Complex Symmetry of Invertible Composition Operators on Weighted Bergman Spaces

Abstract

In this article, we study the complex symmetry of composition operators $C_{\phi }f=f\circ \phi$ induced on the weighted Bergman spaces $A^2_{\beta }(\mathbb {D}),$ by analytic self-maps of the unit disk. One of our main results shows that if $C_\phi$ is complex symmetric then $\phi$ must fix a point in $\mathbb {D}$. From this, we prove that if $\phi$ is neither constant nor an elliptic automorphism of $\mathbb {D}$ and $C_{\phi }$ is complex symmetric then $C_{\phi }$ and $C_{\phi }^*$ are cyclic operators. Moreover, by assuming $\phi$ is an elliptic automorphism of $\mathbb {D}$ which not a rotation and $\beta \in \mathbb {N},$ we show that $C_{\phi }$ is not complex symmetric whenever $\phi$ has order greater than $2(3+\beta ).$