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 {Tj: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⊕T2⊕⋯⊕Tn is hypercyclic for every n.