Sharp noncommutative Hilbert and Riesz inequalities: classification and spectral corrections
Abstract
For every 1<p<∞, with p*=max{p,p/(p−1)}, we determine the sharp Hilbert norm Cp=cot(π/(2p*)) on Lp(ℝ;𝒮p) and the sharp uniform bound Dp=csc(π/p) for retained and strict Riesz projections of maximal subdiagonal algebras in finite tracial von Neumann algebras. The Hilbert value is also the common Fourier--Schur completely bounded norm. For finite matrices, a nonnegative logarithmic determinant defect connects Heinävaara's tracial joint spectral measure to eigenvalues. The resulting transfer identity classifies finite signed real ridge inequalities on nilpotents and identifies their least additive eigenvalue corrections. A dual theorem characterizes weak-* limits of weighted angular measures of finite nilpotents. In general finite tracial algebras, separate positive-kernel and sharp determinant-covariance comparisons yield the Riesz bounds. Consequences include the sharp Volterra constant, Riesz--Titchmarsh and shifted-transform formulas, joint real-pencil norms, and exact triangular and operator-valued Fourier Riesz norms.
Related
BibTeX
@misc{long2026sharp,
title = {{Sharp noncommutative Hilbert and Riesz inequalities: classification and spectral corrections}},
author = {Zhouyu Long},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10158},
eprint = {2610.10158},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10158},
}
RIS
TY - UNPB
TI - Sharp noncommutative Hilbert and Riesz inequalities: classification and spectral corrections
AU - Zhouyu Long
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.10158
UR - https://arxiv.org/abs/2610.10158
AB - For every 1<p<∞, with p^*=max{p,p/(p−1)}, we determine the sharp Hilbert norm C_p=cot(π/(2p^*)) on L^p(ℝ;𝒮_p) and the sharp uniform bound D_p=csc(π/p) for retained and strict Riesz projections of maximal subdiagonal algebras in finite tracial von Neumann algebras. The Hilbert value is also the common Fourier--Schur completely bounded norm. For finite matrices, a nonnegative logarithmic determinant defect connects Heinävaara's tracial joint spectral measure to eigenvalues. The resulting transfer identity classifies finite signed real ridge inequalities on nilpotents and identifies their least additive eigenvalue corrections. A dual theorem characterizes weak-* limits of weighted angular measures of finite nilpotents. In general finite tracial algebras, separate positive-kernel and sharp determinant-covariance comparisons yield the Riesz bounds. Consequences include the sharp Volterra constant, Riesz--Titchmarsh and shifted-transform formulas, joint real-pencil norms, and exact triangular and operator-valued Fourier Riesz norms.
ER -