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 Toeplitz–Hausdorff theorem shows that $W\left(A\right)$ is compact and convex, and $\sigma \left(A\right)\subset W\left(A\right)$.

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 with no pole 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 $p\in \mathbb{C}\left[z\right]$. Then

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

(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]. None of the cited reductions supplies a dimension-free argument that lowers the universal bound from $1+\sqrt{2}$ to 2.

1.1. Where the Earlier Mechanism Loses Sharpness

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 smooth convex domain $\Omega$ 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, not $f\left(B\right)$ by itself. The Crouzeix–Palencia argument couples the two terms in (5), and its abstract norm estimate is sharp at $1+\sqrt{2}$. The examples in [5] show that this abstract mechanism alone cannot give 2. The corresponding general uniform-algebra theorem is likewise sharp at $1+\sqrt{2}$; 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 configuration constant approaches its limiting value, so it yields no smaller universal constant [23]. Separately, continuity and compactness give a constant $C_n<1+\sqrt{2}$ in every fixed matrix dimension n [24].

The missing information is algebraic. The companion in (5) is not an arbitrary operator: for the approximants constructed in Section 5, it belongs to the algebra generated by $f{\left(B\right)}^{*}$. Estimating it as an unrelated summand discards precisely this coupling.

1.2. The New De-Symmetrization Mechanism

Our argument retains the whole Cayley family, rather than the single identity (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. 

At an auxiliary stage, a vanishing perturbation arranges that $T=f\left(B\right)$ and $alg\left(T\right)=alg\left(B\right)$. The double-layer identity then gives the positive-real completion

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

(6) 

3. 

For the approximants with simple spectrum used at that stage, diagonalization turns the unknown term in (6) into a diagonal analytic correction. We sample the Herglotz kernel of H at half the conjugate eigenvalues and add one vector sample at the origin. The latter is chosen to cancel the diagonal correction exactly.

4. 

The surviving positive-matrix inequality compares two weighted Gramians. Its difference is positive, and an eigenvector argument followed by a Stein identity yields $\parallel T\parallel \le 2$.

A perturbation of f that vanishes in the limit enforces algebra generation on each approximant with simple spectrum. We first let the perturbation tend to zero, then let the simple-spectrum matrices tend to A, and finally let the outer domains converge to $W\left(A\right)$. These three limits remove all auxiliary hypotheses.

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

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

(9) 

Indeed, for each $u\in {\mathbb{C}}^n$ the scalar Herglotz theorem gives a positive finite measure ${\mu }_u$ on $\mathbb{T}$ and a real number $c_u$ such that

$$u^{*}F\left(w\right)u=i{c}_u+{\int }_{\mathbb{T}}{\displaystyle \frac{\zeta +w}{\zeta -w}}d{\mu }_u\left(\zeta \right).$$

Uniqueness in the scalar theorem shows that, for every Borel set $E\subset \mathbb{T}$, the map $u\mapsto {\mu }_u\left(E\right)$ is a nonnegative quadratic form. Indeed, uniqueness applied to the corresponding identities for the scalar functions gives, as identities of measures,

$${\mu }_{\alpha u}={\left|\alpha \right|}^2{\mu }_u,{\mu }_{u+v}+{\mu }_{u-v}=2{\mu }_u+2{\mu }_v(\alpha \in \mathbb{C},u,v\in {\mathbb{C}}^n).$$

Its polarization defines

$$x^{*}M\left(E\right)y={\displaystyle \frac{1}{4}}{\displaystyle \sum _{k=0}^3}{i}^{-k}{\mu }_{x+i^{k}y}\left(E\right)(x,y\in {\mathbb{C}}^n).$$

Thus M is a finite ${\mathbb{M}}_n\left(\mathbb{C}\right)$-valued measure and $u^{*}M\left(E\right)u={\mu }_u\left(E\right)\ge 0$, so M is positive. The scalar imaginary constants polarize to the Hermitian matrix $J_F=(F\left(0\right)-F{\left(0\right)}^{*})/\left(2i\right)$, and the scalar representations assemble to (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 $i{J}_F$ 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 an algebraically constrained positive-real completion of a resolvent into the sharp norm bound.

