We study positive shadowing and chain recurrence in the context of linear operators acting on Banach spaces or even normed vector spaces. show that for there is only one recurrent set, this set a closed invariant subspace. prove every transitive dynamical system with property frequently hypercyclic and, as corollary, we obtain hypercyclic.

We study the “smallness” of the set of non-hypercyclic vectors for some classical hypercyclic operators.

We provide a reasonable sufficient condition for a family of operators to have a common hypercyclic subspace. We also extend a result of the third author and A. Montes [22], thereby obtaining a common hypercyclic subspace for certain countable families of compact perturbations of operators of norm no larger than one.

We show that, under suitable conditions, an operator acting like a shift on some sequence space has frequently hypercyclic random vector whose distribution is strongly mixing for the operator. This result will be applied to chaotic weighted shifts. also apply it every satisfying Frequent Hypercyclicity Criterion, recovering of Murillo and Peris.

Suppose that T is a bounded operator on a nonzero Banach space X . Given a vector x ∈ X , we say that x is hypercyclic for T if the orbit OrbTx = {T x}n is dense in X . Similarly, x is said to be weakly hypercyclic if OrbTx is weakly dense in X . A bounded operator is called hypercyclic or weakly hypercyclic if it has a hypercyclic or, respectively, a weakly hypercyclic vector. It is shown in [...

We prove that a continuous linear operator T on a topological vector space X with weak topology is mixing if and only if the dual operator T ′ has no finite dimensional invariant subspaces. This result implies the characterization of hypercyclic operators on the space ω due to Herzog and Lemmert and implies the result of Bayart and Matheron, who proved that for any hypercyclic operator T on ω, ...