Common frequent hypercyclicity

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