The complete Crouzeix inequality

Source and verification

This article explains family 325 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

Scalar numerical-range bounds control $f(A)$ using the values of a scalar function $f$. The complete inequality asks for the same constant for matrix-valued functions, independently of their coefficient dimension. Family 325 presents two claimed proofs of the sharp constant two: a direct polynomial estimate and a common positive boundary representation after a controlled similarity. (OpenAI, 2026) (OpenAI, 2026b)

🏷️ Preliminaries: numerical range and amplification

For a complex matrix $A\in M_n(\mathbb{C})$, its numerical range is

$$\begin{array}{c}W(A)\\ =\{x^{\ast }Ax:x\in {\mathbb{C}}^n,\Vert x{\Vert }_2=1\}.\end{array}$$

It is compact and convex. For a matrix-valued polynomial

$$\begin{array}{c}P(z)=\sum _{j=0}^d{B}_j{z}^j,\\ B_j\in M_m(\mathbb{C}),\end{array}$$

evaluation means

$$P[A]=\sum _{j=0}^d{A}^j\otimes B_j.$$

The base space is the first tensor factor. All norms are Hilbert-space operator norms.

The word complete means that one bound works for every coefficient size $m$. Applying a scalar estimate entry by entry generally loses a factor depending on $m$. Moreover, products of coefficient matrices need not commute, so a scalar argument cannot automatically be amplified.

Manuscript claim

For all positive integers $n,m$ and all matrix polynomials,

$$\Vert P[A]\Vert \le 2\underset{z\in W(A)}{\max }\Vert P(z)\Vert .$$

The constant is uniform in both dimensions and the degree, includes numerical ranges with empty interior, and is sharp. The structural companion also obtains the complete polynomial, rational, and holomorphic bounds for bounded operators on arbitrary complex Hilbert spaces, using the closure of the numerical range where appropriate. (OpenAI, 2026a) (OpenAI, 2026b)

Sharpness has an elementary witness:

$$A=\left(\begin{array}{cc}0& 2\\ 0& 0\end{array}\right),P(z)=z.$$

Here $\Vert A\Vert =2$, while $W(A)$ is the closed unit disk. Indeed, $x^{\ast }Ax=2\overline{x_1}{x}_2$ has modulus at most $|x_1{|}^2+|x_2{|}^2=1$, and every radius between zero and one occurs.

🏷️ A metric that makes the conformal image contractive

Temporarily enclose $W(A)$ strictly inside a bounded convex domain $\Omega$ with regular real-analytic boundary. Let $f:\Omega \to \mathbb{D}$ be conformal and set $T=f(A)$.

A positive definite matrix $H$ defines the squared norm $x^{\ast }Hx$. The inequality $T^{\ast }HT⪯H$ says that $T$ is contractive in this norm. The structural paper minimizes

$${\kappa }^2=\min \left\{\tau :\begin{array}{c}I⪯H⪯\tau I,\\ T^{\ast }HT⪯H\end{array}\right\}.$$

Its claimed theorem is attainment with $\kappa \le 2$. If $S=H^{1/2}$, then $D=ST{S}^{-1}$ is a contraction and

$$\Vert S\Vert \Vert S^{-1}\Vert =\kappa \le 2.$$

Existence of some adapted norm is much easier than the sharp condition-number bound. The proof must control the least distortion of the metric, not merely solve a Lyapunov inequality.

🏷️ One positive density for every test function

Let $G$ parametrize $\partial \Omega$ from the unit circle. The structural companion constructs one continuous positive matrix density $\Lambda$ such that

$$\Lambda (t)⪰0,{\int }_{\mathbb{T}}\Lambda (t)d\sigma (t)=I,$$

where $\sigma$ is normalized angular measure. For $A^{\prime }=SA{S}^{-1}$, it claims

$$v[A^{\prime }]={\int }_{\mathbb{T}}\Lambda (t)\otimes v(G(t))d\sigma (t)$$

for every finite matrix size and every matrix-valued $v$ holomorphic near $\overline{\Omega }$. The same $S$ and $\Lambda$ serve all tests. (OpenAI, 2026b)

The norm deduction can be checked directly. Define

$$(Vb)(t)=(\Lambda (t{)}^{1/2}\otimes I_m)b.$$

The mass-one identity makes $V$ an isometry. The displayed evaluation is the compression $V^{\ast }{M}_{v}V$, where $M_v$ multiplies boundary functions by $I_n\otimes v(G(t))$. Therefore $\Vert v[A^{\prime }]\Vert \le {\sup }_{\partial \Omega }\Vert v\Vert$. Conjugation back costs at most $\kappa$.

