Iterates of Markov operators and their limits

It is well known that iterates of quasi-compact operators converge towards a spectral projection, whereas the explicit construction of the limiting operator is in general hard to obtain. Here, we show a simple method to explicitly construct this projection operator, provided that the fixed points of the operator and its adjoint are known which is often the case for operators used in approximation theory. We use an approach related to Riesz-Schauder and Fredholm theory to analyze the iterates of operators on general Banach spaces, while our main result remains applicable without specific knowledge on the underlying framework. Applications for Markov operators on the space of continuous functions $C(X)$ are provided, where $X$ is a compact Hausdorff space.