on two proofs of Crouzeix's conjecture

Overview

Two independent 2026 preprints prove the sharp Crouzeix inequality. Jin’s first version appeared on July 27; Lorist and Schwenninger’s five-page note followed on arXiv on August 4 (Jin, 2026; Lorist & Schwenninger, 2026). Both proofs begin with the same positive double-layer functional calculus. Their decisive innovation is to retain algebraic information that the earlier $1+\sqrt{2}$ argument discarded. Jin encodes all powers of $f(A)$ in a matrix-valued Carathéodory function, cancels its unknown adjoint-algebra correction by a tailored Herglotz-kernel sample, and compares two weighted Gramians. Lorist and Schwenninger instead turn the powers into uniformly bounded commuting defects of a $2$-dilation and close a scalar recurrence at a norming vector. This post isolates the common reduction, compares the two sharp mechanisms, and records the proof tools that survive beyond the numerical-range problem.

🏷️ Numerical Range and Spectral Sets

Let $A$ be a bounded operator on a complex Hilbert space $H$. Its numerical range is

$$W(A)=\{⟨Ax,x⟩:\Vert x\Vert =1\}.$$

For a matrix, $W(A)$ is compact and convex; in infinite dimension its closure is compact and convex. A compact set $K\supseteq \sigma (A)$ is a $C$-spectral set for $A$ if

$$\Vert r(A)\Vert \le C\underset{z\in K}{\max }|r(z)|$$

for every rational function $r$ with poles off $K$. The conjecture formulated by Crouzeix in 2004 asserted that one may always take $K=\overline{W(A)}$ and $C=2$ (Crouzeix, 2004).

Crouzeix's theorem

For every bounded operator $A$ on a complex Hilbert space and every rational function $f$ with poles off $\overline{W(A)}$,

$$\Vert f(A)\Vert \le 2\underset{z\in W(A)}{\sup }|f(z)|.$$

Equivalently, $\overline{W(A)}$ is a $2$-spectral set for $A$ (Jin, 2026; Lorist & Schwenninger, 2026).

The factor $2$ is optimal already in dimension two. For

$$N=\left(\begin{array}{cc}0& 2\\ 0& 0\end{array}\right),$$

the numerical range is the closed unit disk, while $\Vert N\Vert =2$. Taking $f(z)=z$ gives equality.

Version chronology

Jin submitted version 1 of The Numerical Range Is a 2-Spectral Set on July 24, 2026, and it was posted on July 27. Lorist and Schwenninger submitted A Solution to Crouzeix’s Conjecture to arXiv on August 4, eight days after Jin’s posting; their introduction explicitly notes the independent proof. Jin’s rapidly revised version 4 was posted on August 7 and is the version used below for its mass-parameterized formulation. Both works are recent preprints, so the account here explains the arguments as written rather than substituting for peer review.

🏷️ The Earlier Barrier

The best preceding universal theorem was the Crouzeix–Palencia estimate

$$\Vert f(A)\Vert \le (1+\sqrt{2})\underset{z\in W(A)}{\sup }|f(z)|$$

from 2017 (Crouzeix & Palencia, 2017). Later refinements improved this constant for each fixed numerical-range shape, but thin convex domains showed that those shape-dependent estimates still approach $1+\sqrt{2}$ in the universal limit (Malman et al., 2025).

The One-Function Closure at $1+\sqrt{2}$

The obstruction was not a lack of positivity. The double-layer integral used by Crouzeix and Palencia was already positive and had total mass two. In the notation developed below, let

$$\theta (f)=f(A),g=\alpha (f),S_f=2\Phi (f)=\theta (f)+\theta (g{)}^{\ast }.$$

For $\Vert f{\Vert }_{\infty ,\Omega }\le 1$, their two decisive estimates are

$$\Vert S_f\Vert \le 2,\Vert g{\Vert }_{\infty ,\Omega }\le 1.$$

The first comes from positivity of the double-layer map; the second says that the Cauchy transform $f\mapsto \alpha (f)$ is an antilinear contraction on a convex domain. Let

$$C_{\Omega }=\Vert \theta \Vert =\underset{\Vert h{\Vert }_{\infty ,\Omega }\le 1}{\sup }\Vert h(A)\Vert .$$

