Multiples of hypercyclic operators

Badea, Catalin; Grivaux, Sophie; Müller, Vladimir

doi:10.1090/S0002-9939-08-09696-2

Multiples of hypercyclic operators
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by Catalin Badea, Sophie Grivaux and Vladimir Müller

Proc. Amer. Math. Soc. 137 (2009), 1397-1403

DOI: https://doi.org/10.1090/S0002-9939-08-09696-2

Published electronically: October 27, 2008

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Abstract:

We give a negative answer to a question of Prăjitură by showing that there exists an invertible bilateral weighted shift $T$ on $\ell _2(\mathbb {Z})$ such that $T$ and $3T$ are hypercyclic but $2T$ is not. Moreover, any $G_\delta$ set $M \subseteq (0,\infty )$ which is bounded and bounded away from zero can be realized as $M=\{t>0 \mid tT \textrm { is hypercyclic}\}$ for some invertible operator $T$ acting on a Hilbert space.

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