Narrow orthogonally additive operators (Q2253913)

The aim of this paper is to study narrow operators in the context of nonlinear maps on vector lattices. Recall that given a vector lattice \(E\) and a Banach space \(X\), a map \(f:E\rightarrow X\) is called \textit{narrow} provided that for every \(x\in E_+\) and every \(\varepsilon>0\), there is \(y\in E\) with \(|y|=x\) such that \(\|f(y)\|<\varepsilon\). The results extend those developed for linear operators in [\textit{O. V. Maslyuchenko} et al., Positivity 13, No. 3, 459--495 (2009; Zbl 1183.47033)] to orthogonally additive operators and, in particular, abstract Uryson operators.