The sharp generic 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: they may interlace. For a bounded self-adjoint operator $V$ let $$P = E_A(\sigma), \qquad Q = E_{A+V}\bigl(O_{d/2}(\sigma)\bigr), \qquad O_r(S) = \{x \in \mathbb{R} : \operatorname{dist}(x, S) < r\},$$ where $E_B$ is the spectral measure of $B$; norms are operator norms. If $\|V\| < d/2$, the spectrum of $A + V$ splits into a part in $O_{d/2}(\sigma)$ and a part in $O_{d/2}(\Sigma)$, and the bound $d/2$ is sharp for this; $\arcsin\|P - Q\|$ is the maximal angle between the ranges of $P$ and $Q$. If the convex hull of one of $\sigma$ and $\Sigma$ does not meet the other, $\|V\| < d/2$ is known to imply $\|P - Q\| < 1$. In general, Seelmann proved $\|P - Q\| < 1$ whenever $\|V\| < c_{\mathrm{crit}}\, d$, with $c_{\mathrm{crit}} = 0.4548399\ldots$.
Problem. Does $\|V\| < d/2$ always imply $\|P - Q\| < 1$? Equivalently, is $c_{\mathrm{opt}} = 1/2$, where $c_{\mathrm{opt}}$ is the optimal universal constant for which $\|V\| < c_{\mathrm{opt}}\, d$ guarantees that the maximal angle $\arcsin\|P - Q\|$ is less than $\pi/2$?
Source: Kostrykin, Makarov and Motovilov (2003), §1, for bounded A; for possibly unbounded A, Albeverio and Motovilov (2013), question (i), who conjecture a positive answer.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- FA, NA
- Keywords
- OpenProblemsInNLA, perturbation bounds, self-adjoint operators, spectral subspaces
- Literature checked
- Oct. 7, 2026 by clemens-admin
References
@article{kostrykin2003,
author = {Kostrykin, Vadim and Makarov, Konstantin A. and Motovilov, Alexander K.},
title = {On a subspace perturbation problem},
journal = {Proc. Amer. Math. Soc.},
volume = {131},
number = {11},
year = {2003},
pages = {3469--3476},
doi = {10.1090/S0002-9939-03-06917-X},
eprint = {math/0203240},
archivePrefix = {arXiv},
}
@article{albeverio2013,
author = {Albeverio, Sergio and Motovilov, Alexander K.},
title = {Sharpening the norm bound in the subspace perturbation theory},
journal = {Complex Anal. Oper. Theory},
volume = {7},
number = {4},
year = {2013},
pages = {1389--1416},
doi = {10.1007/s11785-012-0245-7},
eprint = {1112.0149},
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},
}
@article{seelmann2018,
author = {Seelmann, Albrecht},
title = {On an estimate in the subspace perturbation problem},
journal = {J. Anal. Math.},
volume = {135},
number = {1},
year = {2018},
pages = {313--343},
doi = {10.1007/s11854-018-0042-y},
eprint = {1310.4360},
archivePrefix = {arXiv},
}
@article{seelmann2021,
author = {Seelmann, Albrecht},
title = {Unifying the treatment of indefinite and semidefinite perturbations in the subspace perturbation problem},
journal = {Oper. Matrices},
volume = {15},
number = {3},
year = {2021},
pages = {1181--1188},
doi = {10.7153/oam-2021-15-74},
eprint = {2006.16102},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-sp01,
author = {Townsend, Alex},
title = {{SP-01}: The sharp generic 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-01/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-00152, "The sharp generic threshold for spectral-subspace rotation", clopen.solutions.