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 = g ₁ + g ₂ for f, g ₁, g ₂ positive and g ₁ and g ₂ disjoint implies there exist disjoint positive elements f ₁ and f ₂ each less than f with the property that Tf ₁ = g ₁ and Tf ₂ = g ₂. 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.