D. Herrero conjectured in 1991 that a finitely-hypercyclic operator was actually hypercyclic. In 1999 Alfredo Peris proved Herrero's conjecture. Several people asked if there was a countable version of Herrero's conjecture . In this paper we propose a reasonable definition of a countably hypercyclic operator, and establish a Countable Hypercyclicity Criterion very similar to the Hypercyclicity Criterion. Although there are countably hypercyclic operators that are not cyclic, there is good evidence that the following version of Herrero's conjecture may be true.
T is countably hypercyclic if there is a bounded separated sequence with dense orbit.
A sequence is separated if the distance between any two of its terms is uniformly bounded away from zero. Among other things we prove the following theorem. If T is a backward weighted shift and T is countably hypercyclic, then T is hypercyclic. The question for bilateral weighted shifts is still open as well as the general question. The Countably Hypercyclic Criterion says if there are two subspaces Y and Z such that Y is infinite dimensional and Z is dense and forward orbits go to zero on Y and on Z backward orbits go to zero, then our operator is countably hypercyclic.