Theorem 2  

(Positive-real completion). Let $T\in {\mathbb{M}}_n\left(\mathbb{C}\right)$ have n distinct eigenvalues, all in $\overline{\mathbb{D}}$. Let $H:\mathbb{D}\to {\mathbb{M}}_n\left(\mathbb{C}\right)$ be analytic and satisfy

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

(11) 

and

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

(12) 

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

Proof.  

Choose a diagonalization

$$T=S\Lambda S^{-1},\Lambda =diag({\lambda }_1,\dots ,{\lambda }_n),G=S^{*}S.$$

(13) 

The eigenvalues ${\lambda }_j$ are distinct and belong to $\overline{\mathbb{D}}$. Polynomial interpolation shows that

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

(14) 

Set

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

By (14), for each $w\in \mathbb{D}$ there is a unique diagonal matrix $\Theta \left(w\right)$ such that

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

In fact,

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

so $\Theta$ is analytic; moreover, $\Theta \left(0\right)=0$ 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) 

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) 

We first verify that this sample cancels the entire unknown function $\Theta$. 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) 

Between the eigenvalue samples and the origin, the two blocks are $G+Q$, and the block at the origin is $2G$. After the cancellation of $\Theta$, kernel positivity therefore gives

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

(21) 

where in the second line we used $v=-G^{-1}Pu$, and in the last line we used $Q=P+Y$. 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 $e^{*}\widehat{Y}e=0$. Using the series in (25), we obtain

$$0=e^{*}\widehat{Y}e={\displaystyle \sum _{k=1}^{\infty }}(2^{-k}-4^{-k}){\parallel {\tilde{T}}^{k}e\parallel }_2^2\ge {\displaystyle \frac{1}{4}}{\parallel \tilde{T}e\parallel }_2^2.$$

Thus $\tilde{T}e=0$, and 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) 

Finally, the first Gramian satisfies the Stein identity

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

(29) 

Using (28), we obtain

$${\tilde{T}}^{*}\tilde{T}⪯{\tilde{T}}^{*}\widehat{P}\tilde{T}=4(\widehat{P}-I)⪯4I.$$

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 and the Stein identity are what recover the sharp constant. The origin sample removes the unknown analytic correction exactly; no estimate is used at this step.

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 new point is that retaining the full Cayley family yields the algebraic constraint (12).

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 $\overline{\Omega }$,

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

(30) 

where

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

(31) 

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

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

(32) 

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

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

(33) 

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 (33) into the matrix in parentheses in (32).

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

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

(34) 

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

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

(35) 

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

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

(36) 

By (35), $\Phi$ 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 (31) is holomorphic on $\Omega$, so $g_h\left(B\right)$ is defined by the holomorphic functional calculus. To verify the relevant integral formula without assuming that $g_h$ extends across $\partial \Omega$, choose a positively oriented contour $\Gamma \subset \Omega$ enclosing $\sigma \left(B\right)$. Cauchy’s formula and Fubini’s theorem give

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

(37) 

Taking adjoints in (37) and using $d\overline{\zeta }=-i\overline{\nu \left(\zeta \right)}ds$ gives the second layer of $2\Phi \left(h\right)$. This proves (30). □

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

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

(38) 

Put $T=f\left(B\right)$. If

$$alg\left(T\right)=alg\left(B\right),$$

(39) 

then there is an analytic $H:\mathbb{D}\to {\mathbb{M}}_n\left(\mathbb{C}\right)$ satisfying

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

(40) 

and

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

(41) 

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

Proof.  

By (38) 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

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

(42) 

and put

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

(43) 

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 $ReH\left(w\right)=\Phi (Re{c}_w)⪰0$.

The boundary expansion, convergent in $C(\partial \Omega )$ locally uniformly for $w\in \mathbb{D}$,

$$c_w=1+2{\displaystyle \sum _{m=1}^{\infty }}{w}^m{f}^m$$

(44) 

and (30) yield, with norm convergence,

$$H\left(w\right)=I+{\displaystyle \sum _{m=1}^{\infty }}{w}^m\left(T^m+g_{f^m}{\left(B\right)}^{*}\right).$$

(45) 

Since the spectral radius of $wT$ is less than 1, the Neumann series for ${(I-wT)}^{-1}$ converges in norm. Subtracting it from (45) gives

$$H\left(w\right)-{(I-wT)}^{-1}={\displaystyle \sum _{m=1}^{\infty }}{w}^m{g}_{f^m}{\left(B\right)}^{*}.$$

(46) 

