Five classes of monotone linear relations and operators

TL;DR

This paper explores the relationships among five classes of monotonicity for linear operators and relations in Hilbert spaces, providing examples and new results, especially in finite and infinite-dimensional settings.

Contribution

It systematically analyzes the connections between five monotonicity classes for linear operators and relations, including new results for linear relations.

Findings

Identified overlaps among the five classes of monotonicity.

Provided examples illustrating class relationships.

Extended some results to general Hilbert spaces.

Abstract

The relationships between five classes of monotonicity, namely 3^*-, 3-cyclic, strictly, para-, and maximal monotonicity, are explored for linear operators and linear relations in Hilbert space. Where classes overlap, examples are given; otherwise their relationships are noted for linear operators in $\mathcal{R}^2$, $\mathcal{R}^n$, and general Hilbert spaces. Along the way, some results for linear relations are obtained.