Complete Crouzeix conjecture
Note: OpenProblemsInNLA (MF-23) lists this problem as solved in Lean: a Lean proof of the complete Crouzeix inequality, built on OpenAI's formalization, at https://github.com/ajt60gaibb/OpenProblemsInNLA/tree/5487945e5a72abd952b0c89915915de8cc9b3e65/matrix-functions-and-stability/MF-23/lean. It uses Lean 4.34.1, newer than the registry's verifier, so it has not been checked here.
For $A \in \mathbb{C}^{n \times n}$ let $W(A) = \{x^*Ax : x \in \mathbb{C}^n,\ x^*x = 1\}$ be its numerical range. For a matrix polynomial $F = (f_{ij})_{i,j=1}^m \in \mathbb{C}^{m \times m}[z]$ let $F(A) = (f_{ij}(A))_{i,j=1}^m$, an $mn \times mn$ block matrix, and let $\|\cdot\|_2$ be the spectral norm. For $m = 1$ the inequality below is the conjecture Crouzeix posed in 2004 for scalar polynomials, proving it for $n = 2$; the constant $2$ cannot be lowered, as $A = \begin{pmatrix} 0 & 2 \\ 0 & 0 \end{pmatrix}$ and $F(z) = z$ show. Crouzeix (2007) proved the inequality below with $11.08$ in place of $2$ and conjectured it with $2$. Crouzeix and Palencia (2017) lowered the constant to $1 + \sqrt 2$, and Åhag, Czyż and Virtanen (2026) proved the inequality with constant $2$ for $n \le 3$.
Problem. Is it true that, for all $n, m \ge 1$, every $A \in \mathbb{C}^{n \times n}$ and every $F \in \mathbb{C}^{m \times m}[z]$, $$\|F(A)\|_2 \le 2 \max_{z \in W(A)} \|F(z)\|_2?$$
Source: Crouzeix (2007), for matrix-valued polynomials, extending the conjecture for scalar polynomials he posed in 2004; see Åhag, Czyż and Virtanen (2026), (1.2).
Imported from github.com/ajt60gaibb/OpenProblemsInNLA.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- FA — Functional Analysis
- Secondary
- NA
- Keywords
- matrix polynomials, numerical ranges, OpenProblemsInNLA, spectral sets
- Literature checked
- Oct. 8, 2026 by clemens-admin
References
@article{crouzeix2007,
author = {Crouzeix, Michel},
title = {Numerical range and functional calculus in {H}ilbert space},
journal = {J. Funct. Anal.},
volume = {244},
number = {2},
year = {2007},
pages = {668--690},
doi = {10.1016/j.jfa.2006.10.013},
}
@article{crouzeix2004,
author = {Crouzeix, Michel},
title = {Bounds for analytical functions of matrices},
journal = {Integral Equations Operator Theory},
volume = {48},
number = {4},
year = {2004},
pages = {461--477},
doi = {10.1007/s00020-002-1188-6},
}
@article{crouzeix2017,
author = {Crouzeix, Michel and Palencia, C{\'e}sar},
title = {The numerical range is a $(1+\sqrt{2})$-spectral set},
journal = {SIAM J. Matrix Anal. Appl.},
volume = {38},
number = {2},
year = {2017},
pages = {649--655},
doi = {10.1137/17M1116672},
eprint = {1702.00668},
archivePrefix = {arXiv},
}
@misc{ahag2026,
author = {{\AA}hag, Per and Czy{\.z}, Rafa{\l} and Virtanen, Jani},
title = {The Complete {C}rouzeix Conjecture in Dimension Three and the {C}lou{\^a}tre--{O}stermann--{R}ansford conjecture},
year = {2026},
eprint = {2608.27346},
archivePrefix = {arXiv},
}
@misc{openproblemsinnla-mf23,
author = {Townsend, Alex},
title = {{MF-23}: Complete Crouzeix conjecture},
howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
year = {2026},
url = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-functions-and-stability/MF-23/README.md},
note = {Read at commit eb7c9d6, 2026-10-07},
}
Resolution claims
For all positive integers $n$ and $m$, every $A \in M_n(\mathbb{C})$ and every polynomial $P(z) = \sum_k B_k z^k$ with $B_k \in M_m(\mathbb{C})$, the manuscript proves $\|P[A]\| \le 2 \max_{z \in W(A)} \|P(z)\|$, where $P[A] = \sum_k A^k \otimes B_k$ and $W(A)$ is the numerical range. This is the sharp constant-two numerical-range inequality for matrix-valued polynomials in all finite base …
We resolve the complete Crouzeix conjecture by proving the sharp constant-two numerical-range inequality for every bounded operator on a complex Hilbert space and every matrix-valued polynomial. No separability assumption is needed. The closure of the numerical range is a complete $2$-spectral set, and the bound extends to finite matrix-valued functions holomorphic near that closure. For a finite matrix and a …
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-00234, "Complete Crouzeix conjecture", clopen.solutions.