An extension of Sine's counterexample (Q1007098)

This article is about the construction of mean ergodic positive linear contractions whose powers are not mean ergodic. A bounded operator \(T\) on a Banach space \(X\) is called mean ergodic if the sequence \(\{\frac1n \sum_{i=0}^{n-1} T^i \}_n \) converges strongly. It is known that if \(T\) and \(S\) are mean ergodic, without some additional assumptions, it is not always true that \(TS\) is mean ergodic [see \textit{U.\,Krengel}, ``Ergodic theorems'' (Berlin-New York: Walter de Gruyter) (1985; Zbl 0575.28009), for more details]. \textit{R.\,Sine} [Proc.\ Am.\ Math.\ Soc.\ 57, No.\,1, 169--172 (1976; Zbl 0333.54027)] constructed the first example of a mean ergodic positive contraction \(T\) on a \(C(K)\)-space such that \(T^2\) is not mean ergodic. The authors extend the method of R.\,Sine to construct a mean ergodic positive isometry \(T\) on a \(C(K)\)-space such that \(T^k\) is not mean ergodic for any \(k>1. \) This is a well-written article with a lucid exposition of the construction technique.