2013/09/21 by Marat Pliev, Pliev, Marat, M. M. Popov +2
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Primary 47H30 #Secondary 47H99 #math.FA #msc:47H30 #msc:47H99
paper · pdf · doi:10.48550/arxiv.1309.5487
arxiv created 2013/09/21 · openalex publication_date 2013/09/21 · arxiv updated 2013/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We extend 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 (2009), pp. 459--495, 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 Uonlc(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 Unlc(E,F).