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ISSN 1088-6826(online) ISSN 0002-9939(print)



Cyclic Nevanlinna class functions in Bergman spaces

Author: Paul Bourdon
Journal: Proc. Amer. Math. Soc. 93 (1985), 503-506
MSC: Primary 30D35; Secondary 47B38
MathSciNet review: 774012
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Abstract: Let $ f$ be a function which is in both the Bergman space $ {A^p}$ $ (p \geq 1)$ and the Nevanlinna class $ N$. We show that if $ f$ is expressed as the quotient of $ {H^\infty }$ functions, then the inner part of its denominator is cyclic. As a corollary, we obtain that $ f$ is cyclic if and only if the inner part of its numerator is cyclic. These results extend those of Berman, Brown, and Cohn [2]. Using more difficult methods, they have obtained them for the case $ f \in {A^2} \cap N$. Finally, we show that the condition $ \vert f(z)\vert \geq \delta {(1 - \vert z\vert)^c}$ ($ z \in D$; $ \delta ,c$ positive constants) is sufficient for cyclicity for $ f \in {A^p} \cap N$, which answers a question of Aharonov, Shapiro, and Shields [1].

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  • [1] D. Aharonov, H. S. Shapiro and A. L. Shields, Weakly invertiale elements in the space of square summable holomorphic functions, J. London Math. Soc. (2) 9 (1974), 183-192. MR 0365150 (51:1403)
  • [2] R. Berman, L. Brown and W. Cohn, Cyclic vectors of bounded characteristic in Bergman spaces, preprint. MR 767609 (86e:46018)
  • [3] P. Duren, Theory of $ {H^p}$ spaces, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [4] B. Korenblum, An extension of the Nevanlinna theory, Acta Math. 135 (1975), 187-219. MR 0425124 (54:13081)
  • [5] -, A Beurling-type theorem, Acta Math. 138 (1977), 265-293. MR 0447584 (56:5894)
  • [6] -, Cyclic elements in some spaces of analytic functions, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 317-318. MR 628662 (82j:30074)
  • [7] J. Roberts, Cyclic inner functions in the Bergman spaces and weak outer functions in $ {H^p}$, $ 0 < p < 1$, Illinois J. Math. (to appear). MR 769756 (86c:30069)
  • [8] H. S. Shapiro, Some remarks on weighted polynomial approximations by holomorphic functions, Math. U.S.S.R. Sbornik 2 (1967), 285-294.
  • [9] J. Shapiro Cyclic inner functions in Bergman spaces, unpublished seminar notes on the results of H. S. Shapior and J. Roberts (1980).

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Article copyright: © Copyright 1985 American Mathematical Society

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