Coboundaries of irreducible Markov operators on C (K)

Let K be a compact Hausdorff space, and let T be an irreducible Markov N operator on C(K). We show that if g E C(K) satisfies suPN II ~j=0 TJgll < oo, then (and only then) there exists f E C(K) with (I - T)y = g. Generalizing the result to irreducible Markov operator representations of certain semi-groups, we obtain that bounded cocycles are (continuous) coboundaries. For minimal semi-group actions in C(K), no restriction on the semi-group is needed.