# Idéaux fermés de certaines algèbres de beurling et applications aux opérateurs - Ensembles d'unicité

Abstract : In the first part, we study operators with spectrum included in the unit circle $\bbt$. We obtain results concerning growth of $\| T^{-n} \| \, (n \geq 0)$ for operators $T$ with countable spectrum or spectrum satisfying geometric conditions. For this, we need to work in the spaces
$A_{\omega}(\bbt) = \Big\{ f \textrm{ continuous on } \bbt : \, \big\| f \big\|_{\omega} = \sum_{n = -\infty}^{+\infty} | \widehat{f}(n) | \omega(n) < +\infty \Big\},$
where $\omega = \big( \omega(n) \big)_{n \in \bbz}$ is a sequence of non-negative real numbers, and $\widehat{f}(n)$ denotes the $\textrm{n}^{\textrm{th}}$ Fourier coefficient of $f$. When $\omega = \big( \omega(n) \big)_{n \in \bbz}$ is a weight, $\big( A_{\omega}(\bbt), \| \, \|_{\omega} \big)$ is a Banach algebra. We obtain the characterisation of some closed ideals of $A_{\omega}(\bbt)$ for a family of weight.

In the second part, we are interested in closed subset of $\bbt$ which are (or not) sets of uniqueness for $\dsp A_{\omega}^{+}(\bbt) = \Big\{ f \in A_{\omega}(\bbt): \, \widehat{f}(n) = 0 \quad (n < 0) \Big\}$, where $\omega = \big( \omega(n) \big)_{n \in \bbz}$ is a sequence of non-negative real numbers. A closed subset $E$ of $\bbt$ is said to be a set of uniqueness for $X$, a space of continuous functions on $\bbt$, if the zero function is the only function in $X$ that vanishes on $X$. More precesely, we study the link between the fact that a closed subset of $\bbt$ satisfies a given geometric condition, and the fact that it is or not a set of uniqueness for $A_{\omega}^{+}(\bbt)$.
Keywords :
Document type :
Theses
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https://tel.archives-ouvertes.fr/tel-00011141
Contributor : Cyril Agrafeuil <>
Submitted on : Wednesday, December 7, 2005 - 12:47:17 PM
Last modification on : Wednesday, October 10, 2018 - 1:26:43 AM
Long-term archiving on: : Friday, April 2, 2010 - 11:20:03 PM

### Identifiers

• HAL Id : tel-00011141, version 1

### Citation

Cyril Agrafeuil. Idéaux fermés de certaines algèbres de beurling et applications aux opérateurs - Ensembles d'unicité. Mathématiques [math]. Université Sciences et Technologies - Bordeaux I, 2004. Français. ⟨tel-00011141⟩

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