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