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An ergodic theorem of Abelian type. (English) Zbl 0166.40404

Summary: Let \(T\) be a positive linear operator on \(L_1\) of a \(\sigma\)-finite measure space \(\| T|\leq 1\). Let \(\{f_n,\;n\geq 1\}\) be a sequence of non-negative numbers with \(\sum f_i=1\) and set \(u_0=1\), \(u_n=\sum f_\nu u_{n-\nu}\) \((n\geq 1)\). Further set \(S_n=\sum_{\nu=0}^{n-1} u_\nu T^\nu\) and \(U(\lambda T)=\sum_{\nu=0}^{\infty} u_\nu \lambda^\nu T^\nu\) \((0\leq \lambda<1)\). The author proves that \(\lim_{\lambda\to 1}\frac{U(\lambda T)f}{U(\lambda T)p}\) exists a.e. on \(\{p>0\}\) if \(f\in L_1\), \(p\in L_1\), \(p\geq 0\). G. Baxter [J. Math. Mech. 14, 277–288 (1965; Zbl 0139.31001)] proved that \(\lim \frac{S_nf}{S_p}\) exists a.e. on \(\{p>0\}\) and R. V. Chacon [Bull. Am. Math. Soc. 70, 796–797 (1964; Zbl 0196.43902)] reduced this result to the Chacon-Ornstein theorem and replaced the set \(\{p>0\}\) by the set \(\{\sum_{\nu=0}^{\infty} S_np>0\}\). See also U. Krengel [Proc. 5th Berkeley Sympos. Math. Stat. Probab. Univ. Calif. 1965/1966 2, Part 2, 415–429 (1967; Zbl 0236.60051)].
Reviewer: L. Sucheston

MSC:

47A35 Ergodic theory of linear operators
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