# invariant subspace problem

Initially formulated for Banach spaces^{}, the invariant subspace conjecture stated the following:

*Let $X$ be a complex Banach space. Then every bounded operator ^{} $T$ in $X$ has a non-trivial closed (http://planetmath.org/ClosedSet) invariant subspace^{}, i.e. there exists a closed vector subspace $S\mathrm{\subset}X$ such that $S\mathrm{\ne}\mathrm{0}$, $S\mathrm{\ne}X$ and $T\mathit{}\mathrm{(}S\mathrm{)}\mathrm{\subseteq}S$.*

This conjecture was proven to be false when P. Enflo (1975) and . Read (1984) gave examples of bounded operators which did not have the above property.

However, if one considers only Hilbert spaces^{}, this is still an open problem. Today the invariant subspace conjecture is formulated as follows:

*Let $H$ be a complex Hilbert space. Then every bounded operator $T$ in $H$ has a non-trivial invariant subspace, i.e. there exists a closed vector subspace $S\mathrm{\subset}H$ such that $S\mathrm{\ne}\mathrm{0}$, $S\mathrm{\ne}H$ and $T\mathit{}\mathrm{(}S\mathrm{)}\mathrm{\subseteq}S$.*

Title | invariant subspace problem |
---|---|

Canonical name | InvariantSubspaceProblem |

Date of creation | 2013-03-22 17:24:02 |

Last modified on | 2013-03-22 17:24:02 |

Owner | asteroid (17536) |

Last modified by | asteroid (17536) |

Numerical id | 5 |

Author | asteroid (17536) |

Entry type | Conjecture |

Classification | msc 47A15 |

Classification | msc 46-00 |

Synonym | invariant subspace conjecture |