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 on the unit disk is in the little Bloch space if and only if the uniformly closed algebra on the disk generated by and does not contain the complex conjugate of any interpolating Blaschke product. A version of this result is then used to prove that if and are bounded analytic functions on the unit disk such that the commutator (here denotes the operator of multiplication by on the Bergman space of the disk) is compact, then as .

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DOI:
https://doi.org/10.1090/S0002-9947-1988-0961609-9

Article copyright:
© Copyright 1988
American Mathematical Society