On a Lemma by Br\'ezis and Haraux

TL;DR

This paper revisits and expands upon a key lemma from Brézis and Haraux's 1976 work, offering new applications and insights into the properties of monotone operators.

Contribution

It introduces several new applications of a lesser-known part of the original theorem, unifying and extending previous results on monotone operator ranges.

Findings

Unified various existing results on monotone operators.

Extended the scope of the original Brézis-Haraux theorem.

Provided new applications and insights into the theorem's implications.

Abstract

We propose several applications of an often overlooked part of the 1976 paper by Br\'ezis and Haraux, in which the Br\'ezis--Haraux theorem was established. Our results unify and extend various existing ones on the range of a linearly composite monotone operator and provide new insight into their seminal paper.