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Harmonic functions of maximal growth: invertibility and cyclicity in Bergman spaces

Authors: Alexander Borichev and Hå kan Hedenmalm
Journal: J. Amer. Math. Soc. 10 (1997), 761-796
MSC (1991): Primary 30H05, 46E15
MathSciNet review: 1446365
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Abstract: In the theory of commutative Banach algebras with unit, an element generates a dense ideal if and only if it is invertible, in which case its Gelfand transform has no zeros, and the ideal it generates is the whole algebra. With varying degrees of success, efforts have been made to extend the validity of this result beyond the context of Banach algebras. For instance, for the Hardy space $H^{2}$ on the unit disk, it is known that all invertible elements are cyclic (an element is cyclic if its polynomial multiples are dense), but cyclic elements need not be invertible. In this paper, we supply examples of functions in the Bergman and uniform Bergman spaces on the unit disk which are invertible, but not cyclic. This answers in the negative questions raised by Shapiro, Nikolskiĭ, Shields, Korenblum, Brown, and Frankfurt.

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Additional Information

Alexander Borichev
Affiliation: Department of Mathematics, University of Bordeaux I, 351, cours de la Liberation, 33405 Talence, France

Hå kan Hedenmalm
Affiliation: Department of Mathematics, Lund University, Box 118, 22100 Lund, Sweden

Keywords: Bergman spaces, cyclic vectors, canonical divisors
Received by editor(s): July 18, 1996
Received by editor(s) in revised form: March 25, 1997
Additional Notes: The research of both authors was supported in part by the Swedish Natural Science Research Council. The second author was also supported by the 1992 Wallenberg Prize from the Swedish Mathematical Society.
The paper was written when the authors worked at the Uppsala University, Sweden, and visited MSRI, USA
Article copyright: © Copyright 1997 American Mathematical Society