Generalized weighted composition operators on weighted Hardy spaces

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

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