The numerical range of a square complex matrix A in M_n(C) is
W(A) = {u* A u : u in C^n, u* u = 1}. It is compact and convex
(Toeplitz 1918; Hausdorff 1919), and it contains the spectrum of A,
since a unit eigenvector u with A u = lambda u satisfies
u* A u = lambda u* u = lambda.

https://eudml.org/doc/168868
https://eudml.org/doc/168874

For a normal matrix A = U Lambda U* with U unitary and Lambda diagonal,
the quantity u* A u is a convex combination of the eigenvalues, so W(A)
coincides with the convex hull of the spectrum. For a nonnormal matrix
W(A) is strictly larger. The 2-by-2 nilpotent A = [[0,2],[0,0]] has
spectrum {0}, but 2|x1 x2| <= |x1|^2 + |x2|^2 = 1 for every unit vector
(x1, x2) in C^2, so W(A) is the closed unit disk.

Polynomial functional calculus asks for a constant C such that
||P(A)|| <= C max_{z in W(A)} |P(z)|, uniformly over polynomials P and
matrices A. Crouzeix (2004) proved the constant-2 bound for 2-by-2
matrices and conjectured it in general; Crouzeix (2007) obtained a
universal constant 11.08. The scalar case (coefficient dimension 1)
was settled at 2 in 2026 by Jin, by Lorist and Schwenninger, and by
Luo, each with a different certificate. Jin records the underlying
device:

   "The central ingredient is a mass-parameterized positive-real
    completion theorem relative to an auxiliary eigenbasis."

https://doi.org/10.1007/s00020-002-1188-6
https://doi.org/10.1016/j.jfa.2006.10.013
https://www.preprints.org/manuscript/202607.1919/v4
https://www.preprints.org/manuscript/202608.1661/v2
refs/lorist-schwenninger-2026.pdf

The conjectured constant was verified case by case beforehand: for
perturbed Jordan blocks by Greenbaum and Choi (2012), for cyclic
weighted shifts by Choi (2013), and for tridiagonal 3-by-3 matrices
whose elliptical numerical range is centred at an eigenvalue by
Glader, Kurula, and Lindstrom (2018). The survey of Bickel, Gorkin,
Greenbaum, Ransford, Schwenninger, and Wegert (2020) catalogues these
and many later intermediate cases.

https://doi.org/10.1016/j.laa.2011.12.005
https://doi.org/10.1016/j.laa.2012.12.045
https://doi.org/10.1137/17M1110663
https://doi.org/10.1007/s40315-020-00350-9

Positive boundary representations of the numerical range were
developed by Delyon and Delyon (1999). Crouzeix and Palencia (2017)
combined those with the Cauchy transform of conjugate boundary data to
obtain the general bound 1 + sqrt(2), and Ransford and Schwenninger
(2018) simplified the final norm estimate.

refs/delyon-delyon-1999.pdf
refs/crouzeix-palencia-2017.pdf
https://doi.org/10.1137/17M1116672
refs/ransford-schwenninger-2018.pdf
https://doi.org/10.1137/17M1143757

The complete Crouzeix inequality permits matrix coefficients: for
P(z) = sum_{k=0}^d B_k z^k with B_k in M_m(C), the operator
P[A] = sum_{k=0}^d A^k @ B_k in M_n(C) @ M_m(C), acting on
C^n @ C^m, satisfies

   ||P[A]|| <= 2 max_{z in W(A)} ||P(z)||,

uniformly in the base dimension n, coefficient dimension m, and degree
d. The constant 2 is optimal: for A = [[0,2],[0,0]] and P(z) = z,
||P[A]|| = ||A|| = 2 while max_{W(A)} ||P(z)|| = 1.

The complete case is harder than the scalar case because coefficient
matrices B_i and B_j need not commute: F(z) G(z) and G(z) F(z) can
differ, and F[A] G[A] need not equal G[A] F[A]. A complete proof cannot
swap one product for the other without accounting for the error.

For n = 2 the complete inequality at constant 2 was proved by Badea,
Crouzeix, and Delyon (2006) via a similarity of condition number at
most 2 built from the numerical-range ellipse. For n = 3 the complete
inequality at constant 2 was proved by Ahag, Czyz, and Virtanen (2026).

For weighted shift matrices the complete inequality at constant 2 was
proved by Crouzeix and Greenbaum (2025), who also obtain the completely
bounded form of the inequality.

https://doi.org/10.1007/s00209-005-0857-y
refs/ahag-czyz-virtanen-2026.pdf
refs/crouzeix-greenbaum-2025.pdf

The exterior coefficient method traces back to Beckermann (2005), who
connected Faber polynomials to GMRES convergence and the numerical
range, and to Beckermann and Crouzeix (2014), who extended Faber
polynomial bounds beyond convexity:

   "It has been recently shown that ||Fn(A)|| <= 2, where A is a linear
    continuous operator acting in a Hilbert space, and Fn is the Faber
    polynomial of degree n corresponding to some convex compact
    E subset of C containing the numerical range of A."