Each $g_{f^m}$ is holomorphic on a neighborhood of $\sigma \left(B\right)$. Finite-dimensional holomorphic functional calculus, equivalently Hermite interpolation followed by Cayley–Hamilton, gives $g_{f^m}\left(B\right)\in alg\left(B\right)$. Hence

$$g_{f^m}{\left(B\right)}^{*}\in alg\left(B^{*}\right)=alg\left(T^{*}\right),$$

where the equality follows by taking adjoints in (39). Every partial sum on the right of (46) therefore belongs to $alg\left(T^{*}\right)$. This algebra is finite-dimensional and hence norm closed, which proves (41). □

Corollary 1  

(Auxiliary sharp bound). In Proposition 1, if $T=f\left(B\right)$ has distinct eigenvalues, then $\parallel f\left(B\right)\parallel \le 2$.

Proof.  

Apply Theorem 2 to the function H constructed in Proposition 1. □

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

We now remove the hypotheses of simple spectrum, algebra generation, and boundary regularity by successive approximations.

Lemma 3  

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

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

(47) 

Proof.  

The discriminant of the characteristic polynomial is a nonzero polynomial in the matrix entries, as is seen at $diag(1,\dots ,n)$. Its nonvanishing set is therefore 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 (47). □

Lemma 4  

(Outer approximation). If $K\subset \mathbb{C}$ is nonempty, compact, and convex, then there are bounded convex domains ${\Omega }_j$ with continuously differentiable boundaries such that

$$K\subset {\Omega }_j,d_{\mathrm{H}}({\overline{\Omega }}_j,K)⟶0,$$

(48) 

where $d_{\mathrm{H}}$ denotes the Hausdorff distance between compact sets.

Proof.  

For $\delta >0$, let

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

This is a bounded convex domain and its closure is the Minkowski sum $K+\delta \overline{\mathbb{D}}$, so its Hausdorff distance from K is $\delta$. 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 $2\delta$ and hence never vanishes. Since $\partial {\Omega }_{\delta }=\{d_K=\delta \}$, the implicit-function theorem shows that $\partial {\Omega }_{\delta }$ is continuously differentiable. Taking ${\Omega }_j={\Omega }_{1/j}$ for $j\ge 1$ proves the lemma. □

Proof  

(Proof of Theorem 1). Let A and p be arbitrary as in Theorem 1. Choose a bounded convex domain $\Omega$ with continuously differentiable boundary such that

$$W\left(A\right)\subset \Omega .$$

(49) 

We first prove

$$\parallel p\left(A\right)\parallel \le 2\underset{z\in \overline{\Omega }}{max}\left|p\left(z\right)\right|.$$

(50) 

Because $W\left(A\right)$ is compactly contained in $\Omega$,

$$dist\left(W\right(A),\mathbb{C}\setminus \Omega )>0.$$

Lemma 3 and (47) therefore give a sequence of matrices $B_k\to A$ having simple spectrum and satisfying $W\left(B_k\right)\subset \Omega$. Since $p\left(B_k\right)\to p\left(A\right)$, it is enough to prove (50) for an arbitrary such matrix B.

Write the distinct eigenvalues of B as ${\beta }_1,\dots ,{\beta }_n$ and set

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

(51) 

If $m_{\Omega }=0$, the polynomial vanishes on the open set $\Omega$ and is identically zero. In the remaining case $m_{\Omega }>0$, put

$$f={\displaystyle \frac{p}{m_{\Omega }}},R_{\Omega }=\underset{z\in \overline{\Omega }}{max}\left|z\right|.$$

(52) 

For $\eta \in \mathbb{C}$ define

$$f_{\eta }\left(z\right)={\displaystyle \frac{f\left(z\right)+\eta z}{1+\left|\eta \right|R_{\Omega }}}.$$

(53) 

Then ${max}_{z\in \overline{\Omega }}\left|f_{\eta }\left(z\right)\right|\le 1$. For a pair $i\ne j$, the equality

$$f\left({\beta }_i\right)+\eta {\beta }_i=f\left({\beta }_j\right)+\eta {\beta }_j$$

excludes at most one value of $\eta$. There are thus only finitely many exceptional values for which the numbers $f_{\eta }\left({\beta }_1\right),\dots ,f_{\eta }\left({\beta }_n\right)$ fail to be distinct. Choose nonexceptional ${\eta }_{\ell }\to 0$ and put

$$T_{\ell }=f_{{\eta }_{\ell }}\left(B\right).$$

(54) 

