Finite elements in some vector lattices of nonlinear operators

TL;DR

This paper investigates finite elements within the vector lattice of orthogonally additive, order bounded operators, revealing structural properties related to atoms and continuity in various vector lattices.

Contribution

It characterizes finite elements in the lattice of abstract Uryson operators, especially relating to atomic and atomless vector lattices, and describes the structure of finite elements for finite-dimensional cases.

Findings

Finite elements in atomic lattices are supported on finitely many atoms.

In atomless lattices, only the zero element is finite.

The structure of finite elements in finite-dimensional operator spaces is explicitly described.

Abstract

We study the collection of finite elements $\Phi_{1}(\mathcal{U}(E,F))$ in the vector lattice $\mathcal{U}(E,F)$ of orthogonally additive, order bounded (called abstract Uryson) operators between two vector lattices $E$ and $F$, where $F$ is Dedekind complete. In particular, for an atomic vector lattice $E$ it is proved that for a finite element in $\phi\in \mathcal{U}(E,\mathbb{R})$ there is only a finite set of mutually disjoint atoms, where $\phi$ does not vanish and, for an atomless vector lattice the zero-vector is the only finite element in the band of $\sigma$-laterally continuous abstract Uryson functionals. We also describe the ideal $\Phi_{1}(\mathcal{U}(\mathbb{R}^n,\mathbb{R}^m))$ for $n,m\in\mathbb{N}$ and consider rank one operators to be finite elements in $\mathcal{U}(E,F)$.