The Numerical Range Is a 2-Spectral Set

1. Introduction

Let $\mathbb{C}$ denote the complex field and, for $n\ge 1$, let ${\mathbb{M}}_n\left(\mathbb{C}\right)$ denote the algebra of $n\times n$ complex matrices. For $A\in {\mathbb{M}}_n\left(\mathbb{C}\right)$, its numerical range is

$$W\left(A\right)=\{x^{*}Ax:x\in {\mathbb{C}}^n,{\parallel x\parallel }_2=1\}.$$

(1)

Here $x^{*}$ denotes the conjugate transpose, $\sigma \left(A\right)$ denotes the spectrum, ${\parallel x\parallel }_2$ is the Euclidean norm, $\parallel X\parallel$ is the induced operator norm, and scalar moduli are denoted by $|·|$. The set $W\left(A\right)$ is compact, the Toeplitz–Hausdorff theorem makes it convex, and $\sigma \left(A\right)\subset W\left(A\right)$. Functions holomorphic near $\sigma \left(A\right)$ are evaluated using the holomorphic functional calculus.

A compact set $K\subset \mathbb{C}$ containing $\sigma \left(A\right)$ is called a C-spectral set for A if

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

(2)

for every rational function r pole-free on K. Our main result settles Crouzeix’s conjecture by identifying the sharp universal constant for $K=W\left(A\right)$.

Theorem 1 

(Main theorem). Let $n\ge 1$, let $A\in {\mathbb{M}}_n\left(\mathbb{C}\right)$, and let f be holomorphic on a neighborhood of $W\left(A\right)$. Then

$$\parallel f\left(A\right)\parallel \le 2\underset{z\in W\left(A\right)}{max}\left|f\left(z\right)\right|.$$

(3)

Consequently, $W\left(A\right)$ is a 2-spectral set for A.

The factor 2 is optimal: for

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

(4)

one has $W\left(N\right)=\{z\in \mathbb{C}:|z|\le 1\}$ and $\parallel N\parallel =2$.

For later use, write

$$\mathbb{D}=\{z\in \mathbb{C}:|z|<1\},\mathbb{T}=\partial \mathbb{D}.$$

We use I for the identity matrix of the size dictated by context, $X^{*}$ for the adjoint, put $ReX=(X+X^{*})/2$, and let $alg\left(T\right)$ denote the unital algebra generated by T. For Hermitian matrices X and Y, we write $X⪯Y$ (equivalently, $Y⪰X$) when $Y-X$ is positive semidefinite.

Crouzeix proved the estimate with constant 2 for $2\times 2$ matrices and formulated the general conjecture in [1]. His subsequent dimension-free theorem gave the constant $11.08$ [2]. After further structural and numerical work [3], Crouzeix and Palencia proved that $W\left(A\right)$ is a $(1+\sqrt{2})$-spectral set [4]. Their double-layer and Cauchy-transform argument was clarified in [5] and developed in several directions in [6,7].

The intervening literature established the conjectured constant for perturbed Jordan blocks and, more generally, scalar translates of cyclic weighted shifts, as well as for tridiagonal $3\times 3$ matrices with elliptic numerical range centered at an eigenvalue [8,9,10]. Numerical and extremal investigations appear in [11,12], and a broad account of the problem and its variants is given in [13]. Subsequent work has treated Blaschke-product level sets, cyclicity reductions, compressions of shifts, further nilpotent classes, and quantitative bounds for KMS matrices [14,15,16,17]. An alternative proof for a class of weighted shifts appears in [18]. Across these cited reductions, the best dimension-free universal bound remained $1+\sqrt{2}$.

1.1. The Sharpness Barrier in the Symmetrized Calculus

The positive boundary measure used below originates in the convex-domain integral calculus of Delyon–Delyon [19]; see also [4,20]. If B has numerical range in a convex domain $\Omega$ with continuously differentiable boundary and f is bounded by one on $\Omega$, this calculus produces a unital positive map $\Phi$ and a holomorphic Cauchy companion $g_f$ such that

$$2\Phi \left(f\right)=f\left(B\right)+g_f{\left(B\right)}^{*}.$$

(5)

Thus positivity directly controls a symmetrized functional calculus, while the target estimate concerns $f\left(B\right)$. The Crouzeix–Palencia argument couples the two terms in (5); the examples in [5] show that its abstract norm estimate is sharp at $1+\sqrt{2}$. The corresponding general uniform-algebra theorem has the same sharp constant; its additional unital case was proposed as a possible route to 2 in [21]. Schwenninger and de Vries revisited the double-layer method [22]. Malman et al. proved that a configuration-constant refinement gives a strict improvement for every fixed numerical-range shape; they also constructed thin quadrilaterals for which the resulting bounds approach $1+\sqrt{2}$ [23]. Separately, continuity and compactness give a constant $C_n<1+\sqrt{2}$ in every fixed matrix dimension n [24].

