Bounded sections on a Riemann surface
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- by Walter Pranger PDF
- Proc. Amer. Math. Soc. 69 (1978), 77-80 Request permission
Abstract:
Let X denote a hyperbolic Riemann surface, $\zeta$ a unitary line bundle, and ${H^\infty }(\zeta )$ the Banach space of bounded holomorphic sections of $\zeta$. If, for a given point $\xi$ in X, the norm of the evaluation functional on ${H^\infty }(\zeta )$ varies continuously with the bundle $\zeta$, then it is shown that the space of bounded holomorphic sections is dense in the space of holomorphic sections for every unitary line bundle.References
- Harold Widom, ${\cal H}_{p}$ sections of vector bundles over Riemann surfaces, Ann. of Math. (2) 94 (1971), 304β324. MR 288780, DOI 10.2307/1970862
- Harold Widom, The maximum principle for multiple-valued analytic functions, Acta Math. 126 (1971), 63β82. MR 279311, DOI 10.1007/BF02392026
- Joel H. Shapiro and Allen L. Shields, Unusual topological properties of the Nevanlinna class, Amer. J. Math. 97 (1975), no.Β 4, 915β936. MR 390227, DOI 10.2307/2373681
- Morisuke Hasumi, Invariant subspaces on open Riemann surfaces. II, Ann. Inst. Fourier (Grenoble) 26 (1976), no.Β 2, viii, 273β299 (English, with French summary). MR 407283
- Errett Bishop, Subalgebras of functions on a Riemann surface, Pacific J. Math. 8 (1958), 29β50. MR 96818
- Errett Bishop, Analyticity in certain Banach algebras, Trans. Amer. Math. Soc. 102 (1962), 507β544. MR 142015, DOI 10.1090/S0002-9947-1962-0142015-8
- H. L. Royden, Algebras of bounded analytic functions on Riemann surfaces, Acta Math. 114 (1965), 113β142. MR 173763, DOI 10.1007/BF02391819 R. Narasimhan, Analysis on real and complex manifolds, North-Holland, Amsterdam, 1973.
- Lawrence Zalcman, Bounded analytic functions on domains of infinite connectivity, Trans. Amer. Math. Soc. 144 (1969), 241β269. MR 252665, DOI 10.1090/S0002-9947-1969-0252665-2
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 69 (1978), 77-80
- MSC: Primary 46J15; Secondary 30A98
- DOI: https://doi.org/10.1090/S0002-9939-1978-0482224-9
- MathSciNet review: 0482224