REPRESENTATIONS DE GROUPES TOPOLOGIQUES ET ETUDE SPECTRALE D'OPERATEURS DE DECALAGE UNILATERAUX ET BILATERAUX

Abstract : In the first part of this thesis we study the behaviour of a representation $\theta$ of a group $G$ in a Banach algebra $A$ with the behaviour of $\limsup_{u \rightarrow 1}\| \theta(u)-I \|$, where $1$ denotes the unit element of $G$ and $I$ the unit element of $A$. We also obtain automatic continuity results for a large class of group representations.

In the second and third parts we study in some concrete cases the spectrum of the operator $S_M: E/M \rightarrow E/M$ defined by $S(f+M)=Sf +M$, where $E$ is a Banach space, $S:E \rightarrow E$ a bounded operator and $M$ a closed $S$-invariant subspace, i.e. $S(M) \subset M$. We first study the case when $E$ is a Banach space of analytic functions on $\D$ such that the usual shift $S: f \mapsto zf$ and the backward shift $T:f \mapsto \frac{f-f(0)}{z}$ have their spectrum equal to the unit circle and satisfy a non-quasianalytic condition. We show that, if there exists a function $f \in M$ having an analytic extension to $\D \cup D(\zeta,r)$, with $|\zeta|=1$, $f(\zeta)\neq 0$, then $\zeta \notin Spec(S_M)$. We apply this result to the weighted Hardy space $H_{\sigma_{\alpha}}(\D)$, with $\sigma_{\alpha}(n)=e^{-n^{\alpha}}$, $n \geq 0$, $\alpha \in (\frac{1}{2},1)$.

Finally we study a quasianalytic situation in the spaces $l^2(w,\Z)$ , with '$\log$-even" weights. Let $S: (u_n)_{n \in \Z} \mapsto (u_{n-1})_{n \in \Z}$ be the usual bilateral shift on $l^2(w,\Z)$. When $L$ is a closed arc of the unit circle we show that the construction of Y.Domar of translation invariant subspaces in $l^2(w,\Z)$ spaces satisfying a natural regularity condition permits us to construct subspaces $M_L$ such that $Spec (S_{M_L})=L$.}
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• HAL Id : tel-00011971, version 1

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Sébastien Dubernet. REPRESENTATIONS DE GROUPES TOPOLOGIQUES ET ETUDE SPECTRALE D'OPERATEURS DE DECALAGE UNILATERAUX ET BILATERAUX. Mathématiques [math]. Université Sciences et Technologies - Bordeaux I, 2005. Français. ⟨tel-00011971⟩

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