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arXiv · 1508.03984

Finite elements in some vector lattices of nonlinear operators

M. A. Pliev·M. R. Weber
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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 $\mathcal{U}(E,F)$.

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M. A. Pliev, M. R. Weber. 2015-08-17. Finite elements in some vector lattices of nonlinear operators. https://arxiv.org/abs/1508.03984

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