We show that the Hypercyclicity Criterion coincides with other existing hypercyclicity criteria and prove that a wide class of

We provide a reasonable sufficient condition for a countable
family of operators to have a common hypercyclic subspace. We also
extend a result of the third author and A. Montes, thereby obtaining a common hypercyclic subspace for certain countable families of compact perturbations of operators of norm no larger than one.