Bounded Sequences, Orbits and Invariant Subspaces

Let {x(n)} be a sequence of complex numbers and let Delta(n)(xj) = Sigma(n)(k=0)(-1)(k) ((n)(k))x(n-k+j). In this paper, we will show that if vertical bar x(n)vertical bar = O(n(k)), as n -> infinity for some positive integer k, and n vertical bar Delta(n)(xj)vertical bar(1/n) -> 0 as n -> infinity, then Delta(k+1)(xj) = 0. More importantly, applications to the orbits of operators and invariant subspace problem are also given; this helps to improve former results obtained by Gelfand-Hille, Mbekhta-Zemanek and others.