, Volume 22, Issue 1, pp 245–260 | Cite as

Finite elements in some vector lattices of nonlinear operators

  • M. A. Pliev
  • M. R. Weber


We study the collection of finite elements \(\Phi _{1}\big ({\mathcal {U}}(E,F)\big )\) 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 \(\varphi \in {\mathcal {U}}(E,{\mathbb {R}})\) there is only a finite set of mutually disjoint atoms, where \(\varphi \) 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 \(\Phi _{1}\big ({\mathcal {U}}({\mathbb {R}}^n,{\mathbb {R}}^m)\big )\) for \(n,m\in {\mathbb {N}}\) and consider rank one operators to be finite elements in \({\mathcal {U}}(E,F)\).


Finite elements Orthogonally additive order bounded operators Uryson operators Rank-one operators 

Mathematics Subject Classification

Primary 47H07 Secondary 47H99 



The authors thank the referee for valuable remarks and suggestions, which made the presentation shorter and more precise. M. A. Pliev was supported by the Russian Foundation of Fundamental Research, the Grant No. 17-51-12064 and by the Ministry of Education and Science of the Russian Federation (the Agreement No. 02.A03.21.0008).


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Southern Mathematical InstituteRussian Academy of SciencesVladikavkazRussia
  2. 2.RUDN UniversityMoscowRussia
  3. 3.Department of Mathematics, Institute of AnalysisTechnical University DresdenDresdenGermany

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