On the uniform ergodic theorem in invariant subspaces.
Keywords:
Uniform ergodic theorem, Cesàro averages, Decomposition ergodicAbstract
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 ((9), theorem 1), also to the theorem of the Gelfand-Hille type.References
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