The research step is proving $\kappa \le 2$. An optimal metric supplies extremal vectors at its largest and smallest eigenvalues. The source compares two positive boundary block systems and preserves both multiplication orders of analytic tests. Orthogonality of the relevant positive and negative Fourier components, including a zero matrix mean, prevents loss in the final constant.

🏷️ The direct coefficient argument

The direct paper estimates one polynomial at a time. An exterior conformal map $h$ defines Faber polynomials $b_k$ by

$$\frac{\lambda h^{\prime }(\lambda )}{h(\lambda )-z}=\sum _{k\ge 0}{b}_k(z){\lambda }^{-k}.$$

On the boundary, $b_k\circ h$ consists of its positive Fourier mode ${\lambda }^k$ plus strictly negative modes. A nonnegative kernel of mass one in each variable controls the negative coefficients in $L^2$.

For $G={\sum }_k{b}_k{C}_k$, this gives coefficient estimates with weight one on the constant term and weight two on the other terms, using the Hilbert–Schmidt norm. Dual norms consequently have reciprocal weights. A top singular pair for $P[A]$ turns the operator bound into estimates of scalar functionals on matrix-valued polynomials.

Testing on $PG$ and $GP$ separately is essential. The numerical-range condition makes a boundary resolvent plus its adjoint positive; its two-by-two compression bounds the off-diagonal contribution by diagonal ones. The coefficient weights match this comparison and yield two. These are matrix estimates throughout, not an entrywise application of the scalar theorem. (OpenAI, 2026a)

Finally, shrinking analytic convex neighborhoods gives arbitrary compact numerical ranges, including segments and points. For a bounded operator and a polynomial of degree $d$, finite spans of vectors $A^j\xi$, $0\le j\le d$, reproduce the required polynomial action under compression. Taking enough vectors to test the norm transfers the finite-dimensional inequality without assuming separability. The structural companion handles the subsequent holomorphic approximation.

🏷️ Key steps and proof outlines

Faber coefficients encode both analytic and reflected information

Normalize a matrix polynomial $F$ so its boundary operator norm is at most one. Expand an arbitrary matrix polynomial $G$ in the Faber basis $G={\sum }_k{b}_k{C}_k$. On the exterior circle, each $b_k$ contributes one positive Fourier mode and a strictly negative tail. The source proves that the operator producing those negative tails is contractive in the relevant Hilbert–Schmidt $L^2$ norm, using a nonnegative kernel with both angular marginals equal to one.

The constant mode is counted once and nonconstant modes twice in the upper coefficient estimate. Dualizing therefore gives reciprocal weights one and one half. These weights are essential to the sharp constant: they match the positive and negative Fourier pieces of the Hermitian boundary fields. (OpenAI, 2026a)

Ordered products turn a top singular pair into two diagonal bounds

Choose unit vectors $x,y$ with $F[A]x=\gamma y$ and $F[A{]}^{\ast }y=\gamma x$, where $\gamma =\Vert F[A]\Vert$. Multiplicativity of polynomial evaluation gives

$$y^{\ast }(FG)[A]x=\gamma x^{\ast }G[A]x,y^{\ast }(GF)[A]x=\gamma y^{\ast }G[A]y.$$

Both products are required because the coefficient matrices need not commute. Boundary multiplication by $F$ is contractive in Hilbert–Schmidt norm on either side. Applying the coefficient estimates to the two identities bounds the two diagonal functionals with the same weighted norm. Their dual coefficient arrays become the diagonal blocks of a boundary resolvent compression.