The key additional information is algebraic. For the approximants constructed in Section 5, the companion in (5) belongs to the algebra generated by $B^{*}$, where the simple-spectrum matrix B supplies an eigenbasis that also diagonalizes $f\left(B\right)$. Treating $f\left(B\right)$ and the companion jointly preserves this coupling.

1.2. Auxiliary-Basis de-Symmetrization

Our argument uses the whole Cayley family associated with (5). It has four steps.

1.

Applying $\Phi$ to

$$\zeta ⟼{\displaystyle \frac{1+wf\left(\zeta \right)}{1-wf\left(\zeta \right)}},w\in \mathbb{D},$$

produces a matrix-valued Carathéodory function H, meaning that $H\left(0\right)=I$ and $ReH\left(w\right)⪰0$.

2.

For a simple-spectrum auxiliary matrix B, put $T=f\left(B\right)$. The double-layer identity gives the positive-real completion

$$H\left(w\right)-{(I-wT)}^{-1}\in alg\left(B^{*}\right),w\in \mathbb{D}.$$

(6)

3.

The eigenbasis of B diagonalizes $T=f\left(B\right)$ even when some values of f on the spectrum of B coincide, and it turns the unknown term in (6) into a diagonal analytic correction. We sample the Herglotz kernel of H at half the conjugate diagonal entries of T and add one vector sample at the origin. Repeated sampling points are allowed, and the origin sample cancels the diagonal correction’s contribution exactly.

4.

The surviving positive-matrix inequality compares two weighted Gramians. Its difference is positive, and an eigenvector argument followed by the first nonconstant Gramian term yields $\parallel T\parallel \le 2$.

The proof uses two limits: simple-spectrum matrices tend to A, followed by outer domains converging to $W\left(A\right)$. Together they yield the general case.

2. Positive-Real Kernels

Let $\mathcal{K}$ be an ${\mathbb{M}}_n\left(\mathbb{C}\right)$-valued kernel on $\mathbb{D}\times \mathbb{D}$. We call $\mathcal{K}$ positive if, for every finite choice $w_0,\dots ,w_m\in \mathbb{D}$ and ${\xi }_0,\dots ,{\xi }_m\in {\mathbb{C}}^n$,

$${\displaystyle \sum _{i,j=0}^m}{\xi }_i^{*}\mathcal{K}(w_i,w_j){\xi }_j\ge 0.$$

(7)

We shall use the following standard matrix-valued form of the Herglotz theorem. Its kernel consequence is included to fix the normalization.

Lemma 1 

(Matrix Herglotz kernel). Let $F:\mathbb{D}\to {\mathbb{M}}_n\left(\mathbb{C}\right)$ be analytic with $ReF\left(w\right)⪰0$ for every $w\in \mathbb{D}$. Then

$${\mathcal{L}}_F(w,z)={\displaystyle \frac{F\left(w\right)+F{\left(z\right)}^{*}}{1-w\overline{z}}}$$

(8)

is a positive kernel.

Proof. 

The matrix Herglotz representation gives a positive ${\mathbb{M}}_n\left(\mathbb{C}\right)$-valued measure M on $\mathbb{T}$, and a Hermitian matrix $J_F$, such that

$$F\left(w\right)=\mathrm{i}{J}_F+{\int }_{\mathbb{T}}{\displaystyle \frac{\zeta +w}{\zeta -w}}dM\left(\zeta \right).$$

(9)

For $\left|\zeta \right|=1$, direct calculation gives

$${\displaystyle \frac{1}{1-w\overline{z}}}\left({\displaystyle \frac{\zeta +w}{\zeta -w}}+\overline{{\displaystyle \frac{\zeta +z}{\zeta -z}}}\right)={\displaystyle \frac{2}{(1-w\overline{\zeta })(1-\overline{z}\zeta )}}.$$

(10)

The constant $\mathrm{i}{J}_F$ cancels from (8). Substituting (9) and then summing as in (7) yields

$$2{\int }_{\mathbb{T}}{\left(\sum _i{\displaystyle \frac{{\xi }_i}{1-\overline{w_i}\zeta }}\right)}^{*}dM\left(\zeta \right)\left(\sum _j{\displaystyle \frac{{\xi }_j}{1-\overline{w_j}\zeta }}\right)\ge 0.$$

□

3. The Positive-Real Completion Principle

The following completion theorem is the core of the argument: it converts a positive-real completion constrained by the adjoint algebra of an auxiliary simple-spectrum matrix into the sharp norm bound.

