Representation theorems for regular operators

Abstract

We elaborate, strengthen, and generalize known representation theorems by different authors for regular operators on vector and Banach lattices. Our main result asserts, in particular, that every regular linear operator T acting from a vector lattice E with the principal projection property to a Dedekind complete vector lattice F, which is an ideal of some order continuous Banach lattice G, admits a unique representation , where is the sum of an absolutely order summable family of disjointness preserving operators and is an order narrow (= diffuse) operator. Our main contribution is waiver of the order continuity assumption on T. In proofs, we use new techniques that allow obtaining more general results for a wider class of orthogonally additive operators, which has somewhat different order structure than the linear subspace of linear operators.

Diffuse orthogonally additive operators

“…Remark 2. Note that a particular case of Theorem 2, when F is an order ideal of a Banach lattice with order continuous norm, was established earlier by Pliev and Popov (see Theorem 2.6 in [37]).…”