To see exactly where $1+\sqrt{2}$ enters, take a nearly extremal $f$, put $T=f(A)$, and write $\lambda =\Vert T\Vert >1$. Since $\Vert fg{\Vert }_{\infty }\le 1$, the function

$$h=\frac{f}{{\lambda }^2+fg}$$

is holomorphic on $\Omega$ and satisfies

$$\Vert h{\Vert }_{\infty }\le \frac{1}{{\lambda }^2-1}.$$

The identity $S_f^{\ast }T=T^{\ast }T+(fg)(A)$ gives the factorization

$${\lambda }^{2}I-T^{\ast }T=(I-S_f^{\ast }h(A))({\lambda }^{2}I+(fg)(A)).$$

The operator ${\lambda }^{2}I-T^{\ast }T$ is not invertible because ${\lambda }^2=\Vert T^{\ast }T\Vert$ belongs to the spectrum of the positive operator $T^{\ast }T$. The second factor is invertible because $\Vert fg{\Vert }_{\infty }\le 1<{\lambda }^2$. Hence $I-S_f^{\ast }h(A)$ is not invertible, and therefore

$$1\le \Vert S_f^{\ast }h(A)\Vert \le \frac{2C_{\Omega }}{{\lambda }^2-1}.$$

Letting $\lambda$ approach $C_{\Omega }$ yields

$$C_{\Omega }^2\le 2C_{\Omega }+1,C_{\Omega }\le 1+\sqrt{2}.$$

The extra $1$ in the quadratic is precisely the price of bounding the product $fg$ only by $\Vert fg{\Vert }_{\infty }\le 1$. At that point the companion $g$ has been reduced to an arbitrary contractive correction. Ransford and Schwenninger proved that the abstract lemma retaining only

$$\Vert g{\Vert }_{\infty }\le \Vert f{\Vert }_{\infty },\Vert f(A)+g(A{)}^{\ast }\Vert \le 2\Vert f{\Vert }_{\infty }$$

is itself sharp at $1+\sqrt{2}$ (Ransford & Schwenninger, 2018). No rearrangement of those two norm inequalities alone can reach $2$.

The Information Recovered in 2026

Saying that the earlier method loses the sharp constant “because of the antilinear part” is accurate only with an important qualification. Antilinearity is not itself an obstruction. The 2017 factorization does take an adjoint and uses the commutativity of $\alpha (f)(A)$ with $T=f(A)$ once, in the identity $\alpha (f)(A)T=(\alpha (f)f)(A)$. What it retains about the companion afterward is only its contractive size and this one product. It does not retain the coherent family produced by $f,f^2,f^3,\dots$. Because $\alpha$ is antilinear rather than multiplicative, that family cannot be reconstructed from $\alpha (f)$ alone.

Lorist and Schwenninger recover the missing commutativity by applying the double-layer identity to every power $f^n$ and then taking adjoints:

$$E_n:=2\Phi (f^n{)}^{\ast }-T^{\ast n}=\alpha (f^n)(A).$$

The adjoint has moved off the companion. Consequently, both $E_n$ and $T$ lie in the same commutative holomorphic functional calculus, so $E_{n}T=T{E}_n$. Moreover, unital positivity realizes the entire sequence $\Phi (f^n)$ as compression moments $V^{\ast }{Q}^{n}V$ of one contraction. The proof therefore retains a coherent family of bounded commuting defects instead of replacing one companion by the scalar estimate $\Vert g{\Vert }_{\infty }\le 1$. In fact, the new dilation argument needs only boundedness of $\alpha$, not the contractivity that was central to the 2017 bound.

Jin preserves the lost structure in a different form. The Cayley family

$$c_w\circ f=\frac{1+wf}{1-wf}$$

packages all powers of $f$ into one positive-real function. Its double-layer remainder is kept inside the structured algebra $\mathrm{alg}(B^{\ast })$, rather than bounded independently. A tailored origin sample then annihilates the entire correction inside a positive Herglotz kernel. Thus Lorist–Schwenninger use commutativity across powers, while Jin uses algebraic localization and cancellation across kernel samples. Neither proof pays the $\Vert fg{\Vert }_{\infty }\le 1$ penalty responsible for the old quadratic.

This sharper structure also explains a tradeoff. The Crouzeix–Palencia estimates pass directly to matrix-valued functions and hence give a complete $(1+\sqrt{2})$-spectral-set theorem. The new scalar mechanisms rely respectively on commutativity and diagonal scalar cancellation, neither of which survives matrix amplification automatically.

