Composition Operators on a Class of Analytic Function Spaces Related to Brennan's Conjecture

Valentin Matache, Wayne Smith

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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 L 2 ap) 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 ε (-1/3,1). Motivated by this result, we study the boundedness and compactness of composition operators in this setting.

Original languageEnglish (US)
Pages (from-to)139-162
Number of pages24
JournalComplex Analysis and Operator Theory
Volume6
Issue number1
DOIs
StatePublished - Feb 2012

ASJC Scopus subject areas

  • Computational Mathematics
  • Computational Theory and Mathematics
  • Applied Mathematics

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