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