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The numerical range of a complex matrix is $W(A)=\{x^*Ax\colon x\in\C^n,\ x^*x=1\}$.
It is compact, being the continuous image of the unit sphere, and convex: the two-dimensional case is due to Toeplitz (1918), the general case follows by compressing to the span of two unit vectors, and Hausdorff (1919) completed the argument. Crouzeix's 2004 conjecture states that every polynomial $p$ and every $A$ satisfy $\|p(A)\|\le2\max_{W(A)}|p|$; his 2007 paper proves the universal bound $11.08$. Von Neumann's (1929) polynomial inequality for contractions, $\|p(T)\|\le\max_{|z|\le1}|p(z)|$ for every contraction $T$ and every polynomial $p$, is the prototype: the closed unit disk is a complete $1$-spectral set for the shift on the disk. Crouzeix's conjecture replaces the unit disk by the numerical range. Delyon and Delyon (1999) introduced a positive double-layer integral representation, expressing a matrix function as a boundary average against a positive density on the boundary of an enclosing domain. Crouzeix and Palencia (2017) obtained the bound $1+\sqrt{2}$ through a conjugate-data Cauchy transform, and Ransford and Schwenninger (2018) simplified the final step. Scalar constant-two proofs are due to Jin (2026), to Lorist and Schwenninger (2026), and to Luo (2026); \AA hag, Czy{\.{z}}, and Virtanen (2026) prove the constant-two estimate for base matrices of order at most three at arbitrary coefficient size. The matrix-valued case at every finite base dimension, and the resulting operator case on arbitrary complex Hilbert spaces, are proved below.

\url{https://doi.org/10.1090/S0002-9904-1918-01447-8}

\url{https://link.springer.com/article/10.1007/BF01202959}

\url{https://doi.org/10.1515/crll.1929.161.208}

\url{https://link.springer.com/article/10.1007/s00020-002-1212-x}

\url{https://doi.org/10.1016/j.jfa.2006.09.006}

\url{http://www.numdam.org/item/BSMF_1999__127_1_25_0.pdf}

\url{https://arxiv.org/pdf/1702.00668}

\url{https://www.math.uni-hamburg.de/home/schwenninger/hp/RS_SIMAX2018.pdf}

\url{https://www.preprints.org/manuscript/202607.1919}

\url{https://arxiv.org/pdf/2608.03841}

\url{https://www.preprints.org/manuscript/202608.1661}

\url{https://arxiv.org/pdf/2608.27346}

Write $p(z)=\sum_{j=0}^dB_jz^j$ with $B_j\in M_m(\C)$ and set $p[A]=\sum_{j=0}^dA^j\otimes B_j$ (base factor first). Crouzeix's conjecture in the complete form asks whether $\|p[A]\|\le2\max_{W(A)}\|p\|$ holds for every pair of sizes $n,m$ and every degree $d$. It is equivalent, for the applications below, to a single polynomial identity for all such $n,m,d$, and Runge's (1885) polynomial approximation on a compact set with connected complement transports the polynomial statement to matrix-valued functions holomorphic on a neighborhood of $\overline{W(A)}$ and to rational functions with poles outside that closure.

\url{https://link.springer.com/chapter/10.1007/BF02400416}

Let $\Omega\subset\C$ be admissible: bounded, open, convex, with regular real-analytic Jordan boundary. All matrix functions below are evaluated by the finite Jordan Taylor formula on each Jordan block and by change of basis to the diagonal. Two conformal coordinates will be used: the interior map $f\colon\Omega\to\D$ and the exterior map $G\colon\{|t|>1\}\to\C\setminus\overline\Omega$, normalized at infinity by $G(t)=at+b+O(t^{-1})$ with $a\ne0$; both extend univalently through the closures, $G(\T)=\partial\Omega$, and $q_G(t)=tG'(t)$ points outward normal on $\T$. The boundary circle $\T$ carries the normalized arc measure $d\sigma$.

For $A\in M_n(\C)$ with $W(A)\subset\Omega$, put $T=f(A)$ and define
\[
\kappa^2=\min\bigl\{\tau\in\R\colon I\preceq H\preceq\tau I,\ \ T^*HT\preceq H\text{ for some }H=H^*\bigr\}.
\]

\begin{theorem}[optimal metric]
The minimum is attained, $1\le\kappa^2\le4$, and with $S=H^{1/2}$, $A'=SAS^{-1}$, $D=Sf(A)S^{-1}=f(A')$, one has $D^*D\preceq I$, $\rho(D)<1$, $\|S\|\|S^{-1}\|=\kappa\le2$.
\end{theorem}

\begin{theorem}[single positive density]
With the same data, there is a continuous $\Lambda\colon\T\to M_n(\C)$ satisfying $\Lambda(t)\succeq0$ and $\int_\T\Lambda\,d\sigma=I_n$, such that for every $m$ and every $M_m$-valued $v$ holomorphic near $\overline\Omega$,
\[
v[A']=\int_\T\Lambda(t)\otimes v(G(t))\,d\sigma(t),\qquad
\|v[A]\|\le\kappa\max_{\overline\Omega}\|v\|.
\]
\end{theorem}

The same $S$, $\Lambda$, and $\kappa$ serve every $m$ and every $v$; the two theorems say that a single change of inner product and a single positive boundary density encode the entire analytic functional calculus of $A$.

