On the general form of bimonotone operators
Nicolas Hadjisavvas
TL;DR
The paper addresses the general form of bimonotone operators in normed spaces without assuming paramonotonicity. It introduces the reduction $\hat{T}$ on $V=[\operatorname{dom}T']$ via a translated operator $T'(x)=T(x+u)-u^{*}$ and proves that bimonotonicity is equivalent to the existence of a single-valued skew-symmetric $A:V\to V^{*}$ with $\hat{T}(x)=A x$, with the converse also true. When $\operatorname{dom}T$ is dense, the operator takes the affine form $T(x)=A x+v^{*}$ on $\operatorname{dom}T$, extending to the Banach setting where $A$ extends to $\overline{V}$ iff it is continuous. These results generalize the finite-dimensional characterization for single-valued operators on $\mathbb{R}^{n}$ to the broader multivalued setting and have implications for convex feasibility problems and translations of disjoint convex sets.
Abstract
In a recent paper (2024) Camacho, Cánovas, Martínez-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.