Complex symmetric composition operators on weighted Hardy spaces

Let φ \varphi be an analytic self-map of the open unit disk D \mathbb {D} . We study the complex symmetry of composition operators C φ C_\varphi on weighted Hardy spaces induced by a bounded sequence. For any analytic self-map of D \mathbb {D} that is not an elliptic automorphism, we establish that if C φ C_{\varphi } is complex symmetric, then either φ ( 0 ) = 0 \varphi (0)=0 or φ \varphi is linear. In the case of weighted Bergman spaces A α 2 A^{2}_{\alpha } , we find the non-automorphic linear fractional symbols φ \varphi such that C φ C_{\varphi } is complex symmetric.