Sharp noncommutative Hilbert and Riesz inequalities: classification and spectral corrections

Abstract

For every $1<p<\infty$, with $p^*=\max\{p,p/(p-1)\}$, we determine the sharp Hilbert norm $C_p=\cot(\pi/(2p^*))$ on $L^p(\mathbb{R};\mathcal{S}_p)$ and the sharp uniform bound $D_p=\csc(\pi/p)$ 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 Hein\"avaara'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.