A Compact Counterexample to Suárez's Berezin Approximation Question
Abstract
Let A2(𝔻) be the unweighted Bergman space and write Qm(S)=TBm(S) for the map induced by the mth higher-order Berezin transform. Suárez asked in 2005 whether Qm(S) converges to S in operator norm for every S in the full Bergman Toeplitz algebra. We answer this question negatively in a strong form: there is a compact operator S such that supm≥ 0|Qm(S)|=∞. In particular, along a strictly increasing sequence (mn) one has |Qmn(S)−S|⟶∞. The obstruction is a moving matrix edge. The proof uses a moving family of rank-one test operators, each supported in the mth matrix column. Near the corresponding edge, these test operators are sent to weighted Hankel matrices, and the limiting coefficients form an explicit Pascal kernel. We prove a standalone Pascal--Hankel theorem showing that the associated weighted Hankel transformation fails to map ℓ2 boundedly into trace class. Finite-section convergence, trace duality, and the Uniform Boundedness Principle then transfer this instability to the exact Bergman maps.
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BibTeX
@misc{mitkovski2026compact,
title = {{A Compact Counterexample to Suárez's Berezin Approximation Question}},
author = {Mishko Mitkovski and Yuyuan Ouyang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10846},
eprint = {2609.10846},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10846},
}
RIS
TY - UNPB
TI - A Compact Counterexample to Suárez's Berezin Approximation Question
AU - Mishko Mitkovski
AU - Yuyuan Ouyang
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10846
UR - https://arxiv.org/abs/2609.10846
AB - Let A^2(𝔻) be the unweighted Bergman space and write Q_m(S)=T_B_m(S) for the map induced by the mth higher-order Berezin transform. Suárez asked in 2005 whether Q_m(S) converges to S in operator norm for every S in the full Bergman Toeplitz algebra. We answer this question negatively in a strong form: there is a compact operator S such that sup_m≥ 0|Q_m(S)|=∞. In particular, along a strictly increasing sequence (m_n) one has |Q_m_n(S)−S|⟶∞. The obstruction is a moving matrix edge. The proof uses a moving family of rank-one test operators, each supported in the mth matrix column. Near the corresponding edge, these test operators are sent to weighted Hankel matrices, and the limiting coefficients form an explicit Pascal kernel. We prove a standalone Pascal--Hankel theorem showing that the associated weighted Hankel transformation fails to map ℓ^2 boundedly into trace class. Finite-section convergence, trace duality, and the Uniform Boundedness Principle then transfer this instability to the exact Bergman maps.
ER -