🏷️ The Shared Double-Layer Calculus

Let $\Omega$ be a smoothly bounded open convex set containing $\overline{W(A)}$, and let $n_{\Omega }(\sigma )$ denote the outward unit normal at $\sigma \in \partial \Omega$. Define

$$P_{\Omega }(\sigma )=\frac{1}{\pi }\mathrm{Re}\left(n_{\Omega }(\sigma )(\sigma I-A{)}^{-1}\right).$$

Convexity and the inclusion $W(A)\subset \Omega$ imply

$$P_{\Omega }(\sigma )⪰0,{\int }_{\partial \Omega }{P}_{\Omega }(\sigma )|d\sigma |=2I.$$

Consequently,

$$\Phi (h)=\frac{1}{2}{\int }_{\partial \Omega }h(\sigma )P_{\Omega }(\sigma )|d\sigma |$$

defines a unital positive map on continuous boundary functions; below we use its restriction to the holomorphic algebra. If $h$ is holomorphic in $\Omega$ and continuous on its closure, the Cauchy-transform companion $\alpha (h)$ satisfies

$$2\Phi (h)-h(A)=\alpha (h)(A{)}^{\ast }.$$

When the matrix is $B$ rather than $A$, write this map as ${\Phi }_{B,\Omega }$.

This is the common analytic layer. Normalize $f$ by $\Vert f{\Vert }_{\infty ,\Omega }\le 1$ and put $T=f(A)$. The two proofs make different choices for $h$:

• Lorist–Schwenninger use every power $h=f^n$ separately.

• Jin uses the full Cayley family

$$h=c_w\circ f,c_w(\zeta )=\frac{1+w\zeta }{1-w\zeta },w\in \mathbb{D},$$

whose power series packages all $f^n$ into one positive-real function.

Thus the proofs diverge only after the same double-layer reduction. One turns positivity into a dilation; the other turns it into a positive kernel.

🏷️ Lorist–Schwenninger: Commuting Defects of a $2$-Dilation

The positivity of $P_{\Omega }$ has a concrete Stinespring representation. Let

$$K=L^2(\partial \Omega ,|d\sigma |;H),(Vx)(\sigma )=\frac{1}{\sqrt{2}}{P}_{\Omega }(\sigma {)}^{1/2}x.$$

The mass identity makes $V:H\to K$ an isometry. Let $Q$ be multiplication by $f$ on $K$. Since $\Vert f{\Vert }_{\infty }\le 1$, $Q$ is a contraction, and

$$V^{\ast }{Q}^{n}V=\Phi (f^n).$$

Taking adjoints in the double-layer identity gives the defect sequence

$$E_n:=2V^{\ast }{Q}^{\ast n}V-T^{\ast n}=\alpha (f^n)(A).$$

Two properties are decisive:

$$\underset{n\ge 1}{\sup }\Vert E_n\Vert <\infty ,E_{n}T=T{E}_n.$$

If $\theta (h)=h(A)$ denotes the bounded evaluation homomorphism, then

$$\Vert E_n\Vert \le \Vert \theta \Vert \Vert \alpha \Vert \Vert f^n{\Vert }_{\infty }\le \Vert \theta \Vert \Vert \alpha \Vert .$$

This proves uniform boundedness; commutativity follows because both $T=f(A)$ and $E_n=\alpha (f^n)(A)$ belong to the same holomorphic functional calculus.

Commuting $2$-dilation perturbation lemma

Let $H$ be finite-dimensional and $T\in \mathcal{L}(H)$. Suppose there are a Hilbert space $K$, a contraction $Q\in \mathcal{L}(K)$, and an isometry $V:H\to K$ such that

$$E_n=2V^{\ast }{Q}^{\ast n}V-T^{\ast n}$$

is uniformly bounded in $n$ and commutes with $T$. Then $\Vert T\Vert \le 2$ (Lorist & Schwenninger, 2026).

