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We present a spectral theory for a class of operators satisfying a weak“Doeblin–Fortet" condition and apply it to a class of transition operators.This gives the convergence of the series ∑k≥0krPkƒ, $r \in \mathbb{N}$ 
,under some regularity assumptions and implies the central limit theoremwith a rate in $n^{- \frac{1}{2} }$ 
for the corresponding Markov chain.An application to a non uniformly hyperbolic transformation on the interval is also given.
