Absolutely summing Toeplitz operators on weighted Fock spaces
Abstract
We study absolutely r-summing Toeplitz operators Tf:Fφp→ Fφq with complex symbols. The weight has a uniformly positive and bounded real Hessian. The main difficulty is that local distance to holomorphic functions does not detect an entire part of the symbol, while the Toeplitz operator does. Using the complete classification of absolutely summing diagonal maps, we characterize absolutely summing Fock--Carleson embeddings. An IDA decomposition, a Rademacher argument, and local scalar recovery then detect the missing analytic part of a complex symbol. We obtain equivalent criteria in terms of the local Lq mean, the complex ball average, and the Berezin transform for every 1≤ p,q,r<∞. At q=1 the criterion is Lsp, where sp=2p/(3p−2) for p≤2 and sp=1 for p>2. We also obtain a sectorial-symbol criterion without an IDA assumption and prove sharpness by separated models.
Related
- Maximal Spaceability of the set of non-absolutely summing operators in quasi-Banach spaces
- Absolutely summing operators on Bergman spaces over convex domains of finite type
- Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces
- IDA symbols and Schatten--Lorentz theory for Toeplitz operators on weighted Fock spaces
- Two-parameter Toeplitz-type operators between Fock spaces
BibTeX
@misc{chunxu2026absolutely,
title = {{Absolutely summing Toeplitz operators on weighted Fock spaces}},
author = {Xu Chunxu and Dong Jianxiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15424},
eprint = {2609.15424},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15424},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Absolutely summing Toeplitz operators on weighted Fock spaces
AU - Xu Chunxu
AU - Dong Jianxiang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15424
UR - https://arxiv.org/abs/2609.15424
AB - We study absolutely r-summing Toeplitz operators T_f:F_φ^p→ F_φ^q with complex symbols. The weight has a uniformly positive and bounded real Hessian. The main difficulty is that local distance to holomorphic functions does not detect an entire part of the symbol, while the Toeplitz operator does. Using the complete classification of absolutely summing diagonal maps, we characterize absolutely summing Fock--Carleson embeddings. An IDA decomposition, a Rademacher argument, and local scalar recovery then detect the missing analytic part of a complex symbol. We obtain equivalent criteria in terms of the local L^q mean, the complex ball average, and the Berezin transform for every 1≤ p,q,r<∞. At q=1 the criterion is L^s_p, where s_p=2p/(3p−2) for p≤2 and s_p=1 for p>2. We also obtain a sectorial-symbol criterion without an IDA assumption and prove sharpness by separated models.
ER -