Weak disjointness 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.