Nuclear Toeplitz operators between Fock spaces

Abstract: We study Toeplitz operators with measure-valued symbols acting between Fock spaces. Given $1\le p,q\le\infty$ and a Borel measure $μ$ on $\mathbb C$, we investigate when the associated Toeplitz operator [ T_μ: F^p_α\to F^q_α] belongs to the nuclear class. For positive measures $μ$ and in the range $1\le q\le p\le\infty$, we obtain necessary and sufficient conditions for the nuclearity of $T_μ$ in terms of the Berezin transform of $μ$. As a consequence, nuclearity in this setting exhibits a rigidity property: if $T_μ$ is nuclear from $F^p_α$ to $F^q_α$ for some $q\le p$, then it is nuclear for all such $q$. In the case $p<q$, we show that the situation is more delicate. We provide separate necessary and sufficient conditions for nuclearity, indicating that the Berezin transform alone does not yield a complete characterization. The proofs rely on tools from Banach space operator theory combined with kernel estimates on Fock spaces. Our results extend naturally to Fock spaces on $\mathbb C^n$.