Document Type

Article

Publication Date

2012

Publication Title

Complex Analysis and Operator Theory

Volume

6

Issue

1

First Page

139

Last Page

162

Abstract

Brennan’s conjecture in univalent function theory states that if τ is any analytic univalent transform of the open unit disk D onto a simply connected domain G and −1/3 < p < 1, then 1/(τ′) p belongs to the Hilbert Bergman space of all analytic square integrable functions with respect to the area measure. We introduce a class of analytic function spaces L2a(μp) on G and prove that Brennan’s conjecture is equivalent to the existence of compact composition operators on these spaces for every simply connected domain G and all p∈(−1/3,1) . Motivated by this result, we study the boundedness and compactness of composition operators in this setting.

Comments

This is a post-peer-review, pre-copyedit version of an article published in Complex Analysis and Operator Theory. The final authenticated version is available online at: http://dx.doi.org/10.1007/s11785-010-0090-5.

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