Schatten limits for semi-commutators in Toeplitz quantization on Fock space
Abstract
We solve a problem posed by Hu and Virtanen by characterizing, for every 0<p<∞, the bounded symbols f for which Tf(t)Tg(t)−Tfg(t) tends to zero in the unnormalized Schatten p-norm or quasi-norm as t→0+ for every fixed bounded g. The setting is weighted Fock spaces over ℂn with weights whose real Hessians are uniformly positive and bounded. Only constants satisfy the limit when p<2n; at p=2n, it is equivalent to the weak real gradient belonging to L2n(ℝ2n). When p>2n, the limit is characterized by relative weak compactness in L1(ℝ2n) of normalized local mean oscillation densities. For p≥2, uniform convergence over |g|∞≤1 is equivalent to vanishing density mass. A continuous compactly supported counterexample for p>2n shows that bounded density mass and convergence for g=f̅ do not imply convergence for every fixed bounded g. The symbol classes are weight-independent unital algebras, yielding limits for finite Toeplitz products and commutators; quantitative Hölder estimates also give finite-rank approximation bounds. The Gaussian case transfers to anti-Wick quantization by unitary equivalence. On compact Kähler manifolds with positive line bundles, we prove a Hölder sufficient condition, including the endpoint, for each fixed bounded second symbol. A projective-line example shows that trace norm convergence can fail for a smooth first symbol and a fixed bounded second symbol. The proofs combine local holomorphic approximation, Schatten and Sobolev estimates, singular integral methods, and the Dunford--Pettis criterion.
Related
- The Skew Commutators of Toeplitz and Hankel operators in vector-valued Hardy Space
- Single commutators in Schatten ideals
- Endpoint Schatten class properties of commutators on quantum Euclidean spaces
- Schatten--Lorentz properties of commutators with matrix weights
- Endpoint Schatten class properties of commutators on quantum tori
BibTeX
@misc{qin2026schatten,
title = {{Schatten limits for semi-commutators in Toeplitz quantization on Fock space}},
author = {Jie Qin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30787},
eprint = {2610.30787},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30787},
}
RIS
TY - UNPB
TI - Schatten limits for semi-commutators in Toeplitz quantization on Fock space
AU - Jie Qin
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30787
UR - https://arxiv.org/abs/2610.30787
AB - We solve a problem posed by Hu and Virtanen by characterizing, for every 0<p<∞, the bounded symbols f for which T_f^(t)T_g^(t)−T_fg^(t) tends to zero in the unnormalized Schatten p-norm or quasi-norm as t→0^+ for every fixed bounded g. The setting is weighted Fock spaces over ℂ^n with weights whose real Hessians are uniformly positive and bounded. Only constants satisfy the limit when p<2n; at p=2n, it is equivalent to the weak real gradient belonging to L^2n(ℝ^2n). When p>2n, the limit is characterized by relative weak compactness in L^1(ℝ^2n) of normalized local mean oscillation densities. For p≥2, uniform convergence over |g|_∞≤1 is equivalent to vanishing density mass. A continuous compactly supported counterexample for p>2n shows that bounded density mass and convergence for g=f̅ do not imply convergence for every fixed bounded g. The symbol classes are weight-independent unital algebras, yielding limits for finite Toeplitz products and commutators; quantitative Hölder estimates also give finite-rank approximation bounds. The Gaussian case transfers to anti-Wick quantization by unitary equivalence. On compact Kähler manifolds with positive line bundles, we prove a Hölder sufficient condition, including the endpoint, for each fixed bounded second symbol. A projective-line example shows that trace norm convergence can fail for a smooth first symbol and a fixed bounded second symbol. The proofs combine local holomorphic approximation, Schatten and Sobolev estimates, singular integral methods, and the Dunford--Pettis criterion.
ER -