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Crouzeix constant (C2C_{2})

Crouzeix constant: determine C2C_{2} or improve its bounds; the repository records 2 (lower) and 2 (upper).

MSC 47A12 · Numerical range, numerical radiusMSC 15A60 · Norms of matrices, numerical range, applications of functional analysis to matrix theory

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

C2C_{2} is the Crouzeix constant (sometimes denoted QQ). It is the smallest constant CC such that for every n≥1n \ge 1, every complex matrix A∈Cn×nA \in \mathbb{C}^{n \times n}, and every complex polynomial pp one has

∥p(A)∥ ≤ C max⁡z∈W(A)∣p(z)∣,\|p(A)\| \ \le\ C \ \max_{z \in W(A)} |p(z)|,

where ∥⋅∥\|\cdot\| is the operator norm induced by the Euclidean norm (i.e. the spectral norm), and

W(A):={v∗Av:v∈Cn, ∥v∥2=1}W(A) := \{ v^\ast A v : v \in \mathbb{C}^n,\ \|v\|_2 = 1\}

is the numerical range (field of values) of AA.

Equivalently,

C2=sup⁡n≥1 sup⁡A∈Cn×n sup⁡p≢0∥p(A)∥max⁡z∈W(A)∣p(z)∣.C_{2} = \sup_{n \ge 1}\ \sup_{A \in \mathbb{C}^{n\times n}}\ \sup_{p \not\equiv 0} \frac{\|p(A)\|}{\max_{z \in W(A)} |p(z)|}.

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Background

  • Crouzeix conjectured (in [C2004]) that C2=2C_{2}=2. The lower bound 22 matches the upper bounds of [Jin2026] and [LS2026], so the conjecture is (claimed to be) settled: C2=2C_{2} = 2.
  • Priority and independence. [Jin2026] was posted first, on 27 July 2026, and so has priority of public announcement; [LS2026] appeared on arXiv on 4 August 2026. The two proofs are nonetheless independent, and take different routes. [LS2026, §1] has said so since its first version: "a proof of Crouzeix's conjecture appeared independently in [Jin2026] using a different approach involving function-theoretic representations of matrix-valued functions. Our proof instead uses the operator-theoretic Lemma 1." The AI-disclosure statement in [LS2026, v2] dates the interactions that produced its key observation to 25 and 26 July 2026, and records in a footnote that "the authors became aware of [Jin2026] on July 28, 2026, after the interactions described above." Between 12 and 27 August 2026 this page instead described [LS2026] as merely acknowledging [Jin2026]; that read a citation of prior public posting as a statement of dependence, and was incorrect.
  • The conjectured constant 22 is known to hold in a number of special cases; see, for instance, [Cho2013], [GKL2018], [CGL2018].
  • Numerical experiments strongly support the conjecture; see [GO2018].
  • Wikipedia page on Crouzeix’s conjecture
  • AIM workshop page on Crouzeix’s conjecture

References (from the repository page)

  • [C2004] Crouzeix, Michel. Bounds for analytical functions of matrices. Integral Equations and Operator Theory 48 (2004), no. 4, 461–477. DOI: 10.1007/s00020-002-1188-6.
  • [C2007] Crouzeix, Michel. Numerical range and functional calculus in Hilbert space. J. Funct. Anal. 244 (2007), no. 2, 668–690. DOI: 10.1016/j.jfa.2006.10.013.
  • [CP2017] Crouzeix, Michel; Palencia, César. The Numerical Range is a (1+2)(1+\sqrt2)-Spectral Set. SIAM J. Matrix Anal. Appl. 38 (2017), no. 2, 649–655. DOI: 10.1137/17M1116672.
  • [DD1999] Delyon, Bernard; Delyon, François. Generalization of Von Neumann’s spectral sets and integral representation of operators. Bull. Soc. Math. France 127 (1999), 25–42. (See also: https://www.numdam.org/article/BSMF_1999__127_1_25_0.pdf)
  • [Cho2013] Choi, Daeshik. A proof of Crouzeix’s conjecture for a class of matrices. Linear Algebra Appl. 438 (2013), no. 8, 3247–3257. DOI: 10.1016/j.laa.2012.12.045.
  • [GO2018] Greenbaum, Anne; Overton, Michael L. Numerical investigation of Crouzeix’s conjecture. Linear Algebra Appl. 542 (2018), 225–245. DOI: 10.1016/j.laa.2017.04.035.
  • [GKL2018] Glader, Christer; Kurula, Mikael; Lindström, Mikael. Crouzeix’s conjecture holds for tridiagonal 3×33\times 3 matrices with elliptic numerical range centered at an eigenvalue. SIAM J. Matrix Anal. Appl. 39 (2018), no. 1, 346–364. DOI: 10.1137/17M1110663. arXiv:1701.01365.
  • [CGL2018] Caldwell, Trevor; Greenbaum, Anne; Li, Kenan. Some Extensions of the Crouzeix–Palencia Result. SIAM J. Matrix Anal. Appl. 39 (2018). DOI: 10.1137/17M1140832. arXiv:1707.08603.
  • [RS2018] Ransford, Thomas; Schwenninger, Felix L. Remarks on the Crouzeix–Palencia proof that the numerical range is a (1+2)(1+\sqrt2)-spectral set. SIAM J. Matrix Anal. Appl. 39 (2018), no. 1, 342–345. DOI: 10.1137/17M1143757. arXiv:1708.08633.
  • [Jin2026] Jin, Shanmu. The Numerical Range Is a 22-Spectral Set. Preprints.org manuscript 202607.1919, first posted 27 July 2026; version 4 posted 7 August 2026, marked there as not peer-reviewed. DOI: 10.20944/preprints202607.1919.v4.
  • [LS2026] Lorist, Emiel; Schwenninger, Felix L. A solution to Crouzeix's conjecture. arXiv:2608.03841 (2026).

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 2; best known upper bound 2. Values marked with an asterisk have minimal verification. The recorded bounds coincide, so the repository treats the value as determined.

Known upper bounds

Bound Reference Comments
11.0811.08 [C2007] First dimension-free bound. Also holds in the completely bounded (matrix-valued) setting.
1+2≈2.414211+\sqrt{2} \approx 2.41421 [CP2017] Best universal upper bound prior to the [Jin2026] / [LS2026] pair. Also holds in the completely bounded setting.
22 [Jin2026] Jin: proof of Crouzeix's conjecture, first posted on preprints.org on 27 July 2026. The current version 4 is marked there as not peer-reviewed.
22 [LS2026] Lorist–Schwenninger: independent proof by a different route, combining previous tools for weaker estimates with a "simple perturbation lemma for 2-dilations" applied to the iterates fnf^n in the double-layer potential representation. See the note on priority below.

Known lower bounds

Bound Reference Comments
11 Trivial Take p≡1p \equiv 1.
22 [C2007] Achieved by p(z)=zp(z)=z and A=(0200)A=\begin{pmatrix}0 & 2\\\\ 0 & 0\end{pmatrix}, for which W(A)W(A) is the unit disk.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Crouzeix constant ($C_{2}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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