Theorem 2 

(Positive-real completion). Let

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

(11)

where S is invertible, the numbers ${\beta }_1,\dots ,{\beta }_n$ are distinct, and $\left|{\lambda }_i\right|\le 1$ for every i. The numbers ${\lambda }_i$ may repeat. Let $H:\mathbb{D}\to {\mathbb{M}}_n\left(\mathbb{C}\right)$ be analytic and satisfy

$$H\left(0\right)=I,ReH\left(w\right)⪰0(w\in \mathbb{D}),$$

(12)

and

$$H\left(w\right)-{(I-wT)}^{-1}\in alg\left(B^{*}\right)(w\in \mathbb{D}).$$

(13)

Then $\parallel T\parallel \le 2$.

Proof. 

Use the simultaneous diagonalizations in (11) and put $G=S^{*}S\succ 0$. Since the ${\beta }_i$ are distinct, polynomial interpolation shows that

$$alg\left(B^{*}\right)=\left\{{\left(S^{-1}\right)}^{*}D{S}^{*}:D\mathrm{is}\mathrm{diagonal}\right\}.$$

(14)

By (14), there is a unique diagonal analytic function $\Theta$ such that

$$H\left(w\right)-{(I-wT)}^{-1}={\left(S^{-1}\right)}^{*}\Theta \left(w\right)S^{*}.$$

Indeed,

$$\Theta \left(w\right)=S^{*}\left(H\left(w\right)-{(I-wT)}^{-1}\right){\left(S^{*}\right)}^{-1},$$

and $\Theta \left(0\right)=0$ because $H\left(0\right)=I$. Write $\Theta \left(w\right)=diag({\theta }_1\left(w\right),\dots ,{\theta }_n\left(w\right))$. On setting $\tilde{H}\left(w\right)=S^{*}H\left(w\right)S$, we obtain

$$\tilde{H}\left(w\right)=G{(I-w\Lambda )}^{-1}+\Theta \left(w\right)G.$$

(15)

Since $Re\tilde{H}\left(w\right)=S^{*}(ReH\left(w\right))S$, the real part of $\tilde{H}\left(w\right)$ is positive semidefinite. Consequently, Lemma 1 applies to $\tilde{H}$.

Define Hermitian matrices $P,Q,Y\in {\mathbb{M}}_n\left(\mathbb{C}\right)$ by

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

(16)

The denominators are nonzero, and P and Q are Hermitian, because $\left|{\lambda }_i\right|\le 1$. Let $e_1,\dots ,e_n$ be the standard basis of ${\mathbb{C}}^n$, and fix $u={(u_1,\dots ,u_n)}^T\in {\mathbb{C}}^n$. In the positive-kernel inequality use the n points and vectors

$$w_i={\displaystyle \frac{\overline{{\lambda }_i}}{2}},{\xi }_i=u_i{e}_i(1\le i\le n),$$

(17)

and add the point $w_0=0$ with vector

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

(18)

The points $w_i$ may coincide. Kernel positivity applies to arbitrary finite indexed lists of points and vectors, including $w_i=w_0$ when ${\lambda }_i=0$. We first verify that this sample cancels the contribution of $\Theta$ to the sampled kernel quadratic form. The contribution of the second term in (15) to the kernel quadratic form is

$$2Re{\displaystyle \sum _{i=1}^n}\overline{u_i}{\theta }_i\left(w_i\right)\left({\left(Gv\right)}_i+{\displaystyle \sum _{j=1}^n}{\displaystyle \frac{G_{ij}{u}_j}{1-w_i\overline{w_j}}}\right).$$

(19)

Because

$$1-w_i\overline{w_j}=1-\overline{{\lambda }_i}{\lambda }_j/4,$$

the parenthesis in (19) is ${(Gv+Pu)}_i$, which vanishes by (18).

It remains to calculate the contribution of $G{(I-w\Lambda )}^{-1}$. Between $w_i$ and $w_j$ its $(i,j)$ entry is

$${\displaystyle \frac{2G_{ij}}{(1-\overline{{\lambda }_i}{\lambda }_j/2)(1-\overline{{\lambda }_i}{\lambda }_j/4)}}={(4Q-2P)}_{ij}.$$

(20)

After compression by the coordinate vectors $e_i$, the two effective sample–origin blocks are $G+Q$, and the origin block is $2G$. After the cancellation of $\Theta$’s contribution, kernel positivity gives, for the prescribed vector $v=-G^{-1}Pu$,

$$\begin{array}{cc}\hfill 0& \le {\left(\begin{array}{c}u\\ v\end{array}\right)}^{*}\left(\begin{array}{cc}4Q-2P& G+Q\\ G+Q& 2G\end{array}\right)\left(\begin{array}{c}u\\ v\end{array}\right)\hfill \\ \hfill & =u^{*}\left((4Q-2P)-(G+Q)G^{-1}P-P{G}^{-1}(G+Q)+2P{G}^{-1}P\right)u\hfill \\ \hfill & =u^{*}\left(4Y-Y{G}^{-1}P-P{G}^{-1}Y\right)u,\hfill \end{array}$$

(21)

