On the general form of bimonotone operators

TL;DR

This paper characterizes the most general form of bimonotone operators in Banach spaces, showing they can be reduced to skew symmetric linear operators, which aids in understanding their applications.

Contribution

It extends the theory of bimonotone operators by removing paramonotonicity assumptions and identifying their fundamental structure as skew symmetric linear operators.

Findings

Bimonotone operators can be reduced to skew symmetric linear operators.

The general form of bimonotone operators in Banach spaces is characterized.

Facilitates applications involving these operators.

Abstract

In a recent paper (2024) Camacho, C\'{a}novas, Mart\'{\i}nez-Legaz and Parra introduced bimonotone operators, i.e., operators $T$ such that both $T$ and $-T$ are monotone, and found some interesting applications to convex feasibility problems, especially in the case the operator is also paramonotone. In the present paper we drop paramonotonicity and examine the question of finding the most general form of a bimonotone operator in a Banach space. We show that any such operator can be reduced in some sense to a single-valued, skew symmetric linear operator. This facilitates the proof of some results involving these operators in applications.