Schatten norm complement involving Lin's positive definite quantity
For positive definite $A, B \in \mathbb{C}^{n \times n}$ let $$G = A^{1/2}\bigl(A^{-1/2} B A^{-1/2}\bigr)^{1/2} A^{1/2}, \qquad L = A^{1/2}\bigl(B^{1/2} A^{-1} B^{1/2}\bigr)^{1/2} A^{1/2},$$ so that $G = A \# B$ is the geometric mean and $L$ is the positive definite matrix introduced by Lin, with the factors in the order shown. Let $\|X\|_p = (\operatorname{tr}|X|^p)^{1/p}$, with $|X| = (X^*X)^{1/2}$, for $1 \le p < \infty$, and let $\|X\|_\infty$ be the largest singular value of $X$. Lin showed that the eigenvalues of $L$ log-majorize those of $A^{1/2}B^{1/2}$, and hence those of $G$. Ghabries, Abbas, Mourad and Assi proved the inequality below for $p = 1$ and $p = 2$.
Problem. Is $$\|A + B + G + L\|_p \le \|A + B + 2L\|_p$$ for every $n \ge 1$, all positive definite $A, B \in \mathbb{C}^{n \times n}$ and every $1 \le p \le \infty$?
Source: Ghabries, Abbas, Mourad and Assi (2023), Conjecture 4.1 in arXiv:2105.13356; restated in Ghabries (2022), PhD thesis, Open Problems, Problem 5.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- NA
- Keywords
- matrix geometric means, norm inequalities, OpenProblemsInNLA, Schatten norms
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@article{ghabries2023,
author = {Ghabries, Mohammad M. and Abbas, Hassane and Mourad, Bassam and Assi, Abdallah},
title = {New log-majorization results concerning eigenvalues and singular values and a complement of a norm inequality},
journal = {Linear Multilinear Algebra},
volume = {71},
number = {7},
year = {2023},
pages = {1228--1243},
doi = {10.1080/03081087.2022.2059050},
eprint = {2105.13356},
archivePrefix = {arXiv},
}
@phdthesis{ghabries2022,
author = {Ghabries, Mohammad Mahdi},
title = {Contributions to Matrix Inequalities and Some Applications},
school = {Universit{\'e} d'Angers and Lebanese University},
year = {2022},
doi = {10.70675/32bff73dzcb5az430ezbda8ze209c4b1c319},
note = {HAL tel-03936351},
url = {https://theses.hal.science/tel-03936351},
}
@article{lin2017,
author = {Lin, Minghua},
title = {On a determinantal inequality arising from diffusion tensor imaging},
journal = {Commun. Contemp. Math.},
volume = {19},
number = {5},
year = {2017},
pages = {1650044},
doi = {10.1142/S0219199716500449},
eprint = {1604.04141},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mi24,
author = {Townsend, Alex},
title = {{MI-24}: Schatten norm complement involving Lin's positive definite quantity},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-24/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
For positive definite $A, B \in \mathbb{C}^{n \times n}$, let $G = A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$ be their geometric mean and $L = A^{1/2}(B^{1/2}A^{-1}B^{1/2})^{1/2}A^{1/2}$. The note proves $\|A + B + G + L\|_p \le \|A + B + 2L\|_p$ for every Schatten norm, $1 \le p \le \infty$. Write $H = A + B$ and $K = A^{1/2}B^{1/2} + B^{1/2}A^{1/2}$. The Heron …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Needs verification → Solution claimed | clemens-admin | Listed with the record of its solution, unreviewed. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00249, "Schatten norm complement involving Lin's positive definite quantity", clopen.solutions.