Numerical range of weighted composition operators which contain zero

Abstract

In this paper, we study when zero belongs to the numerical range of weighted composition operators $C_{\psi,\varphi}$ on the Fock space $\mathcal{F}^{2}$, where $\varphi(z)=az+b$, $a,b \in \mathbb{C}$ and $|a|\leq 1$. In the case that $|a|<1$, we obtain a set contained in the numerical range of $C_{\psi,\varphi}$ and find the conditions under which the numerical range of $C_{\psi,\varphi}$ contain zero. Then for $|a|=1$, we precisely determine the numerical range of $C_{\psi,\varphi}$ and show that zero lies in its numerical range.