A characterization of positively decomposable non-linear maps between Banach lattices

A map between Banach lattices E and F is called positively decomposable if Tf = g1 + g2 for f, g1, g2 positive and g1 and g2 disjoint implies there exist disjoint positive elements f1 and f2 each less than f with the property that Tf1 = g1 and Tf2 = g2. Recently, the positive decomposability of linear Carleman operators on Banach lattices were characterized using disjointness condition of images of the approximate atoms. This note provides an extension of the characterization for a class of non-linear maps. Further, disjointness preserving maps are studied.