The Existence of Translation Invariant Subspaces of Symmetric Self-Adjoint Sequence Spaces on Z

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 ℓp 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 {〈Anu, 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.