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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
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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

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