Domination Problem for Narrow Orthogonally Additive Operators

TL;DR

This paper extends the 'Up-and-down' theorem to positive abstract Uryson operators on vector lattices, showing that order narrowness is preserved under certain bounds, advancing the understanding of operator structure.

Contribution

It proves an analog of the 'Up-and-down' theorem for positive abstract Uryson operators and demonstrates order narrowness preservation under bounds, a novel result in operator theory.

Findings

Extended the 'Up-and-down' theorem to Uryson operators

Proved that order narrowness is preserved under bounds for these operators

Established new structural insights into positive operators on vector lattices

Abstract

The "Up-and-down" theorem which describes the structure of the Boolean algebra of fragments of a linear positive operator is the well known result of the operator theory. We prove an analog of this theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result we apply to prove that for an order narrow positive abstract Uryson operator $T$ from a vector lattice $E$ to a Dedekind complete vector lattice $F$, every abstract Uryson operator $S:E\to F$, such that $0\leq S\leq T$ is also order narrow.