Binormal and complex symmetric weighted composition operators on the Fock Space over $\mathbb{C}$

In this paper, we give simple characterization of binormal weighted composition operators $C_{\psi, \phi}$ on the Fock space over $\mathbb{C}$ where weight function is of the form $\psi(\zeta) = e^{\langle \zeta, c \rangle}$ for some $c \in \mathbb{C}$. We derive conditions for $C_{\phi}$ to be binormal such that $C^*_{\phi}C_{\phi}$ and $C^*_{\phi} + C_{\phi}$ commute. Finally we give some simple characterization of binormal weighted composition operator to be complex symmetric.