A Birkhoff type transitivity theorem for non-separable completely metrizable spaces with applications to Linear Dynamics

In this note we prove a Birkhoff type transitivity theorem for continuous maps acting on non-separable completely metrizable spaces and we give some applications for dynamics of bounded linear operators acting on complex Fr'{e}chet spaces. Among them we show that any positive power and any unimodular multiple of a topologically transitive linear operator is topologically transitive, generalizing similar results of S.I. Ansari and F. Le'{o}n-Saavedra V. M"{u}ller for hypercyclic operators.