A note on complex symmetric composition operators on the Bergman space $A^2(\mathbb{D})$

Abstract

In this note complex symmetric composition operators $C_\varphi$ on the Bergman space $A^2(\mathbb{D})$ are studied. It is shown that if an operator $C_\varphi$ is complex symmetric on $A^2(\mathbb{D})$ then either $\varphi\colon \mathbb{D}\to\mathbb{D}$ has a Denjoy--Wolff point in $\mathbb{D}$ or is an elliptic automorphism of the disc. Moreover in the latter case $\varphi$ is either a rotation or has an order smaller than six.