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

Borichev, Alexander; kan Hedenmalm, Hå

doi:10.1090/S0894-0347-97-00244-0

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

Authors: Alexander Borichev and H\aa{} kan Hedenmalm
Journal: J. Amer. Math. Soc. 10 (1997), 761-796
MSC (1991): Primary 30H05, 46E15
DOI: https://doi.org/10.1090/S0894-0347-97-00244-0
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, Nikolskii, Shields, Korenblum, Brown, and Frankfurt.

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