\medskip
\textbf{The metric.}  A pair $(H,\tau)$ is strictly feasible when $H-I$, $\tau I-H$, and $H-T^*HT$ are all positive definite. Stein's (1952) criterion for iterated contractions gives strict feasibility: because $\rho(T)<1$, the geometric series $H_0=\sum_{j\ge0}T^{*j}T^j$ converges in norm, satisfies $H_0\succeq I$, and $H_0-T^*H_0T=I$; a large enough multiple of $H_0$ together with a large enough $\tau$ then moves all three slacks into the positive cone. Feasible pairs form a closed bounded subset of $\R\times\Herm_n$, so a minimum is attained at some $H$, and $\tau\ge1$ always. Setting $S=H^{1/2}$ and $D=Sf(A)S^{-1}$, congruence gives $D^*D\preceq I$, similarity gives $\rho(D)=\rho(f(A))<1$ by the spectral mapping theorem for the holomorphic functional calculus, and $I\preceq H\preceq\kappa^2I$ gives $\|S\|\|S^{-1}\|\le\kappa$. The minimizing metric is therefore exactly Paulsen's (1984) similarity to a contraction:
\[
\kappa=\min\bigl\{\|R\|\,\|R^{-1}\|\colon R\text{ invertible},\ \|RTR^{-1}\|\le1\bigr\}.
\]
The optimizer may have $\|D\|=1$; that is allowed.

\url{https://nvlpubs.nist.gov/nistpubs/jres/048/jresv48n1p82_A1b.pdf}

\url{https://www.ams.org/journals/proc/1984-092-02/S0002-9939-1984-0754708-X/}

\medskip
\textbf{The density.}  On $\T$ define the Delyon double-layer
\[
\Phi_D(\zeta)=(I-D/\zeta)^{-1}+[(I-D/\zeta)^{-1}]^*-I.
\]
For $|\zeta|=1$ the invertible factor $E_\zeta=I-D/\zeta$ turns it positive: $E_\zeta^*\Phi_D(\zeta)E_\zeta=I-D^*D\succeq0$. Since $\sum_{j\ge0}\|D^j\|<\infty$ (spectral radius alone suffices), the Laurent series is absolutely uniformly convergent on $\T$ and has the moments $\int_\T\zeta^j\Phi_D(\zeta)\,d\sigma=D^j$ for $j\ge0$. Integration of the uniformly convergent Taylor series of any $M_m$-valued $h$ holomorphic near $\overline\D$ therefore gives the common dilation identity
\[
h[D]=\int_\T\Phi_D(\zeta)\otimes h(\zeta)\,d\sigma(\zeta).
\]
This is the positive form of the Sz.-Nagy dilation; the interior-versus-exterior normalization makes it the Delyon--Delyon representation. Applying $h=v\circ f^{-1}$ and substituting $\zeta=\psi(t)=f(G(t))$ pulls the identity to the exterior boundary. Writing $\psi(e^{i\theta})=e^{i\eta(\theta)}$ gives the positive angular Jacobian
\[
j(t)=\frac{tG'(t)\,f'(G(t))}{\psi(t)}=\eta'(t)>0,
\qquad\Lambda(t)=j(t)\,\Phi_D(\psi(t)),
\]
and $\int j\,d\sigma=1$ together with the zeroth moment $\int\Phi_D=I_n$ gives $\int\Lambda=I_n$. Both identity and inequality in the second theorem follow by inserting $V_m$ as the isometry $b\mapsto(\Lambda^{1/2}\otimes I_m)b$ of $\C^n\otimes\C^m$ into $L^2(\T,d\sigma;\C^n\otimes\C^m)$.

\url{http://www.numdam.org/item/BSMF_1999__127_1_25_0.pdf}

\url{https://link.springer.com/book/10.1007/978-0-387-72476-8}


\medskip
\textbf{The extremal.}  Assume $\kappa>1$. In $\R\oplus\Herm_n^3$ the affine set $\mathcal A_T=\{(s,J-I,sI-J,J-T^*JT)\colon s\in\R,\ J\in\Herm_n\}$ is disjoint from the open convex set $\mathcal O=\{(s,U,V,W)\colon s<\kappa^2,\ U,V,W\succ0\}$: a point of $\mathcal A_T$ would be a strictly feasible pair below the minimizing $\tau$. Writing $\mathcal A_T=a_0+L$ and projecting orthogonally onto $L^\perp$, the set $\mathcal K=\pi(\mathcal O-a_0)$ is nonempty, open, convex, and does not contain $0$. If $0\notin\overline{\mathcal K}$, a nearest point $z_0\in\overline{\mathcal K}$ to $0$ satisfies $\langle z_0,c\rangle\ge\|z_0\|^2>0$ for every $c\in\mathcal K$. If $0\in\overline{\mathcal K}$, fix $c_*\in\mathcal K$ and $z_j=-c_*/j$; the $z_j$ stay outside $\overline{\mathcal K}$ by the interior-point segment argument, the nearest points $y_j$ satisfy $y_j\to0$, and a subsequence of $(y_j-z_j)/\|y_j-z_j\|$ converges to a unit vector $\nu$ with $\langle\nu,c\rangle\ge0$ for every $c\in\mathcal K$. Either case yields a nonzero separating functional $\ell(v)=-\langle\nu,\pi v\rangle$.

