J-class Weighted Shifts on the Space of Bounded Sequences of Complex Numbers

During the last years the dynamics of linear operators on infinite dimensional spaces has been extensively studied, see the survey articles [3], [6], [7], [8], [9], [11]. Let us recall the notion of hypercyclicity. Let X be a separable Banach space and T : X → X be a bounded linear operator. The operator T is said to be hypercyclic provided there exists a vector x ∈ X such that its orbit under T , Orb(T, x) = {T x : n = 0, 1, 2, . . .}, is dense in X. If X is Banach space (possibly nonseparable) and T : X → X is a bounded linear operator then T is called topologically transitive (topologically mixing) if for every pair of non-empty open subsets U, V of X there exists a positive integer n such that T U ∩ V 6= ∅ (TU ∩ V 6= ∅ for every m ≥ n). It is well known, and easy to prove, that if T is a bounded linear operator acting on separable Banach space X then T is hypercyclic if and only if T is topologically transitive. A first step to understand the dynamics of linear operators is to look at particular operators as for example the weighted shifts. Salas [10] was the first who characterized the hypercyclic weighted shifts in terms of their weight sequences. We would like to point out that l(N) and l(Z) do not support hypercyclic operators since they are not separable Banach spaces. In fact they do not support topologically transitive