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Nuclear Toeplitz operators between Fock spaces

Tengfei Ma, Yufeng Lu, Chao Zu

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 $\mathbb C^n$.

Nuclear Toeplitz operators between Fock spaces

Abstract

We study Toeplitz operators with measure-valued symbols acting between Fock spaces. Given and a Borel measure on , we investigate when the associated Toeplitz operator belongs to the nuclear class. For positive measures and in the range , we obtain necessary and sufficient conditions for the nuclearity of in terms of the Berezin transform of . As a consequence, nuclearity in this setting exhibits a rigidity property: if is nuclear from to for some , then it is nuclear for all such . In the case , 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 .
Paper Structure (8 sections, 19 theorems, 107 equations)

This paper contains 8 sections, 19 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nuclear class Toeplitz operator
  4. Trace formula and density theorem
  5. Funding
  6. Data availability statement
  7. Declarations
  8. Conflict of interest

Key Result

Proposition 2.2

For $1<p<\infty$, the Fock space $F_\alpha^p$ is a Banach space that enjoys both the approximation property and the Radon--Nikodým property.

Theorems & Definitions (24)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Theorem 3.5
  • ...and 14 more