## Abstract

We prove that if X is a reflexive translation invariant Banach space of complex sequences on Z that contains all finitely supported sequences, in which the coordinate functionals are continuous, and for every sequence {c(n)} in the space the sequences {c(n)} and {c(-n)} are also in the space, then X has a nontrivial translation invariant subspace. This provides in particular a positive solution to the translation invariant subspace problem for weighted ℓ^{p} spaces on Z with even weights, for 1<p<∞. The proof is based on an intermediate result that asserts that if A is an operator on a reflexive real Banach space of dimension greater than one, and there exist non-zero vectors, u in the space and v in the dual space, such that {〈A^{n}u, v〉}^{∞}_{n=0} is a moment sequence of a finite positive Borel measure on a bounded interval on the real line, then A has a nontrivial invariant subspace.

Original language | English |
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Pages (from-to) | 372-380 |

Number of pages | 9 |

Journal | Journal of Functional Analysis |

Volume | 178 |

Issue number | 2 |

DOIs | |

State | Published - 20 Dec 2000 |