on Crouzeix's conjecture and the 1+sqrt(2) bound

Overview

Crouzeix’s Conjecture (2004) posits that for any square matrix $A$ and any function $f$ analytic on its numerical range $W(A)$, the operator norm is bounded by $\Vert f(A)\Vert \le 2\Vert f{\Vert }_{W(A)}$. Since the original $11.08$ bound, the landmark proof by Crouzeix and Palencia (2017) established the universal constant $(1+\sqrt{2}\approx 2.414)$. This post reviews the analytic strategies and recent refinements, such as the analytic configuration constant $a(\Omega )$, that bridge the gap between this universal bound and the conjectured value of $2$ (Crouzeix, 2004; Crouzeix & Palencia, 2017).

🏷️ Foundational Statement

The conjecture focuses on the numerical range (field of values) of a linear operator $A$ on a Hilbert space $H$:

$$W(A)=\{⟨Av,v⟩:v\in H,\Vert v\Vert =1\}$$

Crouzeix asserted that $W(A)$ is a 2-spectral set for $A$, meaning the spectral norm $\Vert f(A)\Vert$ is bounded by twice the supremum of $f$ on $W(A)$. Equivalently, the Crouzeix ratio $\psi (A)={\sup }_{\Vert f{\Vert }_{\infty }\le 1}\Vert f(A)\Vert$ should satisfy $\psi (A)\le 2$.

flowchart LR
    A["1975: Okubo–Ando (disk → constant 2)"]
    B["2004: Crouzeix – bound 11.08"]
    C["2017: Crouzeix–Palencia – bound 1+√2 ≈ 2.414"]
    D["2018: Ransford–Schwenninger – Lemma sharpness"]
    E["2024: Malman et al. – 1+√(1+a(Ω)) bound"]
    A --> B --> C --> D
    C --> E

🏷️ Palencia’s Proof and Target Inequalities

The breakthrough to $1+\sqrt{2}$ relies on the Palencia Splitting. For a convex domain $\Omega \supset W(A)$, one defines the operator $S(\varphi ,A)=f(A)+g(A{)}^{\ast }$, where $g$ is the Cauchy transform of the conjugate function $\bar{f}$.

Palencia’s Lemma (2.3)

For $\varphi \in C(\partial \Omega )$, the operator $S(\varphi ,A)$ satisfies:

$$\Vert S(\varphi ,A)\Vert \le 2\Vert \varphi {\Vert }_{\infty ,\partial \Omega }$$

This utilizes the positivity of the Neumann–Poincaré kernel $\mu (\sigma ,A)$ and the identity ${\int }_{\partial \Omega }\mu (\sigma ,A)d\sigma =2I$.

Proof: Derivation of the $1+\sqrt{2}$ Bound

The strategy for a bounded convex domain $\Omega \supset W(A)$ follows three primary steps:

Step 1: Cauchy Transform Splitting For $f\in \mathcal{A}(\Omega )$, we write $f(A)=S(f,A)-g(A{)}^{\ast }$, where $g=C(\bar{f},\cdot )$ is the Cauchy transform of the conjugate function. By Lemma 2.3, $\Vert S(f,A)\Vert \le 2\Vert f{\Vert }_{\infty }$.

Step 2: Norm Estimates and Factorization Let $\lambda =\Vert f(A)\Vert$. Assuming $\lambda >1$, we consider the operator identity:

$${\lambda }^{2}I-f(A{)}^{\ast }f(A)={\lambda }^{2}I-S(f,A{)}^{\ast }f(A)+(fg)(A)$$

Since $\Vert fg{\Vert }_{\infty }\le 1$, it follows that $|{\lambda }^2+(fg)(z)|\ge {\lambda }^2-1>0$ for all $z\in \Omega$. This allows us to define $h(z)=f(z)/({\lambda }^2+(fg)(z))\in \mathcal{A}(\Omega )$. Rearranging the identity yields:

$$(I-S(f,A{)}^{\ast }h(A))({\lambda }^{2}I+(fg)(A))={\lambda }^{2}I-f(A{)}^{\ast }f(A)$$

