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Common Cesàro hypercyclic vectors

100%
Costakis G.
Studia Mathematica
|
2010
|
tom 201
|
nr 3
203-226
EN
In this work, which can be seen as a continuation of a paper by Hadjiloucas and the author [Studia Math. 175 (2006)], we establish the existence of common Cesàro hypercyclic vectors for the following classes of operators: (i) multiples of the backward shift, (ii) translation operators and (iii) weighted differential operators. In order to do so, we first prove a version of Ansari's theorem for operators that are hypercyclic and Cesàro hypercyclic simultaneously; then our argument essentially relies on Baire's category theorem. In addition, the minimality of the irrational rotation, Runge's approximation theorem and a common hypercyclicity-universality criterion established by Sambarino and the author [Adv. Math. 182 (2004)], play an important role in the proofs.
2
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Somewhere dense Cesàro orbits and rotations of Cesàro hypercyclic operators

63%
Costakis G., Hadjiloucas D.
Studia Mathematica
|
2006
|
tom 175
|
nr 3
249-269
EN
Let T be a continuous linear operator acting on a Banach space X. We examine whether certain fundamental results for hypercyclic operators are still valid in the Cesàro hypercyclicity setting. In particular, in connection with the somewhere dense orbit theorem of Bourdon and Feldman, we show that if for some vector x ∈ X the set {Tx,T²/2 x,T³/3 x, ... } is somewhere dense then for every 0 < ε < 1 the set (0,ε){Tx,T²/2 x,T³/3 x,...} is dense in X. Inspired by a result of Feldman, we also prove that if the sequence ${n^{-1}Tⁿx}$ is d-dense then the operator T is Cesàro hypercyclic. Finally, following the work of León-Saavedra and Müller, we consider rotations of Cesàro hypercyclic operators and we establish that in certain cases, for any λ with |λ | = 1, T and λT share the same sets of Cesàro hypercyclic vectors.
3
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Common hypercyclic entire functions for multiples of differential operators

63%
Costakis G., Mavroudis P.
Colloquium Mathematicum
|
2008
|
tom 111
|
nr 2
199-203
EN
The purpose of the present note is to prove the existence of common hypercyclic entire functions for the family of differential operators {λp(D): λ ∈ ℂ ∖ {0}}$, where D is the differentiation operator acting on the space of entire functions and p a non-constant polynomial.
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