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arXiv · 2608.15558

A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results

Zijian Zeng·Houde Liu·Kurunathan Ratnavelu
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Abstract

For $m\geq 2$, let $c_p(m)$ be the all-dimensional best constant in $$ \left\|\sum_{k=1}^m A_k\right\|_p \leq c_p(m)\left\|\sum_{k=1}^m |A_k|\right\|_p. $$ Tang and Zhang conjectured an explicit formula for every finite $p>1$. We disprove the conjecture with two explicit real $2\times 2$ rank-one matrices at $p=3/2$. The comparison is certified by seven strict rational inequalities and, in particular, places the attained ratio above $207/200$, while the conjectured constant lies below $207/200$. On the positive side, we prove the conjectured sharp bound for every family of rank-at-most-one summands when $2\leq p<\infty$, and classify all equality cases. We also prove the corresponding endpoint statement for $p=\infty$. Finally, for arbitrary complex matrices, we establish the conjectured sharp constant in the case $m=2$, $p=4$.

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Zijian Zeng, Houde Liu, Kurunathan Ratnavelu. 2026-08-16. A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results. https://arxiv.org/abs/2608.15558

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