##
**The invariant subspace problem for a class of Banach spaces. II: Hypercyclic operators.**
*(English)*
Zbl 0782.47002

Summary: We continue here the line of investigation begun in [Lect. Notes Math. 1317, 1-20 (1988; Zbl 0663.47004)], where we showed that on every Banach space \(X=\ell_ 1\oplus W\) (where \(W\) is separable) there is an operator \(T\) with no nontrivial invariant subspaces. Here, we work on the same class of Banach spaces, and produce operators which not only have no invariant subspaces, but are also hypercyclic. This means that for every nonzero vector \(x\) in \(X\), the translates \(T^ r x\) \((r=1,2,3,\dots)\) are dense in \(X\). This is an interesting result even if stated in a form which disregards the linearity of \(T\): it tells us that there is a continuous map of \(X\backslash\{0\}\) into itself such that the orbit \(\{T^ r x: r\geq 0\}\) of any \(x\in X\backslash\{0\}\) is dense in \(X\backslash\{0\}\). The methods used to construct the new operator \(T\) are similar to those in [loc. cit.], but we need to have somewhat greater complexity in order to obtain a hypercyclic operator.

### MSC:

47A15 | Invariant subspaces of linear operators |

### Citations:

Zbl 0663.47004
Full Text:
DOI

### References:

[1] | B. Beauzamy,Operators without invariant subspaces, simplification of a result of P. Enflo, J. Integral Equations and Operator Theory, January 1985. |

[2] | P. Enflo,On the invariant subspace problem for Banach spaces, Acta Math.158 (1987), 212–313. · Zbl 0663.47003 |

[3] | R. Ovsepian and A. Pelcynski,The existence in every separable Banach space of a fundamental and total bounded biorthogonal sequence and related constructions of uniformly bounded orthonormal systems in L, Studia Math.54 (1975), 149–159. · Zbl 0317.46019 |

[4] | C. J. Read,A solution to the invariant subspace problem, Bull. London Math. Soc.16 (1984), 337–401. · Zbl 0566.47003 |

[5] | C. J. Read,A solution to the invariant subspace problem on the space l 1, Bull. London Math. Soc.17 (1985), 305–317. · Zbl 0574.47006 |

[6] | C. J. Read,A short proof concerning the invariant subspace problem, J. London Math. Soc. (2)33 (1986), 335–348. · Zbl 0664.47006 |

[7] | C. J. Read,The invariant subspace problem on a class of nonreflexive Banach spaces, Israel Seminar on Geometrical Aspects of Functional Analysis, to appear. |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.