Here the block quadratic form is used only on the graph $v=-G^{-1}Pu$; it is not asserted to be positive for independent u and v. The last equality uses $Q=P+Y$. Since u was arbitrary,

$$4Y-Y{G}^{-1}P-P{G}^{-1}Y⪰0.$$

(22)

Balance the diagonalization by defining

$$\begin{array}{cccc}\hfill \tilde{T}& =G^{1/2}\Lambda G^{-1/2},\hfill & \hfill \widehat{P}& =G^{-1/2}P{G}^{-1/2},\hfill \\ \hfill \widehat{Q}& =G^{-1/2}Q{G}^{-1/2},\hfill & \hfill \widehat{Y}& =\widehat{Q}-\widehat{P}.\hfill \end{array}$$

(23)

Expanding the scalar kernels in (16) into geometric series gives

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

(24)

Indeed, ${\tilde{T}}^k=G^{1/2}{\Lambda }^k{G}^{-1/2}$, so the sequence $\left({\tilde{T}}^k\right)$ is bounded even when some $\left|{\lambda }_j\right|=1$; the geometric weights give norm convergence. It follows that

$$\widehat{Y}={\displaystyle \sum _{k=1}^{\infty }}(2^{-k}-4^{-k}){\tilde{T}}^{*k}{\tilde{T}}^k⪰0.$$

(25)

Taking the congruence of (22) by $G^{-1/2}$ yields

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

(26)

Let e be an eigenvector of the Hermitian matrix $\widehat{P}$, say $\widehat{P}e=\alpha e$. If $\alpha >2$, then (26) gives

$$0\le 2(2-\alpha )e^{*}\widehat{Y}e.$$

(27)

Since $\widehat{Y}⪰0$ and $\alpha >2$, (27) forces $e^{*}\widehat{Y}e=0$. The $k=1$ term in (25) is ${\tilde{T}}^{*}\tilde{T}/4$, so $\tilde{T}e=0$. The first series in (24) gives $\widehat{P}e=e$, contradicting $\widehat{P}e=\alpha e$ with $\alpha >2$. The first series in (24) also begins with I, and hence

$$I⪯\widehat{P}⪯2I.$$

(28)

The $k=1$ term of the same series now gives

$${\displaystyle \frac{1}{4}}{\tilde{T}}^{*}\tilde{T}⪯\widehat{P}-I⪯I.$$

Thus $\parallel \tilde{T}\parallel \le 2$. The polar decomposition $S=U{G}^{1/2}$ gives

$$T=U\tilde{T}{U}^{*},$$

where U is unitary. Therefore $\parallel T\parallel \le 2$. □

Remark 1. 

The factor $1/2$ in the sampling points (17) simultaneously produces the weights $1/2$ and $1/4$ in (24). Their positive difference bounds $\widehat{P}$, and the first nonconstant term of $\widehat{P}$ then gives the sharp constant. The origin sample cancels the correction’s contribution by an exact algebraic identity. Distinct auxiliary eigenvalues identify $alg\left(B^{*}\right)$ with the diagonal algebra in the chosen basis, while the diagonal entries of T may repeat.

4. The Double-Layer Completion

We now construct from the numerical range the function required by Theorem 2. The measure and companion transform used here are classical [2,4,19]. The full Cayley family yields the algebraic constraint (13) relative to the auxiliary matrix.

Lemma 2 

(Double-layer positive map). Let $B\in {\mathbb{M}}_n\left(\mathbb{C}\right)$, and let $\Omega \subset \mathbb{C}$ be a bounded convex domain with continuously differentiable boundary and $W\left(B\right)\subset \Omega$. Writing $C(\partial \Omega )$ for the continuous complex-valued functions on $\partial \Omega$, there is a unital positive map

$$\Phi :C(\partial \Omega )⟶{\mathbb{M}}_n\left(\mathbb{C}\right)$$

such that, whenever h is holomorphic on a neighborhood of $\overline{\Omega }$,

$$2\Phi \left(h\right)=h\left(B\right)+g_h{\left(B\right)}^{*},$$

(29)

where

$$g_h\left(z\right)={\displaystyle \frac{1}{2\pi \mathrm{i}}}{\int }_{\partial \Omega }{\displaystyle \frac{\overline{h\left(\zeta \right)}}{\zeta -z}}d\zeta ,z\in \Omega .$$

(30)

Proof. 