The dual pair $X_0,Y_0,Z_0\in\Herm_n$ satisfies $X_0,Z_0\succeq0$, $Y_0\succ0$, $\tr(X_0+Y_0)=1$, and
\[
X_0-Z_0=TZ_0T^*,\quad X_0+SZ_0S=\kappa^{-2}SY_0S,\quad SX_0S+Z_0=\kappa^2Y_0,
\]
with $\tr(Y_0(I-T^*T))=1$ and $\tr(X_0(\kappa^2I-T^*T))=0$. Equality in $\tr(BC)\le\|C^{1/2}B^{1/2}\|_{\HS}^2$ with disjoint support forces $X_0$ to be supported on the $\kappa^2$-eigenspace of $H$ and $Y_0$ on the $1$-eigenspace. Put $XX^*=\kappa^{-2}SX_0S$, $YY^*=\kappa^{-2}SY_0S$, $Q=\kappa^{-2}SZ_0S$; the trace normalizations become $\tr(XX^*)=\tr(YY^*)=1$, the eigendata give $SX=\kappa X$ and $SY=Y$, and the identity $X_0-Z_0=TZ_0T^*$ becomes $XX^*-YY^*=Q-DQD^*$ with orthogonal column ranges (by disjoint support), hence $X^*Y=0$.

Write $L=Q^{1/2}$. The rows $R_1=[L\ Y]$, $R_2=[DL\ X]$ have equal Gram matrices $R_1R_1^*=R_2R_2^*$, so a unitary $U=\begin{pmatrix}a&b\\c&d\end{pmatrix}$ exists with $[L\ Y]=[DL\ X]U$, giving $L=DLa+Xc$ and $Y=DLb+Xd$. Since $\|a\|\le1$ and $\rho(D)<1$, the series
\[
Y=Xd+\sum_{j\ge0}D^{j+1}Xca^jb
\]
converges in norm. Set $R(\zeta)=d+\zeta c(I-\zeta a)^{-1}b$ and $F_\D(\zeta)=R(\zeta)^{\mathsf T}$. Unitarity of $U$ gives, for every $e\in\C^n$ and $v=(I-\zeta a)^{-1}be$, $\|R(\zeta)e\|^2=\|e\|^2-(1-|\zeta|^2)\|v\|^2\le\|e\|^2$, so $F_\D$ is contractive; the rational entries are bounded in $\D$, so poles on $\T$ are removable, and finitely many remaining poles stay outside $\overline\D$. The Taylor series of $F_\D$ has coefficients $d^{\mathsf T}$ and the transposes of $ca^jb$, and the vectorization identity $(D^k\otimes B)\vecop(X)=\vecop(D^kXB^{\mathsf T})$ turns these into $F_\D[D]x=y$ for $x=\vecop(X)$, $y=\vecop(Y)$. Setting $F_\Omega=F_\D\circ f$ and $F=F_\Omega\circ G$ gives $F_\Omega[A']x=y$ and $\|F(t)\|\le1$ on $\T$.

\url{https://www.ams.org/journals/bull/1970-76-05/S0002-9904-1970-12543-X/}

\url{https://doi.org/10.1070/RM1978v033n04ABEH002501}

\url{https://arxiv.org/abs/0706.1896}

\medskip
\textbf{Local boundary tools.}  With $G(t)=at+b+O(t^{-1})$ and $q_G(t)=tG'(t)$, the Faber polynomials $b_k$ (degree $k$, leading coefficient $a^{-k}$, $b_0=1$) are defined by
\[
\frac{q_G(t)}{G(t)-z}=\sum_{k\ge0}b_k(z)\,t^{-k}
\]
at infinity. The mixed kernel
\[
B(w,t)=\frac{q_G(t)}{G(t)-G(w)}-\frac{t}{t-w}=\sum_{i,j\ge1}\beta_{ij}w^{-i}t^{-j}
\]
is holomorphic through the diagonal and both infinities of the exterior product collar, and Cauchy estimates on the reciprocal bidisk give $|\beta_{ij}|\le C_\rho\rho^{i+j}$ for each $r<\rho<1$. The Grunsky-type kernel $m(w,t)=1+B(w,t)+\overline{B(w,t)}$ on $\T^2$ is continuous, nonnegative (convexity of $\Omega$ gives $m(w,t)=2\operatorname{Re}(q_G(t)/(G(t)-G(w)))\ge0$ off the diagonal), and each normalized-circle marginal of $m$ equals one.

