Norms of hypercyclic sequences

In this note, is proved that every member of a wide class of Banach spaces supports a sequence (Tn) of operators with a hereditary hypercyclic subsequence (Tnk) such that (Tn) itself does not satisfy the so-called Hypercyclicity Criterion and such that, in addition, the norms of the Tn's are controlled in a certain natural sense.