Rappelons que la condition (D 0 ) estéquivalentèestéquivalentè a la propriété de ?-mélange (cf lemme 5-(3.9)) condition plus forte que celle de ?-mélange ,
P est donc un opérateur borné défini sur l'espace de Hilbert L 2 (?) et l'existence d'une unique probabilité invariante implique que 1 est une valeur propre de multiplicité 1 (relation entre le spectre de P et de son adjoint) donc que le projecteur propre noté E ? est de rang fini 1. L'espace H = L 2 (?) admet alors la décomposition Le spectre de P M est constitué d'un seul point, donc P M ? I est quasi-nilpotent (i.e. son rayon spectral est nul) La résolvante R(?) de P peut s'´ ecrire comme ? E ? ), o` u R (?) s'appelle la résolvante réduite de P, est isolée. Notons par (P ) la distance de 1 au reste du spectre de P. Supposonségalement Supposonségalement qu ,
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