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

• URL: http://arxiv.org/abs/2608.15558v1
• Date: Sun, 16 Aug 2026 06:10:33 GMT
• Title: A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results
• Abstract summary: Tang and Zhang conjectured an explicit formula for every finite $p>1$.<n>We prove the conjectured sharp bound for every family of rank-at-most-one summands when $2leq pinfty$.
• Score: 1.8051706864605614
• License: http://creativecommons.org/licenses/by/4.0/
• 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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