Maximal zero sequences for Fock spaces

TL;DR

This paper characterizes maximal zero sequences in Fock spaces, showing their unique properties and establishing their maximality with respect to extension and the dimension of the vanishing function space.

Contribution

It introduces the concept of maximal zero sequences for Fock spaces and proves their existence with specific properties, advancing understanding of zero sets in these spaces.

Findings

Existence of zero sequences that cannot be extended by any point

The space of functions vanishing on these sequences is one-dimensional

Maximal zero sequences are characterized by their maximality and uniqueness

Abstract

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