On the uniform ergodic theorem of Lin

Lloyd, Stuart P.

doi:10.1090/S0002-9939-1981-0630042-0

On the uniform ergodic theorem of Lin
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by Stuart P. Lloyd PDF

Proc. Amer. Math. Soc. 83 (1981), 710-714 Request permission

Abstract:

Lin has given necessary and sufficient conditions for convergence in the uniform operator topology of ${A_n} = (1 + T + \cdots + {T^{n - 1}})/n$, $T$ being a Banach space operator satisfying $\left \| {{T^n}} \right \|/n \to 0$. We prove a generalization in which the Cesàro means are replaced by any bounded sequence in the affine hull converging uniformly to invariance. In the case where $T:{C_0}(X) \to {C_0}(X)$ is a transient Feller operator for noncompact locally compact Hausdorff space $X$, we show that $\{ {A_n}\}$ converges strongly but never uniformly.

References

Linear operators

Michael Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc. 43 (1974), 337–340. MR 417821, DOI 10.1090/S0002-9939-1974-0417821-6
Stuart P. Lloyd, On the mean ergodic theorem of Sine, Proc. Amer. Math. Soc. 56 (1976), 121–126. MR 451007, DOI 10.1090/S0002-9939-1976-0451007-6
Ry\B{o}tar\B{o} Sat\B{o}, The Hahn-Banach theorem implies Sine’s mean ergodic theorem, Proc. Amer. Math. Soc. 77 (1979), no. 3, 426. MR 545609, DOI 10.1090/S0002-9939-1979-0545609-9