Positivity of one block matrix yields the factor two

The numerical-range condition makes a boundary resolvent field plus its adjoint positive. Compress it to the columns corresponding to the singular vectors to obtain

$$\Delta =\left(\begin{array}{cc}A& B^{\ast }\\ B& C\end{array}\right)⪰0.$$

Let $J=\mathrm{diag}(I,-I)$. Since $J\Delta J⪰0$, the trace of the product of these two positive matrices is nonnegative. Expanding gives

$$2\Vert B{\Vert }_{\mathrm{HS}}^2\le \Vert A{\Vert }_{\mathrm{HS}}^2+\Vert C{\Vert }_{\mathrm{HS}}^2.$$

Parseval evaluates the integrated diagonal norms using the reciprocal coefficient weights. The off-diagonal field has a lower bound from the singular-pair functional. The source’s exact normalization reduces the comparison to $2a^2\le 8a^2/{\gamma }^2$ for a positive auxiliary norm $a$, hence $\gamma \le 2$. This is a matrix-level comparison uniform in coefficient size.

The optimal metric gives a common representation

The structural companion begins with the semidefinite optimization for $H$. Strict stability of $T=f(A)$ supplies a feasible Lyapunov metric; finite-dimensional compactness gives a minimizing one after bounding its largest eigenvalue. Optimality yields positive dual variables supported at the extreme eigenspaces and a balance identity

$$X{X}^{\ast }-Y{Y}^{\ast }=Q-DQ{D}^{\ast },D=H^{1/2}T{H}^{-1/2}.$$

Writing $L=Q^{1/2}$, the rows $[LY]$ and $[DLX]$ have equal Gram matrices. The induced isometry between their ranges extends to a unitary block matrix. Its transfer function supplies the extremal analytic test needed to compare the optimal metric with the positive numerical-range boundary field. The ordered boundary estimates then force the optimal condition number to be at most two. (OpenAI, 2026b)

For the resulting contraction $D$, define on $|\zeta |=1$

$${\Phi }_D(\zeta )=(I-D/\zeta {)}^{-1}+[(I-D/\zeta {)}^{-1}{]}^{\ast }-I.$$

Congruence by $I-D/\zeta$ gives $I-D^{\ast }D⪰0$, proving positivity. The strict spectral-radius bound makes its Fourier series uniformly convergent, even if $\Vert D\Vert =1$. Its mean is $I$, and its moments reproduce powers of $D$. Transporting this density to the conformal boundary gives one positive representation for every matrix-valued analytic test.

Approximation preserves the sharp constant

Shrink analytic convex neighborhoods onto the numerical range. Polynomial suprema converge to the supremum on the compact limit, so the bound remains two even for a segment or point. For arbitrary bounded operators, finite-dimensional spans containing the required iterates reproduce polynomial action on chosen test vectors. Rational and holomorphic versions then use the companion’s approximation and functional-calculus arguments. The matrix amplification is preserved throughout, rather than recovered by entrywise estimates at the end.

See Also

• on Crouzeix’s conjecture records the earlier numerical-range framework and the $1+\sqrt{2}$ bound.
• on two proofs of Crouzeix’s conjecture compares the scalar proofs; this family addresses the additional uniformity in matrix coefficient size.

🏷️ Manuscripts and verification scope

• A direct proof of the complete Crouzeix inequality — pinned manuscript and companion source.
• The complete Crouzeix theorem: optimal similarity and a common positive boundary representation — pinned manuscript and companion source.

The family scope page describes selected formalization coverage. The documented coverage includes finite matrix amplification and sharpness, the finite-dimensional optimal similarity and positive representation, and the bounded-Hilbert-operator inequality. The finite structural conclusions should not be silently strengthened to an infinite-dimensional similarity construction. No Lean build, axiom audit, or manuscript-to-formal-statement equivalence check was performed for these notes.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026a. A direct proof of the complete Crouzeix inequality.

🐻  OpenAI 2026b. The complete Crouzeix theorem: optimal similarity and a common positive boundary representation.