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Narrow orthogonally additive operators. (English) Zbl 1331.47078

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 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 [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.

MSC:

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
46A40 Ordered topological linear spaces, vector lattices

Citations:

Zbl 1183.47033
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References:

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