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

For m ≥ 2, let cp(m) be the all-dimensional best constant in ||m∑k=1 Ak||p ≤ cp(m) || m ∑k=1|Ak| ||p. Tang and Zhang conjectured an explicit formula for every finite p > 1. We disprove the conjecture with two explicit real 2 × 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 ≤ p < ∞, and classify all equality cases. We also prove the corresponding endpoint statement for p = ∞. Finally, for arbitrary complex matrices we establish the conjectured sharp constant in the case m = 2, p = 4.