Beckermann and Stylianopoulos (2018) related the mixed coefficients of
exterior maps to the Grunsky matrix and derived weighted coefficient
inequalities.

https://doi.org/10.1016/j.crma.2005.04.027
refs/beckermann-crouzeix-2014.pdf
https://doi.org/10.1007/s00365-017-9381-7

Fix a bounded convex open Omega in C with a regular real-analytic
Jordan boundary. Every nonempty compact convex set, in particular W(A),
is a limit in Hausdorff distance of such domains.

By the exterior Riemann mapping, a conformal bijection
h : {|lambda| > 1} -> C \ closure(Omega) exists, extends to a
homeomorphism of closures, and has a Laurent expansion at infinity

   h(lambda) = c lambda + c_0 + sum_{j>=1} c_j lambda^{-j},  c != 0.

The Ahlfors text covers existence and boundary regularity (Chapters
6.1-6.2) and the Schwarz reflection step (Chapters 4.6, 6.3-6.4) used
to continue h across the unit circle and to reach an exterior collar
of some radius r_0 < 1.

https://archive.org/details/complexanalysis0000ahlf

Put q(lambda) = lambda h'(lambda). At lambda = exp(i theta) in
T = {|lambda| = 1}, the vector q(lambda) is an outward normal at
h(lambda) in the boundary of Omega. Since Omega is convex, the
supporting line through h(lambda) with normal q(lambda) has all of
closure(Omega) on its inner side:

   Re( conj(q(lambda)) (h(lambda) - z) ) >= 0   for z in closure(Omega), lambda in T.

The vector i q(lambda) is a positively oriented tangent at h(lambda).

The Faber polynomials b_k(z), k >= 0, of Omega are the polynomials
defined by the expansion at infinity

   q(lambda) / (h(lambda) - z) = sum_{k>=0} b_k(z) lambda^{-k}.

Factoring c lambda from the denominator gives b_0 = 1 and, for
k >= 1, a polynomial b_k of degree k with leading coefficient c^{-k}.
So (b_k)_{k>=0} is a basis of C[z], and polynomial degrees match
indices.

https://doi.org/10.1016/j.crma.2005.04.027
refs/beckermann-crouzeix-2014.pdf

On the unit circle, each Faber polynomial has a clean Fourier form.
There are scalars s_kj, k, j >= 1, with

   b_k(h(mu)) = mu^k + sum_{j>=1} s_kj mu^{-j}   for k >= 1.

Equivalently, the mixed correction kernel

   s(lambda, mu) = q(lambda) / (h(lambda) - h(mu)) - lambda / (lambda - mu)

has a Laurent expansion s(lambda, mu) = sum_{k,j>=1} s_kj lambda^{-k} mu^{-j}
that converges normally on the exterior product collar
|lambda|, |mu| > r_0 for some r_0 < 1. Cauchy estimates on the
reciprocal bidisk of radius rho^{-1} give |s_kj| <= C_rho rho^{k+j}
for every r_0 < rho < 1. Since every term of s and of conj(s) has
strictly mixed modes, both functions have zero integral against
dtau(lambda) at fixed mu and against dtau(mu) at fixed lambda.

https://doi.org/10.1007/s00365-017-9381-7

The kernel that drives the argument lives on T x T:

   K(lambda, mu) = 1 + s(lambda, mu) + conj(s(lambda, mu)).

For distinct lambda, mu in T, using conj(lambda) = 1/lambda and
conj(mu) = 1/mu,

   lambda/(lambda - mu) + conj(lambda)/(conj(lambda) - conj(mu)) = 1,

so K(lambda, mu) = 2 Re( q(lambda) / (h(lambda) - h(mu)) ).

Let W(lambda, mu) = conj(q(lambda)) (h(lambda) - h(mu)). The
supporting-line inequality with z = h(mu) gives Re(W) >= 0. Since
q(lambda) is nonzero, writing q/(h - h(mu)) = |q|^2 / W gives

   Re( q/(h - h(mu)) ) = |q|^2 Re(W) / |W|^2 >= 0.

Hence K >= 0 off the diagonal, and by continuity also on the
diagonal. In particular K is continuous and nonnegative on T x T.

https://doi.org/10.1137/17M1116672
refs/delyon-delyon-1999.pdf

The two marginal identities

   int_T K(lambda, mu) dtau(lambda) = 1   for fixed mu,
   int_T K(lambda, mu) dtau(mu)   = 1   for fixed lambda,

follow from the vanishing boundary integrals of s and conj(s).

