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International audienceLet P be a Markov kernel on a measurable space E with countably generated sigma-algebra, let w: E --> [1, +infinity] such that Pin <= C w with C >= 0, and let B-w be the space of measurable functions on E satisfying parallel to f parallel to(w) = sup{w(x)\, x is an element of E} +infinity. We prove that P is quasi-compact on (B-w, parallel to.parallel to(w)) if and only if, for all f infinity B-w, (1/n Sigma(n)(k=1) P(k)f)(n) contains a subsequence convergaing in B-w to Pi f = Sigma(d)(i=1) mu(i) (f)v(i), where the v(i)'s are non-negative bounded measurable Junctions on E and the mu(i)'s are probability distributions on E. In particular. when the space of P-invariant functions in B-w is finite-dimensional, uniform ergodicity is equivalent to mean ergodicity

Quasi-compactness and mean ergodicity for Markov kernels acting on weighted supremum normed spaces