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Narrow orthogonally additive operators

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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 \({\mathcal U}_n^{lc}(E,F)\).

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Notes

  1. \((C_{1})\) and \((C_{2})\) are called the Carathéodory conditions.

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Authors and Affiliations

  1. South Mathematical Institute, Russian Academy of Sciences, str. Markusa 22, Vladikavkaz, 362027, Russia

    Marat Pliev

  2. Department of Applied Mathematics, Chernivtsi National University, str. Kotsyubyns’koho 2, Chernivtsi, 58012, Ukraine

    Mikhail Popov

Authors
  1. Marat Pliev
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  2. Mikhail Popov
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Correspondence to Mikhail Popov.

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Pliev, M., Popov, M. Narrow orthogonally additive operators. Positivity 18, 641–667 (2014). https://doi.org/10.1007/s11117-013-0268-y

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  • DOI: https://doi.org/10.1007/s11117-013-0268-y

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