Cyclic Nevanlinna class functions in Bergman spaces

Author:
Paul Bourdon

Journal:
Proc. Amer. Math. Soc. **93** (1985), 503-506

MSC:
Primary 30D35; Secondary 47B38

DOI:
https://doi.org/10.1090/S0002-9939-1985-0774012-4

MathSciNet review:
774012

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Abstract: Let be a function which is in both the Bergman space and the Nevanlinna class . We show that if is expressed as the quotient of functions, then the inner part of its denominator is cyclic. As a corollary, we obtain that 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 . Finally, we show that the condition (; positive constants) is sufficient for cyclicity for , which answers a question of Aharonov, Shapiro, and Shields [**1**].

**[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**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*, , 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 U.S.S.R. Sbornik**2**(1967), 285-294.**[9]**J. Shapiro*Cyclic inner functions in Bergman spaces*, unpublished seminar notes on the results of H. S. Shapior and J. Roberts (1980).

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DOI:
https://doi.org/10.1090/S0002-9939-1985-0774012-4

Article copyright:
© Copyright 1985
American Mathematical Society