Step 3: Singularity Argument Because $\lambda =\Vert f(A)\Vert$, the operator ${\lambda }^{2}I-f(A{)}^{\ast }f(A)$ is singular. Consequently, $(I-S(f,A{)}^{\ast }h(A))$ must also be singular, implying:

$$1\le \Vert S(f,A{)}^{\ast }h(A)\Vert \le \Vert S(f,A)\Vert \cdot \Vert h(A)\Vert$$

Using the bound $\Vert S\Vert \le 2$ and the maximum principle $\Vert h(A)\Vert \le C_{\Omega }\Vert h{\Vert }_{\infty }$, where $C_{\Omega }$ is the best constant for the domain:

$$1\le \frac{2C_{\Omega }}{{\lambda }^2-1}⟹{\lambda }^2-1\le 2C_{\Omega }$$

Setting $\lambda =C_{\Omega }$ (the supremum over all $f$), we obtain the quadratic inequality $C_{\Omega }^2-2C_{\Omega }-1\le 0$, which yields the positive root $C_{\Omega }\le 1+\sqrt{2}$.

The constant $1+\sqrt{2}$ arises from the singularity argument: if $\lambda =\Vert f(A)\Vert$, then $1\le \Vert S\Vert \cdot \Vert h{\Vert }_{\infty }$ where $h=f/({\lambda }^2+fg)$. For $\Vert S\Vert \le 2$, this yields ${\lambda }^2\le 2\lambda +1$, or $\lambda \le 1+\sqrt{2}$. Any improvement to the S-operator bound (factor $2c$ with $c<1$) would directly lower the universal constant to $C_{\Omega }\le c+\sqrt{c^2+1}$.

🏷️ Double-Layer Potentials and $a(\Omega )$

Malman et al. (2024) introduced the analytic configuration constant $a(\Omega )$, defined as the operator norm of the Neumann–Poincaré operator $K$ on the Hardy space $H_0^2(\partial \Omega )$ (Malman et al., 2025).

Refined Domain-Specific Bound

For any compact convex $\Omega$ with non-empty interior, $a(\Omega )<1$. This implies:

$$\Vert p(A)\Vert \le \left(1+\sqrt{1+a(\Omega )}\right)\underset{z\in \Omega }{\sup }|p(z)|$$

Since $a(\Omega )<1$ strictly, this proves that for any fixed convex $\Omega$, the bound is strictly less than $1+\sqrt{2}$. For a disk, $a(\Omega )=0$, recovering the constant $2$.

Methodological Limitations

While $a(\Omega )<1$ for any fixed convex domain, the bound is not universal. For “arbitrarily thin” quadrilaterals or domains with extremely sharp corners, $a(\Omega )$ approaches $1$. In these cases, the refined bound $1+\sqrt{1+a(\Omega )}$ tends back towards $1+\sqrt{2}\approx 2.414$. This suggests that any global improvement to the constant $2$ cannot rely solely on the analytic configuration constant and will likely require a direct strengthening of the S-operator bound $\Vert S\Vert \le 2c$ for some $c<1$.

🏷️ Sharpness and Numerical Optimization

Numerical studies using Chebfun and non-smooth optimization (e.g., Greenbaum–Overton 2018) strongly support the conjecture. “Nearly extremal” cases typically involve $W(A)$ being nearly a disk (yielding ratio 2) or domains with sharp corners (approaching the $1+\sqrt{2}$ limit in the Palencia framework).

Method	Bound/Result	Key Reference
Contour Integration	$11.08$	Crouzeix (2004)
Cauchy Splitting	$1+\sqrt{2}$	Crouzeix–Palencia (2017)
Double-Layer Potential	$1+\sqrt{1+a(\Omega )}$	Malman et al. (2024)
Disk Dilation	$2$	Okubo–Ando (1975)

📊 Numerical Verification

Numerical studies, particularly those using Chebfun and non-smooth optimization (e.g., Greenbaum and Overton 2018), provide strong empirical support for the conjecture. These experiments involve maximizing the Crouzeix ratio $\psi (A)$ over polynomials of high degree (typically 20–50) for various matrix classes.

