On a~class of weighted composition operators on Fock space

Namita Das

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 2 (2014), pp. 3-8.

Résumé

Let $T_{\varphi }$ be the Toeplitz operator defined on the Fock space $L_a^2(\mathbb{C})$ with symbol $\varphi \in L^{\mathrm{\infty }}(\mathbb{C})$. Let for $\lambda \in \mathbb{C}$, $k_{\lambda }(z)=e^{\frac{\overline{\lambda }z}{2}-\frac{|\lambda {|}^2}{4}}$, 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 $M{T}_{\psi }H=T_{\psi \circ t_{\alpha }}$ for all $\psi \in L^{\mathrm{\infty }}(\mathbb{C})$, then $M$ and $H$ must be constant multiples of the weighted composition operator $W_{\alpha }$ and its adjoint respectively.

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