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

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 $m=2$, $p=4$.

Combinatorics (math.CO); Machine Learning (cs.LG); Functional Analysis (math.FA)

arXiv:2608.15558 [math.CO]

(or arXiv:2608.15558v1 [math.CO] for this version)

https://doi.org/10.48550/arXiv.2608.15558 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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