Crouzeix's conjecture

Crouzeix's conjecture is a problem in matrix analysis proposed by Michel Crouzeix in 2004,^[1] and it can be stated as follows:

\|f(A)\|\leq 2\sup _{z\in W(A)}|f(z)|,

where the set $W(A)$ is the field of values of a n×n (i.e. square) complex matrix $A$ and $f$ is a complex function that is analytic in the interior of $W(A)$ and continuous up to the boundary of $W(A)$ . Slightly reformulated, the conjecture can also be stated as follows: for all square complex matrices $A$ and all complex polynomials $p$ :

\|p(A)\|\leq 2\sup _{z\in W(A)}|p(z)|

holds, where the norm on the left-hand side is the spectral operator 2-norm.

History

Crouzeix's theorem, proved in 2007, states that:^[2]

\|f(A)\|\leq 11.08\sup _{z\in W(A)}|f(z)|

(the constant $11.08$ is independent of the matrix dimension, thus transferable to infinite-dimensional settings).

Michel Crouzeix and Cesar Palencia proved in 2017 that the result holds for $1+{\sqrt {2}}$ ,^[3] improving the original constant of $11.08$ . More recently, dimension-dependent improvements have been obtained: Malman, Mashreghi, O'Loughlin and Ransford showed that for each fixed dimension $N$ there exists a constant $C_{N}<1+{\sqrt {2}}$ such that the inequality holds for all $N\times N$ matrices.^[4]

Related work connects the constant in Crouzeix-type inequalities to configuration constants arising from the Neumann–Poincaré operator and yields domain-dependent improvements of the Crouzeix–Palencia bound in certain settings.^[5]

The conjecture states that the constant can be refined to $2$ .

Special cases

Before the general case was resolved, the conjecture was shown to hold in several special cases. For instance, it holds for all normal matrices,^[6] for tridiagonal 3×3 matrices with elliptic field of values centered at an eigenvalue^[7] and for general n×n matrices that are nearly Jordan blocks.^[6]. Furthermore, Anne Greenbaum and Michael L. Overton provided numerical support for Crouzeix's conjecture.^[8]

Proof

On July 27, 2026, neurosurgeon Shanmu Jin posted a preprint claiming a proof of Crouzeix's conjecture.^[9] Jin reported that the proof was obtained with the assistance of OpenAI's GPT-5.6 Sol, after which he checked the resulting argument.^[10] Greenbaum and Alex Townsend reported that they and Crouzeix had thoroughly reviewed the proof and believed it to be correct.^[10]

On August 4, 2026, Emiel Lorist and Felix Schwenninger posted an independent proof of the conjecture using a different argument.^[10] They reported that GPT-5.6 in its Sol variant "was used to explore proof strategies for this note".^[11]

References

[1]
Crouzeix, Michel (2004-04-01). "Bounds for Analytical Functions of Matrices". Integral Equations and Operator Theory. 48 (4): 461–477. doi:10.1007/s00020-002-1188-6. ISSN 0378-620X. S2CID 121371601.
[2]
Crouzeix, Michel (2007-03-15). "Numerical range and functional calculus in Hilbert space". Journal of Functional Analysis. 244 (2): 668–690. doi:10.1016/j.jfa.2006.10.013.
[3]
Crouzeix, Michel; Palencia, Cesar (2017-06-07). "The Numerical Range is a $(1+{\sqrt {2}})$ -Spectral Set". SIAM Journal on Matrix Analysis and Applications. 38 (2): 649–655. doi:10.1137/17M1116672.
[4]
Malman, Bartosz; Mashreghi, Javad; O'Loughlin, Ryan; Ransford, Thomas (2024). "On the Crouzeix ratio for $N\times N$ matrices". arXiv:2409.14127 [math.FA].{{cite arXiv}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
[5]
Malman, Bartosz; Mashreghi, Javad; O'Loughlin, Ryan; Ransford, Thomas (2025-04-10). "Double-Layer Potentials, Configuration Constants, and Applications to Numerical Ranges". International Mathematics Research Notices. 2025 (8) rnaf084. doi:10.1093/imrn/rnaf084.{{cite journal}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
[6]
Choi, Daeshik (2013-04-15). "A proof of Crouzeix's conjecture for a class of matrices". Linear Algebra and Its Applications. 438 (8): 3247–3257. doi:10.1016/j.laa.2012.12.045.
[7]
Glader, Christer; Kurula, Mikael; Lindström, Mikael (2018-03-01). "Crouzeix's Conjecture Holds for Tridiagonal 3 x 3 Matrices with Elliptic Numerical Range Centered at an Eigenvalue". SIAM Journal on Matrix Analysis and Applications. 39 (1): 346–364. arXiv:1701.01365. doi:10.1137/17M1110663. S2CID 43922128.
[8]
Greenbaum, Anne; Overton, Michael L. (2017-05-04). "Numerical investigation of Crouzeix's conjecture" (PDF). Linear Algebra and Its Applications. 542: 225–245. doi:10.1016/j.laa.2017.04.035.
[9]
Jin, Shanmu (2026). "The Numerical Range Is a 2-Spectral Set".
[10]
Townsend, Alex; Greenbaum, Anne (2026-08-14). "The Neurosurgery Resident Who Proved Crouzeix's Conjecture" (PDF).
[11]
Lorist, Emiel; Schwenninger, Felix (2026). "A solution to Crouzeix's conjecture". arXiv:2608.03841 [math.CA].

Crouzeix's conjecture

History

Special cases

Proof

See also

References

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