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A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results
Zijian Zeng, Houde Liu, Kurunathan Ratnavelu
TL;DR
The paper tests a conjectured formula for Schatten-norm inequalities, disproves it via an explicit 2×2 rank-one counterexample, and proves sharp results in several restricted settings. It establishes the conjectured bound for rank-at-most-one families when 2≤p<∞ and for arbitrary complex matrices when m=2, p=4.
Problem
The paper examines whether the Tang–Zhang conjectured formula for the all-dimensional Schatten-norm constant is valid.
Method
The proofs use explicit 2×2 spectral calculations and reduce the rank-one matrix problem to a single scalar variable.
Results
The conjecture is false, while the conjectured sharp bound holds for rank-at-most-one families with 2≤p<∞ and is sharp with classified equality cases.
Takeaways & Limitations
The conjectured constant cannot hold universally, but remains valid in the proved rank-one regime.
Takeaways & Limitations
The paper does not completely classify all equality cases for the m=2, p=4 theorem.
Abstract
from arXiv · showhide
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 $m=2$, $p=4$.
1. Introduction
The introduction presents an exact rank-one counterexample disproving the Tang–Zhang conjecture at p = 3/2, then establishes sharp positive results for rank-one families and for the full m = 2, p = 4 case.
- Counterexample: At m = n = 2 and p = 3/2, two real rank-one matrices produce a ratio exceeding the Tang–Zhang conjectured value, so the conjecture is false.The comparison is certified by the stated exact counterexample theorem.
- Rank-one positive result: For m ≥ 2 and 2 ≤ p < ∞, the conjectured sharp bound holds for all nonzero families of complex matrices with rank at most one.The constant is sharp in the dimension-free rank-one problem.
- Rank-one positive result: The rank-one theorem classifies equality cases up to common input and output unitaries, common positive scaling, and phase changes in rank-one factorizations.Equality requires n ≥ m and is attained whenever n ≥ m under the stated characterization.
- Endpoint result: At p = ∞, the rank-one argument yields the sharp constant √m, with orthonormal right vectors.This is the endpoint version of the rank-one result.
- Full m = 2 case: For arbitrary complex matrices with m = 2 and p = 4, the conjectured constant is sharp in the dimension-free problem.The theorem states that the constant is attained for every family specified there.
2. An exact 2 × 2 counterexample
The section constructs two real 2 × 2 rank-one matrices with unit nonzero singular values and certifies an exact counterexample by proving C < 207/200 < R. The comparison uses rational inequalities to place the attained ratio above the conjectured constant.
- Exact spectral computation: The squared singular values of A1 + A2 are obtained as eigenvalues of the product of the two Gram matrices.The Gram matrices are simultaneously diagonalized by (1, 1)^T and (1, −1)^T.
- Exact spectral computation: The comparison denominator comes from the eigenvalues of |A1| + |A2| = V V^T, yielding the expression (13/8)3/2 + (3/8)3/2.This expression is recorded as equation (10).
3. The sharp rank-one problem for p ≥2
For p ≥ 2, the matrix problem reduces to a scalar maximization with a unique optimizer, yielding the sharp rank-one bound. Equality occurs only in a precisely classified configuration, possible when n ≥ m.
- Scalar reduction: The proof reduces the matrix inequality to a single scalar variable and establishes the sharp bound through this reduction.The construction uses Gram matrices and their eigenvalues to connect the scalar problem with singular values.
- Scalar reduction: For p ≥ 2, the scalar objective has a unique maximizer xp,m > 1 because yp − 2y − (m − 1) has exactly one zero on [1, ∞).The function is strictly increasing there, aside from an inessential zero derivative at the left endpoint when p = 2.
- Equality cases: Equality requires the range of L to lie in the one-dimensional top eigenspace, forcing coincident left vectors after phase changes.The resulting Gram-matrix conditions determine the extremizing family, and the converse shows it attains equality at every step.
- Equality cases: The rank-one endpoint constant is sharp dimension-free, with equality possible only when n ≥ m.Equality occurs precisely for equal nonzero singular values, a common one-dimensional range, and pairwise orthogonal right vectors, up to the stated unitary and phase symmetries.
- Equality cases: The equality conditions are equivalent to L having rank one and G being a scalar multiple of the identity, yielding exactly the classified configuration.These conditions follow from R = Tr G ≤ mλ and ∥VV*∥∞ = λ.
4. The full m = 2, p = 4 problem
For arbitrary complex matrices with m = 2 and p = 4, the proof reduces the problem to a one-variable maximization and establishes the conjectured sharp bound. Explicit rank-one extremizers attain the bound, while a complete equality classification is not attempted.
- Reduction: Unitary invariance reduces the numerator to S = H + WK, where H = |A|, K = |B|, and W is unitary.The reduction uses polar decompositions extended to unitaries.
- Upper bound: The trace expansion Tr(H + K)^4 = a + b + 4(e + f) + 4d + 2g, combined with Schatten inequalities, yields the required upper bound.Here a, b, d, e, f, and g are trace quantities defined from H and K.
- Optimization: The associated scalar function has a unique maximizer s0 ∈ (0, 1/2), characterized by the critical-point equation.The competing first branch cannot dominate because f(1/6) > 2.
- Sharpness: Rank-one constructions using unit vectors with inner product s0 attain ratio f(s0), proving sharpness.The denominator identity is 4 = (1 + s0)^4 + (1 − s0)^4 = 2(1 + 6s0^2 + s0^4).
- Equality cases: Orthogonal direct sums of the two-dimensional extremal block provide higher-dimensional equality examples, but the proof does not classify all equality cases.The limitation concerns Theorem 1.3.