On the uniform ergodic theorem in invariant subspaces.
El Bakkali, Abdeslam
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 → ∞ ‖ 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 , theorem 1), also to the theorem of the Gelfand-Hille type.