Trace-norm-small perturbations preserve Hermitian spectral distributions
For matrices $A_n \in \mathbb C^{n \times n}$, $n \ge 1$, and a measurable $f : [0, 1] \to \mathbb R$, write $\{A_n\} \sim_\lambda f$ if $$\lim_{n \to \infty} \frac1n \sum_{j=1}^n F(\lambda_j(A_n)) = \int_0^1 F(f(t))\,dt$$ for every continuous, compactly supported $F : \mathbb C \to \mathbb C$, the eigenvalues $\lambda_j(A_n)$ being counted with algebraic multiplicity. Let $\|E\|_*$ be the trace norm of $E$, the sum of its singular values, and $\|E\|_2$ its spectral norm. For Hermitian $H_n$ with $\{H_n\} \sim_\lambda f$, Golinskii and Serra-Capizzano proved $\{H_n + E_n\} \sim_\lambda f$ when $\|E_n\|_* = o(n)$ and both sequences are bounded in spectral norm. Barbarino and Serra-Capizzano proved it, with no bound on $\|H_n\|_2$, when $\|E_n\|_* = o(n)$ and $\sup_n \|E_n\|_2 < \infty$, and with no spectral-norm bounds at all when the Frobenius norms satisfy $\|E_n\|_F = o(\sqrt n)$. Barbarino and Garoni proved it when $\|E_n\|_* = o(n)$ and the matrices $H_n + E_n$ are normal.
Problem. Is $\{H_n + E_n\} \sim_\lambda f$ for every measurable $f : [0, 1] \to \mathbb R$, all Hermitian $H_n \in \mathbb C^{n \times n}$ with $\{H_n\} \sim_\lambda f$ and all $E_n \in \mathbb C^{n \times n}$ with $\|E_n\|_* / n \to 0$, with no bound on the spectral norms of $H_n$ or $E_n$ and no assumption that $H_n + E_n$ is normal?
Source: Barbarino and Serra-Capizzano (2020), Conjecture 1, for symbols on any domain of finite positive measure; also Barbarino (2018), Conjecture 1, who notes that the question does not depend on the domain of the symbols and then works on [0, 1].
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- NA — Numerical Analysis
- Secondary
- SP
- Keywords
- eigenvalue distributions, matrix sequences, non-Hermitian perturbations, OpenProblemsInNLA
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@article{barbarino2020,
author = {Barbarino, Giovanni and Serra-Capizzano, Stefano},
title = {Non-{H}ermitian perturbations of {H}ermitian matrix-sequences and applications to the spectral analysis of the numerical approximation of partial differential equations},
journal = {Numer. Linear Algebra Appl.},
volume = {27},
number = {3},
year = {2020},
pages = {e2286},
doi = {10.1002/nla.2286},
}
@misc{barbarino2018,
author = {Barbarino, Giovanni},
title = {Conjectures on Perturbations of {H}ermitian Sequences},
year = {2018},
eprint = {1808.05555},
archivePrefix = {arXiv},
}
@article{golinskii2007,
author = {Golinskii, Leonid and Serra-Capizzano, Stefano},
title = {The asymptotic properties of the spectrum of nonsymmetrically perturbed {J}acobi matrix sequences},
journal = {J. Approx. Theory},
volume = {144},
number = {1},
year = {2007},
pages = {84--102},
doi = {10.1016/j.jat.2006.05.002},
eprint = {math/0512222},
archivePrefix = {arXiv},
}
@article{barbarino2025,
author = {Barbarino, Giovanni and Garoni, Carlo},
title = {{GLT} sequences and normal matrices},
journal = {Electron. J. Linear Algebra},
volume = {41},
year = {2025},
pages = {1--20},
doi = {10.13001/ela.2025.8929},
eprint = {1805.08708},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-sp13,
author = {Townsend, Alex},
title = {{SP-13}: Trace-norm-small perturbations preserve Hermitian spectral distributions},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/eigenvalues-and-inverse-problems/SP-13/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
Let $H_n$ be Hermitian $n \times n$ matrices whose eigenvalues are distributed, in the Weyl sense, as a real measurable function $f$ on $[0, 1]$, and let $E_n$ be complex $n \times n$ matrices with trace norm $o(n)$. The manuscript proves that the eigenvalues of $H_n + E_n$ are also distributed as $f$, with no bound on spectral norms and …
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-00254, "Trace-norm-small perturbations preserve Hermitian spectral distributions", clopen.solutions.