A Beginner-Friendly Note on Maximal Monotone Operators

TL;DR

This paper provides an accessible introduction to maximal monotone operators in Hilbert spaces, covering foundational concepts, key properties, and a fundamental theorem with practical implications for beginners.

Contribution

It offers a self-contained, beginner-friendly explanation of maximal monotone operators, including their properties and a core theorem, aimed at graduate students new to the topic.

Findings

Maximal monotone linear operators have dense domains and are closed.

The inverse of (I + λA) is non-expansive for all λ > 0.

The Banach fixed point theorem is used to analyze these operators.

Abstract

We give a self-contained and introductory account of some basic functional analytic tools needed to understand maximal monotone operators in Hilbert spaces. We review domains of (possibly unbounded) operators, closed sets and closed operators, and provide concrete examples of bounded and unbounded operators in both finite and infinite dimensions. We then explain in detail a fundamental result of Br\'ezis: if $A$ is a maximal monotone linear operator, then its domain is dense, $A$ is closed, and $(I+\lambda A)^{-1}$ is a non-expansive mapping for every $\lambda>0$. The Banach fixed point theorem (contraction mapping principle) is stated and used as a key ingredient in the analysis. The presentation is aimed at beginning graduate students and readers seeing these notions for the first time.