# Invariant subspaces on open Riemann surfaces

Annales de l'institut Fourier (1974)

- Volume: 24, Issue: 4, page 241-286
- ISSN: 0373-0956

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topHasumi, Morisuke. "Invariant subspaces on open Riemann surfaces." Annales de l'institut Fourier 24.4 (1974): 241-286. <http://eudml.org/doc/74202>.

@article{Hasumi1974,

abstract = {Let $R$ be a hyperbolic Riemann surface, $d_\chi $ a harmonic measure supported on the Martin boundary of $R$, and $H^\infty (d\chi )$ the subalgebra of $L^\infty (d\chi )$ consisting of the boundary values of bounded analytic functions on $R$. This paper gives a complete classification of the closed $H^\infty (d\chi )$-submodules of $L^p(d\chi )$, $1\le p\le \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.},

author = {Hasumi, Morisuke},

journal = {Annales de l'institut Fourier},

language = {eng},

number = {4},

pages = {241-286},

publisher = {Association des Annales de l'Institut Fourier},

title = {Invariant subspaces on open Riemann surfaces},

url = {http://eudml.org/doc/74202},

volume = {24},

year = {1974},

}

TY - JOUR

AU - Hasumi, Morisuke

TI - Invariant subspaces on open Riemann surfaces

JO - Annales de l'institut Fourier

PY - 1974

PB - Association des Annales de l'Institut Fourier

VL - 24

IS - 4

SP - 241

EP - 286

AB - Let $R$ be a hyperbolic Riemann surface, $d_\chi $ a harmonic measure supported on the Martin boundary of $R$, and $H^\infty (d\chi )$ the subalgebra of $L^\infty (d\chi )$ consisting of the boundary values of bounded analytic functions on $R$. This paper gives a complete classification of the closed $H^\infty (d\chi )$-submodules of $L^p(d\chi )$, $1\le p\le \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.

LA - eng

UR - http://eudml.org/doc/74202

ER -

## References

top- [1] A. BEURLING, On two problems concerning linear transformations in Hilbert space, Acta Math., 81 (1949), 239-255. Zbl0033.37701MR10,381e
- [2] M. BRELOT et G. CHOQUET, Espaces et lignes de Green, Ann. Inst. Fourier, Grenoble, 3 (1952), 199-264. Zbl0046.32701MR16,34e
- [3] C. CONSTANTINESCU and A. CORNEA, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihrer Grenzgebiete, 32, Springer, Berlin, 1963. Zbl0112.30801MR28 #3151
- [4] F. FORELLI, Bounded holomorphie functions and projections, Illinois J. Math., 10 (1966), 367-380. Zbl0141.31401MR33 #1754
- [5] M. HASUMI, Invariant subspace theorems for finite Riemann surfaces, Canad. J. Math., 18 (1966), 240-255. Zbl0172.41603MR32 #8200
- [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. Zbl0266.46040MR46 #364
- [8] C. NEVILLE, Invariant subspaces of Hardy classes on infinitely connected open surfaces (to appear). Zbl0314.46052
- [9] A. READ, A converse of Cauchy's theorem and applications to extremal problems, Acta Math., 100 (1958), 1-22. Zbl0142.04503MR20 #4640
- [10] H. ROYDEN, Boundary values of analytic and harmonic functions, Math. Z., 78 (1962), 1-24. Zbl0196.33701MR25 #2190
- [11] L. RUBEL and A. SHIELDS, The space of bounded analytic functions on a region, Ann. Inst. Fourier, Grenoble, 16 (1966), 235-277. Zbl0152.13202MR33 #6440
- [12] L. SARIO and M. NAKAI, Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, 164, Springer, Berlin, 1970. Zbl0199.40603MR41 #8660
- [13] T. P. SRINIVASAN, Doubly invariant subspaces, Pacific J. Math., 14 (1964), 701-707. Zbl0136.10904MR29 #1528
- [14] T. P. SRINIVASAN, Simply invariant subspaces and generalized analytic functions, Proc. Amer. Math. Soc., 16 (1965), 813-818. Zbl0136.11002MR34 #8219
- [15] M. VOICHICK, Ideals and invariant subspaces of analytic functions, Trans. Amer. Math. Soc., 111 (1964), 493-512. Zbl0147.11502MR28 #4129
- [16] M. VOICHICK, Invariant subspaces on Riemann surfaces, Canad. J. Math., 18 (1966), 399-403. Zbl0147.11501MR32 #8201
- [17] H. WIDOM, The maximum principle for multiple-valued analytic functions, Acta Math., 126 (1971), 63-82. Zbl0203.38302MR43 #5034
- [18] H. WIDOM, Hp sections of vector bundles over Riemann surfaces, Ann. of Math., 94 (1971), 304-324. Zbl0238.32014MR44 #5976
- [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. Zbl0086.30603MR20 #6608

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