The Scalar Recurrence

The proof of the lemma is the technical core of the five-page note. Put $\kappa =\Vert T\Vert$ and suppose $\kappa >1$. Finite-dimensionality gives a unit vector $x$ such that

$$T^{\ast }Tx={\kappa }^{2}x.$$

Write

$$S_n^{\ast }=2V^{\ast }{Q}^{\ast n}V,m_n=\mathrm{Re}⟨E_n{T}^{n}x,x⟩,$$

and define

$$y_n=(S_{n+1}^{\ast }T-\kappa S_n^{\ast })x.$$

Commutativity of $E_n$ with $T$, followed by completing the square, gives

$$\kappa m_n-m_{n+1}\ge -\frac{\Vert y_n{\Vert }^2}{4({\kappa }^2-\kappa )}.$$

The dilation structure factors the troublesome vector:

$$y_n=2V^{\ast }{Q}^{\ast n}(Q^{\ast }VT-\kappa V)x.$$

If

$$d=\Vert Q^{\ast }VTx-\kappa Vx\Vert ,$$

then $\Vert y_n\Vert \le 2d$. Dividing the recurrence by powers of $\kappa$ and summing yields

$$m_1\ge -\frac{d^2}{\kappa (\kappa -1{)}^2}.$$

Indeed, the defect identity, uniform boundedness of $(E_n)$, and contractivity of $Q$ first make $(T^n)$ uniformly bounded, hence also $(E_n{T}^n)$; therefore the terminal term vanishes. A second expansion, using that $V$ is isometric and $Q$ contractive, gives

$$d^2\le 2{\kappa }^2-\kappa m_1-{\kappa }^3.$$

Combining the last two inequalities produces

$$d^2\left(1-\frac{1}{(\kappa -1{)}^2}\right)\le {\kappa }^2(2-\kappa ).$$

If $\kappa >2$, the left-hand side is nonnegative and the right-hand side is negative. Hence $\kappa \le 2$.

Applying the lemma to the double-layer defects proves the normalized estimate $\Vert f(A)\Vert \le 2$. Outer approximation of $W(A)$ and standard finite-dimensional compression give the rational and infinite-dimensional forms.

🏷️ Jin: Positive-Real Completion and Gramian Cancellation

Jin keeps the full Cayley family. For a simple-spectrum auxiliary matrix $B$ with $W(B)\subset \Omega$, put $T=f(B)$ and define

$$H(w)={\Phi }_{B,\Omega }(c_w\circ f),w\in \mathbb{D}.$$

Because $\mathrm{Re}{c}_w(\zeta )\ge 0$ whenever $|w|<1$ and $|\zeta |\le 1$,

$$H(0)=I,\mathrm{Re}H(w)⪰0.$$

Keeping both layers of the boundary integral, rather than estimating the companion, gives the algebraic completion

$$H(w)-(I-wT{)}^{-1}\in \mathrm{alg}(B^{\ast }).$$

The role of simple spectrum is now transparent. If

$$B=S\mathrm{diag}({\beta }_1,\dots ,{\beta }_n)S^{-1},T=S\Lambda S^{-1},\Lambda =\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n),$$

then polynomial interpolation gives

$$\mathrm{alg}(B^{\ast })=\{(S^{-1}{)}^{\ast }D{S}^{\ast }:D\text{ is diagonal}\}.$$

Thus, after multiplication on the left by $S^{\ast }$ and on the right by $S$, the adjoint-algebra correction has the form $\Psi (w)G$, where $\Psi$ is diagonal. The values ${\lambda }_i=f({\beta }_i)$ are allowed to repeat. With $G=S^{\ast }S$, the completion becomes

$$\tilde{H}(w):=S^{\ast }H(w)S=G(I-w\Lambda {)}^{-1}+\Psi (w)G,$$

where $\Psi (w)$ is diagonal, analytic, and satisfies $\Psi (0)=0$.

Herglotz-Kernel Sampling

Positive real part of $\tilde{H}$ implies positivity of the matrix Herglotz kernel

$$L_{\tilde{H}}(w,z)=\frac{\tilde{H}(w)+\tilde{H}(z{)}^{\ast }}{1-w\overline{z}}.$$

The unknown diagonal function $\Psi$ cannot be bounded sharply term by term. Jin instead chooses a finite kernel sample that annihilates it. Set

$$P_{ij}=\frac{G_{ij}}{1-\overline{{\lambda }_i}{\lambda }_j/4},Q_{ij}=\frac{G_{ij}}{1-\overline{{\lambda }_i}{\lambda }_j/2},Y=Q-P.$$

For an arbitrary $u=(u_1,\dots ,u_n{)}^{\mathsf{T}}$, sample at

$$w_i=\frac{\overline{{\lambda }_i}}{2},{\xi }_i=u_i{e}_i,$$

and add the origin sample

$$w_0=0,{\xi }_0=v=-G^{-1}Pu.$$

The contribution of $\Psi$ to the kernel quadratic form contains the factor

$$Gv+Pu,$$

so it vanishes exactly. The origin sample is therefore not an auxiliary estimate; it is the device that removes the entire unknown completion.

