Common frequent hypercyclicity

TL;DR

This paper develops new criteria for families of operators to share common frequently hypercyclic vectors, extending classical results and applying to various operator classes, with implications for weighted shifts and $ extit{ extbf{α}}$-frequent hypercyclicity.

Contribution

It introduces variants of the Frequent Hypercyclicity Criterion for families of operators and characterizes common frequent hypercyclic vectors for specific operator classes, including C-type operators.

Findings

Criteria for common frequent hypercyclicity of operator families

Necessary and sufficient conditions for scalar multiples of an operator to share a hypercyclic vector

Existence of frequent hypercyclic weighted shifts without common vectors

Abstract

We provide with criteria for a family of sequences of operators to share a frequently universal vector. These criteria are variants of the classical Frequent Hypercyclicity Criterion and of a recent criterion due to Grivaux, Matheron and Menet where periodic points play the central role. As an application, we obtain for any operator T in a specific class of operators acting on a separable Banach space, a necessary and sufficient condition on a subset $\Lambda$ of the complex plane for the family {$\lambda$T : $\lambda$ $\in$ $\Lambda$} to have a common frequently hypercyclic vector. In passing, this permits us to easily exhibit frequent hypercyclic weighted shifts which do not possess common frequent hypercyclic vectors. We also provide with criteria for families of the recently introduced operators of C-type to share a common frequently hypercyclic vector. Further, we prove that the same problem of common $\alpha$-frequent hypercyclicity may be vacuous, where the notion of $\alpha$-frequent hypercyclicity extends that of frequent hypercyclicity replacing the natural density by more general weighted densities. Finally, it is already known that any operator satisfying the classical Frequent Universality Criterion is $\alpha$-frequently universal for any sequence $\alpha$ satisfying a suitable condition. We complement this result by showing that for any such operator, there exists a vector x which is $\alpha$-frequently universal for T , with respect to all such $\alpha$.