The Dodds-Fremlin type theorem for abstract Uryson operators

TL;DR

This paper extends the Dodds-Fremlin theorem to abstract Uryson operators in vector lattices, demonstrating that domination preserves AM-compactness for positive operators between Banach lattices.

Contribution

It proves a Dodds-Fremlin type theorem for AM-compact abstract Uryson operators, establishing that domination preserves AM-compactness in this context.

Findings

Domination preserves AM-compactness for positive abstract Uryson operators.

The theorem applies to operators between Banach lattices with specific properties.

Extension of classical results to nonlinear, orthogonally additive operators.

Abstract

We continue the investigation of abstract Uryson operators in vector lattices. Using the recently proved Up-and-down theorem for order bounded, orthogonally additive operators, we consider the domination problem for AM-compact abstract Uryson operators. We obtain the Dodds-Fremlin type theorem and prove that for an AM- compact positive abstract Uryson operator T from a Banach lattice E to a order continuous Banach lattice F, every abstract Uryson operator S from E to F included between 0 and T is also AM-compact.