Sharp subquadratic Lee constants for sums of matrices
For $X \in \mathbb{C}^{n \times n}$ write $|X| = (X^*X)^{1/2}$, and let $\|\cdot\|_p$ be the Schatten $p$-norm. For an integer $m \ge 2$ and $1 \le p < \infty$ let $$C_p(m) = \sup_{n \ge 1} \, \sup \frac{\|A_1 + \cdots + A_m\|_p}{\bigl\| |A_1| + \cdots + |A_m| \bigr\|_p},$$ the inner supremum over all $A_1, \dots, A_m \in \mathbb{C}^{n \times n}$ with $|A_1| + \cdots + |A_m| \ne 0$. Lee posed the question for $m = 2$ and conjectured $C_2(2) \le \sqrt{(1+\sqrt2)/2}$, which Lin and Zhang proved. Tang and Zhang showed $C_2(m) = \sqrt{(1+\sqrt m)/2}$ and $C_p(m) \le m^{(1-1/p)/2}$, and conjectured a closed formula for $C_p(m)$. Qiu proved the formula for $p \ge 2$. For $1 < p < 2$ it fails: Zeng, Liu and Ratnavelu gave a counterexample with $m = 2$ and $p = 3/2$, Qiu one for $m = 2$ and every such $p$, and Pang one for every $m$ and every such $p$.
Problem. Determine $C_p(m)$ for every integer $m \ge 2$ and every real $1 < p < 2$.
Source: Lee (2010), Section 3, for two matrices, restated as Question 1 in Lee (2012); for m matrices, Tang and Zhang (2026), Question 1.3; for 1 < p < 2, where their conjectured formula fails, Qiu (2026), Question 5.1, and Li (2026), Problem 4.2.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- NA
- Keywords
- matrix absolute values, norm inequalities, OpenProblemsInNLA, Schatten norms
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{lee2010,
author = {Lee, Eun-Young},
title = {Rotfel'd type inequalities for norms},
journal = {Linear Algebra Appl.},
volume = {433},
number = {3},
year = {2010},
pages = {580--584},
doi = {10.1016/j.laa.2010.03.029},
}
@article{lee2012,
author = {Lee, Eun-Young},
title = {How to compare the absolute values of operator sums and the sums of absolute values?},
journal = {Oper. Matrices},
volume = {6},
number = {3},
year = {2012},
pages = {613--619},
doi = {10.7153/oam-06-42},
}
@article{tang2026,
author = {Tang, Quanyu and Zhang, Shu},
title = {Generalizing {L}ee's conjecture on the sum of absolute values of matrices},
journal = {Linear Algebra Appl.},
volume = {731},
year = {2026},
pages = {196--204},
doi = {10.1016/j.laa.2025.11.015},
eprint = {2510.16846},
archivePrefix = {arXiv},
}
@misc{qiu2026,
author = {Qiu, Hongsen},
title = {Sharp Quasi-Reverse {M}inkowski Inequality for {S}chatten Norms},
year = {2026},
eprint = {2608.17565},
archivePrefix = {arXiv},
}
@misc{li2026,
author = {Li, Xing},
title = {Sharp Concave-Function Transfer for {L}ee-Type {S}chatten Norm Inequalities},
year = {2026},
eprint = {2608.25989},
archivePrefix = {arXiv},
}
@article{lin2022,
author = {Lin, Jianfu and Zhang, Yun},
title = {A proof of {L}ee's conjecture on the sum of absolute values of matrices},
journal = {J. Math. Anal. Appl.},
volume = {516},
number = {2},
year = {2022},
pages = {126542},
doi = {10.1016/j.jmaa.2022.126542},
}
@misc{zeng2026,
author = {Zeng, Zijian and Liu, Houde and Ratnavelu, Kuru},
title = {A Counterexample to the {T}ang--{Z}hang {S}chatten Norm Conjecture and Sharp Positive Results},
year = {2026},
eprint = {2608.15558},
archivePrefix = {arXiv},
}
@misc{pang2026,
author = {Pang, Zhekai},
title = {Counterexamples to a conjectured {S}chatten norm inequality},
year = {2026},
eprint = {2608.18112},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mi20,
author = {Townsend, Alex},
title = {{MI-20}: Sharp subquadratic Lee constants for sums of matrices},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-20/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Resolution claims
For the dimension-independent constant $C_p(m)=\sup_n\sup\|\sum_jA_j\|_p/\|\sum_j|A_j|\|_p$, we prove the computer-assisted bound $C_{4/3}(2)\le(861/800)^{1/4}$. An exact rational certificate excludes an entire parameter interval in a positive cubic system, using 184 coefficient identities and 44 positive-definite Gram multipliers; the eight-stage certificate chain behind the preceding bound $(43983/40000)^{1/4}$ is included as a proof dependency. We also give a positive power-system characterization of $C_p(m)$ at every …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Open → Partially resolved | The registry's curators | An Exact Universal Bound and a Positive Power-System Reduction v1 recorded from OpenProblemsInNLA @ eb7c9d6, MI-20/report.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00144, "Sharp subquadratic Lee constants for sums of matrices", clopen.solutions.