# Journal of Operator Theory

Volume 56, Issue 2, Fall 2006 pp. 423-449.

Composition operators on embedded disks**Authors**: Michael Stessin (1) and Kehe Zhu (2)

**Author institution:**(1) Department of Mathematics, SUNY, Albany, New York 12222, USA

(2) Department of Mathematics, SUNY, Albany, New York 12222, USA

**Summary:**Let $\D$ be the open unit disk in $\C$ and let $\Omega$ be a domain in $\cn$. Every holomorphic map $\varphi:\D\to\Omega$ induces a composition operator $C_\varphi:H(\Omega)\to H(\D)$, where $H(\Omega)$ and $H(\D)$ are the spaces of holomorphic functions in $\Omega$ and $\D$, respectively. We study the action of $C_\varphi$ on the Hardy spaces $H^p(\Omega)$ and the weighted Bergman spaces $A^p_\alpha(\Omega)$ when $\Omega$ is the unit ball or the polydisc. More specifically, we determine the optimal range spaces, prove the boundedness of $C_\varphi$, and characterize the compactness of $C_\varphi$ on these spaces.

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