Extending Families of Disjoint Hypercyclic Operators

Abstract: In the present note, we solve two open questions posed by Salas in [H. Salas, The strong disjoint blow-up/collapse property, J. Funct. Spaces Appl., 2013, Article ID 146517, 6 pages] about disjoint hypercyclic operators. First, we show that given any family $T_1, \dots, T_N$ of disjoint hypercyclic operators, one can always select an operator $T$ such that the extended family $T_1, \dots, T_N, T$ of operators remains disjoint hypercyclic. In fact, we prove that the set of operators $T$ which can extend the family of disjoint hypercyclic operators is dense in the strong operator topology in the algebra of bounded operators. Second, we show the existence of two disjoint weakly mixing operators that fail to possess a dense d-hypercyclic manifold. Thus, these operators satisfy the Disjoint Blow-up/Collapse property but fails to satisfy the Strong Disjoint Blow-up/Collapse property, a notion which was introduced by Salas as a sufficient condition for having a dense linear manifold of disjoint hypercyclic vectors.