Numerical range of weighted composition operators on the Fock space

Abstract

We consider the numerical range of a bounded weighted composition operator $C_{\psi ,\phi }$ on the Fock space ${\mathcal{F}}^2$, where the entire function φ must have the form az + b with a and b in $\mathbb{C}$ and $|a|\le 1$. We obtain necessary and sufficient conditions for $\{\psi (p)a^n:n\text{is a non-negative integer}\}$ to be a subset of the interior of the numerical range of $C_{\psi ,\phi }$, where p is the fixed point of φ. We characterize when 0 belongs to the numerical range of a weighted composition operator and determine which weighted composition operators have numerical ranges with no corner points. Furthermore, we describe the corner points of the closure of the numerical range of a compact weighted composition operator. Moreover, we precisely determine the numerical range of $C_{\psi ,\phi }$ when $|a|=1$.

Keywords:

• Fock space
• weighted composition operator
• numerical range

AMS Subject Classifications:

• 47B33
• 30H20
• 46E22
• 47A12

Disclosure statement

No potential conflict of interest was reported by the author(s).