Nuclear Toeplitz operators between Fock spaces

TL;DR

This paper characterizes when Toeplitz operators between Fock spaces are nuclear, using Berezin transforms for certain cases and providing new necessary and sufficient conditions, extending results to higher dimensions.

Contribution

It offers a comprehensive characterization of nuclear Toeplitz operators on Fock spaces, including new conditions and extensions to complex Euclidean spaces.

Findings

Necessary and sufficient conditions for nuclearity when p ≥ q.

Rigidity property for nuclearity across p and q.

Extension of results to Fock spaces on c2b5^n.

Abstract

We characterize the nuclearity of Toeplitz operators $T_\mu: F_\alpha^p \to F_\alpha^q$ with Borel measure symbols for $1\leq p,q\leq \infty$. For positive measures $\mu$ and $q\leq p$, we provide necessary and sufficient conditions in terms of the Berezin transform and establish a rigidity property for nuclearity across this range. In the case $p<q$, we obtain separate necessary and sufficient conditions, indicating that the Berezin transform alone is insufficient for a complete characterization. Our results extend to Fock spaces on $\mathbb{C}^n$.