Transitive and hypercyclic operators on locally convex spaces

Solutions are provided to several questions concerning topologically transitive and hypercyclic continuous linear operators on Hausdorff locally convex spaces that are not Frechet spaces. Among others, the following results are presented. (1) There exist transitive operators on the space phi of all finite sequences endowed with the finest locally convex topology (it was already known that there is no hypercyclic operator on phi). (2) The space of all test functions for distributions, which is also a complete direct sum of Frechet spaces, admits hypercyclic operators. (3) Every separable infinite-dimensional Frechet space contains a dense hyperplane that admits no transitive operator.