Orient $\partial \Omega$ counterclockwise. Let $\nu \left(\zeta \right)$ be the outward unit normal, let $ds$ be arclength, and put

$$Z_{\zeta }=\zeta I-B,{\mathcal{R}}_{\zeta }=Z_{\zeta }^{-1}.$$

This resolvent exists because $\sigma \left(B\right)\subset W\left(B\right)\subset \Omega$. Define the matrix-valued boundary measure

$$d{\mu }_B\left(\zeta \right)={\displaystyle \frac{1}{2\pi }}\left(\nu \left(\zeta \right){\mathcal{R}}_{\zeta }+\overline{\nu \left(\zeta \right)}{\mathcal{R}}_{\zeta }^{*}\right)ds.$$

(31)

It is positive. In fact, the supporting-line property of the convex set $\overline{\Omega }$ gives

$$\nu \left(\zeta \right)Z_{\zeta }^{*}+\overline{\nu \left(\zeta \right)}{Z}_{\zeta }=2Re\left(\overline{\nu \left(\zeta \right)}{Z}_{\zeta }\right)⪰0.$$

(32)

Indeed, for every unit vector $x\in {\mathbb{C}}^n$, the quadratic form on the left is

$$2Re\left(\overline{\nu \left(\zeta \right)}(\zeta -x^{*}Bx)\right)\ge 0,$$

because $x^{*}Bx\in W\left(B\right)\subset \Omega$. Multiplication on the left by ${\mathcal{R}}_{\zeta }^{*}$ and on the right by ${\mathcal{R}}_{\zeta }$ turns the left side of (32) into the matrix in parentheses in (31).

Along the counterclockwise boundary, $d\zeta =\mathrm{i}\nu \left(\zeta \right)ds$. The matrix Cauchy formula gives

$${\displaystyle \frac{1}{2\pi }}{\int }_{\partial \Omega }\nu \left(\zeta \right){\mathcal{R}}_{\zeta }ds=I.$$

(33)

Taking adjoints gives the same identity for the second layer. Hence

$${\int }_{\partial \Omega }d{\mu }_B=2I.$$

(34)

For $\phi \in C(\partial \Omega )$, define

$$\Phi \left(\phi \right)={\displaystyle \frac{1}{2}}{\int }_{\partial \Omega }\phi \left(\zeta \right)d{\mu }_B\left(\zeta \right).$$

(35)

By (34), $\Phi$ is unital and positive, hence *-preserving; moreover, $\parallel \Phi \left(\phi \right)\parallel \le {\parallel \phi \parallel }_{\infty }$, where

$${\parallel \phi \parallel }_{\infty }=\underset{\zeta \in \partial \Omega }{max}\left|\phi \left(\zeta \right)\right|.$$

For holomorphic h, the first layer of $2\Phi \left(h\right)$ is $h\left(B\right)$ by Cauchy’s formula. The function $g_h$ in (30) is holomorphic on $\Omega$, so $g_h\left(B\right)$ is defined by the holomorphic functional calculus. To verify the relevant integral formula using only the holomorphy of $g_h$ on $\Omega$, choose a positively oriented contour $\Gamma \subset \Omega$ enclosing $\sigma \left(B\right)$. The compact contours $\Gamma$ and $\partial \Omega$ are positively separated, so Cauchy’s formula and Fubini’s theorem give

$$\begin{array}{cc}\hfill g_h\left(B\right)& ={\displaystyle \frac{1}{2\pi \mathrm{i}}}{\int }_{\Gamma }{g}_h\left(z\right){(zI-B)}^{-1}dz\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi \mathrm{i}}}{\int }_{\partial \Omega }\overline{h\left(\zeta \right)}{(\zeta I-B)}^{-1}d\zeta .\hfill \end{array}$$

(36)

Taking adjoints in (36) and using $d\overline{\zeta }=-\mathrm{i}\overline{\nu \left(\zeta \right)}ds$ gives the second layer of $2\Phi \left(h\right)$. This proves (29). □

Proposition 1 

(Cayley completion for an auxiliary pair). Let B and Ω be as in Lemma 2, and let f be holomorphic on a neighborhood of $\overline{\Omega }$ and satisfy

$$\underset{\zeta \in \overline{\Omega }}{max}\left|f\left(\zeta \right)\right|\le 1.$$

(37)

Put $T=f\left(B\right)$. Then there is an analytic $H:\mathbb{D}\to {\mathbb{M}}_n\left(\mathbb{C}\right)$ satisfying

$$H\left(0\right)=I,ReH\left(w\right)⪰0(w\in \mathbb{D}),$$

(38)

and

$$H\left(w\right)-{(I-wT)}^{-1}\in alg\left(B^{*}\right)(w\in \mathbb{D}).$$

(39)

Moreover, $\sigma \left(T\right)\subset \overline{\mathbb{D}}$.

Proof. 

By (37) and spectral mapping from $\sigma \left(B\right)\subset \Omega$, we have $\sigma \left(T\right)\subset \overline{\mathbb{D}}$. For $w\in \mathbb{D}$, define on the boundary

$$c_w\left(\zeta \right)={\displaystyle \frac{1+wf\left(\zeta \right)}{1-wf\left(\zeta \right)}}$$

(40)

and put

$$H\left(w\right)=\Phi \left(c_w\right).$$

(41)

The map $w\mapsto c_w$ is analytic with values in $C(\partial \Omega )$, locally uniformly on $\mathbb{D}$. Hence H is analytic, and $H\left(0\right)=I$. Furthermore,

$$Re{c}_w\left(\zeta \right)={\displaystyle \frac{1-{\left|wf\left(\zeta \right)\right|}^2}{{|1-wf\left(\zeta \right)|}^2}}\ge 0.$$

The positivity and *-preservation of $\Phi$ give $ReH\left(w\right)=\Phi (Re{c}_w)⪰0$.

