Widom's canonical distribution conjecture for Toeplitz eigenvalues
Let $\mathbb T$ be the unit circle and $a : \mathbb T \to \mathbb C$ continuous, and let $T_n(a)$ be the $n \times n$ Toeplitz matrix with symbol $a$. Say that $a$ extends analytically to an inner annulus if, for some $0 < r < 1$, some function holomorphic on $r < |z| < 1$ extends continuously to $|z| = 1$ with boundary values $a$, and to an outer annulus if the same holds on $1 < |z| < R$ for some $R > 1$. The eigenvalues $\lambda_1(T_n(a)), \dots, \lambda_n(T_n(a))$ of $T_n(a)$, repeated according to algebraic multiplicity, are canonically distributed if $$\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^{n} F(\lambda_j(T_n(a))) = \frac{1}{2\pi} \int_0^{2\pi} F(a(e^{it}))\,dt$$ for every continuous compactly supported $F : \mathbb C \to \mathbb C$. This holds when $a(\mathbb T)$ has empty interior and connected complement (Tilli), in particular when $a$ is real-valued, and Widom proved it for certain symbols whose range is a Jordan curve and which are smooth except at one point.
Problem. Are the eigenvalues of $T_n(a)$ canonically distributed for every continuous $a : \mathbb T \to \mathbb C$ that extends analytically to neither an inner nor an outer annulus?
Source: Widom (1990), as restated for continuous symbols by Bogoya, Böttcher and Grudsky (2012), §1; in this form, OpenProblemsInNLA.
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- FA, NA
- Keywords
- eigenvalue distributions, OpenProblemsInNLA, Toeplitz matrices
- Literature checked
- Oct. 7, 2026 by clemens-admin
References
@incollection{widom1990,
author = {Widom, Harold},
title = {Eigenvalue distribution of nonselfadjoint {T}oeplitz matrices and the asymptotics of {T}oeplitz determinants in the case of nonvanishing index},
booktitle = {Topics in Operator Theory: {E}rnst {D}. {H}ellinger Memorial Volume},
series = {Oper. Theory Adv. Appl.},
volume = {48},
publisher = {Birkh{\"a}user},
address = {Basel},
year = {1990},
pages = {387--421},
}
@incollection{bogoya2012,
author = {Bogoya, J. M. and B{\"o}ttcher, A. and Grudsky, S. M.},
title = {Asymptotics of individual eigenvalues of a class of large {H}essenberg {T}oeplitz matrices},
booktitle = {Recent Progress in Operator Theory and Its Applications},
series = {Oper. Theory Adv. Appl.},
volume = {220},
publisher = {Springer Basel},
year = {2012},
pages = {77--95},
doi = {10.1007/978-3-0348-0346-5_5},
}
@article{bogoya2024,
author = {Bogoya, Manuel and Ekstr{\"o}m, Sven-Erik and Serra-Capizzano, Stefano and Vassalos, Paris},
title = {Matrix-less methods for the spectral approximation of large non-{H}ermitian {T}oeplitz matrices: A concise theoretical analysis and a numerical study},
journal = {Numer. Linear Algebra Appl.},
volume = {31},
number = {4},
year = {2024},
pages = {e2545},
doi = {10.1002/nla.2545},
}
@article{tilli1999,
author = {Tilli, Paolo},
title = {Some results on complex {T}oeplitz eigenvalues},
journal = {J. Math. Anal. Appl.},
volume = {239},
number = {2},
year = {1999},
pages = {390--401},
doi = {10.1006/jmaa.1999.6572},
}
@article{basor2022,
author = {Basor, Estelle and B{\"o}ttcher, Albrecht and Ehrhardt, Torsten},
title = {Harold {W}idom's work in {T}oeplitz operators},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {59},
number = {2},
year = {2022},
pages = {175--190},
doi = {10.1090/bull/1758},
}
@misc{openproblemsinnla-sp14,
author = {Townsend, Alex},
title = {{SP-14}: Widom's canonical distribution conjecture for Toeplitz eigenvalues},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/eigenvalues-and-inverse-problems/SP-14/README.md},
note = {Read at commit 7336302, 2026-10-06},
}
Resolution claims
We construct a continuous complex scalar symbol with neither an inner nor an outer annular holomorphic extension, whose Toeplitz sections are not canonically distributed. Along an infinite sequence of orders, each of two points outside the symbol range is an eigenvalue whose multiplicity is at least a fixed positive fraction of the order. The construction uses exact finite spectral corrections, …
For a continuous nonvanishing zero-index symbol, geometric decay of all sufficiently large top-right inverse corners of its Toeplitz matrices is equivalent to an inner-annulus extension. The proof of the difficult implication uses an exact bordering identity: summable corner increments control the first inverse row. Exponential control gives an analytic row-generating function, and a Hardy-space localization argument forces exponential decay of …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-10 | Partially resolved → Solution claimed | Clemens Thalhammer | A counterexample to canonical Toeplitz eigenvalue distribution v1 announced. |
| 2026-10-09 | Open → Partially resolved | The registry's curators | Inverse corners and deterministic canonical subsequences v1 recorded from OpenProblemsInNLA @ eb7c9d6, SP-14/report.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00159, "Widom's canonical distribution conjecture for Toeplitz eigenvalues", clopen.solutions.