For $A\in M_n(\C)$ with $W(A)\subset\Omega$, put
\[
K_A(t)=q_G(t)(G(t)I-A)^{-1}=\sum_{k\ge0}b_k(A)t^{-k},\qquad E_A(t)=K_A(t)+K_A(t)^*\succeq0.
\]
Every eigenvalue of $A$ lies in $W(A)$, so $G(t)I-A$ is invertible on and outside $\T$; Cauchy estimates in the reciprocal disk give geometric decay. Positivity follows from the congruence $R_t^*(K_A+K_A^*)R_t=q_GR_t^*+\overline{q_G}R_t$ with $R_t=G(t)I-A$ and the supporting half-plane of $\Omega$ at $G(t)$.

\url{https://math.univ-lille1.fr/~bbecker/bb_pub.html}

\url{https://arxiv.org/pdf/1310.1356}

\url{https://arxiv.org/abs/1610.05768}

For $k\ge1$ use $\cH_k=L^2(\T,d\sigma;M_k(\C))$ with $\langle u,v\rangle=\int\tr(u^*v)\,d\sigma$. Let $P_0,P_+,P_-$ be the constant, strictly positive, strictly negative Fourier projections and $\M u(w)=\int_\T m(w,t)u(t)\,d\sigma(t)$; then $\M$ sends constants to constants, positive to strictly negative, negative to strictly positive, preserves the mean, and is contractive on every $\cH_k$, with $\M^\dagger$ acting by the exchanged kernel $m(t,w)$. Define
\[
\cC=P_0+P_++\M P_+,\qquad\widetilde\cC u=(\cC(u^*))^*.
\]
For finite Laurent data $u$, the physical Cauchy integral $c_u(z)=\int_\T u(t)q_G(t)/(G(t)-z)\,d\sigma(t)$ is a polynomial in $z$, and substitution $z=G(w)$ gives $\cC u=c_u\circ G$; hence $\cC$ extends to a bounded projection on each $\cH_k$, $\cC^2=\cC$, and for $F=F_\Omega\circ G$ with $F_\Omega$ holomorphic near $\overline\Omega$ the two module identities $\cC(F\cC u)=F\cC u$, $\cC((\cC u)F)=(\cC u)F$ hold. Adjoints:
\[
\cC^\dagger u=u_0+u_++\M^\dagger u_-,\qquad \widetilde\cC^\dagger u=u_0+u_-+\M^\dagger u_+,
\]
and the weighted inequality
\[
\|\cC^\dagger u\|_2^2+\|(\cC^\dagger u)_0\|_2^2\le2\|u\|_2^2
\]
follows from orthogonality of the three frequency pieces and $\|\M^\dagger\|\le1$. For finite Laurent data $u$,
\[
\int_\T E_A\otimes u\,d\sigma=c_u[A]+c_{u^*}[A]^*,\qquad \int_\T E_A\,d\sigma=2I_n.
\]


\medskip
\textbf{The bridge.}  Assume $\kappa>1$. Let $x=\vecop(X)$, $y=\vecop(Y)$, $R_S=S\otimes I_n$. The endpoint relations are $\|x\|=\|y\|=1$, $R_Sx=\kappa x$, $R_Sy=y$, $F_\Omega[A']x=y$. With the isometry $V=V_n$ from the density theorem, set $x_\Lambda=(\Lambda^{1/2}\otimes I_n)x$ and $y_\Lambda=(\Lambda^{1/2}\otimes I_n)y$. The interpolation identity $v[A']=V^\dagger M_vV$ at $v=F$ gives
\[
1=\|V^\dagger M_FVx\|\le\|M_FVx\|\le\|Vx\|=1.
\]
Equality in the first inequality puts $M_FVx$ in the range of $V$, so $M_FVx=Vy$; equality in the second makes $I-M_F^\dagger M_F$ annihilate $Vx$. Hence, almost everywhere,
\[
y_\Lambda=(I_n\otimes F(t))x_\Lambda,\qquad x_\Lambda=(I_n\otimes F(t)^*)y_\Lambda.
\]

Take the base partial trace (sum of diagonal base blocks). Set $p=\operatorname{Tr}_{\rm base}(x_\Lambda x_\Lambda^*)$, $q=\operatorname{Tr}_{\rm base}(y_\Lambda y_\Lambda^*)$, $r=\operatorname{Tr}_{\rm base}(y_\Lambda x_\Lambda^*)$; then $p,q\succeq0$ and $p,q,r\in\cH_n$. Writing $x_\Lambda=\sum_a e_a\otimes\xi_a$ and $y_\Lambda=\sum_a e_a\otimes\eta_a$ gives $r=\sum_a\eta_a\xi_a^*$, and the fiber identity gives $\eta_a=F\xi_a$, $\xi_a=F^*\eta_a$. Hence
\[
p=F^*r,\qquad r=Fp,\qquad q=rF^*,
\]
with both multiplication orders preserved. The same convention gives $p=(X^*\Lambda X)^{\mathsf T}$, $q=(Y^*\Lambda Y)^{\mathsf T}$, $r=(X^*\Lambda Y)^{\mathsf T}$, and integrating against $\int\Lambda=I_n$ with $X^*Y=0$ yields the matrix identity
\[
r_0=0,\qquad \tr(p_0)=1.
\]
The full matrix mean-zero, not just $\tr(r_0)=0$, is what makes the two projected cross densities Fourier-orthogonal below.

