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 für Mathematik, Universität Zürich, CH-8057 Zürich, 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 $\cA^p_\al$, $-1<\al<\iy$, $0 < p < \iy$. Let $X$ be any of the spaces $\cA_\al^p$, $\cN_\al$ and $Y$ any of the spaces $\cA_\be^q$, $\cN_\be$, $\be>-1$, $0 < q < \iy$. 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.

Keywords: Composition operators, continuity, compactness, order boundedness, weighted Nevanlinna classes, weighted Bergman spaces

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