Dual disjoint hypercyclic operators

Let E be a separable Fréchet space. The operators T 1 , … , T m are disjoint hypercyclic if there exists x ∈ E such that the orbit of ( x , … , x ) under ( T 1 , … , T m ) 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 T 1 ∗ , … , T m ∗ are also disjoint hypercyclic.