For fixed $w\in \mathbb{D}$, the same formula defines $c_w$ holomorphically on a neighborhood of $\overline{\Omega }$: on $\overline{\Omega }$ one has $|1-wf|\ge 1-\left|w\right|>0$. Hence Lemma 2 gives

$$2H\left(w\right)=c_w\left(B\right)+g_{c_w}{\left(B\right)}^{*}.$$

Since $T=f\left(B\right)$, the holomorphic functional calculus gives

$$c_w\left(B\right)=(I+wT){(I-wT)}^{-1}=2{(I-wT)}^{-1}-I.$$

Consequently,

$$H\left(w\right)-{(I-wT)}^{-1}={\displaystyle \frac{1}{2}}\left(g_{c_w}{\left(B\right)}^{*}-I\right).$$

(42)

The function $g_{c_w}$ is holomorphic on a neighborhood of $\sigma \left(B\right)$, so finite-dimensional holomorphic functional calculus gives $g_{c_w}\left(B\right)\in alg\left(B\right)$. Thus the right-hand side of (42) belongs to $alg\left(B^{*}\right)$, proving (39). □

Corollary 1 

(Auxiliary sharp bound). In Proposition 1, if B has simple spectrum, then $\parallel f\left(B\right)\parallel \le 2$.

Proof. 

Write

$$B=Sdiag({\beta }_1,\dots ,{\beta }_n)S^{-1}.$$

Then

$$T=f\left(B\right)=Sdiag\left(f\left({\beta }_1\right),\dots ,f\left({\beta }_n\right)\right)S^{-1}.$$

Since $\sigma \left(B\right)\subset \Omega$, the bound on f gives $\left|f\left({\beta }_i\right)\right|\le 1$ for every i. Apply Theorem 2 to these simultaneous diagonalizations and to the function H constructed in Proposition 1. The values $f\left({\beta }_i\right)$ may repeat. □

5. Passage to Arbitrary Matrices and Proof of the Main Theorem

Successive approximations pass from the auxiliary setting to arbitrary matrices.

Lemma 3 

(Simple spectrum and numerical range). Matrices with n distinct eigenvalues are dense in ${\mathbb{M}}_n\left(\mathbb{C}\right)$. Moreover, for $A,B\in {\mathbb{M}}_n\left(\mathbb{C}\right)$,

$$W\left(B\right)\subset W\left(A\right)+\{z\in \mathbb{C}:|z|\le \parallel B-A\parallel \}.$$

(43)

Proof. 

The discriminant of the characteristic polynomial is a nonzero polynomial in the matrix entries, as is seen at $diag(1,\dots ,n)$. The zero set of a nonzero complex polynomial has empty interior, so its nonvanishing set is dense in ${\mathbb{M}}_n\left(\mathbb{C}\right)$, proving the first assertion. For a unit vector x,

$$\left|x^{*}(B-A)x\right|\le \parallel B-A\parallel ,$$

which proves (43). □

Lemma 4 

(Parallel convex domains). Let $K\subset \mathbb{C}$ be nonempty, compact, and convex. For $\delta >0$, the set

$${\Omega }_{\delta }=\{z\in \mathbb{C}:dist(z,K)<\delta \}$$

is a bounded convex domain with continuously differentiable boundary, and ${\overline{\Omega }}_{\delta }=K+\delta \overline{\mathbb{D}}$.

Proof. 

The set ${\Omega }_{\delta }$ is bounded and convex, and its closure is the Minkowski sum $K+\delta \overline{\mathbb{D}}$. Identify $\mathbb{C}$ with ${\mathbb{R}}^2$. The metric projection ${\pi }_K:\mathbb{C}\to K$ is unique and continuous. Put $d_K\left(z\right)=dist(z,K)$. For $z,h\in \mathbb{C}$, comparison first with ${\pi }_K\left(z\right)$ and then with ${\pi }_K(z+h)$ gives

$$\begin{array}{cc}\hfill 2Re\left(\overline{z+h-{\pi }_K(z+h)}h\right)-{\left|h\right|}^2& \le d_K{(z+h)}^2-d_K{\left(z\right)}^2\hfill \\ & \le 2Re\left(\overline{z-{\pi }_K\left(z\right)}h\right)+{\left|h\right|}^2.\hfill \end{array}$$

The continuity of ${\pi }_K$ now shows that $d_K^2$ is continuously differentiable, with real gradient $2(z-{\pi }_K\left(z\right))$. On the level set $d_K\left(z\right)=\delta$ this gradient has norm $2\delta >0$. Since $\partial {\Omega }_{\delta }=\{d_K=\delta \}$, the implicit-function theorem shows that $\partial {\Omega }_{\delta }$ is continuously differentiable. □

Proof of Theorem 1. 

