A sharp numerical-range bound for matrix functions

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Family 325 · Functional analysis
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-25

1. The question

A matrix is more than its eigenvalues. A normal matrix, one that commutes with its conjugate transpose, behaves particularly well: evaluating a polynomial at the matrix has norm equal to the largest absolute value of that polynomial on its spectrum. For a nonnormal matrix, eigenvalues alone can badly underestimate this norm. Even a nonzero nilpotent matrix has only the eigenvalue zero.

The numerical range provides a larger picture of a matrix. Instead of asking only which scalars are eigenvalues, it records the quadratic expressions obtained from every unit vector. Crouzeix’s question asks whether this region always controls polynomial evaluation up to the universal factor two.

Family 325 concerns the complete version: the polynomial can itself have matrix coefficients of any finite size. The two manuscripts claim the sharp factor two in this setting. One takes a direct route to the inequality; the other constructs a single change of inner product and a single positive boundary representation that work for every coefficient size and every admissible analytic function.

2. Background and essential definitions

For a complex matrix $A$, its numerical range is

$$W(A)=\{x^*Ax:\ x^*x=1\}.$$

Here $x^*$ is conjugate transpose. This is a convex, compact subset of the complex plane containing the eigenvalues. For a bounded operator on an infinite-dimensional complex Hilbert space, the same definition applies, but one generally uses its closure when making compact-set statements.

Write a matrix polynomial as

$$P(z)=\sum_{j=0}^{d}z^jB_j, \qquad P[A]=\sum_{j=0}^{d}A^j\otimes B_j.$$

The coefficients $B_j$ are $m$-by-$m$ matrices. The tensor product means that $A$ acts on the original Hilbert-space coordinate and $B_j$ on an additional $m$-dimensional coordinate. All norms are operator norms for the Hilbert-space norm.

A complete numerical-range estimate is uniform in $m$. Merely proving the scalar case $m=1$ does not automatically give it: matrix coefficients need not commute, and a scalar rearrangement of factors can cease to be legitimate.

A similarity replaces $A$ by $SAS^{-1}$, with $S$ invertible. Its cost is the condition number $\|S\|\|S^{-1}\|$. Equivalently, a positive definite matrix $H$ defines a new inner product, and $S=H^{1/2}$ converts that inner product back to the usual one. A contraction is an operator of norm at most one.

3. Development history and previous partial results

The classical spectral theorem explains polynomial evaluation for normal operators. Von Neumann’s inequality gives another model: a contraction is controlled by the supremum of its polynomial on the unit disk. Numerical-range estimates seek an analogous principle for arbitrary operators, using a region adapted to the particular operator.

Crouzeix posed the factor-two question in 2004 and obtained a universal complete bound of 11.08 in 2007. Positive boundary integral formulas, including work of Delyon and Delyon, supplied an important mechanism. Crouzeix and Palencia subsequently proved the complete bound $1+\sqrt2$, bringing the universal constant close to the conjectured endpoint. Their argument also made the role of a positive boundary kernel especially clear.

The source manuscripts situate themselves after several 2026 scalar claims by Jin, Lorist and Schwenninger, and Luo. In particular, Lorist and Schwenninger’s August 17 version states the scalar result for bounded Hilbert-space operators and discusses the obstruction to simply transferring its argument to matrix coefficients. A further cited 2026 preprint establishes the complete factor-two estimate when the base matrix has order at most three.

Thus the family should be read as an all-dimensions complete result, together with a structural representation theorem. It is not presented here as the first scalar factor-two claim. The distinction matters because controlling every coefficient size is precisely what connects the inequality to powerful similarity and dilation principles.

4. What the new manuscripts claim

The complete inequality

M1 states that for every finite complex matrix $A$ and every finite matrix polynomial,

$$\|P[A]\|\leq 2\max_{z\in W(A)}\|P(z)\|.$$

There is no dependence on the size of $A$, the coefficient size, or the degree. Numerical ranges that are points or line segments are included.

M2 obtains the same estimate and extends it to every bounded operator on an arbitrary complex Hilbert space, including nonseparable spaces. Its closed numerical range is a complete 2-spectral set: the factor-two bound also holds for finite matrix-valued functions holomorphic on a neighborhood of that closed set. In particular, it covers rational functions whose poles lie outside it.

The constant cannot be improved universally. For

$$A=\begin{pmatrix}0&1\\0&0\end{pmatrix}, \qquad P(z)=z,$$

one has $\|A\|=1$, while $W(A)$ is the disk of radius $1/2$. This scalar example already requires the factor two.

A common metric and boundary density

The additional structural statement is finite-dimensional. Enclose $W(A)$ strictly inside a bounded convex domain $\Omega$ with regular real-analytic boundary, and choose a conformal bijection $f$ from $\Omega$ to the unit disk. Set $T=f(A)$.

