Hypercyclic Operators on Non-normable Fréchet Spaces

Every infinite dimensional separable non-normable Fréchet space admits a continuous hypercyclic operator. A large class of separable countable inductive limits of Banach spaces with the same property is given, but an example of a separable complete inductive limit of Banach spaces which admits no hypercyclic operator is provided. It is also proved that no compact operator on a locally convex space is hypercyclic.