The complete Crouzeix conjecture

What OpenAI claims

Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has ‖ P[A]‖ ≤ 2 sup_{z ∈ W(A)}‖ P(z)‖, where W(A) is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions. (Lean)

Papers (2)

1. A direct proof of the complete Crouzeix inequality

   26 September 2026 · Preprint, not peer reviewed · PDF

   We give a direct proof of the sharp constant-two numerical-range inequality for matrix-valued polynomials in all finite base and coefficient dimensions. This resolves the complete Crouzeix conjecture in its matrix formulation, including matrices whose numerical ranges are points or line segments.

2. The complete Crouzeix theorem: optimal similarity and a common positive boundary representation

   23 September 2026 · Preprint, not peer reviewed · PDF

   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 bounded convex domain containing its numerical range with regular real-analytic Jordan boundary, the optimal similarity making its conformal disk image contractive is attained with condition number at most two. For the similar matrix, one continuous positive boundary density of mass the identity represents the evaluation of every matrix-valued function holomorphic near the closed domain.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.