Operators on Fock-type and weighted spaces of entire functions

Abstract

Let us denote by $\mathcal F_1^\phi(\mathbb C)$ the space of entire functions $f\in \mathcal H(\C)$ such that $\int_\mathbb C |f(z)|e^{-\phi(|z|)}dm(z)<\infty$, where $\phi:(0,\infty)\to \mathbb R^+$ is assumed to be continuous and non-decreasing. Also given a continuous non-increasing function $v:(0,\infty)\to \mathbb R^+$ and a complex Banach space $X$, we write $H^\infty_v(\mathbb C,X)$ for the space of $X$-valued entire functions $F$ such that $\sup_{z\in \C} v(z)\|F(z)\|<\infty$. We find a very general class of weights $\phi$ and $v$ for which the space of bounded operators $\mathcal L(\mathcal F_1^\phi(\mathbb C),X)$ can be identified with $H^\infty_v(\mathbb C, X)$.