Audenaert's two-row Schatten norm-compression conjecture
For $1 \le p < \infty$ let $\|\cdot\|_p$ be the Schatten $p$-norm. Partition a complex matrix into $2 \times N$ blocks, $$T = \begin{bmatrix} A_1 & \cdots & A_N \\ B_1 & \cdots & B_N \end{bmatrix},$$ and let $C_p(T)$ be the $2 \times N$ matrix with entries $\|A_j\|_p$ and $\|B_j\|_p$. The conjecture contains Hanner's inequality for matrices as a special case, which Heinävaara has since proved. Audenaert proved special cases, and showed that the analogue for $3 \times 3$ and larger partitions fails.
Problem. Is $\|T\|_p \ge \|C_p(T)\|_p$ for $1 \le p \le 2$, and $\|T\|_p \le \|C_p(T)\|_p$ for $2 \le p < \infty$, for every such partition?
Source: Audenaert (2008), Conjecture 1.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- NA
- Keywords
- block matrices, norm inequalities, OpenProblemsInNLA, Schatten norms
- Literature checked
- Oct. 2, 2026 by clemens-admin
References
@article{audenaert2008,
author = {Audenaert, Koenraad M. R.},
title = {On a norm compression inequality for $2\times N$ partitioned block matrices},
journal = {Linear Algebra Appl.},
volume = {428},
number = {4},
year = {2008},
pages = {781--795},
doi = {10.1016/j.laa.2007.08.007},
eprint = {math/0702186},
archivePrefix = {arXiv},
}
@article{heinavaara2025,
author = {Hein{\"a}vaara, Otte},
title = {Tracial joint spectral measures},
journal = {Invent. Math.},
volume = {239},
number = {2},
year = {2025},
pages = {505--526},
doi = {10.1007/s00222-024-01309-6},
eprint = {2310.03227},
archivePrefix = {arXiv},
}
@misc{zhang2024,
author = {Zhang, Teng},
title = {Clarkson--{M}c{C}arthy type inequalities, part {I}: $\ell_p$--$\ell_p$ and $\ell_q$--$\ell_p$ {S}chatten $p$-estimates},
year = {2024},
eprint = {2410.21961},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mi01,
author = {Townsend, Alex},
title = {{MI-01}: Audenaert's two-row Schatten norm-compression 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-01/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 It will not change if the problem is solved, reclassified or withdrawn.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-00071, "Audenaert's two-row Schatten norm-compression conjecture", clopen.solutions.