# Maximally monotone operators with ranges whose closures are not convex and an answer to a recent question by Stephen Simons

## TL;DR

This paper investigates the convexity properties of ranges of certain maximally monotone operators, providing a negative answer to a question about the scalar value 4 and extending results to the Fitzpatrick-Phelps operator.

## Contribution

It proves that the range closure of the sum of the Gossez operator and a multiple of the duality map is nonconvex for all scalars greater than or equal to 4, answering a recent open question.

## Key findings

- Range closure is nonconvex for scalar ≥ 4.
- Results apply to the Fitzpatrick-Phelps integral operator.
- Provides an abstract framework for analyzing such operators.

## Abstract

In his recent Proceedings of the AMS paper "Gossez's skew linear map and its pathological maximally monotone multifunctions", Stephen Simons proved that the closure of the range of the sum of the Gossez operator and a multiple of the duality map is nonconvex whenever the scalar is between 0 and 4. The problem of the convexity of that range when the scalar is equal to 4 was explicitly stated. In this paper, we answer this question in the negative for any scalar greater than or equal to 4. We derive this result from an abstract framework that allows us to also obtain a corresponding result for the Fitzpatrick-Phelps integral operator.