Result 325, Functional analysis

The complete Crouzeix conjecture

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]∥≤2sup⁡z∈W(A)∥P(z)∥\lVert P[A]\rVert\le2\sup_{z\in W(A)}\lVert P(z)\rVert, where W(A)W(A) is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions.

Lean formalization Proof

The bigger picture

Why it matters

Applying a polynomial to a linear operator can produce amplification that is hard to estimate directly. These unreviewed manuscripts claim a sharp factor-two bound that controls this amplification using a geometric set associated with the operator.

What changes?

The reported bound covers every bounded operator A on a complex Hilbert space, including infinite-dimensional spaces without a separability assumption. Its numerical range consists of inner products of A v with v over unit vectors v. For every finite matrix whose entries are polynomials, substituting A for the variable gives an operator with norm at most twice the supremum of the polynomial matrix's norm on that range. The constant two is sharp and independent of the coefficient matrix size.

What does that help mathematicians do?

The result would let researchers bound an entire matrix of polynomial operator expressions through values at ordinary complex numbers, without an extra penalty for the matrix size. Sharpness rules out any smaller universal constant. The second abstract also reports the same control for finite matrix-valued functions holomorphic near the numerical range's closure, extending the deduction beyond polynomials to functions that are complex differentiable nearby.

Are there practical applications?

The immediate value is foundational in operator theory: it provides a uniform rule for transferring function bounds on a geometric region into bounds after operator substitution. This could support analyses involving coupled operator expressions, but the supplied sources establish no computational speedup or deployment. Their claimed contribution is the exact, dimension-independent estimate itself.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

A direct proof of the complete Crouzeix inequality

September 26, 2026 12 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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

September 23, 2026 22 pages Main result formalized in Lean

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/325.md.

The complete Crouzeix conjecture

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 22 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

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.