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Hypercyclicity in omega


Author: Henrik Petersson
Journal: Proc. Amer. Math. Soc. 135 (2007), 1145-1149
MSC (2000): Primary 47A16
DOI: https://doi.org/10.1090/S0002-9939-06-08584-4
Published electronically: October 4, 2006
MathSciNet review: 2262918
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Abstract: A sequence $ \mathbb{T}=(T_n)$ of operators $ T_n :\mathscr{X}\to \mathscr{X}$ is said to be hypercyclic if there exists a vector $ x\in \mathcal X$, called hypercyclic for $ \mathbb{T}$, such that $ \{ T_n x : n\geq 0 \}$ is dense. A hypercyclic subspace for $ \mathbb{T}$ is a closed infinite-dimensional subspace of, except for zero, hypercyclic vectors. We prove that if $ \mathbb{T}$ is a sequence of operators on $ \omega $ that has a hypercyclic subspace, then there exist (i) a sequence $ (p_n)$ of one variable polynomials $ p_n $ such that $ (p_n (\xi))\in \omega $ is hypercyclic for every fixed $ \xi$ and (ii) an operator $ S:\omega \to \omega $ that maps nonzero vectors onto hypercyclic vectors for $ \mathbb{T}$.

We complement earlier work of several authors.


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Additional Information

Henrik Petersson
Affiliation: School of Mathematical Sciences, Chalmers/Göteborg University, SE-412 96 Göteborg, Sweden
Email: henripet@math.chalmers.se

DOI: https://doi.org/10.1090/S0002-9939-06-08584-4
Keywords: Hypercyclic subspace, omega, backward shift.
Received by editor(s): November 11, 2005
Published electronically: October 4, 2006
Additional Notes: This work was supported by the The Royal Swedish Academy of Sciences.
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society

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