Order properties of nonlinear superposition operators

This article is devoted to orthogonally additive operators in vector lattices. Orthogonally additive operators acting between vector lattices were introduced and studied in 1990 by Mazón and Segura de León. Today the theory of orthogonally additive operators is an active area of Functional Analysis. Let $E$ be a vector lattice and $F$ a real linear space. We say that an operator $T:E\rightarrow F$ is called orthogonally additive if $T(x+y)=T(x)+T(y)$ whenever $x, y\in E$ are disjoint. It follows from the definition that $T(0)=0$. It is immediate that the set of all orthogonally additive operators is a real vector space with respect to the natural linear operations. Let $E$ and $F$ be vector lattices. We say that an orthogonally additive operator $T:E\rightarrow F$ is order bounded if $T$ maps order bounded sets in $E$ to order bounded sets in $F$. The aim of this notes is to continue this line of investigation. In this paper, we prove some new results for abstract Nemytskii operators, an important subclass of orthogonally additive operators. We say that an orthogonally additive operator $T:E\to E$ defined on a vector lattice $E$ is called an abstract Nemytskii operator if the equality $T\pi=\pi T$ holds for any order projection on $E$. We've showws that any abstract Nemytskii operator $T:E\to E$ defined on a vector lattice with the principal projection property $E$ has a module. We've proved that the set of all abstract Nemytskii operators defined on a Dedekind complete vector lattice $E$ is a band in the vector lattice of all order bounded orthogonally additive operators acted on $E$. We've got the formula for the order projection onto this band.