Sharp Schatten triangle constants for the arithmetic symmetric modulus
For $X \in \mathbb{C}^{n \times n}$ let $S(X) = \tfrac12\bigl((X^*X)^{1/2} + (XX^*)^{1/2}\bigr)$ be its arithmetic symmetric modulus, and let $\|\cdot\|_p$ be the Schatten $p$-norm, $1 \le p \le \infty$, with $\|\cdot\|_\infty$ the operator norm. For integers $m, n \ge 2$ let $$c_p^{\rm sym}(m,n) = \sup \frac{\|S(A_1 + \cdots + A_m)\|_p}{\|S(A_1) + \cdots + S(A_m)\|_p},$$ the supremum over all $(A_1, \dots, A_m) \ne (0, \dots, 0)$ with $A_1, \dots, A_m \in \mathbb{C}^{n \times n}$. Bourin and Lee proved $c_p^{\rm sym}(m,n) \le \sqrt2$. Zhang showed that $c_1^{\rm sym}(m,n) = 1$, that $c_\infty^{\rm sym}(m,n) = \sqrt2$ for $n \ge 3$, and that $c_p^{\rm sym}(m,n) \le 2^{1-1/p}$, and gave lower bounds.
Problem. Determine $c_p^{\rm sym}(m,n)$ for all integers $m, n \ge 2$ and all $1 \le p \le \infty$.
Source: Zhang (2026), Problem 1.17.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- NA
- Keywords
- OpenProblemsInNLA, Schatten norms, symmetric moduli, triangle inequalities
- Literature checked
- Oct. 6, 2026 by clemens-admin
References
@article{zhang2026,
author = {Zhang, Teng},
title = {Operator symmetric moduli and sharp triangle inequalities},
journal = {J. Lond. Math. Soc. (2)},
volume = {114},
number = {2},
year = {2026},
pages = {e70672},
doi = {10.1112/jlms.70672},
eprint = {2603.01046},
archivePrefix = {arXiv},
}
@article{bourin2026a,
author = {Bourin, Jean-Christophe and Lee, Eun-Young},
title = {Triangle inequalities for the operator symmetric modulus},
journal = {Proc. Amer. Math. Soc.},
year = {2026},
doi = {10.1090/proc/17861},
eprint = {2602.19607},
archivePrefix = {arXiv},
}
@article{bourin2026b,
author = {Bourin, Jean-Christophe and Lee, Eun-Young},
title = {Some hybrid matrix triangle inequalities},
journal = {Linear Algebra Appl.},
volume = {749},
year = {2026},
pages = {197--207},
doi = {10.1016/j.laa.2026.06.033},
eprint = {2606.29188},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mi09,
author = {Townsend, Alex},
title = {{MI-09}: Sharp Schatten triangle constants for the arithmetic symmetric modulus},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-inequalities-and-norms/MI-09/README.md},
note = {Read at commit e375a6f, 2026-09-30},
}
Resolution claims
Write $\mathcal S(T)=(|T|+|T^*|)/2$ for the arithmetic symmetric modulus. For every $m\ge2$, all $T_1,\dots,T_m\in M_2(\mathbb C)$ and every unitarily invariant norm, $|||\mathcal S(\sum_jT_j)|||\le\sqrt{6\sqrt3-9}\,|||\sum_j\mathcal S(T_j)|||$. The constant $\sqrt{6\sqrt3-9}=1.17995\ldots$ is optimal when the inequality is required for all such norms, and is attained in the operator norm by two real rank-one matrices. In particular $c_\infty^{\rm sym}(m,2)=\sqrt{6\sqrt3-9}$ for every $m\ge2$. The operator-norm case reduces …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-08 | Open → Partially resolved | The registry's curators | An exact sharp universal norm constant in dimension two v1 recorded from OpenProblemsInNLA @ eb7c9d6, MI-09/solution.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00140, "Sharp Schatten triangle constants for the arithmetic symmetric modulus", clopen.solutions.