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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On invertible hypercyclic operators


Authors: Domingo A. Herrero and Carol Kitai
Journal: Proc. Amer. Math. Soc. 116 (1992), 873-875
MSC: Primary 47A65; Secondary 47A15, 47B99
MathSciNet review: 1123653
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Abstract: Let $ A$ be an invertible (bounded linear) operator acting on a complex Banach space $ \mathcal{X}$. $ A$ is called hypercyclic if there is a vector $ y$ in $ \mathcal{X}$ such that the orbit $ \operatorname{Orb}(A;y): = \{ y,Ay,{A^2},y, \ldots \} $ is dense in $ \mathcal{X}$. ( $ \mathcal{X}$ is necessarily separable and infinite dimensional.)

Theorem 1. The following are equivalent for an invertible operator A acting on $ \mathcal{X}:({\text{i}})A$ or $ {A^{ - 1}}$ is hypercyclic; (ii) $ A$ and $ {A^{ - 1}}$ are hypercyclic; (iii) there is a vector $ z$ such that $ \operatorname{Orb}(A;z)^ - = \operatorname{Orb}({A^{ - 1}}{\text{;z}})^ - = \mathcal{X}$ (the upper bar denotes norm-closure); (iv) there is a vector $ y$ in $ \mathcal{X}$ such that

$\displaystyle [\operatorname{Orb}(A;y) \cup \operatorname{Orb}({A^{ - 1}};y)]^ - = \mathcal{X}.$

.

Theorem 2. If $ A$ is not hypercyclic, then $ A$ and $ {A^{ - 1}}$ have a common nontrivial invariant closed subset.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1123653-7
PII: S 0002-9939(1992)1123653-7
Keywords: Invertible hypercyclic operator, orbits, common nontrivial invariant closed subset
Article copyright: © Copyright 1992 American Mathematical Society