Applying quantitative perturbation theory for linear operators, we prove non-asymptotic limit theorems for Markov chains whose averaging operator has a spectral gap in a suitable function space. The main results are concentration inequalities and Berry-Esséen bounds, with some flexibility in the choice of the function space and no "warm start" hypothesis nor burn-in: the law of the first term of the chain can be arbitrary. The constants are completely explicit and reasonable enough to make the results usable in practice, notably in MCMC methods.