# Invertibility of frame/sampling operator on Bargmann-Fock spaces

Let $F_\alpha ^p (\mathbb{C}^n)$ for $1 < p < \infty$ and $\alpha > 0$ be the Bargmann-Fock space defined as the Banach space of entire functions $f$ such that $f(\cdot) e^{- \frac{\alpha}{2} |\cdot|^2} \in L^p(\mathbb{C}^n, dv)$ where $dv$ is ordinary Lebesgue volume measure and where $F_\alpha ^p (\mathbb{C}^n)$ is given it's natural Banach space norm.

For $\epsilon > 0$ small, treat $\epsilon \mathbb{Z}^{2n}$ canonically as a lattice in $\mathbb{C}^n$. It is well known that $\epsilon \mathbb{Z}^{2n}$ is a "sampling set" for $F_\alpha ^p (\mathbb{C}^n)$, so that there exists constants $C_1, C_2 > 0$ independent of $f \in F_\alpha ^p (\mathbb{C}^n)$ where

\begin{align} C_1 \sum_{\sigma \in \epsilon \mathbb{Z}^{2n} } |f(\sigma)|^p e^{- \frac{\alpha p |\sigma|^2}{2} } \leq \|f\|_ {F_\alpha ^p (\mathbb{C}^n) } ^p \leq C_2 \sum_{\sigma \in \epsilon \mathbb{Z}^{2n} } |f(\sigma)|^p e^{- \frac{\alpha p |\sigma|^2}{2} }. \end{align}

The question is then whether or not the "sampling operator" mapping $F_\alpha ^p (\mathbb{C}^n)$ to $F_\alpha ^p (\mathbb{C}^n)$ given by \begin{align} f \mapsto \sum_{ \sigma \in \epsilon \mathbb{Z}^{2n} } f(\sigma) e^{ \alpha (z \cdot \overline{\sigma} ) - \alpha |\sigma|^2 } \nonumber \end{align} is invertible for small enough $\epsilon$.

Elementary Hilbert space theory says that for $p = 2$ it is, but I don't find where in the literature it is shown for $p \neq 2$, or shown not to be invertible. It's easy to show that this operator is injective and has dense range, but proving surjectivity seems to no easy task.