Ordered Weighted Gramians

After this cancellation, kernel positivity gives

$$4Y-Y{G}^{-1}P-P{G}^{-1}Y⪰0.$$

Balance the nonorthogonal eigenbasis by setting

$$\tilde{T}=G^{1/2}\Lambda G^{-1/2},\hat{P}=G^{-1/2}P{G}^{-1/2},\hat{Q}=G^{-1/2}Q{G}^{-1/2},\hat{Y}=\hat{Q}-\hat{P}.$$

The Cauchy denominators expand into two Gramians at different scales:

$$\hat{P}=\sum _{k=0}^{\infty }{4}^{-k}{\tilde{T}}^{\ast k}{\tilde{T}}^k,\hat{Q}=\sum _{k=0}^{\infty }{2}^{-k}{\tilde{T}}^{\ast k}{\tilde{T}}^k.$$

There is no circular convergence assumption here: ${\tilde{T}}^k=G^{1/2}{\Lambda }^k{G}^{-1/2}$ is uniformly bounded because $|{\lambda }_i|\le 1$, so both geometrically weighted series converge in norm.

Thus

$$\hat{Y}⪰\hat{P}-I.$$

The balanced kernel inequality is

$$4\hat{Y}-\hat{Y}\hat{P}-\hat{P}\hat{Y}⪰0.$$

If $\hat{P}e=\alpha e$ for a unit vector $e$ and $y=⟨\hat{Y}e,e⟩$, then

$$0\le (4-2\alpha )y,y\ge \alpha -1.$$

Since $\alpha \ge 1$, these inequalities force $\alpha \le 2$. Hence

$$I⪯\hat{P}⪯2I.$$

The first nonconstant term of the Gramian series now gives

$$\frac{1}{4}{\tilde{T}}^{\ast }\tilde{T}⪯\hat{P}-I⪯I,$$

so $\Vert \tilde{T}\Vert \le 2$. The polar decomposition of $S$ shows that $T$ is unitarily equivalent to $\tilde{T}$, and therefore $\Vert T\Vert \le 2$.

Jin removes the auxiliary hypotheses by approximating an arbitrary matrix by simple-spectrum matrices and replacing $W(A)$ by the parallel convex domains

$${\Omega }_{\epsilon }=\{z:\mathrm{dist}(z,W(A))<\epsilon \}.$$

The current formulation also records a mass parameter: if $m\ge 2$ and the completion has the form

$$H(w)-\frac{2}{m}(I-wT{)}^{-1}\in \mathrm{alg}(B^{\ast }),$$

then the same sampling argument, at scale $m^{-1}$, yields $\Vert T\Vert \le m$ (Jin, 2026).

🏷️ Comparison of the Proof Mechanisms

Feature	Crouzeix–Palencia (2017)	Jin (2026)	Lorist–Schwenninger (2026)
Information retained from $f$	One pair $f,\alpha (f)$	The Cayley family $(1+wf)/(1-wf)$	Every power $f^n$
Use of the positive map	The norm estimate $\Vert 2\Phi (f)\Vert \le 2\Vert f\Vert$	A matrix-valued Herglotz kernel	Compression moments $V^{\ast }{Q}^{n}V$ of one contraction
Treatment of the remainder	Contractivity and the single product $f\alpha (f)$	A diagonalizable adjoint-algebra correction	Uniformly bounded defects commuting with $T$
Closing mechanism	Extremal factorization and noninvertibility	An origin sample cancels the correction	A norming-vector recurrence absorbs the defects
Final inequality	$C_{\Omega }^2\le 2C_{\Omega }+1$	Two ordered weighted Gramians	Opposite signs when $\Vert T\Vert >2$
Role of $\alpha$	Its contractivity is essential	Its full Cayley-family remainder is retained	Boundedness suffices; contractivity is unnecessary
Resulting constant	$1+\sqrt{2}$	$2$	$2$
Matrix amplification	Passes directly; the result is complete	The cancellation does not pass directly	The commuting-defect hypothesis does not pass directly
Natural abstraction	Contractive-companion lemma	Positive-real completion of mass $m$	Completely positive unital symmetrization of a homomorphism

The two sharp proofs are therefore neither unrelated nor cosmetic variants of the 2017 argument. All three analytic front ends use the same double-layer symmetrization, but they retain different amounts of information when removing the antiholomorphic layer. Jin exploits the full Pick-kernel geometry of the Cayley transform. Lorist–Schwenninger exploit only the power moments, but their recurrence needs much less coordinate structure.

