The sharp off-diagonal threshold for spectral-subspace rotation
Let $A$ be a self-adjoint, possibly unbounded operator on a separable complex Hilbert space, and let $\sigma(A) = \sigma \cup \Sigma$ with $\sigma$ and $\Sigma$ nonempty and closed and $d = \operatorname{dist}(\sigma, \Sigma) > 0$; no ordering of the two sets is assumed. Let $P = E_A(\sigma)$, where $E_B$ is the spectral measure of $B$, and let $V$ be a bounded self-adjoint operator that is off-diagonal for this decomposition: $$PVP = 0, \qquad (I - P)V(I - P) = 0.$$ Let $Q = E_{A+V}\bigl(O_{d/2}(\sigma)\bigr)$, where $O_r(S) = \{x \in \mathbb{R} : \operatorname{dist}(x, S) < r\}$; norms are operator norms. If $\|V\| < \tfrac{\sqrt3}{2} d$, the spectrum of $A + V$ splits into a part in $O_{d/2}(\sigma)$ and a part in $O_{d/2}(\Sigma)$, and this bound is sharp for that conclusion. Seelmann proved $\|P - Q\| < 1$ whenever $\|V\| < c\, d$, for an explicit constant $c > 0.6940725$.
Problem. Does $\|V\| < \tfrac{\sqrt3}{2} d$ always imply $\|P - Q\| < 1$?
Source: Kostrykin, Makarov and Motovilov (2007), conjectured in the introduction, for bounded A; restated for possibly unbounded A by Seelmann (2016), §1.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- FA, NA
- Keywords
- off-diagonal perturbations, OpenProblemsInNLA, perturbation bounds, spectral subspaces
- Literature checked
- Oct. 7, 2026 by clemens-admin
References
@article{kostrykin2007,
author = {Kostrykin, Vadim and Makarov, Konstantin A. and Motovilov, Alexander K.},
title = {Perturbation of spectra and spectral subspaces},
journal = {Trans. Amer. Math. Soc.},
volume = {359},
number = {1},
year = {2007},
pages = {77--89},
doi = {10.1090/S0002-9947-06-03930-4},
eprint = {math/0306025},
archivePrefix = {arXiv},
}
@article{seelmann2016,
author = {Seelmann, Albrecht},
title = {Notes on the subspace perturbation problem for off-diagonal perturbations},
journal = {Proc. Amer. Math. Soc.},
volume = {144},
number = {9},
year = {2016},
pages = {3825--3832},
doi = {10.1090/proc/13118},
eprint = {1412.6294},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-sp02,
author = {Townsend, Alex},
title = {{SP-02}: The sharp off-diagonal threshold for spectral-subspace rotation},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/eigenvalues-and-inverse-problems/SP-02/README.md},
note = {Read at commit 7336302, 2026-10-06},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00153, "The sharp off-diagonal threshold for spectral-subspace rotation", clopen.solutions.