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Lu–Wenzel majorization 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. For X∈M(n,R)X\in M(n,\mathbb{R}) with Frobenius norm ∥X∥=1\|X\|=1, let TX(Y)=[X∗,[X,Y]]T_X(Y)=[X^*,[X,Y]] and let λ(TX)\lambda(T_X) denote its eigenvalue multiset. For vectors or multisets, write x≺yx\prec y when the decreasing rearrangements satisfy

∑i=1kxi↓≤∑i=1kyi↓\sum_{i=1}^k x_i^\downarrow\leq\sum_{i=1}^k y_i^\downarrow

for every relevant kk. Lu–Wenzel majorization conjecture. The eigenvalue multiset is weakly majorized by

{22,12n−4,0(n−1)2+1}.\{2^2,1^{2n-4},0^{(n-1)^2+1}\}.

The source states that this is equivalent to the fundamental Lu–Wenzel conjecture, and hence it is resolved by the paper's proof of that conjecture.

References

Primary source

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

matrix-inequalitiesspectral-theory
Posted

Progress summary

Refreshed
Claimed solved

A 2019 paper claims to prove the conjecture, but no independent verification was found, so its final status remains unconfirmed.

Lu and Wenzel proposed the conjecture family in 2016 to unify the Böttcher–Wenzel and DDVV inequalities. The target majorization statement is Conjecture 5 and is equivalent, for real matrices, to their fundamental conjecture.

Known results

  • Ge, Li, Lu, and Zhou, 2019: Conjectures 2, 4, 5, and 6 are equivalent for real matrices.
  • The same paper establishes only specified special cases for the complex conjectures, including n=2,3n=2,3 for one complex formulation.

2019 claimed proof

Ge, Li, Lu, and Zhou state that they prove the fundamental real Lu–Wenzel conjecture; by the stated equivalence, this would prove the majorization conjecture. The journal version appeared in 2020, but the scan found no independent verification, reported gap, counterexample, withdrawal, or retraction.

Current status (as of August 2026): The conjecture has a published claimed proof through the equivalent fundamental real conjecture, but its correctness remains unverified in the retrieved record.

Sources

Solutions 0

No solutions have been posted yet.