Normal complex symmetric weighted composition operators on the Hardy space

In this paper, we investigate the normal weighed composition operators $W_{\psi,\varphi}$ which is $\mathcal{J}-$symmetric, $\mathcal{C}_1-$symmetric and $\mathcal{C}_2-$symmetric on the Hardy space $H^2(\mathbb{D})$ respectively. Firstly, equivalent conditions of the normality of $\mathcal{C}_1-$symmetric and $\mathcal{C}_2-$symmetric weighted composition operators on $H^2(\mathbb{D})$ is given. Furthermore, the normal $\mathcal{J}-$symmetric, $\mathcal{C}_1-$symmetric and $\mathcal{C}_2-$symmetric weighted composition operators on $H^2(\mathbb{D})$ when $\varphi$ has an interior fixed point, $\varphi$ is of hyperbolic type or parabolic type are respectively investigated.