Each $T_{\ell }$ has distinct eigenvalues. It also generates the same unital algebra as B. Indeed, $T_{\ell }\in alg\left(B\right)$, while Lagrange interpolation at the distinct points $f_{{\eta }_{\ell }}\left({\beta }_i\right)$ supplies a polynomial $q_{\ell }$ satisfying

$$q_{\ell }\left(f_{{\eta }_{\ell }}\left({\beta }_i\right)\right)={\beta }_i(1\le i\le n).$$

Since B is diagonalizable, this implies $q_{\ell }\left(T_{\ell }\right)=B$. Consequently,

$$alg\left(T_{\ell }\right)=alg\left(B\right).$$

(55) 

Corollary 1, applied to B and $f_{{\eta }_{\ell }}$, gives

$$\parallel f_{{\eta }_{\ell }}\left(B\right)\parallel \le 2.$$

Letting $\ell \to \infty$ yields $\parallel f\left(B\right)\parallel \le 2$, or

$$\parallel p\left(B\right)\parallel \le 2m_{\Omega }.$$

(56) 

Letting the matrices $B_k$ with simple spectrum tend to A proves (50).

Apply Lemma 4 to the compact convex set $K=W\left(A\right)$, obtaining ${\Omega }_j$. All these closures eventually lie in one fixed compact neighborhood of K. Uniform continuity of p there and (48) give

$$\underset{z\in {\overline{\Omega }}_j}{max}\left|p\left(z\right)\right|⟶\underset{z\in W\left(A\right)}{max}\left|p\left(z\right)\right|.$$

(57) 

Apply (50) to each ${\Omega }_j$ and pass to the limit. This proves (3).

It remains to justify the spectral-set conclusion. Let r be rational with no pole on $W\left(A\right)$. Since $W\left(A\right)$ is compact and disjoint from the finite set of poles of r, choose $\delta >0$ so that the closed $\delta$-neighborhood $K_{\delta }$ of $W\left(A\right)$ contains no pole. Thus r is holomorphic on an open neighborhood of the compact convex set $K_{\delta }$. Since $\mathbb{C}\setminus K_{\delta }$ is connected, polynomial Runge approximation gives polynomials $p_j$ converging uniformly to r on $K_{\delta }$. As $\sigma \left(A\right)\subset W\left(A\right)\subset int{K}_{\delta }$, choose a positively oriented contour $\Gamma \subset int{K}_{\delta }$ enclosing $\sigma \left(A\right)$. Uniform convergence on $\Gamma$, together with the Cauchy integral formula for the holomorphic functional calculus, gives $p_j\left(A\right)\to r\left(A\right)$. Also

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

Apply the polynomial estimate to $p_j$ and pass to the limit to obtain (2) with $K=W\left(A\right)$ and $C=2$. □

6. Consequences and Scope

Theorem 1 immediately gives the corresponding estimate for functions holomorphic on a neighborhood of the numerical range.

Corollary 2.  

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

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

(58) 

Proof.  

Choose $\delta >0$ so that the closed $\delta$-neighborhood of $W\left(A\right)$ lies in the open set on which f is holomorphic. Polynomial Runge approximation on this compact convex neighborhood, followed by the Cauchy integral formula for the holomorphic functional calculus, gives the assertion. □

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

Corollary 3  

(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 $p\in \mathbb{C}\left[z\right]$,

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

(59) 

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

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

(60) 

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 (59).

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 without poles on K. The polynomial-approximation argument in the proof of Theorem 1, now applied to K, 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 (59) 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 with no pole on $W\left(A\right)$. Then $f-s$ is holomorphic on a neighborhood of $W\left(A\right)$, and Corollary 2 gives

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

(61) 

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 original, 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

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

(62) 

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 original conjecture 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 and their relation to numerical radius are studied in [28,29]; 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 need not reduce to a single diagonal 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 an adjoint-generated algebra. A direct infinite-dimensional version 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 obstacle is to replace the finite diagonal correction and Lagrange generation (55) by a stable functional-calculus condition without relying on a diagonal spectral model. 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 last operator-inequality chain in the proof of Theorem 2 forces

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

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, $q\in [0,1]$, and propose associated spectral-set problems [37]. These provide a natural testing ground for positive-real completion, although no constant-2 analogue is asserted here.