Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-24T06:16:27.283Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Divisibility in H∞(X)

Published online by Cambridge University Press:  20 November 2018

Frank Forelli
Frank Forelli*
Affiliation:
University of Wisconsin-Madison, Madison, Wisconsin
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a Riemann surface, and H∞(X) the ring of bounded holomorphic functions in X. We offer here a question on divisibility in H∞(X), and then give in Section 2 a condition in which the answer is yes (Corollary 2 to Lemma 1). In Section 3 we use part 2 to prove a theorem on the separation of points by H∞(X). In Section 4 we study X/H∞(X).

If f is meromorphic in X and z ∈ X, then by o(f, z) we mean the order of f at z. (We agree that o(f, z) = ∞ if f ≡ 0.) Let h be memomorphic in X; then h might be said to be of bounded type if h = f/g where f,g ∈ H∞(X), g ≠ 0.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 3 , 01 June 1984 , pp. 458 - 469
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Forster, O., Riemannsche Flachen (Springer-Verlag, Berlin, Heidelberg, New York, 1977).CrossRefGoogle Scholar
2. Heins, M., Algebraic structure and conformai mapping, Trans. Amer. Math. Soc. 89 (1958), 267–276.Google Scholar
3. Royden, H., Algebras of bounded analytic functions on Riemann surfaces, Acta Math. 114 (1965), 113–142.Google Scholar
4. Segawa, S., Bounded analytic functions on unbounded covering surfaces, Pacific J. Math. 79 (1978), 183–187.Google Scholar
5. Selberg, H., Ein Satz uber beschrankte endlichvieldeutige analytische Funktionen, Comm. Math Helv. 9 (1937), 104–108.Google Scholar
6. Stanton, C., Bounded analytic functions on a class of open Riemann surfaces, Pacific J. Math. 59 (1975), 557–565.Google Scholar