The brevity of the second proof comes from the strength of its perturbation lemma: once the double-layer map is written as a compression, the geometry of $\partial \Omega$ disappears. Jin’s longer proof exposes more of the finite-dimensional mechanism and retains a variable mass parameter, which makes its kernel argument portable to other positive-layer spectral-constant problems.

🏷️ Reusable Proof Tools

Bounded commuting defects

A dilation identity need not be exact. If

$$T^{\ast n}=2V^{\ast }{Q}^{\ast n}V-E_n$$

with $Q$ contractive and the errors uniformly bounded and commuting with $T$, then the errors cannot push $\Vert T\Vert$ past $2$. This is a robust perturbative substitute for constructing an exact $2$-dilation.

Algebraic cancellation in a positive kernel

When a positive-real function is known only modulo a diagonal or semisimple correction, a carefully chosen extra sample can cancel the correction before any norm estimate is taken. The vector $v=-G^{-1}Pu$ is a Schur-complement choice: it restricts the positive block kernel to the graph on which the nuisance term vanishes.

Multiscale Gramians

Resolvent kernels at two radii generate weighted observability Gramians

$$\sum _{k\ge 0}{a}^k{T}^{\ast k}{T}^k.$$

Comparing the weights coefficientwise gives an operator order, while the first nonconstant term recovers a norm bound for $T$. This separates positivity, which controls the whole series, from sharpness, which is read from the $k=1$ coefficient.

Positive mass as a spectral constant

The identity

$${\int }_{\partial \Omega }{P}_{\Omega }|d\sigma |=2I$$

is the structural source of the constant. Jin’s mass-$m$ theorem makes this explicit; Lorist–Schwenninger encode the same mass in the coefficient $2V^{\ast }{Q}^{\ast n}V$. This suggests looking for a normalized positive layer and a compatible algebraic remainder whenever a sharp spectral-set constant is expected.

The theorem also gives the sharp universal error-transfer principle used in numerical matrix-function approximation. Let $A\in M_n(\mathbb{C})$, let $f$ be holomorphic on a neighborhood of $W(A)$, and let $s$ be a polynomial or a rational function without poles on $W(A)$. Then

$$\Vert f(A)-s(A)\Vert \le 2\underset{z\in W(A)}{\max }|f(z)-s(z)|.$$

This follows by applying the theorem to $f-s$ (Jin, 2026). Thus a scalar uniform approximation on the numerical range becomes an operator-norm approximation with no dimension-dependent loss.

🏷️ Scope and Remaining Boundary

The proofs settle the scalar-valued Crouzeix conjecture. Neither argument directly gives the corresponding complete, matrix-valued $2$-spectral-set statement. In the dilation proof, scalar functional-calculus elements commute, whereas matrix amplification does not preserve the commutativity required of the defects. In the Herglotz proof, the diagonal scalar correction becomes a block-valued correction for which the same origin-sample cancellation is not automatic. The complete numerical-range problem therefore remains outside both mechanisms.

The simple-spectrum hypothesis in Jin’s proof is only a coordinate device and is removed by approximation. Likewise, finite-dimensional norm attainment in Lorist–Schwenninger’s lemma is removed when the operator theorem is recovered through finite-dimensional compression. Neither is a restriction in the final statement.

🔗 See Also

• on Crouzeix’s conjecture --- The earlier seminar note develops the pre-solution $1+\sqrt{2}$ argument, configuration-constant refinements, and the numerical program that preceded the 2026 proofs.

• on interlacing families and Kadison-Singer --- Both arguments turn positivity of a matrix-valued family into an operator-norm bound. Interlacing uses real stability and barrier functions, while the present proofs use dilation moments or a Herglotz kernel.

📚 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.

🐻  Jin, S. 2026. The Numerical Range Is a 2-Spectral Set.

🐻  Lorist, E. & Schwenninger, F.L. 2026. A Solution to Crouzeix’s Conjecture.

🐻  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.

🐻  Ransford, T. & Schwenninger, F.L. 2018. Remarks on the Crouzeix–Palencia Proof That the Numerical Range Is a (1+\sqrt2)-Spectral Set. SIAM Journal on Matrix Analysis and Applications 39(1), 342–345.