Invariant Subspace Theorems for Finite Riemann Surfaces

1966 ◽  
Vol 18 ◽  
pp. 240-255 ◽  
Author(s):  
Morisuke Hasumi
Keyword(s):  
Riemann Surface ◽  
Invariant Subspace ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Invariant Subspaces ◽  
Multiply Connected Domains ◽  
Disk Algebra ◽  
Multiply Connected ◽  
Beurling Theorem ◽  
Finite Riemann Surface

The purpose of this paper is to extend various invariant subspace theorems for the circle group to multiply connected domains. Such attempts are not new. Actually, Sarason (4) studied the invariant subspaces of annulus operators acting on L2 and showed certain parallelisms between the unit disk case and the annulus case. Voichick (8) observed analytic functions on a finite Riemann surface and generalized the Beurling theorem on the closed invariant subspaces of H2 as well as the Beurling–Rudin theorem on the closed ideals of the disk algebra. Here we shall consider LP(Γ) and C(Γ) defined on the boundary Γ of a finite orientable Riemann surface R.

scholarly journals Analytic Functions on Some Riemann Surfaces

1963 ◽  
Vol 22 ◽  
pp. 211-217 ◽  
Author(s):  
Nobushige Toda ◽  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Interesting Theorem ◽  
Hyperbolic Riemann Surface

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

Invariant Subspaces on Riemann Surfaces

1966 ◽  
Vol 18 ◽  
pp. 399-403 ◽  
Author(s):  
Michael Voichick
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Finite Number ◽  
Harmonic Measure ◽  
Riemann Surfaces ◽  
Invariant Subspaces ◽  
Riemann Surface With Boundary ◽  
Analytic Curves

In this paper we generalize to Riemann surfaces a theorem of Helson and Lowdenslager in (2) describing the closed subspaces of L2(﹛|z| = 1﹜) that are invariant under multiplication by eiθ.Let R be a region on a Riemann surface with boundary Γ consisting of a finite number of disjoint simple closed analytic curves such that R ⋃ Γ is compact and R lies on one side of Γ. Let dμ be the harmonic measure on Γ with respect to a fixed point t0 on R. We shall consider the closed subspaces of L2(Γ, dμ) that are invariant under multiplication by functions in A (R) = ﹛F|F continuous on , analytic on R}.

scholarly journals On Level Curves of Harmonic and Analytic Functions on Riemann Surfaces

1969 ◽  
Vol 34 ◽  
pp. 77-87
Author(s):  
Shinji Yamashitad
Keyword(s):  
Riemann Surface ◽  
Subharmonic Function ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Vector Lattice ◽  
Level Curves ◽  
Harmonic Majorant ◽  
Harmonic And Analytic Functions ◽  
Hyperbolic Riemann Surface

In this note we shall denote by R a hyperbolic Riemann surface. Let HP′(R) be the totality of harmonic functions u on R such that every subharmonic function | u | has a harmonic majorant on R. The class HP′(R) forms a vector lattice under the lattice operations:

scholarly journals Open Riemann Surface with Null Boundary

1951 ◽  
Vol 3 ◽  
pp. 73-79 ◽  
Author(s):  
Kiyoshi Noshiro
Keyword(s):  
Riemann Surface ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Capacity Zero

Recently the writer has obtained some results concerning meromorphic or algebroidal functions with the set of essential singularities of capacity zero, with an aid of a theorem of Evans. In the present paper, suggested from recent interesting papers of Sario and Pfluger, the writer will extend his results to single-valued analytic functions defined on open abstract Riemann surfaces with null boundary in the sense of Nevanlinna, using a lemma instead of Evans’ theorem.

On a Uniqueness Theorem

1979 ◽  
Vol 31 (5) ◽  
pp. 1072-1076
Author(s):  
Mikio Niimura
Keyword(s):  
Riemann Surface ◽  
Uniqueness Theorem ◽  
Analytic Functions ◽  
Unit Disc ◽  
Riemann Surfaces ◽  
Boundary Behavior ◽  
Uniqueness Theorems ◽  
Ideal Boundary ◽  
Open Riemann Surface ◽  
Hyperbolic Riemann Surfaces

The classical uniqueness theorems of Riesz and Koebe show an important characteristic of boundary behavior of analytic functions in the unit disc. The purpose of this note is to discuss these uniqueness theorems on hyperbolic Riemann surfaces. It will be necessary to give additional hypotheses because Riemann surfaces can be very nasty. So, in this note the Wiener compactification will be used as ideal boundary of Riemann surfaces. The Theorem, Corollaries 1, 2 and 3 are of Riesz type, Riesz-Nevanlinna type, Koebe type and Koebe-Nevanlinna type respectively. Corollaries 4 and 5 are general forms of Corollaries 2 and 3 respectively.Let f be a mapping from an open Riemann surface R into a Riemann surface R′.

scholarly journals The Riemann surface of a ring

1969 ◽  
Vol 10 (1-2) ◽  
pp. 251-256
Author(s):  
M. G. Stanley
Keyword(s):  
Riemann Surface ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Contravariant Functor ◽  
Trivial Representation ◽  
Analytic Mapping ◽  
Algebra Of Functions ◽  
The Right ◽  
Analytic Mappings

The contravariant functor F from the category of Riemann surfaces and analytic mappings to the category of complex algebras and homomorphisms which takes each surface Ω to the algebra of analytic functions on Ω does not have an adjoint on the right; but it nearly does. To each algebra A there is associated a surface Σ1 (A) and a homomorphism A from A into FΣ1 (A), indeed onto an algebra of functions not all of which are constant on any component of Σ1 (A), such that every such non-trivial representation A A → F(Ω) is induced by a unique analytic mapping Ω → Σ1(A)

scholarly journals Projective structures on an annulus and Hankel forms

1991 ◽  
Vol 33 (3) ◽  
pp. 247-266 ◽  
Author(s):  
Jaak Peetre ◽  
Genkai Zhang
Keyword(s):  
General Theory ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Parameter Family ◽  
Multiply Connected Domains ◽  
Projective Structures ◽  
Multiply Connected ◽  
Integral Metrics ◽  
Several Variables ◽  
Quadratic Integral

A general theory of Hankel forms over domains in one or several variables has been set forth in [6]. In [7] the study of Hankel forms over an annulus in the complex plane ℂ was begun. (An extension of the results of [7] to multiply connected domains was given in [4].) The present paper amplifies the results of [7] in various respects. First of all we define and study more general Hankel forms associated with a one parameter family of projective structures on the annulus. This displays several new features. For instance, we are now dealing with quadratic integral metrics which do not correspond to integration of the square of the function with respect to a weight. Furthermore, whereas in [7] essentially only the issue of the boundedness of Hankel forms was studied, we obtain here rather satisfactory Sp-results, even for 0 < p < 1. The question which remains is, of course, to which extent all this extends to multiply connected domains (or more general (open) Riemann surfaces).

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