The complete Crouzeix conjecture

2 papersLean

Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has \(\lVert P[A]\rVert\le2\sup_{z\in W(A)}\lVert P(z)\rVert\), where \(W(A)\) is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions.

Watch

▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

A direct proof of the complete Crouzeix inequality

OpenAI · September 26, 2026

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.

Read PDF Download Source

Cite

@misc{OAI:A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026,
  author = {{OpenAI}},
  title = {{A direct proof of the complete Crouzeix inequality}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026/paper.pdf}{OAI:A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026}},
  year = {2026}
}

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

OpenAI · September 23, 2026

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.

Read PDF Download Source

Cite

@misc{OAI:The-complete-Crouzeix-theorem-September-23-2026,
  author = {{OpenAI}},
  title = {{The complete Crouzeix theorem: optimal similarity and a common positive boundary representation}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-complete-Crouzeix-theorem-September-23-2026/paper.pdf}{OAI:The-complete-Crouzeix-theorem-September-23-2026}},
  year = {2026}
}

What was formalized in Lean

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The complete Crouzeix inequality bounds a matrix-valued polynomial evaluated at a matrix by its maximum norm on the numerical range. The formalized result proves the bound with constant \(2\) for every positive matrix and coefficient size, every polynomial degree, and arbitrary complex coefficients. It also proves that no smaller universal constant works. Normality of the matrix and nonempty interior of its numerical range are not assumed.

The formalization proves the complete Crouzeix inequality with sharp constant two for every bounded operator on an arbitrary complex Hilbert space. For every matrix-valued polynomial, its operator evaluation has norm at most twice its supremum norm on the numerical range. The same bound holds for finite matrix-valued functions holomorphic near the closure of the numerical range and for rational functions with poles outside that closure. No separability assumption is required.

For finite matrices in a bounded convex domain with regular real-analytic Jordan boundary, the structural result gives an attained optimal similarity with condition number at most two and one continuous positive boundary density of mass the identity representing every matrix-valued analytic test. The linked supporting statements include numerical-range geometry and finite-compression identities.

Comparator links

Result	Comparator statement
Complete Crouzeix inequality and sharpness	DirectCrouzeix.lean
Complete polynomial inequality and sharpness	CompleteCrouzeix.lean
Optimal similarity and boundary representation	StructuralCrouzeix.lean
Complete sharp Crouzeix theorem on arbitrary Hilbert spaces	CrouzeixHilbert.lean
Numerical-range geometry and finite-compression support	HilbertCrouzeix.lean

Scope note on GitHub →