Comparing positive Toeplitz operators on holomorphic and pluriharmonic Fock spaces
Abstract
We prove sharp norm comparisons between positive Toeplitz operators associated with the same positive Borel measure, acting respectively on holomorphic functions and on sums of holomorphic and antiholomorphic functions, both equipped with the Gaussian-weighted L2 norm. For every positive Borel measure μ on ℂn and 0<p<∞, both operators are compact with summable pth powers of their singular values (i.e. Schatten) exactly when z↦μ(B(z,r)) belongs to Lp(ℂn) for some fixed radius r>0. For other weights, this agreement can fail dramatically. We construct a weight e−2ϕ and a finite positive measure μ on ℂ for which the operator on holomorphic functions has the same singular values as with the Gaussian weight and the same measure μ. These values have summable pth powers for every p>0, yet the quadratic form on sums of holomorphic and antiholomorphic functions is unbounded in weighted L2. The example still satisfies Δϕ=2, as for the Gaussian choice ϕ(z)=|z|2/2.
Related
- Positive Toeplitz Operators Between Gaussian Mixed-Norm Fock Spaces
- Hölder Continuity and Boundedness of Composition Operators on Pluriharmonic α-Bloch Spaces
- Siu's analyticity theorem for positive pluriharmonic currents
- Block Toeplitz Operators, Matrix Valued Truncated Toeplitz operators, and Conjugations
- Characterizations of Toeplitz operators on the annulus
BibTeX
@misc{looi2026comparing,
title = {{Comparing positive Toeplitz operators on holomorphic and pluriharmonic Fock spaces}},
author = {Sam Looi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.13550},
eprint = {2608.13550},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.13550},
}
RIS
TY - UNPB
TI - Comparing positive Toeplitz operators on holomorphic and pluriharmonic Fock spaces
AU - Sam Looi
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2608.13550
UR - https://arxiv.org/abs/2608.13550
AB - We prove sharp norm comparisons between positive Toeplitz operators associated with the same positive Borel measure, acting respectively on holomorphic functions and on sums of holomorphic and antiholomorphic functions, both equipped with the Gaussian-weighted L^2 norm. For every positive Borel measure μ on ℂ^n and 0<p<∞, both operators are compact with summable pth powers of their singular values (i.e. Schatten) exactly when z↦μ(B(z,r)) belongs to L^p(ℂ^n) for some fixed radius r>0. For other weights, this agreement can fail dramatically. We construct a weight e^−2ϕ and a finite positive measure μ on ℂ for which the operator on holomorphic functions has the same singular values as with the Gaussian weight and the same measure μ. These values have summable pth powers for every p>0, yet the quadratic form on sums of holomorphic and antiholomorphic functions is unbounded in weighted L^2. The example still satisfies Δϕ=2, as for the Gaussian choice ϕ(z)=|z|^2/2.
ER -