On invertible hypercyclic operators
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- by Domingo A. Herrero and Carol Kitai PDF
- Proc. Amer. Math. Soc. 116 (1992), 873-875 Request permission
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 \[ [\operatorname {Orb}(A;y) \cup \operatorname {Orb}({A^{ - 1}};y)]^ - = \mathcal {X}.\]. Theorem 2. If $A$ is not hypercyclic, then $A$ and ${A^{ - 1}}$ have a common nontrivial invariant closed subset.References
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- Domingo A. Herrero and Zong Yao Wang, Compact perturbations of hypercyclic and supercyclic operators, Indiana Univ. Math. J. 39 (1990), no. 3, 819–829. MR 1078739, DOI 10.1512/iumj.1990.39.39039 C. Kitai, Invariant subsets for linear operators, Dissertation, University of Toronto, 1982.
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 873-875
- MSC: Primary 47A65; Secondary 47A15, 47B99
- DOI: https://doi.org/10.1090/S0002-9939-1992-1123653-7
- MathSciNet review: 1123653