Let H be a separable Hilbert space over the complex field. The class S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by σ (S) and then Bram proved that Halmos. Halmos proved that σ(Ň) (S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S. Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H: Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions. Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2. If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)? The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.

Cette thèse s'inscrit dans la théorie des matrices aléatoires, à l'intersection avec la théorie des probabilités libres et des algèbres d'opérateurs. Elle s'insère dans une démarche générale qui a fait ses preuves ces dernières décennies : importer les techniques et les concepts de la théorie des probabilités non commutatives pour l'étude du spectre de grandes matrices aléatoires. On s'intéresse ici à des généralisations du théorème de liberté asymptotique de Voiculescu. Dans les Chapitres 1 et 2, nous montrons des résultats de liberté asymptotique forte pour des matrices gaussiennes, unitaires aléatoires et déterministes. Dans les Chapitres 3 et 4, nous introduisons la notion de fausse liberté asymptotique pour des matrices déterministes et certaines matrices hermitiennes à entrées sous diagonales indépendantes, interpolant les modèles de matrices de Wigner et de Lévy. / The thesis fits into the random matrix theory, in intersection with free probability and operator algebra. It is part of a general approach which is common since the last decades: using tools and concepts of non commutative probability in order to get general results about the spectrum of large random matrices. Where are interested here in generalization of Voiculescu's asymptotic freeness theorem. In Chapter 1 and 2, we show some results of strong asymptotic freeness for gaussian, random unitary and deterministic matrices. In Chapter 3 and 4, we introduce the notion of asymptotic false freeness for deterministic matrices and certain random matrices, Hermitian with independent sub-diagonal entries, interpolating Wigner and Lévy models.

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