Disjointly non-singular operators and various topologies on Banach lattices

TL;DR

This paper explores the properties of disjointly non-singular operators on Banach lattices, providing new characterizations and connecting them with phase retrieval, using topological methods to deepen understanding of these operators.

Contribution

It offers a simplified proof that DNS operators in order continuous Banach lattices are n-DNS for some n, and characterizes Banach lattices with order continuous duals via dispersed subspaces.

Findings

Operators are DNS iff n-DNS for some n in order continuous Banach lattices.

Characterization of Banach lattices with order continuous duals.

Connection between DNS operators and phase retrieval in Banach lattices.

Abstract

We continue the study of dispersed subspaces and disjointly non-singular (DNS) operators on Banach lattices using topological methods. In particular, we provide a simple proof of the fact that in an order continuous Banach lattice an operator is DNS if and only if it is $n$-DNS, for some $n\in\mathbb{N}$. We characterize Banach lattices with order continuous dual in terms of dispersed subspaces and absolute weak topology. We also connect these topics with the recently launched study of phase retrieval in Banach lattices.