Hypercyclic and Cyclic Vectors

Let X denote an arbitrary separable Banach space over the field of complex numbers and B(X) the Banach algebra of all bounded linear operators on X. We prove the following results. (1) An element of the space X is hypercyclic (supercyclic) for all positive powers Tn of an operator T in B(X) if it is hypercyclic (supercyclic) for T. (2) Under some condition on the spectrum of the adjoint of a cyclic operator, the set of all cyclic vectors of the operator is dense. This result extends to any cyclic commutative subset of B(X). (3) Under a mild condition on the spectrum of a cyclic operator T the set of all separating vectors for the commutant {T}′ of T is dense. This also extends to any cyclic commutative subset of B(X). (4) A slightly stronger version of a theorem of K. F. Clancey and D. D. Rogers on cyclic vectors. Finally, we define and discuss hereditarily hypercyclic operators.