Compact Toeplitz Operators and the Geometry of Fixed-Radius Averages
Abstract
We characterize compact Toeplitz operators with bounded complex-valued symbols on the Bergman space of a bounded strongly pseudoconvex domain with C2 boundary. For any fixed R>0, the operator Tf is compact if and only if the Euclidean-volume averages of f over Bergman geodesic balls of radius R tend to zero at the boundary. This answers Zhu's fixed-radius question affirmatively on the unit ball. On the polydisc 𝔻n, n≥2, averages over products of Bergman discs of fixed radii characterize compactness, whereas averages over genuine Bergman geodesic balls need not. For every sufficiently small fixed radius, we construct a bounded real-valued symbol, smooth in the domain, whose geodesic-ball averages tend to zero along every sequence approaching the boundary, while its Berezin transform converges to a positive value along one such sequence. Boundary scaling reduces the strongly pseudoconvex case to a uniqueness problem for fixed-radius ball averages on complex hyperbolic space. We prove the required uniqueness theorem under a sharp decay condition expressed in terms of a Busemann height function.
Related
- Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces
- Block Toeplitz Operators, Matrix Valued Truncated Toeplitz operators, and Conjugations
- Characterizations of Toeplitz operators on the annulus
- Hyponormal Toeplitz operators with operator valued normal symbols
- Commuting Toeplitz operators with harmonic symbols on the Fock space
BibTeX
@misc{ma2026compact,
title = {{Compact Toeplitz Operators and the Geometry of Fixed-Radius Averages}},
author = {Pan Ma and Zhaoxi Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22893},
eprint = {2610.22893},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22893},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compact Toeplitz Operators and the Geometry of Fixed-Radius Averages
AU - Pan Ma
AU - Zhaoxi Zhang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.22893
UR - https://arxiv.org/abs/2610.22893
AB - We characterize compact Toeplitz operators with bounded complex-valued symbols on the Bergman space of a bounded strongly pseudoconvex domain with C^2 boundary. For any fixed R>0, the operator T_f is compact if and only if the Euclidean-volume averages of f over Bergman geodesic balls of radius R tend to zero at the boundary. This answers Zhu's fixed-radius question affirmatively on the unit ball. On the polydisc 𝔻^n, n≥2, averages over products of Bergman discs of fixed radii characterize compactness, whereas averages over genuine Bergman geodesic balls need not. For every sufficiently small fixed radius, we construct a bounded real-valued symbol, smooth in the domain, whose geodesic-ball averages tend to zero along every sequence approaching the boundary, while its Berezin transform converges to a positive value along one such sequence. Boundary scaling reduces the strongly pseudoconvex case to a uniqueness problem for fixed-radius ball averages on complex hyperbolic space. We prove the required uniqueness theorem under a sharp decay condition expressed in terms of a Busemann height function.
ER -