QUASI-COMPACT POSITIVE OPERATORS - APPLICATIONS TO THE PERRON-FRONEBIUS OPERATORS

Let X be a compact metric space, {eta(x,.), x is-an-element-of X} a family of non negative measures on X. and P the positive operator defined by Pf(x) = integral-X f(y) eta (x, dy), which we suppose to be bounded on C(X) and quasi-compact on the space E(alpha) of alpha-holderian functions where 0 < alpha less-than-or-equal-to 1. We known that there exists an integer nu such that or l=1+infinity Ker (P' - rho)l = Ker (P' - rho)nu, where rho is the spectral radius of P, and P' = P\E(alpha). The object of this work is to give a necessary and sufficient condition for the existence of a gamma > 0 in C(X) such that (P - rho)nu gamma = 0. In this case we describe, when nu = 1 the space Ker (P - rho), and for rho = 1 and nu greater-than-or-equal-to 1 the P-invariant measures. We give applications to transfert operators.