Tang–Zhang Schatten norm conjecture

For $m\ge 2$ and $1<p<\infty$ , define the all-dimensional best constant $c_p(m)$ by

c_p(m)=\inf\left\{c\ge 0:\ \left\|\sum_{k=1}^m A_k\right\|_p\le c\left\|\sum_{k=1}^m |A_k|\right\|_p\ \text{for every finite family of complex matrices }A_1,\ldots,A_m\right\}.

The Tang–Zhang conjecture asserts that $c_p(m)$ equals the explicit closed-form expression proposed by Tang and Zhang for every finite $p>1$ . The supplied source does not state that expression explicitly. In particular, the conjectured equality is claimed to fail for $m=2$ and $p=3/2$ .

Progress summary

Solved

A new preprint claims to disprove the formula with a rationally certified example, while proving it in several restricted settings; the disproof has not yet been independently verified.

The conjecture asks for a universal sharp formula controlling the Schatten norm of a sum by the Schatten norm of the sum of absolute values. A 2025 paper formulated the closed-form expression as a conjecture and recorded several special cases.

Known results

The cases $p=1$ , $p=2$ , and $p=\infty$ were known in the earlier formulation.
A general bound $c_p(m)\le \sqrt{m}^{\,1-1/p}$ was proved.
The 2026 preprint claims sharp results for rank-at-most-one summands when $2\le p<\infty$ , plus the endpoint $p=\infty$ .
It also claims the unrestricted case $m=2$ , $p=4$ .

August 2026 counterexample

Zeng, Liu, and Ratnavelu claim an explicit real rank-one counterexample at $m=n=2$ and $p=3/2$ , certified by strict rational inequalities, so the universal formula is false if the preprint is correct. The same paper says OpenAI Codex assisted with exploration, search, checking, and preparation; the authors retain responsibility for correctness.

Current status (as of August 2026): A preprint claims a rationally certified counterexample, but independent verification is absent, while its restricted positive cases remain claims of that paper.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

A Counterexample to the Tang Zhang Schatten Norm Conjecture and Sharp Positive Results — arXiv — Zijian Zeng, Houde Liu, Kurunathan Ratnavelu

functional-analysis matrix-analysis

2 days ago