The complex Lu–Wenzel spectral conjecture
For $X \in \mathbb{C}^{n \times n}$ let $T_X$ be the linear map $Y \mapsto [X^*, [X, Y]]$ on $\mathbb{C}^{n \times n}$, where $[P, Q] = PQ - QP$. It is positive semidefinite for the Frobenius inner product $\langle Y, Z \rangle = \operatorname{tr}(Y^* Z)$; let $\lambda_1(T_X) \ge \cdots \ge \lambda_{n^2}(T_X) \ge 0$ be its eigenvalues, with multiplicity. The case $k = 1$ below is equivalent to the Böttcher–Wenzel inequality $\|XY - YX\|_F \le \sqrt2 \|X\|_F \|Y\|_F$. Ge, Li, Lu and Zhou showed that the assertion below is equivalent to the complex form of Lu and Wenzel's fundamental conjecture on commutators, proved it when $X$ is normal, when $X$ has rank one, and when $n = 2, 3$, and proved the weaker bound $\sum_{i=1}^{2k} \lambda_i(T_X) \le 2k + 1 + 2\sqrt{k}$ in general.
Problem. Is $$\sum_{i=1}^{2k} \lambda_i(T_X) \le 2k + 2$$ for every integer $n \ge 2$, every $X \in \mathbb{C}^{n \times n}$ with $\|X\|_F = 1$, and every integer $1 \le k \le \lfloor n^2/2 \rfloor$?
Source: Ge, Li, Lu and Zhou (2020), Conjecture 4 over ℂ (Conjecture 7 in arXiv:1908.06624), equivalent to the complex form, by Ge et al., of Lu and Wenzel's fundamental conjecture (2016).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- DG, NA
- Keywords
- Böttcher–Wenzel inequality, commutators, eigenvalue inequalities, OpenProblemsInNLA
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{ge2020,
author = {Ge, Jianquan and Li, Fagui and Lu, Zhiqin and Zhou, Yi},
title = {On some conjectures by {L}u and {W}enzel},
journal = {Linear Algebra Appl.},
volume = {592},
year = {2020},
pages = {134--164},
doi = {10.1016/j.laa.2020.01.027},
eprint = {1908.06624},
archivePrefix = {arXiv},
}
@incollection{lu2016,
author = {Lu, Zhiqin and Wenzel, David},
title = {The normal {R}icci curvature inequality},
booktitle = {Recent Advances in the Geometry of Submanifolds},
series = {Contemp. Math.},
volume = {674},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2016},
pages = {99--110},
doi = {10.1090/conm/674/13557},
}
@incollection{lu2017,
author = {Lu, Zhiqin and Wenzel, David},
title = {Commutator estimates comprising the {F}robenius norm -- looking back and forth},
booktitle = {Large Truncated {T}oeplitz Matrices, {T}oeplitz Operators, and Related Topics},
editor = {Bini, Dario A. and Ehrhardt, Torsten and Karlovich, Alexei Yu. and Spitkovsky, Ilya},
series = {Oper. Theory Adv. Appl.},
volume = {259},
publisher = {Birkh{\"a}user/Springer},
address = {Cham},
year = {2017},
pages = {533--559},
doi = {10.1007/978-3-319-49182-0_22},
}
@article{bottcher2008,
author = {B{\"o}ttcher, Albrecht and Wenzel, David},
title = {The {F}robenius norm and the commutator},
journal = {Linear Algebra Appl.},
volume = {429},
number = {8-9},
year = {2008},
pages = {1864--1885},
doi = {10.1016/j.laa.2008.05.020},
}
@article{ge2024,
author = {Ge, Jianquan and Li, Fagui and Tang, Zizhou and Zhou, Yi},
title = {A survey on the {DDVV}-type inequalities},
journal = {Adv. Math. (China)},
volume = {53},
number = {3},
year = {2024},
pages = {449--467},
doi = {10.11845/sxjz.2024001a},
eprint = {2402.01085},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mi14,
author = {Townsend, Alex},
title = {{MI-14}: The complex Lu–Wenzel spectral conjecture},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-14/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00142, "The complex Lu–Wenzel spectral conjecture", clopen.solutions.