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Positivity

, Volume 18, Issue 4, pp 641–667 | Cite as

Narrow orthogonally additive operators

  • Marat Pliev
  • Mikhail PopovEmail author
Article

Abstract

We generalize the notion of narrow operators to nonlinear maps on vector lattices. The main objects are orthogonally additive operators and, in particular, abstract Uryson operators. Most of the results extend known theorems obtained by O. Maslyuchenko, V. Mykhaylyuk and the second named author published in Positivity 13:459–495 (2009) for linear operators. For instance, we prove that every orthogonally additive laterally-to-norm continuous C-compact operator from an atomless Dedekind complete vector lattice to a Banach space is narrow. Another result asserts that the set \({\mathcal U}_{on}^{lc}(E,F)\) of all order narrow laterally continuous abstract Uryson operators is a band in the vector lattice of all laterally continuous abstract Uryson operators from an atomless vector lattice \(E\) with the principal projection property to a Dedekind complete vector lattice \(F\). The band generated by the disjointness preserving laterally continuous abstract Uryson operators is the orthogonal complement to \({\mathcal U}_n^{lc}(E,F)\).

Keywords

Narrow operators C-compact operators Orthogonally additive operators Abstract Uryson operators Banach lattices 

Mathematics Subject Classification (1991)

Primary 47H30 Secondary 47H99 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.South Mathematical InstituteRussian Academy of SciencesVladikavkazRussia
  2. 2.Department of Applied MathematicsChernivtsi National UniversityChernivtsiUkraine

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