Invariant subspaces on open Riemann surfaces

Morisuke Hasumi

Annales de l'institut Fourier (1974)

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

Abstract

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Let R be a hyperbolic Riemann surface, d χ a harmonic measure supported on the Martin boundary of R , and H ∞ ( d χ ) the subalgebra of L ∞ ( d χ ) consisting of the boundary values of bounded analytic functions on R . This paper gives a complete classification of the closed H ∞ ( d χ ) -submodules of L p ( d χ ) , 1 ≤ p ≤ ∞ (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.

How to cite

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Hasumi, 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

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