Banach lattices of orthogonally additive operators

Abstract

Using new notions of consistent sets and levels in Riesz spaces, we study the relationship between linear and orthogonally additive operators on Banach lattices. We define a norm on the Riesz space U(E,F) of order bounded orthogonally additive operators acting between Banach lattices E,F and prove that if F is Dedekind complete then the set UB(E,F) of all bounded with respect to this norm elements of U(E,F) is a Dedekind complete Banach lattice which is a sublattice of U(E,F). One of the results asserts that if E is an AL-space, F a Banach lattice with a Fatou-Levi norm (in particular, a KB-space) then the Banach lattice L(E,F)=Lr(E,F) of all continuous linear (regular) operators from E to F is a 1-complemented subspace of UB(E,F). If, moreover, both E and F are separable then for every S∈L(E,F)+ there exists T∈UB(E,F) such that for every S1∈L(E,F) with |S1|≤S there is a contractive projection Φ of UB(E,F) onto L(E,F) with Φ(T)=S1. Note that, being a subspace of UB(E,F), the Banach lattice Lr(E,F) is not a sublattice of UB(E,F), because the order on Lr(E,F) differs from the order on UB(E,F).