The existence of translation invariant subspaces of symmetric self-adjoint sequence spaces on Z

摘要

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))over bar>} 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 l < p < infinity. 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. (C) 2000 Academic Press. [References: 17]