Crouzeix's conjecture

About 22 years old · traced to 2004

Let $n \ge 1$ and let $A \in \mathbb{C}^{n \times n}$ . Write

W(A) = \{\, x^{*}Ax : x \in \mathbb{C}^{n},\ \|x\|_{2} = 1 \,\}

for the field of values (numerical range) of $A$ , a compact convex subset of $\mathbb{C}$ , and let $\|\cdot\|$ denote the operator norm induced by the Euclidean norm on $\mathbb{C}^{n}$ , i.e. the spectral $2$ -norm.

Let $f$ be a complex-valued function that is analytic in the interior of $W(A)$ and continuous on $W(A)$ , and let $f(A)$ be the corresponding matrix, defined for polynomials by substitution and extended by continuity. Then

\|f(A)\| \le 2 \sup_{z \in W(A)} |f(z)|.

Equivalently, for every $n \ge 1$ , every $A \in \mathbb{C}^{n \times n}$ and every polynomial $p \in \mathbb{C}[z]$ ,

\|p(A)\| \le 2 \sup_{z \in W(A)} |p(z)|.

The constant $2$ is independent of $n$ , of $A$ , and of $f$ .

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Crouzeix's conjecture
Crouzeix's conjecture is an unsolved problem in matrix analysis. It was proposed by Michel Crouzeix in 2004, and it can be stated as follows:
source: Wikipedia

References

Primary source

Wikipedia

Additional references

Wikipedia, Crouzeix's conjecture, the article this problem comes from.

algebra unsolved-problem

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2026 preprint claims to prove the conjecture, but no independent verification has appeared.

Crouzeix posed the conjecture in 2004: the norm of a polynomial in a matrix should be bounded by twice its maximum on the matrix’s numerical range. A new preprint claims the bound in full generality, replacing the former open status with an unverified claimed proof.

Known results

Crouzeix obtained the bound $11.08$ (2004); Crouzeix and Palencia improved it to $1+\sqrt{2}$ .
The conjecture holds for normal matrices and all $2\times2$ matrices.
Crouzeix proved it for $3\times3$ nilpotent matrices; other results cover specialized tridiagonal, contraction, and non-cyclic classes.
Reductions connect the general problem to cyclic matrices, differentiation operators, and analytic truncated Toeplitz operators.

August 2026 claimed proof

The preprint “A solution to Crouzeix’s conjecture” claims a perturbation lemma for $2$ -dilations proves the estimate for every bounded Hilbert-space operator, even for rational functions. A separate report links a candidate proof to GPT-5.6-Sol; neither claim has independent verification, a published proof, or a documented error.

Current status (as of August 2026): The conjecture is claimed solved by a new preprint, but the claim remains unverified; the established special cases and previous bounds remain the only confirmed results.

GPT-5.6 SolOpenAIsolved2026-08-18evidence
A candidate proof of Crouzeix’s conjecture is under review

Sources

No solutions have been posted yet.