发表机构
Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明Hilbert变换在Schatten类上的范数为$C_p$，解决多个猜想，并通过谱缺陷恒等式分类迹不等式，提出最优谱修正。
AI 中文摘要
对于每一个$1<p<\infty$，在$L^p(\mathbb{R};\mathcal{S}_p)$上的Hilbert变换和在$\mathcal{S}_p$上的有序Hilbert Schur乘子具有范数$C_p=\cot(\pi/(2p^*))$，其中$p^*=\max\{p,p/(p-1)\}$。标量Hilbert变换和Schur乘子具有相同的典范完全有界范数。这证明了Gohberg-Krupnik锐利Volterra猜想。一个同时的有限维实现也解决了Riesz-Titchmarsh猜想和Laeng猜想5.7对于所有位移的情况。范数恒等式扩展到具有$\ell^p$直和范围的有限族实Hilbert变换铅笔。证明使用了一个具有两个非负余项的谱缺陷恒等式，基于Heinävaara的迹联合谱测度。它分类了有限符号实脊和迹不等式，并确定最小齐次上调和优控函数为逐点最小加性特征值修正。相关的角锥由非负势刻画，并通过有限幂零矩阵的加权角测度的弱-*极限实现。
英文摘要
For every $1<p<\infty$, the Hilbert transform on $L^p(\mathbb{R};\mathcal{S}_p)$ and the ordered Hilbert Schur multiplier on $\mathcal{S}_p$ have norm $C_p=\cot(π/(2p^*))$, where $p^*=\max\{p,p/(p-1)\}$. The scalar Hilbert transform and the Schur multiplier have the same canonical completely bounded norm. This proves the Gohberg-Krupnik sharp Volterra conjecture. A simultaneous finite-dimensional realization also resolves the Riesz-Titchmarsh conjecture and Laeng's Conjecture 5.7 for all shifts. The norm identities extend to finite families of real Hilbert-transform pencils with $\ell^p$ direct-sum ranges. The proof uses a spectral-defect identity with two nonnegative remainders, based on Heinävaara's tracial joint spectral measure. It classifies trace inequalities for finite signed real ridge sums and identifies the least homogeneous superharmonic majorant as the pointwise least additive eigenvalue correction. The associated angular cone is characterized by nonnegative potentials and realized by weak-* limits of weighted angular measures of finite nilpotent matrices.
Comments35 pages
