Non-quasianalytic classes of functions and existence of invariant subspaces

We prove that some classes of functions defined on a compact set in the com plex plane with planar Lebesgue measure zero, are non-quasianalytic. We par ticularly treat the Carleman classes and classes of functions having asymto tically holomorphic continuation. Combining this with Dyn 'kin's functional calculus based on the Cauchy-Green formula, we establish the existence of invariant subspaces for operators for which a part of the spectrum is of pl anar Lebesgue measure zero, provided that the resolvent has a moderate grow th near this part of the spectrum. (C) Academie des Sciences/Elsevier, Pari s.