Closed-range composition operators on $\mathbb{A}^{2}$

Abstract

For analytic self-maps $\varphi$ of the unit disk, we develop a necessary and sufficient condition for the composition operator $C_{\varphi}$ to be closed-range on the classical Bergman space $\mathbb{A}^2$. This condition is relatively easy to apply. Particular attention is given to the case that $\varphi$ is an inner function. Included are observations concerning angular derivatives of Blaschke products. In the case that $\varphi$ is univalent, it is shown that $C_{\varphi}$ is closed-range on $\mathbb{A}^2$ only if $\varphi$ is an automorphism of the disk.