Narrow orthogonally additive operators

TL;DR

This paper generalizes the concept of narrow operators to nonlinear maps on vector lattices, focusing on orthogonally additive and Uryson operators, and establishes new properties and structural results for these classes.

Contribution

It extends known linear operator theorems to nonlinear orthogonally additive operators, introducing new classifications and structural insights.

Findings

Every orthogonally additive laterally-to-norm continuous C-compact operator from an atomless Dedekind complete vector lattice to a Banach space is narrow.

The set of all order narrow laterally continuous abstract Uryson operators forms a band in the lattice of all such operators.

The band generated by disjointness preserving operators is the orthogonal complement to the band of narrow operators.

Abstract

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 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 U_{n}^{lc}(E,F).