Complex symmetry of composition operators on weighted Bergman spaces

In this article, we study the complex symmetry of compositions operators $C_{\varphi }f=f\circ \varphi$ induced on weighted Bergman spaces $A_{\beta }^2(\mathbb{D}),\beta \ge -1,$ by analytic self-maps of the unit disk. One of ours main results shows that $\varphi$ has a fixed point in $\mathbb{D}$ whenever $C_{\varphi }$ is complex symmetric. Our works establishes a strong relation between complex symmetry and cyclicity. By assuming $\beta \in \mathbb{N}$ and $\varphi$ is an elliptic automorphism of $\mathbb{D}$ which not a rotation, we show that $C_{\varphi }$ is not complex symmetric whenever $\varphi$ has order greater than $2(3+\beta )\mathrm{.}$