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Proceedings of the American Mathematical Society

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A mean ergodic theorem in Banach spaces

Author: Takeshi Yoshimoto
Journal: Proc. Amer. Math. Soc. 99 (1987), 115-118
MSC: Primary 47A35
MathSciNet review: 866439
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Abstract: Let $ \Gamma = \left\{ {{U_t}:t \in \Lambda } \right\}\left( {\Lambda = {{\mathbf{Z}... ... \left\{ 0 \right\}{\text{or }}{{\mathbf{R}}^ + } - \left\{ 0 \right\}} \right)$ be a commuting family of nonexpansive affine operators in a Banach space $ X$ satisfying the following conditions:

(i) there is a function $ M\left( {x\left\vert \Gamma \right.} \right) \geq 0$ defined on $ X$ such that

$\displaystyle \left\Vert {{U_{t + s}}x - {U_t}{U_s}x} \right\Vert \leq M\left( ... ...ft\vert \Gamma \right.} \right)\quad \left( {s,t \in \Lambda ,x \in X} \right),$


$\displaystyle \sup\left\{ {{t^{ - 1}}\left\Vert {{U_t}x} \right\Vert:t \in \Lam... ...K\left( {x\left\vert \Gamma \right.} \right) < \infty \left( {x \in X} \right).$

Then it is proved that if $ \left\{ {{t^{ - 1}}{U_t}x:t \in \Lambda } \right\}$ is relatively compact for $ x \in X$, the limit $ X -\lim_{t \to \infty }{t^{ - 1}}{U_t}x = \bar x$ exists in $ X$ and $ \overline {{U_t}x} = \overline x \left( {t \in \Lambda } \right)$.

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

  • [1] M. A. Akcoglu and U. Krengel, A differentiation theorem in $ {L_p}$, Math. Z. 169 (1979), 31-40. MR 546991 (80m:47005)
  • [2] N. Dunford, A mean ergodic theorem, Duke Math. J. 5 (1939), 635-646. MR 0000098 (1:18f)
  • [3] N. Dunford and J. T. Schwartz, Linear operators, Part 1, Interscience, New York, 1958.
  • [4] G. Rodé, An ergodic theorem for convex operators, Arch. Math. 40 (1983), 447-451. MR 707734 (85e:47008)

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