Order Unbounded Orthogonally Additive Operators in Vector Lattices

Abstract

In this paper, we introduce a new class of operators in vector lattices. We say that orthogonally additive operator T from vector lattice E to vector lattice F is laterally-to-order bounded if for any element x of E an operator T maps the set $$\mathcal {F}_{x}$$ of all fragments of x onto an order bounded subset of F. We get a lattice calculus of orthogonally additive laterally-to-order bounded operators defined on a vector lattice and taking values in a Dedekind complete vector lattice. It turns out that these operators, in general, are not order bounded. We investigate the band of laterally continuous orthogonally additive operators and obtain formulas for the order projection onto this band. We consider the procedure of the extension of an orthogonally additive operator from a lateral ideal to the whole space. Finally we obtain conditions on the integral representability for a laterally-to-order bounded orthogonally additive operator.