# Iterates of Markov operators and their limits

## TL;DR

This paper presents a straightforward method to explicitly construct the spectral projection for iterates of quasi-compact operators, especially Markov operators, using known fixed points, with broad applicability in Banach spaces.

## Contribution

It introduces a simple, explicit construction technique for spectral projections of quasi-compact operators based on fixed points, applicable in approximation theory and Banach spaces.

## Key findings

- Explicit construction of spectral projection from fixed points.
- Method applicable to Markov operators on continuous functions.
- Analysis based on Riesz-Schauder and Fredholm theory.

## Abstract

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.