Put $K=W\left(A\right)$, and let U be an open neighborhood of K on which f is holomorphic. Choose ${\epsilon }_0>0$ such that the closed ${\epsilon }_0$-neighborhood of K lies in U. For $0<\epsilon <{\epsilon }_0$, let

$${\Omega }_{\epsilon }=\{z\in \mathbb{C}:dist(z,K)<\epsilon \}.$$

Lemma 4 makes this a bounded convex domain with continuously differentiable boundary.

By Lemma 3, choose simple-spectrum matrices $B_k\to A$. For fixed $\epsilon$, all sufficiently large k satisfy $\parallel B_k-A\parallel <\epsilon /2$, and then (43) gives

$$W\left(B_k\right)\subset K+(\epsilon /2)\overline{\mathbb{D}}\subset {\Omega }_{\epsilon }.$$

Set

$$m_{\epsilon }=\underset{z\in {\overline{\Omega }}_{\epsilon }}{max}\left|f\left(z\right)\right|.$$

If $m_{\epsilon }=0$, then f vanishes on a neighborhood of $\sigma \left(A\right)$, so $f\left(A\right)=0$ and there is nothing to prove. Otherwise, apply Corollary 1 to $B_k$ and $f/m_{\epsilon }$ to obtain

$$\parallel f\left(B_k\right)\parallel \le 2m_{\epsilon }.$$

Continuity of the holomorphic functional calculus, obtained from a fixed contour inside U, gives $f\left(B_k\right)\to f\left(A\right)$. Hence

$$\parallel f\left(A\right)\parallel \le 2m_{\epsilon }.$$

(44)

Put $m_0={max}_{z\in K}\left|f\left(z\right)\right|$. Uniform continuity of f on the closed ${\epsilon }_0$-neighborhood of K gives a modulus of continuity ${\omega }_f$ there. Since every point of ${\overline{\Omega }}_{\epsilon }$ lies within $\epsilon$ of K,

$$0\le m_{\epsilon }-m_0\le {\omega }_f\left(\epsilon \right).$$

Letting $\epsilon \downarrow 0$ in (44) proves (3).

Finally, a rational function pole-free on the compact set $W\left(A\right)$ is holomorphic on a neighborhood of $W\left(A\right)$. Applying (3) to such functions proves (2) with $K=W\left(A\right)$ and $C=2$. □

6. Consequences and Scope

The Hilbert-space extension follows by the standard finite-dimensional compression argument; compare [13].

Corollary 2 

(Hilbert-space form). Let $\mathcal{H}\ne \left\{0\right\}$ be a complex Hilbert space, with inner product linear in the first variable, and let $\mathcal{B}\left(\mathcal{H}\right)$ denote its algebra of bounded operators. For $A\in \mathcal{B}\left(\mathcal{H}\right)$, put

$$W\left(A\right)=\{\langle Ax,x\rangle :x\in \mathcal{H},\parallel x\parallel =1\}.$$

Then, for every $p\in \mathbb{C}\left[z\right]$,

$$\parallel p\left(A\right)\parallel \le 2\underset{z\in W\left(A\right)}{sup}\left|p\left(z\right)\right|.$$

(45)

Consequently, $\overline{W\left(A\right)}$ is a 2-spectral set for A.

Proof. 

The assertion is immediate if $p=0$. Otherwise, write $d=degp$. For a unit vector $x\in \mathcal{H}$, let

$${\mathcal{M}}_x=span\{x,Ax,\dots ,A^{d}x\},A_x={\left.{\Pi }_{x}A\right|}_{{\mathcal{M}}_x},$$

where ${\Pi }_x$ is the orthogonal projection onto ${\mathcal{M}}_x$. Induction gives

$$A_x^{k}x=A^{k}x(0\le k\le d),$$

(46)

because $A^{k+1}x\in {\mathcal{M}}_x$ whenever $k<d$. Moreover, $W\left(A_x\right)\subset W\left(A\right)$. Theorem 1, applied on the finite-dimensional space ${\mathcal{M}}_x$, therefore gives

$${\parallel p\left(A\right)x\parallel }_{\mathcal{H}}={\parallel p\left(A_x\right)x\parallel }_{\mathcal{H}}\le \parallel p\left(A_x\right)\parallel \le 2\underset{z\in W\left(A\right)}{sup}\left|p\left(z\right)\right|.$$

Taking the supremum over unit vectors x proves (45).

