On the uniform ergodic theorem in invariant subspaces.

SUMMARY

Let T be a bounded linear operator on a Banach space X into itself. In this paper, we study the uniform ergodicity of the operator T|Y when Y is a closed subspace invariant under T. We show that if T satisfies, lim n ? 8 ? T n ? n = 0 , then T is uniformly ergodic on X if and only if the restriction of T to some closed subspace Y ? X, invariant under T and R[(I - T)k] ? Y for some integer k = 1, is uniformly ergodic. Consequently, we obtain other equivalent conditions concerning the theorem of Mbekhta and Zemànek [9], theorem 1), also to the theorem of the Gelfand-Hille type.