Weak disjointness of hypercyclic operators

TL;DR

This paper explores the weak disjointness property of hypercyclic operators, extending topological dynamics concepts to linear operators, and provides characterizations and examples for various classes of hypercyclic operators.

Contribution

It establishes an analogue of the Weiss-Akin-Glasner Theorem for linear dynamics, characterizing weak disjointness of mixing operators via Furstenberg families.

Findings

Characterization of weak disjointness for classes of mixing operators

Extension of Weiss-Akin-Glasner Theorem to linear operators

Examples distinguishing classes of hypercyclic operators

Abstract

We study the weak disjointness of hypercyclic operators to advance the classifications of hypercyclic operators. We establish an analogue of the Weiss-Akin-Glasner Theorem from topological dynamics within the framework of linear dynamics, which gives a characterization of the weak disjointness of each class of mixing operators with respect to a given Furstenberg family. The key ingredient is the analogues of Weiss-Akin-Glasner Lemma from topological dynamics, which gives a characterization of subsets of non-negative integers which can be realized by the return time sets of mixing operators with respect to a given Furstenberg family. We also provide several examples to distinguish some classes of hypercyclic operators and end with the characterization of the weak disjointness of backward shifts on Fr\'echet sequence spaces.