For $u\in\cH_n$, the testing orientation is
\[
\langle r,u\rangle=\int_\T y_\Lambda^*(I_n\otimes u)x_\Lambda\,d\sigma=y^*\Bigl(\int_\T\Lambda\otimes u\,d\sigma\Bigr)x,
\]
which reads $\tr(r^*u)=\sum_a\eta_a^*u\xi_a$ with $(x,x)$ for $p$ and $(y,y)$ for $q$.

Using $E_A\succeq0$, put $x_e=(E_A^{1/2}\otimes I_n)x$ and $y_e=(E_A^{1/2}\otimes I_n)y$, and define $p_e,q_e,r_e$ by the same base partial trace. Writing $x_e=\sum_a e_a\otimes\alpha_a$ and $y_e=\sum_a e_a\otimes\beta_a$,
\[
\Delta=\begin{pmatrix}p_e&r_e^*\\ r_e&q_e\end{pmatrix}=\sum_a\binom{\alpha_a}{\beta_a}\binom{\alpha_a}{\beta_a}^{\!*}\succeq0.
\]
Similarity gives $y^*v[A]x=\kappa\, y^*v[A']x$ and $y^*v[A]^*x=\kappa^{-1}y^*v[A']^*x$; diagonal pairings have no $\kappa$. For $u\in\cH_n$ Laurent polynomial with physical representative $c_u$,
\[
\langle r_e,u\rangle=y^*(c_u[A]+c_{u^*}[A]^*)x=\kappa\langle r,\cC u\rangle+\kappa^{-1}\langle r,\widetilde\cC u\rangle,
\]
where the second term uses the analytic adjoint: $c_{u^*}[A']^*$ corresponds to $\widetilde\cC u$, not to an independent analytic function. Since Laurent polynomials are dense in $\cH_n$,
\[
p_e=\cC^\dagger p+\widetilde\cC^\dagger p,\qquad q_e=\cC^\dagger q+\widetilde\cC^\dagger q,\qquad r_e=\kappa\cC^\dagger r+\kappa^{-1}\widetilde\cC^\dagger r.
\]

\medskip
\textbf{The comparison.}  Put $g=\cC^\dagger r$, $P_1=\cC^\dagger p$, $P_2=\cC^\dagger q$. The Hilbert adjoint of the module identities gives $\cC^\dagger L_{F^*}\cC^\dagger=\cC^\dagger L_{F^*}$ and $\cC^\dagger R_{F^*}\cC^\dagger=\cC^\dagger R_{F^*}$, where $L_{F^*}$, $R_{F^*}$ are left and right multiplication by $F^*$. Hence
\[
P_1=\cC^\dagger(F^*r)=\cC^\dagger(F^*g),\qquad P_2=\cC^\dagger(rF^*)=\cC^\dagger(gF^*).
\]
If $g=0$, then $P_1=0$, contradicting $(P_1)_0=p_0$ and $\tr(p_0)=1$; hence $g\ne0$.

Since $\|F\|\le1$ on $\T$, $\|F^*g\|_2\le\|g\|_2$ and $\|gF^*\|_2\le\|g\|_2$; the weighted inequality gives
\[
\|P_j\|_2^2+\|(P_j)_0\|_2^2\le2\|g\|_2^2\qquad(j=1,2).
\]
Fourier orthogonality for a matrix field with only constant and positive modes gives $\|P_j+P_j^*\|_2^2=2(\|P_j\|_2^2+\|(P_j)_0\|_2^2)$. Since $p=p^*$ and $q=q^*$, the antiunitary $u\mapsto u^*$ gives $\widetilde\cC^\dagger p=P_1^*$ and $\widetilde\cC^\dagger q=P_2^*$, hence
\[
p_e=P_1+P_1^*,\quad q_e=P_2+P_2^*,\quad \|p_e\|_2^2\le4\|g\|_2^2,\quad \|q_e\|_2^2\le4\|g\|_2^2.
\]
The matrix identity $r_0=0$ implies $g$ has strictly positive (and no constant) frequencies only, and $h=\widetilde\cC^\dagger r$ has strictly negative (and no constant) frequencies only; their inner product vanishes. Hence
\[
\|r_e\|_2^2=\kappa^2\|g\|_2^2+\kappa^{-2}\|h\|_2^2\ge\kappa^2\|g\|_2^2.
\]
Positivity of $\Delta$ and $J\Delta J$ with $J=\diag(I_n,-I_n)$ makes the Hermitian square $(\Delta^{1/2}J\Delta^{1/2})^2$ positive semidefinite; its trace equals
\[
0\le\tr(\Delta J\Delta J)=\|p_e\|_2^2+\|q_e\|_2^2-2\|r_e\|_2^2.
\]
Combining with the two preceding bounds,
\[
2\kappa^2\|g\|_2^2\le2\|r_e\|_2^2\le\|p_e\|_2^2+\|q_e\|_2^2\le8\|g\|_2^2,
\]
so $\kappa\le2$. This closes both theorem statements: the density theorem supplies the representation and $\|v[A]\|\le\kappa\max_{\overline\Omega}\|v\|$, while $\kappa\le2$ follows from the extremal-to-comparison chain.


