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The Critical Value for the Beurling-Type Theorem in Weighted Bergman Spaces
Abstract
For every α>1 we construct a finite set A⊂𝔻 such that the zero-based invariant subspace IA fails the wandering subspace property. Consequently, the Beurling-type theorem holds on the weighted Bergman space A2α if and only if −1<α≤1, confirming a conjecture of Shimorin. As a further application, we extend the prescribed-curvature construction of Hedenmalm and Perdomo to all α>1, thereby replacing the constant α0≈1.04 in their result by $1$.
Related
- The Beurling-type theorem for powers of weighted Bergman shifts
- Norm of the generalized Hilbert operator on weighted Bergman spaces
- Compact linear combination of composition operators on weighted Bergman-Orlicz spaces induced by doubling weights
- Localization and weighted theory on the Bergman spaces
- Characterizations for quasiconformal weighted Bergman spaces
BibTeX
@misc{lin2026critical,
title = {{The Critical Value for the Beurling-Type Theorem in Weighted Bergman Spaces}},
author = {Zhaopeng Lin and Shibo Xu and Tao Yu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.36723},
eprint = {2609.36723},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36723},
}
RIS
TY - UNPB
TI - The Critical Value for the Beurling-Type Theorem in Weighted Bergman Spaces
AU - Zhaopeng Lin
AU - Shibo Xu
AU - Tao Yu
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2609.36723
UR - https://arxiv.org/abs/2609.36723
AB - For every α>1 we construct a finite set A⊂𝔻 such that the zero-based invariant subspace I_A fails the wandering subspace property. Consequently, the Beurling-type theorem holds on the weighted Bergman space A^2_α if and only if −1<α≤1, confirming a conjecture of Shimorin. As a further application, we extend the prescribed-curvature construction of Hedenmalm and Perdomo to all α>1, thereby replacing the constant α_0≈1.04 in their result by $1$.
ER -