We investigate composition operators between spaces of analytic functions on the unit disk $\De$ in the complex plane. The spaces we consider are the weighted Nevanlinna class $\cN_\al$, which consists of all analytic functions $f$ on $\De$ such that $\int\limits_\De\log^+ |f(z)|(1-|z|^2)^\al {\, {\rm d}x \, {\rm d}y}<\iy$, and the corresponding weighted Bergman spaces $\cA^p_\al$, $-1<\al<\iy$, $0-1$, $0<q<\iy$. We characterize, in function theoretic terms, when the composition operator $\Cf:f\mt f\ci\vf$ induced by an analytic function $\vf:\De\to\De$ defines an operator $X\to Y$ which is continuous, respectively compact, respectively order bounded.

Jarchow, H; Xiao, J (2001). *Composition operators between Nevanlinna classes and Bergman spaces with weights.* Journal of Operator Theory, 46(3, suppl.):605-618.

## Abstract

We investigate composition operators between spaces of analytic functions on the unit disk $\De$ in the complex plane. The spaces we consider are the weighted Nevanlinna class $\cN_\al$, which consists of all analytic functions $f$ on $\De$ such that $\int\limits_\De\log^+ |f(z)|(1-|z|^2)^\al {\, {\rm d}x \, {\rm d}y}<\iy$, and the corresponding weighted Bergman spaces $\cA^p_\al$, $-1<\al<\iy$, $0-1$, $0<q<\iy$. We characterize, in function theoretic terms, when the composition operator $\Cf:f\mt f\ci\vf$ induced by an analytic function $\vf:\De\to\De$ defines an operator $X\to Y$ which is continuous, respectively compact, respectively order bounded.

## Citations

## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Uncontrolled Keywords: | order boundedness; weighted Nevanlinna classes; weighted Bergman spaces; composition operators; spaces of analytic functions |

Language: | English |

Date: | 2001 |

Deposited On: | 29 Nov 2010 16:27 |

Last Modified: | 05 Apr 2016 13:25 |

Publisher: | Theta Foundation |

ISSN: | 0379-4024 |

Free access at: | Related URL. An embargo period may apply. |

Official URL: | http://www.mathjournals.org/jot/2001-046-003/2001-046-003-009.html |

Related URLs: | http://www.ams.org/mathscinet-getitem?mr=1897157 http://www.zentralblatt-math.org/zbmath/search/?q=an%3A0996.47031 |

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