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Invariant Subspaces on Riemann Surfaces

Published online by Cambridge University Press:  20 November 2018

Michael Voichick
Michael Voichick*
Affiliation:
Dartmouth College, University of Wisconsin
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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}.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 399 - 403
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Hasumi, M., Invariant subspaces for finite Riemann spaces (ditto copy, 1964).Google Scholar
2. Helson, H. and Lowdenslager, D., Invariant subspaces, Proc. Intern. Symposium on Linear Spaces, Jerusalem (New York, 1961), pp. 251–262.Google Scholar
3. Hoffman, K., Banach spaces of analytic functions (Englewood Cliffs, N.J., 1962).Google Scholar
4. Pfluger, A., Theorie der Riemannschen Flächen (Berlin-Gottingen-Heidelberg, 1957).Google Scholar
5. Royden, H. L., The boundary values of analytic and harmonic functions, Math. Z., 78 (1962), 1–24.Google Scholar
6. Sarason, D., The Hp spaces of annuli (Dissertation, University of Michigan, 1963).Google Scholar
7. Sarason, D., Doubly invariant subspaces of annulus operators, Bull. Amer. Math. Soc., 69(1963), 593–596.Google Scholar
8. Voichick, M., Ideals and invariant subspaces of analytic functions, Trans. Amer. Math. Soc, 111 (1964), 493–512.Google Scholar