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

DOI:
https://doi.org/10.1090/S0002-9939-1992-1123653-7

MathSciNet review:
1123653

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Abstract: Let be an invertible (bounded linear) operator acting on a complex Banach space . is called *hypercyclic* if there is a vector in such that the orbit is dense in . ( is necessarily separable and infinite dimensional.)

**Theorem 1.** *The following are equivalent for an invertible operator A acting on* *or* *is hypercyclic*; (ii) and *are hypercyclic*; (iii) *there is a vector* *such that* (*the upper bar denotes norm-closure*); (iv) *there is a vector* in such that

**Theorem 2.** *If* *is not hypercyclic, then* and have a common nontrivial invariant closed subset.

**[1]**G. Godefroy and J. H. Shapiro,*Operators with dense, invariant, linear cyclic vector manifolds*, J. Funct. Anal.**98**(1991), 229-269. MR**1111569 (92d:47029)****[2]**D. A. Herrero,*Limits of hypercyclic and supercyclic operators*, J. Funct. Anal.**99**(1991), 179-190. MR**1120920 (92g:47026)****[3]**D. A. Herrero and Z.-Y. Wang,*Compact perturbations of hypercyclic and supercyclic operators*, Indiana Univ. Math. J.**39**(1990), 819-830. MR**1078739 (91k:47042)****[4]**C. Kitai,*Invariant subsets for linear operators*, Dissertation, University of Toronto, 1982.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1992-1123653-7

Keywords:
Invertible hypercyclic operator,
orbits,
common nontrivial invariant closed subset

Article copyright:
© Copyright 1992
American Mathematical Society