Binomial sums, moments and invariant subspaces

The main result of this paper is that if a sequence of complex numbers (a(n ))(n greater than or equal to 0) satisfies [GRAPHICS] for some integer r greater than or equal to 0, then a(n) = 0 for all n > r. As an application, we deduce a localized form of a theorem of Allan about nilpotent elements in Banach algebras, and this in turn leads to an invaria nt-subspace theorem. As a further application, we prove a Variant of Carlem an's theorem on the unique determination of probability distributions by th eir moments. The paper concludes with a quantitative form of the main resul t.