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Propriétés spectrales de l'opérateur solution canonique du d-bar et des opérateurs de Hankel de symbole antiholomorphe.

Abstract : This thesis deals with the spectral properties of the canonical solution operator of the dbar-equation in connexion with Hankel operators in the several complex variables context.
First, we study the spectral properties of the dbar-canonical solution operator on (0,1)-forms with holomorphic coefficients. In this doctoral dissertation, we consider a Hilbert space H of (0,1)-forms and a space L2 of square integrable functions with respect to the measure mu on a rotation invariant open set Omega in C^n. We give necessary and sufficient conditions, in terms of the moments of the measure mu, for the canonical solution operator of the dbar-equation to be bounded, compact and in the Schatten p-class from H into L2. Examples of H can be chosen to be the space of (0,1)-forms with coefficients in one of the classical Hilbert spaces of holomorphic functions such as weighted Bergman spaces, Fock spaces, Hardy-Sobolev spaces, Sobolev spaces of holomorphic functions or the Möbius invariant space.


Secondly, we are interested in the existence in a given Schatten class of a non-zero Hankel operator with antiholomorphic symbol on a Hilbert space of holomorphic functions and we study the connection between the growth of a function f and the size of the singular values of the Hankel operator of symbol f-bar. In this work, we consider the big Hankel operators with antiholomorphic symbol on the Hardy space on the unit disc in C, the Dirichlet space or Sobolev spaces of holomorphic functions on the unit disc. We give a necessary et sufficient condition on p for the Schatten p-class to contain a non-zero Hankel operator with antiholomorphic symbol. Then, we characterize the functions f such that the Hankel operator of symbol f-bar is a Hilbert-Schmidt operator. Moreover, we give necessary conditions on f for the Hankel operator of symbol f-bar to be bounded, compact and in the Schatten p-class, except in the case of the Dirichlet space.
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Submitted on : Monday, March 20, 2006 - 10:56:04 AM
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Stéphanie Lovera. Propriétés spectrales de l'opérateur solution canonique du d-bar et des opérateurs de Hankel de symbole antiholomorphe.. Mathématiques [math]. Université de Provence - Aix-Marseille I, 2005. Français. ⟨tel-00011986⟩

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