In the present paper, transitivity of a linear operators acting on a reflexive Banach space with the weak topology is investigated. We show any bounded operator, transitive on an open bounded subset of $X$ with the weak topology, is weakly hypercyclic.
"On weakly transitive operators." Proc. Japan Acad. Ser. A Math. Sci. 87 (5) 88 - 90, May 2011. https://doi.org/10.3792/pjaa.87.88