The two-sided coefficient estimate comes from the operator (Tu)(mu) =
int_T K(lambda, mu) u(lambda) dtau(lambda) on L^2(T; M_m(C)).

For each fixed mu, Cauchy-Schwarz with respect to the probability
measure K(lambda, mu) dtau(lambda) and the marginal identity in
lambda gives

   ||(Tu)(mu)||_2^2 <= int_T K(lambda, mu) ||u(lambda)||_2^2 dtau(lambda).

Integrating in mu and using the other marginal identity yields
||Tu||_2^2 <= ||u||_2^2: the operator T is an L^2 contraction.

Apply T to u(lambda) = sum_{k=1}^N C_k lambda^k with C_k in M_m(C).
The Fourier form of K splits the integral into the 1 part (zero mode
integral) and the s and conj(s) parts (only strictly mixed modes).
Only the s part contributes, giving

   (Tu)(mu) = sum_{j>=1} ( sum_{k=1}^N s_kj C_k ) mu^{-j}.

Normal convergence of the double series for s on the exterior product
collar justifies the interchange. Orthonormality of the modes mu^{-j} gives
||Tu||_2^2 = sum_{j>=1} || sum_{k=1}^N s_kj C_k ||_2^2,
and the contraction inequality reads

   sum_{j>=1} || sum_{k=1}^N s_kj C_k ||_2^2 <= sum_{k=1}^N ||C_k||_2^2.

Now let G(z) = sum_{k=0}^N b_k(z) C_k be an M_m(C)-valued polynomial
in the Faber basis of Omega. On the unit circle the boundary value
splits into three pieces with disjoint Fourier supports:

   G(h(mu)) = C_0 + sum_{k=1}^N C_k mu^k + sum_{j>=1} (sum_{k=1}^N s_kj C_k) mu^{-j}.

Parseval and the contraction inequality then give the two-sided bound

   sum_{k=0}^N ||C_k||_2^2 <= int_T ||G(h(mu))||_2^2 dtau(mu) <= ||C_0||_2^2 + 2 sum_{k=1}^N ||C_k||_2^2.

In words: passing from coefficients to boundary values can enlarge the
coefficient Hilbert-Schmidt norm by a factor of at most 2, and never
shrinks it. The factor of 2 on the right is the only source of the
constant 2 in the main inequality.

The exterior resolvent of A relative to h and Omega is defined on the
unit circle by

   R(lambda) = q(lambda) ( h(lambda) I - A )^{-1}  in M_n(C).

Since the spectrum of A is contained in W(A) subset Omega, the
resolvent is defined and continuous on and outside T, with Laurent
expansion at infinity

   R(lambda) = sum_{k>=0} b_k(A) lambda^{-k},

where b_k(A) is the same Faber polynomial evaluated entrywise at A.
Cauchy estimates on a larger annulus give geometric decay
||b_k(A)|| <= C theta^k for some 0 < theta < 1, so the series
converges absolutely and uniformly on T. Taking constant terms gives
int_T R(lambda) dtau(lambda) = I.

https://doi.org/10.1016/j.jfa.2006.10.013

The matrix field R + R* is positive semidefinite on T. Write
B(lambda) = h(lambda) I - A so that R = q B^{-1} and

   R + R* = (B*)^{-1} ( conj(q) B + q B* ) B^{-1}.

Both outer factors are adjoint congruence matrices, so positivity of
R + R* follows from positivity of conj(q) B + q B*. For any unit v in
C^n the quadratic form of the latter at v is

   2 Re( conj(q(lambda)) (h(lambda) - v* A v) ),

and v* A v lies in W(A) subset Omega, so the supporting-line
inequality applies. Positivity of R + R* on T follows.

https://doi.org/10.1137/17M1116672
refs/delyon-delyon-1999.pdf

Now the main estimate. Let F be an M_m(C)-valued polynomial with
||F(z)|| <= 1 on the boundary of Omega. The goal is ||F[A]|| <= 2.

Put gamma = ||F[A]||. If gamma = 0 the estimate is trivial, so suppose
gamma > 0. Choose unit vectors x, y in C^n @ C^m with

   F[A] x = gamma y,  F[A]* y = gamma x.

Write x = sum_{i=1}^m X e_i @ e_i and y = sum_{i=1}^m Y e_i @ e_i
for n-by-m matrices X, Y, where (e_i) is the standard basis of C^m.
The unit conditions read ||X||_2 = ||Y||_2 = 1.

