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

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.

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.

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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},
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Hasumi, Morisuke. Invariant subspaces on open Riemann surfaces. Annales de l'Institut Fourier, Volume 24 (1974) no. 4, pp. 241-286. doi : 10.5802/aif.541. http://www.numdam.org/articles/10.5802/aif.541/

[1] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math., 81 (1949), 239-255. | MR | Zbl

[2] M. Brelot et G. Choquet, Espaces et lignes de Green, Ann. Inst. Fourier, Grenoble, 3 (1952), 199-264. | Numdam | MR | Zbl

[3] C. Constantinescu and A. Cornea, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihrer Grenzgebiete, 32, Springer, Berlin, 1963. | MR | Zbl

[4] F. Forelli, Bounded holomorphie functions and projections, Illinois J. Math., 10 (1966), 367-380. | MR | Zbl

[5] M. Hasumi, Invariant subspace theorems for finite Riemann surfaces, Canad. J. Math., 18 (1966), 240-255. | MR | Zbl

[6] C. Neville, Ideals and submodules of analytic functions on infinitely connected plane domains. Thesis, University of Illinois at Urbana-Champaign, 1972.

[7] C. Neville, Invariant subspaces of Hardy classes on infinitely connected plane domains, Bull. Amer. Math. Soc., 78 (1972), 857-860. | MR | Zbl

[8] C. Neville, Invariant subspaces of Hardy classes on infinitely connected open surfaces (to appear). | Zbl

[9] A. Read, A converse of Cauchy's theorem and applications to extremal problems, Acta Math., 100 (1958), 1-22. | MR | Zbl

[10] H. Royden, Boundary values of analytic and harmonic functions, Math. Z., 78 (1962), 1-24. | MR | Zbl

[11] L. Rubel and A. Shields, The space of bounded analytic functions on a region, Ann. Inst. Fourier, Grenoble, 16 (1966), 235-277. | Numdam | MR | Zbl

[12] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, 164, Springer, Berlin, 1970. | MR | Zbl

[13] T. P. Srinivasan, Doubly invariant subspaces, Pacific J. Math., 14 (1964), 701-707. | MR | Zbl

[14] T. P. Srinivasan, Simply invariant subspaces and generalized analytic functions, Proc. Amer. Math. Soc., 16 (1965), 813-818. | MR | Zbl

[15] M. Voichick, Ideals and invariant subspaces of analytic functions, Trans. Amer. Math. Soc., 111 (1964), 493-512. | MR | Zbl

[16] M. Voichick, Invariant subspaces on Riemann surfaces, Canad. J. Math., 18 (1966), 399-403. | MR | Zbl

[17] H. Widom, The maximum principle for multiple-valued analytic functions, Acta Math., 126 (1971), 63-82. | MR | Zbl

[18] H. Widom, Hp sections of vector bundles over Riemann surfaces, Ann. of Math., 94 (1971), 304-324. | MR | Zbl

[19] L. Naïm, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. inst. Fourier, Grenoble, 7 (1957), 183-281. | Numdam | MR | Zbl

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