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 ...

An important result of León-Saavedra and Müller says that the rotations of hypercyclic operators remain hypercyclic. We provide extensions of this result for orbits of operators which are rotated by unimodular complex numbers with polynomial phases. On the other hand, we show that this fails for unimodular complex numbers whose phases grow to infinity too quickly, say at a geometric rate. A fur...

A continuous operator acting on a topological vector space X is called hypercyclic provided there exists a vector x ∈ X such that its orbit {T nx; n ≥ 1} is dense in X. Such a vector is called a hypercyclic vector for T . The set of hypercyclic vectors will be denoted by HC(T ). The first example of hypercyclic operator was given by Birkhoff, 1929 [3], who shows that the operator of translation...