Schwach irreduzible Markoff-Operatoren. (Weakly irreducible Markov operators) (Q1100712)

Let T be a Markoff operator on the space C(X) of continuous functions on a compact space X. If T is irreducible then we show that for any \(f\in C(X)\) the pointwise convergence of (T nf) to an arbitrary function g implies already uniform convergence. For \(g=0\) this was previously shown by \textit{B. Jamison} [Proc. Am. Math. Soc. 24, 366-370 (1970; Zbl 0195.414)]. This is only a very special case of the main result which yields also similar results for averages \(\frac{1}{n}\sum ^{n=1}_{i=1}T\) i. As a further application a characterization of uniform ergodicity due to \textit{H. P. Lotz} [Math. Z. 178, 145-156 (1981; Zbl 0533.47010)] is improved. Finally a new characterization of uniform ergodicity of irreducible Markoff operators in terms of a Harris type condition is given. All the above results are shown for the more general class of weakly irreducible operators introduced in this paper.