Complex Symmetry of Invertible Composition Operators on Weighted Bergman Spaces

In this article, we study the complex symmetry of composition operators C(phi)f = f circle phi induced on the weighted Bergman spaces A(beta)(2) (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 D. From this, we prove that if phi is neither constant nor an elliptic automorphism of 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 D which not a rotation and beta is an element of N, we show that C-phi is not complex symmetric whenever phi has order greater than 2(3 + beta).