Weighted Frames and the Douglas-Foias problem
Abstract
Let T=(T1,…,Tn) be a commuting tuple of bounded linear operators on a separable Hilbert space 𝒦, and let {aα}α∈ℤ+n be strictly positive weights. Under mild assumptions on the weights, we characterize the weighted frame vectors and weighted Parseval frame vectors, that is, those vectors ξ for which {aαTαξ}α∈ℤ+n is a frame or a Parseval frame. We reduce the problem to characterizing surjective module maps from the reproducing analytic Hilbert module associated with the weights onto the module 𝒦, equipped with the module structure defined by T. It is shown that weighted Parseval frame vectors differ by a unimodular scalar whenever they exist. If the kernel defined by the weights has the complete Nevanlinna--Pick property, then any two associated weighted frame vectors are similar whenever they exist. We also focus on the case of the Hardy kernel on the polydisc, that is, the case aα=1 for all α∈ℤn+. This is related to the famous Douglas--Foias problem: whether two similar quotient modules of the Hardy spaces H2(𝔻n) (n>1) are equal. Our result shows that the answer is negative. This leads to the conclusion that frame vectors for representations of ℤ+n (n>1) may fail to be similar. It means that, for the case aα=1, there exists a tuple T=(T1,…,Tn) with n>1 on 𝒦 and two distinct frame vectors for T which are not similar.
Related
- A Convex Feasibility Problem for Which Douglas-Rachford Converges Only Weakly
- Nonexistence of Approximately Dual Frames via Weighted Composition Operators
- Combinatorial Algorithms for Partial Inverse Min-Max Spanning Tree Problem under the Weighted Sum Hamming Distance
- A problem of Yang and Chen on weighted representation functions
- The Reciprocal Problem on Weighted Bergman Spaces
BibTeX
@misc{dan2026weighted,
title = {{Weighted Frames and the Douglas-Foias problem}},
author = {Hui Dan and Kunyu Guo and Dilong Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15646},
eprint = {2610.15646},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15646},
}
RIS
TY - UNPB
TI - Weighted Frames and the Douglas-Foias problem
AU - Hui Dan
AU - Kunyu Guo
AU - Dilong Li
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15646
UR - https://arxiv.org/abs/2610.15646
AB - Let T=(T_1,…,T_n) be a commuting tuple of bounded linear operators on a separable Hilbert space 𝒦, and let {a_α}_α∈ℤ_+^n be strictly positive weights. Under mild assumptions on the weights, we characterize the weighted frame vectors and weighted Parseval frame vectors, that is, those vectors ξ for which {a_αT^αξ}_α∈ℤ_+^n is a frame or a Parseval frame. We reduce the problem to characterizing surjective module maps from the reproducing analytic Hilbert module associated with the weights onto the module 𝒦, equipped with the module structure defined by T. It is shown that weighted Parseval frame vectors differ by a unimodular scalar whenever they exist. If the kernel defined by the weights has the complete Nevanlinna--Pick property, then any two associated weighted frame vectors are similar whenever they exist. We also focus on the case of the Hardy kernel on the polydisc, that is, the case a_α=1 for all α∈ℤ^n_+. This is related to the famous Douglas--Foias problem: whether two similar quotient modules of the Hardy spaces H^2(𝔻^n) (n>1) are equal. Our result shows that the answer is negative. This leads to the conclusion that frame vectors for representations of ℤ_+^n (n>1) may fail to be similar. It means that, for the case a_α=1, there exists a tuple T=(T_1,…,T_n) with n>1 on 𝒦 and two distinct frame vectors for T which are not similar.
ER -