\medskip
\textbf{Matrices.}  For $A\in M_n(\C)$, $K=W(A)$ is compact (continuous image of the unit sphere) and convex. The two-vector compression reduces the general convexity claim to dimension at most two: if $z_j=x_j^*Ax_j$ for unit $x_j$ and $V=\operatorname{span}(x_1,x_2)$, then $A_V=P_VA|_V$ satisfies $z_j\in W(A_V)\subset W(A)$. In dimension two, rank-one projections are
\[
\pi(u,v,w)=\tfrac12\begin{pmatrix}1+u&v-iw\\ v+iw&1-u\end{pmatrix},\quad u^2+v^2+w^2=1,
\]
so the map $(u,v,w)\mapsto\tr(A_V\pi(u,v,w))$ is real-affine into $\C\simeq\R^2$. For rank-2 affine $g\colon\R^3\to\R^2$, $g(S^2)=g(B^3)$ because every line fiber of $g$ through a point of the solid ball meets the sphere; the image is a filled ellipse and is convex. Dimension one is trivial.

\url{https://doi.org/10.1090/S0002-9904-1918-01447-8}

\url{https://link.springer.com/article/10.1007/BF01202959}

\begin{corollary}[sharp polynomial bound]
For all $n,m\ge1$, $A\in M_n(\C)$, $d\ge0$, and $B_0,\dots,B_d\in M_m(\C)$, the polynomial $P(z)=\sum_{j=0}^dB_jz^j$ satisfies
\[
\|P[A]\|=\Bigl\|\sum_{j=0}^dA^j\otimes B_j\Bigr\|\le2\max_{W(A)}\|P\|,
\]
and the constant is optimal.
\end{corollary}
\begin{proof}
For a nonempty compact convex $K$ and $Z\sim N(0,I_2)$ with $c=\mathbb E|Z|$, the distance $d_K$ is convex and $1$-Lipschitz, so $|\mathbb E\,d_K(x-\delta Z)-d_K(x)|\le c\delta$. For $\delta>0$ small, $h_\delta(x)=\mathbb E\,d_K(x-\delta Z)+\delta|x|^2$ is real-analytic (complexify $x$; the integral is dominated by Gaussian decay), strictly convex with Hessian $\succeq2\delta I$, and proper. Choose $\delta>0$ with $c\delta+\delta R_K^2<\epsilon/2$ and $c\delta<\epsilon/2$. For $x\in K$, $h_\delta(x)\le c\delta+\delta R_K^2<\epsilon/2$; for $d_K(x)\ge\epsilon$, $h_\delta(x)\ge\epsilon-c\delta>\epsilon/2$. Hence $\Omega_\epsilon=\{h_\delta<\epsilon/2\}$ is admissible with $K\subset\Omega_\epsilon$ and $\overline\Omega_\epsilon\subset\{d_K<\epsilon\}$.

The two theorems applied at $\Omega_\epsilon$ (with $\kappa\le2$) give $\|P[A]\|\le2\max_{\overline\Omega_\epsilon}\|P\|$. Uniform continuity of $P$ on a fixed compact neighborhood of $K$ makes the right side tend to $2\max_K\|P\|$ as $\epsilon\downarrow0$.

Optimality: $A_0=\binom{0\ 2}{0\ 0}$ has $\|A_0\|=2$, while for $x=(x_1,x_2)$ with $|x|^2=1$, $|x^*A_0x|=2|x_1x_2|\le|x_1|^2+|x_2|^2=1$ with equality at $x_1=x_2=1/\sqrt2$. Thus $W(A_0)=\{|z|\le1\}$ and $P(z)=z$ gives $2=2\max_{W(A_0)}|z|$.
\end{proof}

\medskip
\textbf{Bounded operators.}  Let $A$ be a bounded operator on a complex Hilbert space $\mathcal H$ (with no separability assumption) and $K=\overline{W(A)}$. Compactness follows from $\|W(A)\|\le\|A\|$; convexity follows from the two-vector compression, which adapts verbatim to any Hilbert space. Spectral containment $\spec(A)\subset K$: if $\lambda\notin K$ and $\delta=\dist(\lambda,K)>0$, Cauchy-Schwarz gives $\|(\lambda I-A)x\|\ge\delta\|x\|$ and the same with $A^*$ since $W(A^*)=\{\bar z\colon z\in W(A)\}$; the first makes $\lambda I-A$ injective with closed range, the second makes its range dense.

For $\xi\in\mathcal H\otimes\C^m$, write $\xi=\sum_{j=1}^m\xi_j\otimes e_j$, set $E=\operatorname{span}\{A^k\xi_j\colon0\le k\le d\}$, and $A_E=Q_EA|_E$. Induction gives $A_E^k\xi_j=A^k\xi_j$ for $0\le k\le d$, hence $P[A]\xi=P[A_E]\xi$ as vectors in $E\otimes\C^m$. For unit $x\in E$, $\langle x,A_Ex\rangle=\langle x,Ax\rangle$, so $W(A_E)\subset W(A)$. The matrix bound at every finite size gives
\[
\|P[A]\xi\|\le2\max_{W(A_E)}\|P\|\,\|\xi\|\le2\max_K\|P\|\,\|\xi\|,
\]
and the supremum over unit $\xi$ completes the polynomial inequality.

