$C$-normal weighted composition operators on $H^2$

A bounded linear operator $T$ on a separable complex Hilbert space $H$ is called $C$-normal if there is a conjugation $C$ on $H$ such that $ CT^\ast TC=TT^\ast$\mathrm{.}Let$\varphi$bealinearfractionalself-mapof$$\mathbb{D}$$. In this paper, we characterize the necessary and sufficient condition for the composition operator $C_{\phi }$ and weighted composition operator $W_{\psi ,\phi }$ to be $C$-normal with some conjugations $C$ and a function $\psi$.