Hypercyclic operators on countably dimensional spaces

Abstract: According to Grivaux, the group of invertible linear operators on a separable infinite dimensional Banach space acts transitively on the set of countable dense linearly independent subsets of . As a consequence, each is an orbit of a hypercyclic operator on . Furthermore, every countably dimensional normed space supports a hypercyclic operator. Recently Albanese extended this result to Fréchet spaces supporting a continuous norm. We show that for a separable infinite dimensional Fréchet space , acts transitively on if and only if possesses a continuous norm. We also prove that every countably dimensional metrizable locally convex space supports a hypercyclic operator. [Copyright &y& Elsevier]