunctions

TL;DR

This paper generalizes Gossez's example of a maximally monotone multifunction with a non-convex range closure, introduces new properties of Gossez's skew linear operator, and connects these findings with Stone-Cech compactification techniques.

Contribution

It provides a more elementary generalization of Gossez's example and explores new properties of the associated skew linear operator and its adjoint.

Findings

Range closure of the multifunction is not convex

New properties of Gossez's skew linear operator identified

Connections made with Stone-Cech compactification methods

Abstract

In this note, we give a generalization of Gossez's example of a maximally monotone multifunction such that the closure of its range is not convex, using more elementary techniques than in Gossez's original papers. We also discuss some new properties of Gossez's skew linear operator and its adjoint. While most of this paper uses elementary functional analysis, we correlate our results with those obtained by using the Stone-Cech compactification of the integers.