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

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

Problem Statement
The paper investigates the sharp constant $c_p(m)$ in the inequality relating the Schatten $p$-norm of a sum of matrices to the Schatten $p$-norm of the sum of their absolute values. Specifically, for $m \ge 2$ and $1 \le p \le \infty$, the authors consider the inequality:
$$\left\| \sum_{k=1}^m A_k \right\|_p \le c_p(m) \left\| \sum_{k=1}^m |A_k| \right\|_p$$
where $A_k \in M_n(\mathbb{C})$, $|A| = (A^*A)^{1/2}$, and $\|A\|_p = (\text{Tr}|A|^p)^{1/p}$. The constant $c_p(m)$ is defined as the supremum over all dimensions $n$ and non-zero families $(A_1, \dots, A_m)$.

Tang and Zhang previously established exact values for $p=1, 2, \infty$ and conjectured an explicit formula for all finite $p > 1$. Their conjecture posits that the sharp constant is given by $C^{TZ}_{p,m}$, derived from the unique root $x_{p,m} > 1$ of the equation $x^p - 2x - (m-1) = 0$. This lower bound is known to be attained by rank-one equiangular families. The central question addressed is whether this conjectured formula represents the true sharp constant for general matrices and general $p$.

Methodology
The authors employ a combination of explicit algebraic construction, rational arithmetic verification, and variational analysis:

1. Explicit Counterexample Construction: To disprove the conjecture, the authors construct a specific family of real $2 \times 2$ rank-one matrices ($m=2, n=2$) at $p=3/2$. They define vectors $e, u, v$ and matrices $A_1 = ee^T$ and $A_2 = uv^T$.
2. Rational Certification: Instead of relying on floating-point approximations, the authors compute the ratio of the norms for their counterexample and the conjectured constant using exact rational arithmetic. They establish seven strict rational inequalities to prove that the attained ratio strictly exceeds the conjectured constant.
3. Reduction to Scalar Optimization: For the positive results, the authors reduce the matrix inequality problem to a scalar optimization problem. By expressing rank-one matrices as $A_k = r_k u_k v_k^*$, they relate the Schatten norms to the eigenvalues of Gram matrices $L = U^*U$ and $G = V^*V$. This allows the application of convexity arguments and eigenvalue majorization.
4. Variational Analysis for $p=4$: For the case of arbitrary matrices with $m=2$ and $p=4$, the authors utilize the polar decomposition and unitary invariance. They expand the trace of $(H+K)^4$ (where $H=|A|, K=|B|$) and apply Hölder and Cauchy–Schwarz inequalities to bound the numerator and denominator, reducing the problem to maximizing a specific function of two variables.

Key Contributions and Results

1. Disproof of the Tang–Zhang Conjecture:
   The paper provides a definitive counterexample showing that the Tang–Zhang conjecture is false for general matrices.

   • Theorem 1.1: For $m=n=2$ and $p=3/2$, there exist real rank-one matrices $A_1, A_2$ such that the ratio $\frac{\|A_1+A_2\|_{3/2}}{\||A_1|+|A_2|\|_{3/2}}$ is strictly greater than $207/200$.
   • The authors rigorously prove that the conjectured constant $C^{TZ}_{3/2, 2}$ is strictly less than $207/200$. Thus, the conjectured formula does not yield the sharp constant, even within the restricted class of rank-one matrices.

2. Sharp Bound for Rank-One Summands ($p \ge 2$):
   Despite the failure of the conjecture in general, the authors prove that the Tang–Zhang formula is sharp when restricted to families of rank-at-most-one matrices, provided $p \ge 2$.

   • Theorem 1.2: For $m \ge 2$ and $2 \le p < \infty$, if $A_1, \dots, A_m$ have rank at most one, then $\|\sum A_k\|_p \le C^{TZ}_{p,m} \|\sum |A_k|\|_p$.
   • The constant is sharp in the dimension-free setting and is attained when $n \ge m$. The authors provide a complete classification of the equality cases, which require specific inner product structures among the right singular vectors ($v_k$) and a common range for the left singular vectors.
   • Corollary 3.1: The endpoint case $p=\infty$ is also resolved for rank-one matrices, yielding the sharp constant $\sqrt{m}$.

3. Sharp Constant for Arbitrary Matrices ($m=2, p=4$):
   The authors establish the sharp constant for the case of two arbitrary complex matrices when $p=4$, removing the rank-one restriction.

   • Theorem 1.3: For $A, B \in M_n(\mathbb{C})$, the inequality $\|A+B\|_4^4 \le x^2(x+1)^2 \||A|+|B|\|_4^4$ holds, where $x > 1$ is the solution to $x^4 - 2x - 1 = 0$.
   • This constant equals $(C^{TZ}_{4,2})^4$, indicating that for this specific case ($m=2, p=4$), the Tang–Zhang formula does provide the correct sharp bound, even for full-rank matrices. The bound is attained for $n \ge 2$.

Significance and Claims
The paper claims to resolve the status of the Tang–Zhang conjecture by demonstrating that while the conjectured formula is not universally valid (failing at $p=3/2$), it remains the correct sharp bound in significant specific regimes:

• It is sharp for all rank-one families when $p \ge 2$.
• It is sharp for the full matrix case when $m=2$ and $p=4$.

The work highlights the complexity of Schatten norm inequalities, showing that the behavior of the sharp constant depends critically on both the exponent $p$ and the rank constraints of the summands. The authors emphasize that their counterexample is certified by exact rational arithmetic, avoiding numerical uncertainty. The paper concludes that the Tang–Zhang formula, while a valid lower bound attained by specific rank-one configurations, is not the general solution for all $p > 1$.