Disjoint hypercyclicity, Sidon sets and weakly mixing operators

Abstract

We prove that a finite set of natural numbers J satisfies that J ∪ {0} is not Sidon if and only if for any operator T , the disjoint hypercyclicity of {T j : j ∈ J } implies that T is weakly mixing. As an application we show the existence of a non weakly mixing operator T such that T ⊕ T 2 ... ⊕ T n is hypercyclic for every n.