M2 claims that the optimization problem

$$\kappa^2=\min\{\tau:\ I\leq H\leq\tau I, \ T^*HT\leq H\}$$

has an attained minimum with $1\leq\kappa^2\leq4$. The inequalities compare Hermitian matrices by positivity. For a minimizing $H$, the matrix $S=H^{1/2}$ makes $f(SAS^{-1})$ contractive, and its condition number is $\kappa\leq2$.

For this same similarity, the manuscript constructs a continuous positive matrix density $\Lambda$ on a boundary parameter circle. Its integral is the identity. If $G$ parametrizes the boundary and $d\sigma$ is normalized angular measure, then

$$v[SAS^{-1}]=\int_{\mathbb T}\Lambda(t)\otimes v(G(t))\,d\sigma(t).$$

The same density works for every finite coefficient size and every matrix-valued $v$ holomorphic near the closed domain. The metric is chosen once, rather than separately for each test function.

5. Why the advance matters

Importance: 5/5. The family claims the sharp answer to a central operator inequality in its complete form. Its strength is both quantitative and structural: a universal best constant is paired with a representation that explains how all matrix-valued evaluations can be controlled together.

For readers coming from numerical analysis, the numerical range is a way to retain information lost by looking only at eigenvalues. Bounds on matrix functions underlie estimates for approximations, evolution operators, and iterative methods. The theorem provides a universal norm bound; exploiting it efficiently in a particular computation is a further problem.

For operator theory, the complete estimate is particularly consequential. Complete boundedness is the language in which controlled similarities and normal dilations become available. M2 makes this relationship concrete through an optimal metric and a positive density, rather than only asserting an abstract existence principle.

The rating concerns these claimed mathematical consequences. It does not assign a reliability score to the manuscripts or determine priority among the surrounding preprints.

6. The main idea

Both approaches turn complex-analytic boundary information into positivity, then use positivity to control the size of matrix evaluation. The difficult part is preserving the order of matrix multiplications throughout.

M1 begins with a polynomial and vectors witnessing its largest singular value. It uses an exterior conformal coordinate to express boundary functions through a Faber-type expansion, a version of power-series expansion adapted to the exterior of the numerical range. A positive kernel controls the associated boundary operation in an $L^2$ norm.

The proof then tests products in both orders. Its resolvent expressions involve the original matrix, and the positive block matrix built from them controls the off-diagonal terms responsible for the norm. The manuscript attributes the sharp factor two to this paired comparison. Looking only at a scalar expression would conceal the order issue that the complete problem requires one to retain.

M2 reverses the starting point. It first chooses an optimal inner product making the disk-coordinate operator contractive. From that contraction it constructs the common positive density. Positivity and total mass equal to the identity make evaluation a compression of multiplication on a boundary function space: embed a vector into boundary functions, multiply pointwise, and project back.

The remaining task is to show that the optimal change of metric costs at most two. When its cost exceeds one, finite-dimensional separation supplies extremal vectors and an auxiliary matrix-valued analytic function linking them. Boundary identities turn these extremal data into positive block matrices. Orthogonality of Fourier components separates certain cross terms, and comparison of their squared norms yields the stated bound on $\kappa$.

Finally, enclosing domains shrink toward the numerical range. Finite compressions and holomorphic approximation transfer the resulting inequality to arbitrary Hilbert spaces. These passages extend the norm estimate; they should not be confused with a claim that the finite-dimensional continuous density itself has the same form in every infinite-dimensional setting.

7. Scope, limitations, and open questions

The main inequality concerns complex Hilbert-space operator norms, with finite matrix coefficient sizes. It is not a theorem for arbitrary Banach-space norms. The sharp example shows optimality over all operators, not that every individual operator needs the factor two; normal operators already have constant one.

The common continuous boundary density and the attained matrix optimization are stated for finite matrices and regular analytic convex enclosing domains. Degenerate numerical ranges enter the final inequality by approximation. One should retain this distinction between a convenient domain for the construction and the exact compact set in the endpoint estimate.

The results do not by themselves give a cheap algorithm for computing the optimal metric, evaluating a conformal map, or estimating the numerical range in very high dimensions. More refined constants for special classes, equality and stability phenomena, and computational versions remain separate questions.

The claims and their mechanisms are reported as stated in the two manuscripts. The earlier scalar and small-dimension results remain part of the development of the problem.

8. Sources and further reading

• Crouzeix and Palencia, the complete $1+\sqrt2$ bound.
• Lorist and Schwenninger, A solution to Crouzeix’s conjecture, version 2.
• The cited small-dimensional predecessor, Square functions and the complete Crouzeix conjecture in dimension three.
• Optional background: numerical range.

Source manuscripts

• M1. OpenAI, A direct proof of the complete Crouzeix inequality (September 26, 2026).
• M2. OpenAI, The complete Crouzeix theorem: optimal similarity and a common positive boundary representation (September 23, 2026).

Sources consulted on 25 October 2026 at repository revision 1581d4c. Download the manuscript citations.