Epsilon-hypercyclic operators

Abstract

Abstract. For each fixed number ε in (0, 1) we construct a bounded linear operator on the Banach space 1 having a certain orbit which intersects every cone of aperture ε, but with every orbit avoiding a certain ball of radius d, for every d > 0. This answers a question from [8]. On the other hand, if T is an operator on the Banach space X such that for every ε > 0 there is a point in X whose orbit under the action of T meets every cone of aperture ε, then T has a dense orbit. IntroductionThe aim of this paper is to study some (variations of) density properties of orbits of bounded linear operators acting on a real or complex separable Banach space X. Using a Functional Analysis terminology, an operator T ∈ B(X) is said to be hypercyclic if there exists a vector x ∈ X such that the orbit Orb(x, T ) = {T n x ; n ≥ 0} of x under the action of T is dense in X. A vector x with dense orbit is called a hypercyclic vector for T .While the first examples of Banach and Hilbert space hypercyclic operators are relatively recent ([10]), there is now an important literature on hypercyclicity properties and the dynamics of bounded linear operators. We refer the reader to the recent book [1] for more on this topic. It is natural in this context to investigate which properties of the orbit of a vector, weaker than denseness, imply either that the orbit itself is in fact dense, or that the operator is hypercyclic (i.e. some other orbit is dense in X). Let us mention here some of the results in this direction:

Epsilon-hypercyclic operators on a Hilbert space

“…They are just useful inside the inductive process. Observe also that the deduction of Theorem 1.3 from properties (Px) is completely similar to the process followed in [1].…”

“…(3) Even if T admits a weakly dense orbit, T does not need to be hypercyclic; examples are given in [4], [6] and in [2]. In a recent paper [1], C. Badea, S. Grivaux and V. Müller have investigated a weaker version of Feldman's result: Definition 1.1. Let ε ∈ (0, 1).…”

“…Indeed this space often plays an extremal role in linear dynamics (see for instance Chapters 2, 4 and 12 of [2]). In [1], the authors asked for the existence of an ε-hypercyclic operator, yet not hypercyclic, on the separable Hilbert space. The aim of this paper is to answer this question positively.…”

“…The rest of this paper will be devoted to the proof of Theorem 1.3. It follows the lines of [1], with several modifications due to the Hilbertian nature of the example. However, because of the technical difficulties of these proofs, we provide a completely self-contained argument.…”

“…However, because of the technical difficulties of these proofs, we provide a completely self-contained argument. See Remark 5.1 for an overview of the differences with [1].…”