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A counterexample concerning the coefficient formula for de Branges-Rovnyak norms
Abstract
Let ℋ(b) be the de Branges-Rovnyak space associated to a non-extreme point b of the unit ball of H∞, and let ϕ=b/a, where a is the Pythagorean mate of b. In an earlier article, the author showed that, if ϕ∈ H2, then f∈ℋ(b) ⟺ (f∈ H2 and ∑m≥0|∑n≥0ϕ̂(n)̅f̂(m+n)|2<∞), in which case we have |f|ℋ(b)2=∑m≥0|f̂(m)|2 +∑m≥0|∑n≥0ϕ̂(n)̅f̂(m+n)|2. It was also shown that there exists a space ℋ(b) for which ϕ∈∩p<2Hp, yet f∈ℋ(b)not⇒ ∑m≥0|∑n≥0ϕ̂(n)̅f̂(m+n)|2<∞. Here we complete this circle of ideas by showing that there exists a space ℋ(b) for which ϕ∈∩p<1Hp, yet (f∈ H2 and ∑m≥0|∑n≥0ϕ̂(n)̅f̂(m+n)|2<∞) not⇒ f∈ℋ(b).
Related
BibTeX
@misc{ransford2026counterexample,
title = {{A counterexample concerning the coefficient formula for de Branges-Rovnyak norms}},
author = {Thomas Ransford},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15019},
eprint = {2610.15019},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15019},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A counterexample concerning the coefficient formula for de Branges-Rovnyak norms
AU - Thomas Ransford
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15019
UR - https://arxiv.org/abs/2610.15019
AB - Let ℋ(b) be the de Branges-Rovnyak space associated to a non-extreme point b of the unit ball of H^∞, and let ϕ=b/a, where a is the Pythagorean mate of b. In an earlier article, the author showed that, if ϕ∈ H^2, then f∈ℋ(b) ⟺ (f∈ H^2 and ∑_m≥0|∑_n≥0ϕ̂(n)̅f̂(m+n)|^2<∞), in which case we have |f|_ℋ(b)^2=∑_m≥0|f̂(m)|^2 +∑_m≥0|∑_n≥0ϕ̂(n)̅f̂(m+n)|^2. It was also shown that there exists a space ℋ(b) for which ϕ∈∩_p<2H^p, yet f∈ℋ(b)not⇒ ∑_m≥0|∑_n≥0ϕ̂(n)̅f̂(m+n)|^2<∞. Here we complete this circle of ideas by showing that there exists a space ℋ(b) for which ϕ∈∩_p<1H^p, yet (f∈ H^2 and ∑_m≥0|∑_n≥0ϕ̂(n)̅f̂(m+n)|^2<∞) not⇒ f∈ℋ(b).
ER -