On a class of weighted composition operators on Fock space

Let $T_\phi$ be the Toeplitz operator defined on the Fock space $L_a^2(\mathbb C)$ with symbol $\phi\in L^\infty(\mathbb C)$. Let for $\lambda\in\mathbb C$, $k_\lambda(z)=e^{\frac{\bar\lambda z}2-\frac{|\lambda|^2}4}$, the normalized reproducing kernel at $\lambda$ for the Fock space $L_a^2(\mathbb C)$ and $t_\alpha(z)=z-\alpha,$ $z,\alpha\in\mathbb C$. Define the weighted composition operator $W_\alpha$ on $L_a^2(\mathbb C)$ as $(W_\alpha f)(z)=k_\alpha(z)(f\circ t_\alpha)(z)$. In this paper we have shown that if $M$ and $H$ are two bounded linear operators from $L_a^2(\mathbb C)$ into itself such that $MT_\psi H=T_{\psi\circ t_\alpha}$ for all $\psi\in L^\infty(\mathbb C)$, then $M$ and $H$ must be constant multiples of the weighted composition operator $W_\alpha$ and its adjoint respectively.