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 left|sumk=1m Akright|p ≤ cp(m)left|sumk=1m |Ak|right|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.