Existence of disjoint weakly mixing operators that fail to satisfy the Disjoint Hypercyclicity Criterion

Recently, Bès, Martin, and Sanders [11] provided examples of disjoint hypercyclic operators which fail to satisfy the Disjoint Hypercyclicity Criterion. However, their operators also fail to be disjoint weakly mixing. We show that every separable, infinite dimensional Banach space admits operators T1,T2,…,TNT1,T2,…,TN with N⩾2N⩾2 which are disjoint weakly mixing, and still fail to satisfy the Disjoint Hypercyclicity Criterion, answering a question posed in [11]. Moreover, we provide examples of disjoint hypercyclic operators T1T1, T2T2 whose corresponding set of disjoint hypercyclic vectors is nowhere dense, answering another question posed in [11]. In fact, we explicitly describe their set of disjoint hypercyclic vectors. Those same disjoint hypercyclic operators fail to be disjoint topologically transitive. Lastly, we create examples of two families of d-hypercyclic operators which fail to have any d-hypercyclic vectors in common.