Crouzeix’s conjecture (numerical range spectral set)

Prove that for every square matrix A and every complex polynomial p, the operator norm satisfies ∥p(A)∥_op ≤ 2 sup_{z∈W(A)} |p(z)|, i.e., that the optimal constant C_Crouzeix equals 2.

Background

Crouzeix’s conjecture asserts a sharp spectral set bound for the numerical range W(A), strengthening known inequalities. While solved in special cases (e.g., degree-2 minimal polynomials, powers, disks), the general case remains open.

A resolution has implications for functional calculus of matrices and sharp operator inequalities.

References

Crouzeix conjectured that the lower bound is sharp, thus $$ | p(A) |{op} \leq 2 \sup{z \in W(A)} |p(z)|$$ for all $p$: this is known as the Crouzeix conjecture.

— Mathematical exploration and discovery at scale  (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “Crouzeix’s conjecture” (Section 4.11)

The preceding results leave one equality question for the family. It is not known from the arguments here whether

\psi(A_t)=\psi_{\mathrm{cb}}(A_t)=2

for some t\in(0,1/2)\setminus{1/3}.

— Square Functions and the Complete Crouzeix Conjecture in Dimension Three  (2608.27346 - Åhag et al., 27 Aug 2026) in Final paragraph of Section 9, immediately following Corollary 9.10 (the family A_t)