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Vector-Valued Sub-Bergman Spaces: Range Inclusions and Nonrigidity
Abstract
Let ℰ be a separable Hilbert space, B∈ H∞1(L(ℰ)), and α>−1. Associated with the Toeplitz operator TB on the vector-valued weighted Bergman space A2α,ℰ are the sub-Bergman spaces 𝒜α(B) and 𝒜α(B*). In this paper, we prove that 𝒜α(B)⊃ A2α−1,ℋ(B(0)) and 𝒜α(B*)⊃ A2α−1,ℋ(B(0)*). This confirms a conjecture of \cite{GLM26}. We also construct an infinite-dimensional two-sided inner function B such that |B(0)|<1 and 𝒜α(B)≈ 𝒜α(B*)≈ A2α−1,ℰ, but B is not a finite Blaschke-Potapov product. This example shows that the finite-dimensional rigidity theorem of \cite{GLM26} does not extend to infinite-dimensional coefficient spaces.
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BibTeX
@misc{luo2026vectorvalued,
title = {{Vector-Valued Sub-Bergman Spaces: Range Inclusions and Nonrigidity}},
author = {Shuaibing Luo and Jiming Shen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34694},
eprint = {2609.34694},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34694},
}
RIS
TY - UNPB
TI - Vector-Valued Sub-Bergman Spaces: Range Inclusions and Nonrigidity
AU - Shuaibing Luo
AU - Jiming Shen
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34694
UR - https://arxiv.org/abs/2609.34694
AB - Let ℰ be a separable Hilbert space, B∈ H^∞_1(L(ℰ)), and α>−1. Associated with the Toeplitz operator T_B on the vector-valued weighted Bergman space A^2_α,ℰ are the sub-Bergman spaces 𝒜_α(B) and 𝒜_α(B^*). In this paper, we prove that 𝒜_α(B)⊃ A^2_α−1,ℋ(B(0)) and 𝒜_α(B^*)⊃ A^2_α−1,ℋ(B(0)^*). This confirms a conjecture of \cite{GLM26}. We also construct an infinite-dimensional two-sided inner function B such that |B(0)|<1 and 𝒜_α(B)≈ 𝒜_α(B^*)≈ A^2_α−1,ℰ, but B is not a finite Blaschke-Potapov product. This example shows that the finite-dimensional rigidity theorem of \cite{GLM26} does not extend to infinite-dimensional coefficient spaces.
ER -