Put $K=\overline{W\left(A\right)}$. The standard inclusion $\sigma \left(A\right)\subset K$ shows that $r\left(A\right)$ is defined for every rational function r pole-free on K. Choose a closed convex neighborhood $K_{\delta }$ of K that contains no pole of r. Its complement is connected, so polynomial Runge approximation on $K_{\delta }$, followed by the Cauchy integral formula for the functional calculus, gives polynomials $p_j$ such that $p_j\left(A\right)\to r\left(A\right)$ and

$$\underset{z\in W\left(A\right)}{sup}\left|p_j\left(z\right)\right|⟶\underset{z\in W\left(A\right)}{sup}\left|r\left(z\right)\right|.$$

Passing to the limit in (45) proves the rational estimate and hence the spectral-set assertion. □

The conclusion also controls polynomial and rational approximation of matrix functions. Let $A\in {\mathbb{M}}_n\left(\mathbb{C}\right)$, let f be holomorphic on a neighborhood of $W\left(A\right)$, and let s be either a polynomial or a rational function pole-free on $W\left(A\right)$. Then $f-s$ is holomorphic on a neighborhood of $W\left(A\right)$, and Theorem 1 gives

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

(47)

This is the sharp universal form of the numerical-range estimate that underlies applications to GMRES and rational Krylov methods; see [7,25,26] for the role of spectral sets in those settings.

7. Discussion and Further Directions

Theorem 1 proves the scalar-valued Crouzeix conjecture. The most immediate strengthening is the uniform matrix-polynomial question: for every $m\ge 1$ and every matrix polynomial $\mathcal{F}=\left[f_{ij}\right]\in {\mathbb{M}}_m\left(\mathbb{C}\left[z\right]\right)$, does one have

$$\parallel \mathcal{F}\left(A\right)\parallel \le 2\underset{z\in W\left(A\right)}{max}\parallel \mathcal{F}\left(z\right)\parallel ,\mathcal{F}\left(A\right)=\left[f_{ij}\left(A\right)\right].$$

(48)

Here $\parallel \mathcal{F}\left(A\right)\parallel$ is the operator norm on ${\left({\mathbb{C}}^n\right)}^m$, whereas $\parallel \mathcal{F}\left(z\right)\parallel$ is the operator norm on ${\mathbb{C}}^m$. The case $m=1$ is precisely the scalar theorem proved above; uniformity in m is an additional operator-algebraic requirement. The foundational matrix-level disk estimate is due to Okubo and Ando [27]. Uniform matrix-level spectral-set bounds are studied in [28], and their relation to numerical radius in [29]; the constant $1+\sqrt{2}$ at every matrix level follows from [4], while sharp estimates for $\rho$-contractions with $1\le \rho \le 2$ are developed in [30]. The extension is known with constant 2 for weighted shift matrices [18]. For block-valued inputs, the coefficients in the completion may occupy a larger operator algebra, so the cancellation in (19) would require a genuinely operator-algebraic formulation.

The positive-real completion principle is independent of numerical ranges: it starts with a Carathéodory function whose resolvent defect lies in the adjoint algebra of an auxiliary simple-spectrum matrix sharing an eigenbasis with the matrix in the resolvent. A direct infinite-dimensional extension of Theorem 2, based for example on spectral measures or Riesz bases, could reveal equality cases or uniform matrix-level functional-calculus structure. The main challenge is a stable functional-calculus formulation of the auxiliary-basis correction, valid beyond finite diagonal spectral models. Related questions for unbounded numerical ranges are studied in [31], and related abstract spectral-constant problems in [32,33].

Double-layer methods beyond numerical ranges were developed in [7], while configuration-constant refinements were introduced in [23]; see also [22] for a comparison of double-layer approaches. Annular double-layer kernels are studied in [34], while [35] develops a complementary annular realization and norm theory; elliptic domains admit structured dilation models [36]. Whenever a companion transform lands in a controlled adjoint algebra, the kernel-sampling argument may replace a triangle-inequality estimate and thereby offer a route to sharp constants for other geometric spectral sets.

With the notation of the proof of Theorem 2, if $\parallel \tilde{T}\parallel =2$ and $e\in {\mathbb{C}}^n$ satisfies ${\parallel e\parallel }_2=1$ and ${\parallel \tilde{T}e\parallel }_2=2$, then equality throughout the final Gramian inequality in the proof of Theorem 2 forces

$$\widehat{P}e=2e,{\tilde{T}}^{2}e=0,\widehat{P}\tilde{T}e=\tilde{T}e.$$

Analyzing these vector saturation conditions may classify extremal pairs $(A,p)$ and quantify stability near the nilpotent example (4). Such a classification must allow the nonuniqueness phenomena found in [12].

Finally, O’Loughlin and Rani extend the Crouzeix–Palencia framework to scaled q-numerical ranges, $0<\left|q\right|\le 1$, and propose associated spectral-set problems [37]. These provide a natural testing ground for investigating a constant-2 analogue.