Local ergodicity of nonpositive contractions on 𝐶(𝑋)

Atalla, Robert E.

doi:10.1090/S0002-9939-1983-0699406-5

Local ergodicity of nonpositive contractions on $C(X)$
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by Robert E. Atalla PDF

Proc. Amer. Math. Soc. 88 (1983), 419-425 Request permission

Abstract:

Let $T$ be an operator on $C(X)$, $X$ compact, with $\left \| T \right \| \leqslant 1$, and suppose $T$ has a nowhere vanishing invariant function ${\psi ^{ - 1}}$. The operator $R$ defined by $Rf = T(f{\psi ^{ - 1}})\psi$ is (a) "locally" a Markov operator, and (b) (locally) strongly ergodic iff $T$ is. This is used to prove Sine’s local strong ergodicity theorem without assuming that $T$ is positive.

References

Robert E. Atalla, On the mean convergence of Markov operators, Proc. Edinburgh Math. Soc. (2) 19 (1974/75), no. 2, 205–209. MR 415355, DOI 10.1017/S0013091500010324
Yaakov Friedman and Bernard Russo, Contractive projections on $C_{0}(K)$, Trans. Amer. Math. Soc. 273 (1982), no. 1, 57–73. MR 664029, DOI 10.1090/S0002-9947-1982-0664029-4
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
Robert Sine, Geometric theory of a single Markov operator, Pacific J. Math. 27 (1968), 155–166. MR 240281, DOI 10.2140/pjm.1968.27.155
Robert Sine, A mean ergodic theorem, Proc. Amer. Math. Soc. 24 (1970), 438–439. MR 252605, DOI 10.1090/S0002-9939-1970-0252605-X
Robert Sine, On local uniform mean convergence for Markov operators, Pacific J. Math. 60 (1975), no. 2, 247–252. MR 397882, DOI 10.2140/pjm.1975.60.247