Compactness of Toeplitz Operators on the Bergman Space
Abstract
Let φ∈ L∞(D). We study compactness criteria for Tφ on the Bergman space A2(D). Axler and Zheng~\cite{AZ1998} established a necessary and sufficient condition for compactness in terms of the Berezin transform. For a general bounded measurable function φ, however, its Berezin transform φ̃ does not readily reveal the intrinsic properties of φ. Motivated by a characterization in terms of the symbol itself, Zhu~\cite{ZhuSlides} proposed a conjecture on compact Toeplitz operators. In this paper, we characterize compactness of Tφ on the unweighted Bergman space in terms of local averages of the symbol. We prove that compactness is equivalent to the vanishing of averages over Bergman disks of any prescribed fixed radius. We also establish an equivalent criterion in terms of Carleson box averages that tend to zero uniformly in the angular variable. Finally, we construct a nonnegative bounded symbol whose Carleson box averages tend to zero at every fixed angle, although the associated Toeplitz operator is not compact.
Related
- Commuting analytic and co-analytic Toeplitz operators on the harmonic Bergman space
- The spectral structure of Toeplitz operators with polynomial symbols on the harmonic Bergman space
- Zero products of Toeplitz operators with harmonic symbols on the Bergman space of the unit ball
- Hypercyclic Bergman-Toeplitz operators with some harmonic symbols
- Toeplitz and Hankel operators on Bergman spaces with doubling weights
BibTeX
@misc{cao2026compactness,
title = {{Compactness of Toeplitz Operators on the Bergman Space}},
author = {Guangfu Cao and Li He and Shuqing Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24614},
eprint = {2609.24614},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24614},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Compactness of Toeplitz Operators on the Bergman Space
AU - Guangfu Cao
AU - Li He
AU - Shuqing Zhang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.24614
UR - https://arxiv.org/abs/2609.24614
AB - Let φ∈ L^∞(D). We study compactness criteria for T_φ on the Bergman space A^2(D). Axler and Zheng~\cite{AZ1998} established a necessary and sufficient condition for compactness in terms of the Berezin transform. For a general bounded measurable function φ, however, its Berezin transform φ̃ does not readily reveal the intrinsic properties of φ. Motivated by a characterization in terms of the symbol itself, Zhu~\cite{ZhuSlides} proposed a conjecture on compact Toeplitz operators. In this paper, we characterize compactness of T_φ on the unweighted Bergman space in terms of local averages of the symbol. We prove that compactness is equivalent to the vanishing of averages over Bergman disks of any prescribed fixed radius. We also establish an equivalent criterion in terms of Carleson box averages that tend to zero uniformly in the angular variable. Finally, we construct a nonnegative bounded symbol whose Carleson box averages tend to zero at every fixed angle, although the associated Toeplitz operator is not compact.
ER -