Hereditarily Hypercyclic Operators

We show that a continuous linear operator T on a Fréchet space satisfies the so-called Hypercyclicity Criterion if and only if it is hereditarily hypercyclic, and if and only if the direct sum T⊕T is hypercyclic. In particular, hypercyclic operators with either a dense generalized kernel or a dense set of periodic points (i.e., chaotic in the sense of R. L. Devaney (1989, “An Introduction to Chaotic Dynamical Systems,” Addison–Wesley, Reading, MA)) must satisfy the Criterion. Finally, we provide a characterization of those weighted shifts T that are hereditarily hyper-cyclic with respect to a given sequence (nk) of positive integers, as well as conditions under which T and {Tnk}k⩾1 share the same set of hypercyclic vectors.