Atomic operators in vector lattices

Abstract

In this paper we introduce a new class of operators on vector lattices. We say that a linear or nonlinear operator $T$ from a vector lattice $E$ to a vector lattice $F$ is atomic if there exists a Boolean homomorphism $\Phi$ from the Boolean algebra $\mathfrak{B}(E)$ of all order projections on $E$ to $\mathfrak{B}(F)$ such that $T\pi=\Phi(\pi)T$ for every order projection $\pi\in\mathfrak{B}(E)$. We show that the set of all atomic operators defined on a vector lattice $E$ with the principal projection property and taking values in a Dedekind complete vector lattice $F$, is a band in the vector lattice of all regular orthogonally additive operators from $E$ to $F$. We give the formula for the order projection onto this band, and we obtain an analytic representation for atomic operators between spaces of measurable functions. Finally, we consider the procedure of the extension of an atomic map from a lateral ideal to the whole space.