Existence of disjoint frequently hypercyclic operators which fail to be disjoint weakly mixing

TL;DR

This paper demonstrates the existence of disjoint frequently hypercyclic operators that are not disjoint weakly mixing, answering a question in operator theory and exploring the density of such operators in the bounded operator algebra.

Contribution

It provides a counterexample to the implication from disjoint hypercyclicity to disjoint weakly mixing and analyzes the density of these operators in the operator algebra.

Findings

Disjoint frequently hypercyclic operators can fail to be disjoint weakly mixing.

Operators with $T igoplus T$ frequently hypercyclic have a dense set of disjoint frequently hypercyclic partners.

The set of operators failing the Disjoint Hypercyclicity Criterion is SOT dense in the algebra of bounded operators.

Abstract

In this short note, we answer a question of Martin and Sanders [Integr. Equ. Oper. Theory, 85 (2) (2016), 191-220] by showing the existence of disjoint frequently hypercyclic operators which fail to be disjoint weakly mixing and, therefore, fail to satisfy the Disjoint Hypercyclicity Criterion. We also show that given an operator $T$ such that $T \oplus T$ is frequently hypercyclic, the set of operators $S$ such that $T, S$ are disjoint frequently hypercyclic but fail to satisfy the Disjoint Hypercyclicity Criterion is SOT dense in the algebra of bounded linear operators.