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 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]**Gilles Godefroy and Joel H. Shapiro,*Operators with dense, invariant, cyclic vector manifolds*, J. Funct. Anal.**98**(1991), no. 2, 229–269. MR**1111569**, 10.1016/0022-1236(91)90078-J**[2]**Domingo A. Herrero,*Limits of hypercyclic and supercyclic operators*, J. Funct. Anal.**99**(1991), no. 1, 179–190. MR**1120920**, 10.1016/0022-1236(91)90058-D**[3]**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**, 10.1512/iumj.1990.39.39039**[4]**C. Kitai,*Invariant subsets for linear operators*, Dissertation, University of Toronto, 1982.

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DOI:
http://dx.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