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 | References | Similar Articles | Additional Information
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 closed, if
)
-submodules, say
, of
,
, where
is the harmonic measure on the Martin boundary of a Riemann surface
, and
is the set of boundary functions of all bounded analytic functions on
. Our main result is stated roughly as follows. Let
be of Parreau-Widom type, that is, the space
of bounded analytic sections contains a nonzero element for every complex flat line bundle
. We may assume, without loss of generality, that the Green's function of
vanishes at the infinity. Set
for a fixed point
of
. Then, a necessary and sufficient condition in order that every such an
takes either the form
, where
is the characteristic function of a set
, or the form
, where
a.e. and
is some element of
is that
is continuous for the variable
.
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1983-0709581-6
Keywords:
Invariant subspace,
spaces,
Riemann surfaces
Article copyright:
© Copyright 1983
American Mathematical Society