# Narrow orthogonally additive operators

Narrow orthogonally additive operators 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 $${\mathcal U}_{on}^{lc}(E,F)$$ U o n l c ( 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 $$E$$ E with the principal projection property to a Dedekind complete vector lattice $$F$$ F . The band generated by the disjointness preserving laterally continuous abstract Uryson operators is the orthogonal complement to $${\mathcal U}_n^{lc}(E,F)$$ U n l c ( E , F ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Narrow orthogonally additive operators

, Volume 18 (4) – Dec 15, 2013
27 pages

Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-013-0268-y
Publisher site
See Article on Publisher Site

### 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 $${\mathcal U}_{on}^{lc}(E,F)$$ U o n l c ( 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 $$E$$ E with the principal projection property to a Dedekind complete vector lattice $$F$$ F . The band generated by the disjointness preserving laterally continuous abstract Uryson operators is the orthogonal complement to $${\mathcal U}_n^{lc}(E,F)$$ U n l c ( E , F ) .

### Journal

PositivitySpringer Journals

Published: Dec 15, 2013

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