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Transactions of the American Mathematical Society

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Invariant subspaces on Riemann surfaces of Parreau-Widom type


Author: Mikihiro Hayashi
Journal: Trans. Amer. Math. Soc. 279 (1983), 737-757
MSC: Primary 30D55; Secondary 30F25, 46J15
DOI: https://doi.org/10.1090/S0002-9947-1983-0709581-6
MathSciNet review: 709581
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Abstract: In this paper we generalize Beurling's invariant subspace theorem to the Hardy classes on a Riemann surface with infinite handles. The problem is to classify all closed ( weak$ ^{\ast}$ closed, if $ p = \infty $) $ {H^\infty }(d\chi )$-submodules, say $ \mathfrak{m}$, of $ {L^p}(d\chi )$, $ 1 \leqslant p \leqslant \infty $, where $ d\chi $ is the harmonic measure on the Martin boundary of a Riemann surface $ R$, and $ {H^\infty }(d\chi )$ is the set of boundary functions of all bounded analytic functions on $ R$. Our main result is stated roughly as follows. Let $ R$ be of Parreau-Widom type, that is, the space $ {H^\infty }(R,\gamma )$ of bounded analytic sections contains a nonzero element for every complex flat line bundle $ \gamma \in \pi {(R)^{\ast}}$. We may assume, without loss of generality, that the Green's function of $ R$ vanishes at the infinity. Set $ {m^\infty }(\gamma ) = \sup \{ \vert f({\mathbf{O}})\vert:f \in {H^\infty }(R,\gamma ),\vert f\vert \leqslant 1\} $ for a fixed point $ {\mathbf{O}}$ of $ R$. Then, a necessary and sufficient condition in order that every such an $ \mathfrak{m}$ takes either the form $ \mathfrak{m} = {C_E}{L^p}(d\chi )$, where $ {C_E}$ is the characteristic function of a set $ E$, or the form $ \mathfrak{m} = q{H^p}(d\chi ,\gamma )$, where $ \vert q\vert = 1$ a.e. and $ \gamma $ is some element of $ \pi {(R)^{\ast}}$ is that $ {m^\infty }(\gamma )$ is continuous for the variable $ \gamma \in \pi {(R)^{\ast}}$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0709581-6
Keywords: Invariant subspace, $ {H^p}$ spaces, Riemann surfaces
Article copyright: © Copyright 1983 American Mathematical Society

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