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...

We study multiply recurrent and hypercyclic operators as a special case of $\mathcal F$-hypercyclicity, where F$ is the family subsets natural numbers containing arbitrarily long arithmetic progressions. prove several properties operators, we characterize those which are weakly mixing recurrent, show that there but not mixing.

The backward shift B on the Bergman space of the unit disc is known to be hypercyclic (meaning: it has a dense orbit). Here we ask: “Which operators that commute with B inherit its hypercyclicity?” We show that the problem reduces to the study of operators of the form φ(B) where φ is a holomorphic self-map of the unit disc that multiplies the Dirichlet space into itself, and that the question o...

We study the frequency of hypercyclicity of hypercyclic, non–weakly mixing linear operators. In particular, we show that on the space `(N), any sublinear frequency can be realized by a non–weakly mixing operator. A weaker but similar result is obtained for c0(N) or `(N), 1 < p <∞. Part of our results is related to some Sidon-type lacunarity properties for sequences of natural numbers.