Dual disjoint hypercyclic operators

Let E be a separable Fréchet space. The operators T1,…,Tm are disjoint hypercyclic if there exists x∈E such that the orbit of (x,…,x) under (T1,…,Tm) is dense in E×⋯×E. We show that every separable Banach space E admits an m-tuple of bounded linear operators which are disjoint hypercyclic. If, in addition, its dual E∗ is separable, then they can be constructed such that are also disjoint hypercyclic.