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A mean ergodic theorem for families of contractions in Hilbert space

Authors: J. R. Blum and J. I. Reich
Journal: Proc. Amer. Math. Soc. 61 (1976), 183-185
MSC: Primary 47A35
MathSciNet review: 0473875
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Abstract: Let $ G$ be an LCA group and $ h$ a Hilbert space, and let $ T(g)$ be a function on $ G$ into the contractions on $ H$. Let $ \{ {\sigma _n}\} $ be a sequence of probability measures on $ G$. Under suitable conditions on $ T(g)$ and the sequence $ \{ {\sigma _n}\} $ we prove the strong convergence of the sequence $ {T_n} = \smallint T(g){\sigma _n}(dg)$. In certain cases we identify the limiting operator.

References [Enhancements On Off] (What's this?)

  • [1] J. R. Blum and B. Eisenberg, Generalized summing sequences and the mean ergodic theorem, Proc. Amer. Math. Soc. 42 (1974), 423-429. MR 48 #8749. MR 0330412 (48:8749)
  • [2] C. Foiaş, and Béla Sz.-Nagy, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam; American Elsevier, New York; Akadémiai Kiadó, Budapest, 1970. MR 43 #947. MR 0275190 (43:947)

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