Generalized weighted composition operators on weighted Hardy spaces

In this paper, we investigate the complex symmetric structure of generalized weighted composition operators $D_{m,\psi ,\phi }$ on the weighted Hardy space $H^2(\beta )$. We obtain explicit conditions for $D_{m,\psi ,\phi }$ to be complex symmetric with the conjugation $J_w$. Under the assumption that $D_{m,\psi ,\phi }$ is $J_w$-symmetric, some sufficient and necessary conditions for $D_{m,\psi ,\phi }$ to be Hermitian and normal are given.