Common frequent hypercyclicity

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 $\alpha$.