For a matrix-valued holomorphic $F$ on a neighborhood $U$ of $K$, choose $\epsilon>0$ with $K_\epsilon=\{z\colon\dist(z,K)\le\epsilon\}\subset U$; the complement of $K_\epsilon$ is connected, so Runge approximation gives matrix polynomials $P_\ell\to F$ uniformly on $K_\epsilon$. The Gaussian step supplies admissible $\Omega_\epsilon$ with $K\subset\Omega_\epsilon$ and $\overline\Omega_\epsilon\subset K_\epsilon$. Using the entrywise holomorphic functional calculus $F[A]=(2\pi i)^{-1}\int_\Gamma(zI-A)^{-1}\otimes F(z)\,dz$ on the positively oriented $\Gamma=\partial\Omega_\epsilon$,
\[
\|F[A]-P_\ell[A]\|\le\frac{|\Gamma|}{2\pi}\max_{z\in\Gamma}\|(zI-A)^{-1}\|\max_{z\in\Gamma}\|F(z)-P_\ell(z)\|\to0.
\]
Passing to the limit in the polynomial bound gives $\|F[A]\|\le2\max_K\|F\|$. Rational functions with poles outside $K$ are holomorphic on a neighborhood excluding those poles; the resolvent identity identifies the contour evaluation of $(z-a)^{-1}$ with $(A-aI)^{-1}$ for $a\notin\overline\Omega$, and differentiation plus partial fractions handles the general rational case. The example $A_0=\binom{0\ 2}{0\ 0}$ with $F(z)=z$ shows the constant is optimal.

\url{https://link.springer.com/chapter/10.1007/BF02400416}


\medskip
\textbf{Conformal coordinates.}  The Riemann-mapping theorem, Schwarz reflection, and a collar patching yield both conformal maps to and from $\D$ used above. Let $U\subset\C$ be a bounded simply connected domain and $z_*\in U$. The family of injective holomorphic $\phi\colon U\to\D$ with $\phi(z_*)=0$ is nonempty (small multiples of $z-z_*$ belong). Its supremum $\alpha=\sup|\phi'(z_*)|$ is positive and finite: a closed disk $B(z_*,r)\subset U$ gives $|\phi'(z_*)|\le1/r$ by Cauchy's derivative estimate, so $\alpha\ge|\phi'_{\text{small}}(z_*)|>0$.

An extremizing sequence $\phi_j$ converges locally uniformly on a subsequence to holomorphic $\phi$ by Cauchy's estimates and diagonal compactness. Then $|\phi|\le1$ with $|\phi'(z_*)|=\alpha>0$, so $\phi$ is nonconstant and $\phi(U)\subset\D$ by the maximum principle; injectivity follows from the argument principle (if $\phi(a)=\phi(b)$ for distinct $a,b$, disjoint neighborhoods $U_a$, $U_b$ force $\phi_j$ to fail injectivity for large $j$).

Surjectivity: suppose $w\in\D\setminus\phi(U)$. Then $w\ne0$. The function $q(z)=(\phi(z)-w)/(1-\bar w\phi(z))$ is holomorphic, nonvanishing, and $\D$-valued on the simply connected $U$; it has a holomorphic square root $h$ with $|h|<1$. Set $\beta=h(z_*)$ (so $|\beta|=\sqrt{|w|}$) and $\psi(z)=(h(z)-\beta)/(1-\bar\beta h(z))$. Then $\psi\colon U\to\D$ is injective holomorphic with $\psi(z_*)=0$, and
\[
|\psi'(z_*)|=\frac{(1-|w|^2)\,|\phi'(z_*)|}{2\sqrt{|w|}\,(1-|w|)}=\frac{1+|w|}{2\sqrt{|w|}}|\phi'(z_*)|>\alpha,
\]
contradicting maximality of $\alpha$. So $\phi$ is a conformal bijection.

\url{https://link.springer.com/book/10.1007/978-1-4612-6938-8}

\textbf{Boundary reflection.}  Suppose $\partial U$ is regular real-analytic Jordan. For $0<r<1$, $\phi^{-1}(r\overline\D)$ is compact in $U$ (conformal inverse continuous on $\D$), so $|\phi(z)|\to1$ as $z\to\partial U$ from inside.

Fix $p\in\partial U$. A regular real-analytic parametrization extends holomorphically with nonzero derivative; its holomorphic inverse straightens the boundary, giving a chart $\chi$ sending $0$ to $p$, a real diameter to the boundary arc, and the upper half of a small disk to the corresponding piece of $U$. Choose the neighborhood disjoint from $z_*$. On the upper half disk the function $u(\xi)=\log|\phi(\chi(\xi))|$ is harmonic, strictly negative, and extends continuously with $u=0$ on the real diameter.

