On band preserving orthogonally additive operators

In this paper, we investigate a new class of operators on vector lattices. We say that an orthogonally additive operator $T:E\to E$ on a vector lattice $E$ is band preserving if $T(D)\subset \{D\}^{\perp\perp}$ for every subset $D$ of $E$. We show that the set of all band preserving operators on a Dedekind complete vector lattice $E$ is a band in the vector lattice of all regular orthogonally additive operators on $E$ which coincides with the band generated by the identity operator. We present a formula for the order projection onto this band and obtain an analytical representation for order continuous band preserving operators on the space of all measurable functions. Finally, we consider the procedure of extending a band preserving map from a lateral band to the whole space.