  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.
