Maximal zero sequences for Fock spaces

Abstract

A sequence $Z$ in the complex plane $\C$ is called a zero sequence for the Fock space $F_{\alpha }^p$ if there exists a function $f\in F_{\alpha }^p$, not identically zero, such that $Z$ is the zero set of $f$, counting multiplicities. We show that there exist zero sequences $Z$ for $F_{\alpha }^p$ with the following properties: (1) For any $a\in\C$ the sequence $Z\cup \{a\}$ is no longer a zero sequence for $F_{\alpha }^p$; (2) the space $I_Z$ consisting of all functions in $F_{\alpha }^p$ that vanish on $Z$ is one dimensional. These $Z$ are naturally called maximal zero sequences for $F_{\alpha }^p$.

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