In these notes we provide a new proof of the existence of a hypercyclic uniformly continuous semigroup of operators on any separable infinitedimensional Banach space that is very different from—and considerably shorter than—the one recently given by Bermúdez, Bonilla and Martinón. We also show the existence of a strongly dense family of topologically mixing operators on every separable infinite...

In this paper we characterize hypercyclic translation operators on $$B_{0}(\mathcal {H})$$ , the space of all compact linear a Hilbert $$\mathcal {H}$$ . Also, give some sufficient condition for related cosine operator function to be chaotic or topologically transitive.

Motivated by a recent investigation of Costakis et al. on the notion recurrence in linear dynamics, we study various stronger forms for operators, particular that frequent recurrence. We study, among other things, relationship between type and corresponding hypercyclicity, influence power boundedness, interplay spectral properties. obtain, particular, Ansari- L\'eon-M\"uller-type theorems $\mat...