Harmonic reflection of $u$ (odd in the real axis) gives a full-disk harmonic $\widehat u$ agreeing with $u$ on the upper half (uniqueness of the Dirichlet problem on the half disk). Its first nonvanishing homogeneous term at $0$ has the form $c\operatorname{Im}(\xi^k)$ (harmonic, vanishing on the diameter); if $k\ge2$, $\sin(k\theta)$ has both signs on $(0,\pi)$ and two small rays give two signs for $u$, contradiction. So $k=1$ and $\nabla\widehat u(0)\ne0$.

A harmonic conjugate $\widehat v$ makes $\Phi=\exp(\widehat u+i\widehat v)$ holomorphic, nonvanishing, nonzero derivative at $0$. On the upper half disk $|\Phi|=e^{\widehat u}=e^u=|\phi\circ\chi|$, so $\Phi/(\phi\circ\chi)$ is holomorphic of constant modulus one, hence constant $C$ with $|C|=1$; therefore $\phi\circ\chi=\Phi/C$ extends holomorphically across the real diameter with nonzero derivative.

\textbf{Collar patching.}  For each $p\in\partial U$ the local extension is holomorphic on a disk $U_p=B(p,R_p)$. Choose $p_1,\dots,p_N$ and radii $R_j=R_{p_j}$ such that the smaller disks $B(p_j,R_j/3)$ cover $\partial U$. Let $\phi_j\colon U_j=B(p_j,R_j)\to\C$ be the local extensions. If $V_i\cap V_j\ne\emptyset$ and $R_i\ge R_j$, then $|p_i-p_j|<(R_i+R_j)/3\le2R_i/3<R_i$, so $U_i\cap U_j$ is a nonempty open lens meeting $U$ in an open set; since both $\phi_i,\phi_j$ agree with $\phi$ there, the identity theorem extends the agreement to all of $U_i\cap U_j$. The result is a global holomorphic extension $\Phi$ on a neighborhood of $\overline U$.

The boundary image is a local diffeomorphism onto $\T$ (nonzero derivative, extended continuously), hence open in $\T$; being nonempty and compact it is all of $\T$. Injectivity on $\overline U$: interior injectivity comes from $\phi$; if two distinct boundary points map to the same $\eta\in\T$, two disjoint local inverses restricted to a small ball around $\eta$ give two preimages of nearby $\eta'\in\D$ in $U$, contradicting interior injectivity; interior-boundary collisions are excluded because interior values lie in $\D$ and boundary values in $\T$. Injectivity on a neighborhood: distinct $z_j,w_j\to\overline U$ with $\Phi(z_j)=\Phi(w_j)$ would have subsequential limits $z,w\in\overline U$ with $\Phi(z)=\Phi(w)$, so $z=w$, and $z_j,w_j$ eventually lie in a single local inverse neighborhood, contradiction. Therefore $\Phi|_{W_1}$ is injective for some open $W_1\supset\overline U$; the image is open and contains $\overline\D$; its inverse is holomorphic on a neighborhood of $\overline\D$.

\textbf{Exterior via inversion.}  Fix $z_0\in\Omega$, put $\iota(z)=1/(z-z_0)$ with $\iota(\infty)=0$. The image $U_{\mathrm{ext}}=\iota(\C\setminus\overline\Omega)$ is a bounded domain containing zero. Convexity of $\Omega$ and $z_0\in\Omega$ make each ray from $z_0$ meet $\partial\Omega$ once at a positive distance $R(\theta)$, so
\[
U_{\mathrm{ext}}=\{0\}\cup\{s e^{-i\theta}:0<s<R(\theta)^{-1}\},
\]
star-shaped about zero, hence simply connected. Inversion preserves the regular real-analytic Jordan boundary (its pole $z_0$ is interior to $\Omega$).

The preceding steps give a conformal $\phi_{\mathrm{ext}}\colon U_{\mathrm{ext}}\to\D$ with $\phi_{\mathrm{ext}}(0)=0$; its inverse $h$ extends univalently to $|\zeta|<\rho$ for some $\rho>1$, with $h(0)=0$ simple. Then
\[
G(t)=z_0+\frac{1}{h(1/t)},\qquad |t|>r_0:=\rho^{-1},
\]
is univalent (univalence of $h^{-1}$ and $\iota$), maps $\{|t|>1\}$ onto $\C\setminus\overline\Omega$, and extends as a sphere map to $\{|t|>r_0\}\cup\{\infty\}$ with a simple pole at $\infty$; its Laurent expansion is $G(t)=at+b+O(t^{-1})$ with $a=1/h'(0)\ne0$.

\textbf{Orientation.}  For $t=e^{i\theta}$, radial and angular derivatives of $G$ at $r=1$ are $tG'(t)$ and $itG'(t)$, nonzero and perpendicular; increasing $r$ enters the exterior of $\overline\Omega$, so $tG'(t)$ is outward normal and $itG'(t)$ is positive boundary tangent. The interior conformal map $f$ preserves boundary orientation, so $f\circ G\colon\T\to\T$ is a smooth orientation-preserving diffeomorphism.

\url{https://link.springer.com/book/10.1007/978-1-4612-6938-8}

\url{https://link.springer.com/book/10.1007/978-3-642-85408-1}

Together with the polynomial corollary, the operator corollary, and the sharpness example, the construction of both conformal coordinates above completes what is needed.

\end{document}