Adhérence faible étoile d'algèbres de fractions rationnelles

Chaumat, Jacques

Annales de l'Institut Fourier, Tome 24 (1974), p. 93-120 / Harvested from Numdam

Résumé

Étant donnés un compact $K$ du plan complexe, et une mesure non nulle sur $K$, on étudie $H^\infty (\mu )$, l’adhérence dans $L^\infty (\mu )$, pour la topologie $\sigma (L^\infty (\mu ),\, L^1(\mu ))$, de l’algèbre des fractions rationnelles d’une variable complexe, à pôles hors de $K$. Le résultat principal obtenu est qu’il existe un sous-ensemble $E_\mu $ de $K$, éventuellement vide, mesurable pour la mesure de Lebesgue plane, et une mesure $\mu _s$, éventuellement nulle, absolument continue par rapport à la mesure $\mu $, tels que : $H^\infty (\mu )$ soit isométriquement isomorphe à $H^\infty (\lambda _{E_\mu })\oplus L^\infty (\mu _s)$, où $\lambda _{E_\mu }$ désigne la restriction à $E_\mu $ de la mesure de Lebesgue plane.

Abstract

Let $K$ be a compact subset of the complex plane, and $\mu $ a measure on $K$; we study $H^\infty (\mu )$, the weak star closure in $L^\infty (\mu )$, of the algebra of rational functions with poles off $K$. The main result is the following: there exists a subset $E_\mu $ of $K$, possibly empty, measurable with respect to the Lebesgue measure, and a measure $\mu _s$, possibly equal to zero, absolutely continuous with respect to the measure $\mu $, such that: $H^\infty (\mu )$ is isometrically isomorphic to $H^\infty (\lambda _{E_\mu })\oplus L^\infty (\mu _s)$, with $\lambda _{E_\mu }$ the restriction to $E_\mu $ of the Lebesgue measure.

Détail

Publié le : 1974-01-01

MR 53 #14141 | Zbl 0287.46065 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.533

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