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Journal of Operator Theory

Volume 74, Issue 1, Summer 2015  pp. 149-175.

Constructing Frostman-Blaschke products and applications to operators on weighted Bergman spaces

Authors:  John R. Akeroyd (1) and Pamela Gorkin (2)
Author institution:(1) Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, U.S.A.
(2) Department of Mathematics, Bucknell University, Lewisburg, PA, 17837, U.S.A.

Summary: We give an example of a uniform Frostman--Blaschke product $B$, whose spectrum is a Cantor set, such that the composition operator $C_B$ is not closed-range on any weighted Bergman space $\mathbb{A}_{\alpha}^p$, answering two questions posed in recent papers. We include some general observations about these Blaschke products. Using methods developed in our first example, we improve upon a theorem of V.I. Vasjunin concerning the rate at which the zeros of a uniform Frostman--Blaschke product approach the unit circle.

Keywords: Bergman space, Frostman-Blaschke product, composition operator, harmonic measure

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