Existence and nonexistence of hypercyclic semigroups

In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinite-dimensional Banach space that is very different from - and considerably shorter than - the one recently given by Bermudez, Bonilla and Martinon. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite-dimensional Frechet space. This complements recent results due to Bes and Chan. Moreover, we discuss the Hypercyclicity Criterion for semigroups and we give an example of a separable infinite-dimensional locally convex space which supports no supercyclic strongly continuous semigroup of operators.