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Algebras on the disk and doubly commuting multiplication operators


Authors: Sheldon Axler and Pamela Gorkin
Journal: Trans. Amer. Math. Soc. 309 (1988), 711-723
MSC: Primary 46J15; Secondary 47B35
DOI: https://doi.org/10.1090/S0002-9947-1988-0961609-9
MathSciNet review: 961609
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Abstract: We prove that a bounded analytic function $ f$ on the unit disk is in the little Bloch space if and only if the uniformly closed algebra on the disk generated by $ {H^\infty }$ and $ \overline f $ does not contain the complex conjugate of any interpolating Blaschke product. A version of this result is then used to prove that if $ f$ and $ g$ are bounded analytic functions on the unit disk such that the commutator $ {T_f}T_g^{\ast} - T_g^{\ast}{T_f}$ (here $ {T_f}$ denotes the operator of multiplication by $ f$ on the Bergman space of the disk) is compact, then $ (1 - \vert z{\vert^2})\min \{ \vert f' (z)\vert,\;\vert g' (z)\vert\} \to 0$ as $ \vert z\vert \uparrow 1$.


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DOI: https://doi.org/10.1090/S0002-9947-1988-0961609-9
Article copyright: © Copyright 1988 American Mathematical Society

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