Compact Weighted Composition Operators Between Generalized Fock Spaces

Abstract

Let $\psi$ be an entire self-map of the $n$-dimensional Euclidean complex space $\mathbb{C}^n$ and $u$ be an entire function on $\mathbb{C}^n$. A weighted composition operator induced by $\psi$ with weight $u$ is given by $(uC_{\psi}f)(z)= u(z)f(\psi(z))$, for $z \in \mathbb{C}^n$ and $f$ is the entire function on $\mathbb{C}^n$. In this paper, we study weighted composition operators acting between generalized Fock-types spaces. We characterize the boundedness and compactness of these operators act between $\mathcal{F}_{\phi}^{p}(\mathbb{C}^n)$ and $\mathcal{F}_{\phi}^{q}(\mathbb{C}^n)$ for $0\lt p, q\leq\infty$. Moreover, we give estimates for the Fock-norm of $uC_{\psi}: \mathcal{F}_{\phi}^{p}\rightarrow \mathcal{F}_{\phi}^{q}$ when $0\lt p, q\lt \infty$, and also when $p=\infty$ and $0\lt q\lt \infty$.