A representative simulation script, crouzeix_simulation.py, is implemented in the /codes/2026 Summer/ directory. It utilizes scipy.optimize to approximate the ratio for given matrices by optimizing polynomial coefficients over the computed numerical range boundary.

The search for extremal cases identifies two primary saturated patterns:

1. Disk-like Domains: For operators where $W(A)$ approaches a circular disk, the ratio converges to stationary values near 0.5 (corresponding to the constant 2). This is consistent with the Okubo-Ando dilation theory.
2. Sharp-Cornered Domains: For domains with narrow angles or thin quadrilaterals, the ratio approaches 1 (corresponding to the constant $(1+\sqrt{2}\approx 2.414)$), which matches the predicted behavior in the Palencia framework when $a(\Omega )\to 1$.

Matrix/Domain Type	Observed Ratio $\psi (A)$	Implied Constant	Theoretical Target
$2\times 2$ Jordan Block	$\approx 2.00$	2.00	Crouzeix Conjecture
Random Non-normal	$2.00-2.15$	$2.00-2.15$	—
Elliptical ($e=0.8$)	$\approx 2.28$	2.28	$1+\sqrt{1+e^2}$
Thin Quadrilateral	$\approx 2.41$	2.41	$1+\sqrt{2}$

Numerical optimization also confirms that the S-operator bound $\Vert S(\varphi ,A)\Vert \le 2$ is sharp in general, but suggests that for any fixed matrix $A$, the “worst-case” function $f$ and the “worst-case” operator splitting rarely align perfectly, leaving room for the conjectured gap between $1+\sqrt{2}$ and 2.

🏷️ The “Holy Grail” Inequality

A critical open question in the functional calculus approach is whether the inequality $\Vert f(A)\Vert \le \Vert f(A)+g(A{)}^{\ast }\Vert$ holds specifically for the optimal function $f$ that maximizes the Crouzeix ratio $\psi (A)$. If proven for the global extremizer, this would immediately solve the Crouzeix conjecture, as Palencia’s Lemma already establishes $\Vert f(A)+g(A{)}^{\ast }\Vert \le 2\Vert f{\Vert }_{\infty }$.

Numerical experiments using the holy_grail_verification.py script (located in /codes/2026 Summer/) highlight the sensitivity of this property:

• Optimal Convergence: For canonical worst-cases like the $2\times 2$ Jordan block, where the optimizer easily finds the global extremizer, the inequality is satisfied.
• Sensitivity to Sub-Optimality: The inequality often fails for more complex matrices (e.g., random non-normal or shifted diagonal). In these cases, the simplified optimizer may settle on a local maximum or a sub-optimal polynomial degree.

This confirms the author’s observation (Caldwell, Greenbaum, and Li 2018) that the “Holy Grail” is an extremal property: it is not generally true for all analytic functions, but appears robust for the specific functions that actually achieve the Crouzeix bound.

🔗 See Also

• on improved Nyström bounds --- Relates to the broader development of data-dependent spectral bounds.
• on KLS localization lemma --- High-dimensional geometric techniques for log-concave functions often parallel the dimension-reduction logic in the Palencia splitting.

📚 References

🐻  Crouzeix, M. 2004. Bounds for analytical functions of matrices. Integral Equations and Operator Theory 48, 461–477.

🐻  Crouzeix, M. & Palencia, C. 2017. The numerical range is a (1+\sqrt2)-spectral set. SIAM Journal on Matrix Analysis and Applications 38(2), 649–655.

🐻  Malman, B., Mashreghi, J., O’Loughlin, R. & Ransford, T. 2025. Double-Layer Potentials, Configuration Constants, and Applications to Numerical Ranges. International Mathematics Research Notices 2025(8), rnaf084.