Invariant subspaces on open Riemann surfaces

Hasumi, Morisuke

Annales de l'Institut Fourier, Tome 24 (1974), p. 241-286 / Harvested from Numdam

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Abstract

Soient $R$ une surface de Riemann hyperbolique, $d_{χ}$ une mesure harmonique à support dans la frontière de Martin de $R$ , et $H^{\infty} (d χ)$ la sous-algèbre de $L^{\infty} (d χ)$ formée des valeurs frontières de fonctions holomorphes bornées sur $R$ . On donne une classification complète des $H^{\infty} (d χ)$ -sous-modules fermés de $L^{p} (d χ)$ , $1 \leq p \leq \infty$ ( $σ (L^{\infty}, L^{1})$ -fermés, si $p = \infty$ ), lorsque $R$ est régulière et admet une famille suffisamment grande de fonctions analytiques multiplicatives bornées satisfaisant une condition d’approximation. On en déduit un résultat correspondant pour les espaces de Hardy sur $R$ . Pour établir le résultat principal, on démontre et utilise un théorème de Cauchy généralisé et sa réciproque pour $R$ . La théorie des lignes de Green est aussi utilisée effectivement.

Let $R$ be a hyperbolic Riemann surface, $d_{χ}$ a harmonic measure supported on the Martin boundary of $R$ , and $H^{\infty} (d χ)$ the subalgebra of $L^{\infty} (d χ)$ consisting of the boundary values of bounded analytic functions on $R$ . This paper gives a complete classification of the closed $H^{\infty} (d χ)$ -submodules of $L^{p} (d χ)$ , $1 \leq p \leq \infty$ (weakly $^{*}$ closed, if $p = \infty$ , when $R$ is regular and admits a sufficiently large family of bounded multiplicative analytic functions satisfying an approximation condition. It also gives, as a corollary, a corresponding result for the Hardy spaces on $R$ . A generalized Cauchy theorem and its converse for $R$ are proved in the course of establishing the main result. The theory of Green lines is also used effectively.

Publié le : 1974-01-01

MR 51 #901 | Zbl 0287.46066 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.541

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     author = {Hasumi, Morisuke},
     title = {Invariant subspaces on open Riemann surfaces},
     journal = {Annales de l'Institut Fourier},
     volume = {24},
     year = {1974},
     pages = {241-286},
     doi = {10.5802/aif.541},
     mrnumber = {51 \#901},
     zbl = {0287.46066},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1974__24_4_241_0}
}

Hasumi, Morisuke. Invariant subspaces on open Riemann surfaces. Annales de l'Institut Fourier, Tome 24 (1974) pp. 241-286. doi : 10.5802/aif.541. http://gdmltest.u-ga.fr/item/AIF_1974__24_4_241_0/

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