Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces

@misc{li2026compact,
  title = {{Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces}},
  author = {Yuerang Li and Zipeng Wang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.23733},
  eprint = {2608.23733},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.23733},
}

TY  - UNPB
TI  - Compact Toeplitz operators via the Berezin transform on radial weighted Bergman spaces
AU  - Yuerang Li
AU  - Zipeng Wang
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.23733
UR  - https://arxiv.org/abs/2608.23733
AB  - Let ω be a radial 𝒟̂-weight and u be a bounded function on the unit disk 𝔻. We prove that the Toeplitz operator T_ω,u is compact on A_ω^2 if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for A_ω^2 and a detailed localization analysis of the resulting infinite matrix representation of T_ω,u. Even in the unweighted Bergman space A^2, our argument is new and does not rely on the classical translation operators. We further show that this Axler--Zheng compactness characterization does not extend, in general, to products of Toeplitz operators on 𝒟̂-weighted Bergman spaces, and hence to the corresponding Toeplitz algebra generated by bounded symbols. More precisely, we construct a radial log-subharmonic 𝒟̂-weight ω and bounded symbols u,v such that the product T_ω,vT_ω,u is noncompact, whereas its Berezin transform vanishes at the boundary.
ER  -