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Spaces that admit hypercyclic operators with hypercyclic adjoints

Author: Henrik Petersson
Journal: Proc. Amer. Math. Soc. 134 (2006), 1671-1676
MSC (2000): Primary 47A15, 47A16, 47A05
Published electronically: December 14, 2005
MathSciNet review: 2204278
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Abstract: A continuous linear operator $ T:X\to X $ is hypercyclic if there is an $ x\in X$ such that the orbit $ \{ T^n x\}_{n\geq 0}$ is dense. A result of H. Salas shows that any infinite-dimensional separable Hilbert space admits a hypercyclic operator whose adjoint is also hypercyclic. It is a natural question to ask for what other spaces $ X$ does $ \mathcal{L}(X)$ contain such an operator. We prove that for any infinite-dimensional Banach space $ X$ with a shrinking symmetric basis, such as $ c_0$ and any $ \ell_p$ $ (1<p<\infty)$, there is an operator $ T:X \to X$, where both $ T$ and $ T':X'\to X'$ are hypercyclic.

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Additional Information

Henrik Petersson
Affiliation: School of Mathematical Sciences, Chalmers/Göteborg University, SE-412 96, Göteborg, Sweden

Keywords: Hypercyclic, adjoint, Schauder basis, symmetric and shrinking basis
Received by editor(s): July 4, 2004
Received by editor(s) in revised form: January 3, 2005
Published electronically: December 14, 2005
Additional Notes: The author was supported by the The Royal Swedish Academy of Sciences
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2005 American Mathematical Society

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