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

Abstract

For $m\ge 2$, let $c_p(m)$ be the all-dimensional best constant in $${\Vert \sum _{k=1}^m{A}_k\Vert }_p\le c_p(m){\Vert \sum _{k=1}^m|A_k|\Vert }_p.$$ 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 $2\le p<\infty$, 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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