Cyclic Nevanlinna class functions in Bergman spaces
Author:
Paul Bourdon
Journal:
Proc. Amer. Math. Soc. 93 (1985), 503506
MSC:
Primary 30D35; Secondary 47B38
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].
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 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), 183192. 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), 187219. MR 0425124 (54:13081)
 [5]
 , A Beurlingtype theorem, Acta Math. 138 (1977), 265293. MR 0447584 (56:5894)
 [6]
 , Cyclic elements in some spaces of analytic functions, Bull. Amer. Math. Soc. (N.S.) 5 (1981), 317318. 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), 285294.
 [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 J. Shapiro Cyclic inner functions in Bergman spaces, unpublished seminar notes on the results of H. S. Shapior and J. Roberts (1980).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198507740124
PII:
S 00029939(1985)07740124
Article copyright:
© Copyright 1985
American Mathematical Society
