### 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} _{a}(μ _{p}) 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 language | English (US) |
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Pages (from-to) | 139-162 |

Number of pages | 24 |

Journal | Complex Analysis and Operator Theory |

Volume | 6 |

Issue number | 1 |

DOIs | |

State | Published - Feb 1 2012 |

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### ASJC Scopus subject areas

- Computational Mathematics
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

**Composition Operators on a Class of Analytic Function Spaces Related to Brennan's Conjecture.** / Matache, Valentin; Smith, Wayne.

Research output: Contribution to journal › Article

*Complex Analysis and Operator Theory*, vol. 6, no. 1, pp. 139-162. https://doi.org/10.1007/s11785-010-0090-5

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TY - JOUR

T1 - Composition Operators on a Class of Analytic Function Spaces Related to Brennan's Conjecture

AU - Matache, Valentin

AU - Smith, Wayne

PY - 2012/2/1

Y1 - 2012/2/1

N2 - 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 a(μ p) 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.

AB - 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 a(μ p) 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.

UR - http://www.scopus.com/inward/record.url?scp=84856214956&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84856214956&partnerID=8YFLogxK

U2 - 10.1007/s11785-010-0090-5

DO - 10.1007/s11785-010-0090-5

M3 - Article

AN - SCOPUS:84856214956

VL - 6

SP - 139

EP - 162

JO - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

SN - 1661-8254

IS - 1

ER -