On the uniform ergodic theorem in invariant subspaces.

• Tajmouati, Abdelaziz ^[1] ; El Bakkali, Abdeslam ^[2] ; Barki, Fatih ^[1]
  1. [1] Sidi Mohamed Ben Abdellah University

     Sidi Mohamed Ben Abdellah University

     Fes-Medina, Marruecos

     
  2. [2] Université Chouaib Doukkali

     Université Chouaib Doukkali

     Marruecos

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 38, Nº. 2, 2019, págs. 315-324
• Idioma: inglés
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• Resumen

  • 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.