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Disjointly non-singular operators and various topologies on Banach lattices

Eugene Bilokopytov

TL;DR

This work develops a topological framework for dispersed subspaces and disjointly non-singular operators on Banach lattices, proving that on order continuous lattices DNS operators are tauberian and, in this setting, there are only finitely many such operators. It introduces and analyzes $n$-DNS and $n$-dispersed subspaces, and links these notions to stable phase retrieval (SPR); by employing derivative topologies (un, aw, aaw, uaw) and various convergences (uo, polar), the paper elucidates when DNS is equivalent to complementing specific topologies. The results also connect the structure of the order continuous dual with the behavior of the aw-topology, characterizing how duality, dispersion, and topologies govern DNS and dispersion phenomena. Overall, the article advances a cohesive, topology-driven theory of DNS operators and dispersed subspaces in Banach lattices, with implications for phase retrieval and tauberian analysis in operator theory.

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.

Disjointly non-singular operators and various topologies on Banach lattices

TL;DR

This work develops a topological framework for dispersed subspaces and disjointly non-singular operators on Banach lattices, proving that on order continuous lattices DNS operators are tauberian and, in this setting, there are only finitely many such operators. It introduces and analyzes -DNS and -dispersed subspaces, and links these notions to stable phase retrieval (SPR); by employing derivative topologies (un, aw, aaw, uaw) and various convergences (uo, polar), the paper elucidates when DNS is equivalent to complementing specific topologies. The results also connect the structure of the order continuous dual with the behavior of the aw-topology, characterizing how duality, dispersion, and topologies govern DNS and dispersion phenomena. Overall, the article advances a cohesive, topology-driven theory of DNS operators and dispersed subspaces in Banach lattices, with implications for phase retrieval and tauberian analysis in operator theory.

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 -DNS, for some . 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.
Paper Structure (8 sections, 28 theorems, 8 equations)

This paper contains 8 sections, 28 theorems, 8 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Dispersed subspaces and disjointly non-singular operators
  3. $n$-dispersed subspaces and $n$-DNS operators
  4. The topological approach
  5. The role of the absolute weak topology
  6. The role of various convergences
  7. The role of the Positive Schure Property
  8. Acknowledgements

Key Result

Proposition 1.1

For a semi-norm $\rho$, the following conditions are equivalent: $\rho$ complements $\tau$; There is $\delta>0$ such that $0_{F}$ is $\tau$-separated from $\left\{f\in \mathrm{S}_{F},~ \rho\left(f\right)\le\delta\right\}$; If a net is simultaneously $\tau$-null and $\rho$-null, then it is norm-null.

Theorems & Definitions (52)

  • Proposition 1.1: Part of Theorem 2.2, erz
  • proof
  • Proposition 1.2
  • proof
  • Corollary 1.3
  • proof
  • Theorem 2.1: Theorems 2.8 and 2.10, gmm
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 42 more