Numerical range of weighted composition operators which contain zero

TL;DR

This paper investigates the conditions under which zero is in the numerical range of weighted composition operators on Fock space, providing explicit characterizations for cases where the symbol's linear coefficient has magnitude less than or equal to one.

Contribution

It offers a detailed analysis of the numerical range of weighted composition operators on Fock space, including exact descriptions when the symbol's linear coefficient has magnitude one.

Findings

Zero belongs to the numerical range when |a|<1 under certain conditions.

The numerical range is precisely determined for |a|=1, including the presence of zero.

Conditions for zero to be in the numerical range are explicitly characterized.

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.