MathDB

Lu–Wenzel fundamental conjecture for real matrices

About 7 years old · traced to

Let M(n,R)M(n,\mathbb{R}) be the space of real n×nn\times n matrices. Let B,B2,…,Bm∈M(n,R)B,B_2,\ldots,B_m\in M(n,\mathbb{R}), with Bα∗B_\alpha^* denoting transpose and with the Frobenius inner product Tr⁡(BαBβ∗)\operatorname{Tr}(B_\alpha B_\beta^*). Assume

Tr⁡(BαBβ∗)=0\operatorname{Tr}(B_\alpha B_\beta^*)=0

for every distinct pair among the matrices, and assume

Tr⁡(Bα[B,Bβ])=0\operatorname{Tr}\bigl(B_\alpha[B,B_\beta]\bigr)=0

for all 2≤α,β≤m2\leq\alpha,\beta\leq m. Lu–Wenzel fundamental conjecture. Then

∑α=2m∥[B,Bα]∥2≤(max⁡2≤α≤m∥Bα∥2+∑α=2m∥Bα∥2)∥B∥2.\sum_{\alpha=2}^m\|[B,B_\alpha]\|^2\leq\left(\max_{2\leq\alpha\leq m}\|B_\alpha\|^2+\sum_{\alpha=2}^m\|B_\alpha\|^2\right)\|B\|^2.

This is the central real Lu–Wenzel conjecture and is intended to unify the BW and DDVV inequalities. The source abstract states that this conjecture is proved in the paper, so its status is solved.

References

Primary source

Jianquan Ge, Fagui Li, Zhiqin Lu and Yi Zhou, “On some conjectures by Lu and Wenzel”, arXiv:1908.06624 (2019).

commutatorsmatrix-inequalities
Posted

Progress summary

Refreshed
Claimed solved

A paper from 2019 claims to prove the Lu–Wenzel conjecture, but the available record does not independently verify the proof.

Lu and Wenzel proposed this family of matrix inequalities in 2016. The problem here is their central real-matrix conjecture, stated as Conjecture 22 in the cited paper.

August 2019 claimed proof

The paper On some conjectures by Lu and Wenzel claims to prove Conjecture 22, exactly matching this problem, along with equivalent Conjectures 44–66; it also derives Conjectures 11 and 33. The supplied record gives no independent verification or referee assessment.

Current status (as of September 2026): The conjecture is claimed proved in the 2019 paper, but that claim remains unverified in the available record.

Sources

Solutions 0

No solutions have been posted yet.