Invariant subspaces on open Riemann surfaces
Annales de l'Institut Fourier, Tome 24 (1974) no. 4, pp. 241-286.

Soient R une surface de Riemann hyperbolique, d χ une mesure harmonique à support dans la frontière de Martin de R, et H (dχ) la sous-algèbre de L (dχ) formée des valeurs frontières de fonctions holomorphes bornées sur R. On donne une classification complète des H (dχ)-sous-modules fermés de L p (dχ), 1p (σ(L ,L 1 )-fermés, si p=), 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 (dχ) the subalgebra of L (dχ) consisting of the boundary values of bounded analytic functions on R. This paper gives a complete classification of the closed H (dχ)-submodules of L p (dχ), 1p (weakly * closed, if p=, 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.

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     author = {Hasumi, Morisuke},
     title = {Invariant subspaces on open {Riemann} surfaces},
     journal = {Annales de l'Institut Fourier},
     pages = {241--286},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {24},
     number = {4},
     year = {1974},
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     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.541/}
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Hasumi, Morisuke. Invariant subspaces on open Riemann surfaces. Annales de l'Institut Fourier, Tome 24 (1974) no. 4, pp. 241-286. doi : 10.5802/aif.541. http://archive.numdam.org/articles/10.5802/aif.541/

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