Perturbations of hypercyclic vectors

Feldman, Nathan S.

doi:10.1016/s0022-247x(02)00207-x

Cited by 22 publications

(26 citation statements)

References 5 publications

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“…Observe that under the hypothesis of Corollary 3.9 it is not true in general that Orb(T, x + y) = X (see [15]). In Proposition 3.8 one can also replace the sequence {m −1 } by the interval (0, 1) and get a similar result.…”

Section: Using (1) We Havementioning

confidence: 99%

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Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators

Costakis

Hadjiloucas

2006

Studia Math.

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Section: Using (1) We Havementioning

confidence: 99%

“…Take x to be the vector constructed in the proof of Theorem 2.6 in [15]. Then the first coordinate of (2B) n x belongs to the set F = {z ∈ C : |z| ≥ 1} ∪ {0}.…”

Section: Using (1) We Havementioning

confidence: 99%

Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators

Costakis

Hadjiloucas

2006

Studia Math.

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“…is not necessarily dense; however, T is hypercyclic, so that there exists a dense orbit. This a result of N. Feldman [5]. (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].…”

Section: Introductionmentioning

confidence: 96%

Epsilon-hypercyclic operators on a Hilbert space

Bayart

2010

Proc. Amer. Math. Soc.

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“…-suppose that for some positive number d the orbit of x ∈ X meets every open ball B(y, d) of radius d. Then Orb(x, T ) is not necessarily dense in X, but T must be hypercyclic [6].…”

Section: Introductionmentioning

confidence: 99%

“…We investigate in this paper a weaker version of Feldman's result [6] already mentioned above: it states that if given a positive ε there exists a vector x such that for every y ∈ X ||T n x − y|| ≤ ε for some integer n, then T is hypercyclic.…”

Section: Introductionmentioning

confidence: 99%

Epsilon-hypercyclic operators

Badea¹,

Grivaux²,

Müller³

2009

Ergod. Th. Dynam. Sys.

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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:

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Perturbations of hypercyclic vectors

Cited by 22 publications

References 5 publications

Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators

Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators

Epsilon-hypercyclic operators on a Hilbert space

Epsilon-hypercyclic operators

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