Schwach irreduzible Markoff-Operatoren

For any weakly irreducible Markov operatorT on the space of continuous functions on a compact space and any continuous functionf we show that pointwise convergence of (T ⁿf) already implies uniform convergence. This improves a result ofJamison 2]. We also study convergence whenf is only measurable and give results related to the zero-two law. Finally two characterizations of uniform ergodicity for this class of operators are given. In particular a question ofLotz 4] is answered.