Journal of Operator Theory

Volume 46, Issue 3, Supplementary 2001 pp. 605-618.

Composition operators between Nevanlinna classes and Bergman spaces with weights

Authors: Hans Jarchow (1), and Jie Xiao (2)
Author institution: (1) Institut f\"ur Mathematik, Universit\"at Z\"urich, CH-8057 Z\"urich, Switzerland
(2) Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneue Blvd. West, H3G 1M8 Montreal, Qc, Canada

Summary: 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.

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