Define four M_m(C)-valued fields on T:

   P = X* R X,  Q = Y* R Y,  M = Y* R X,  L = X* R Y,

with coefficient matrices M_k, P_k, Q_k, L_k in M_m(C) at lambda^{-k}.
Since b_0(A) = I,

   P_0 = X* X,  Q_0 = Y* Y,  L_0* = M_0.

Set a^2 = sum_{k>=0} ||M_k||_2^2, finite by geometric decay.

For any M_m(C)-valued polynomial G(z) = sum b_k(z) C_k, the pairing
between the coefficient array (M_k) and the evaluation y* G[A] x is

   y* G[A] x = sum_{k,i,j} (M_k)_ij (C_k)_ij.

Indeed, G[A] x = sum_{k,j} (b_k(A) X e_j) @ (C_k e_j), and pairing
with y = sum Y e_i @ e_i telescopes. Cauchy-Schwarz on the entry
pairing gives

   |y* G[A] x| <= a ( sum_k ||C_k||_2^2 )^{1/2}.  (cs)

Applying (cs) to G = F and using y* F[A] x = gamma gives a > 0.

Apply (cs) to the two products FG and GF. Matrix-polynomial
evaluation respects the order of multiplication:
(FG)[A] = F[A] G[A] and (GF)[A] = G[A] F[A]. Paired with y and x, and
using the singular-vector relations,

   y* (FG)[A] x = gamma x* G[A] x,
   y* (GF)[A] x = gamma y* G[A] y.

Define the boundary norm ||H||_bdry^2 = int_T ||H(h(mu))||_2^2 dtau(mu),
where the inner norm is Hilbert-Schmidt. Pointwise
submultiplicativity of the Hilbert-Schmidt norm over the operator norm
gives, since ||F(z)|| <= 1 on the boundary,

   ||FG||_bdry <= ||G||_bdry,   ||GF||_bdry <= ||G||_bdry.  (mult)

Combine (cs), the lower coefficient estimate, and (mult):

   gamma |x* G[A] x| <= a ||G||_bdry,
   gamma |y* G[A] y| <= a ||G||_bdry.  (diag)

Take the coefficient array C_0 = conj(P_0) and
C_k = conj(P_k)/2 (bars entrywise conjugation). The paired value of
x* G[A] x is d = ||P_0||_2^2 + (1/2) sum_{k>=1} ||P_k||_2^2. The upper
coefficient estimate bounds ||G||_bdry by sqrt(d). Substituting into
(diag),

   d <= (a / gamma) sqrt(d).

The case d = 0 is trivial, and otherwise this gives d <= a^2 / gamma^2.
So

   ||P_0||_2^2 + (1/2) sum_{k>=1} ||P_k||_2^2 <= a^2 / gamma^2.  (Pd)

The same argument with Q in place of P, using y* G[A] y, gives

   ||Q_0||_2^2 + (1/2) sum_{k>=1} ||Q_k||_2^2 <= a^2 / gamma^2.  (Qd)

These are the two reciprocal-weighted coefficient bounds.

Now compute the boundary norms of the three fields. By Parseval on T,
using P_0 = P_0* and the conjugate symmetry between the coefficients of
lambda^k and lambda^{-k} in P + P*,

   int_T ||P + P*||_2^2 dtau
     = 4 ||P_0||_2^2 + 2 sum_{k>=1} ||P_k||_2^2
     = 4 d
     <= 4 a^2 / gamma^2.  (Pbdry)

The same argument with Q gives

   int_T ||Q + Q*||_2^2 dtau <= 4 a^2 / gamma^2.  (Qbdry)

For the cross field M + L*, the constant coefficient is
M_0 + L_0* = 2 M_0, so

   int_T ||M + L*||_2^2 dtau
     = 4 ||M_0||_2^2 + sum_{k>=1} ||M_k||_2^2 + sum_{k>=1} ||L_k||_2^2
     >= sum_{k>=0} ||M_k||_2^2
     = a^2.  (Mbdry)

The positive block comparison closes the argument. Because
R + R* >= 0 on T, the 2m-by-2m matrix

   Delta = [[ P + P*,  L + M*], [ M + L*,  Q + Q*]]

is positive semidefinite at every lambda. Its conjugate J Delta J by
J = diag(I_m, -I_m) is also positive semidefinite. Since the trace of
a product of two positive semidefinite matrices is nonnegative,

   0 <= tr(Delta J Delta J)
     = ||P + P*||_2^2 + ||Q + Q*||_2^2 - 2 ||M + L*||_2^2.  (block)

Integrate (block) over T and use (Pbdry), (Qbdry), (Mbdry):

   0 <= 8 a^2 / gamma^2 - 2 a^2.

Dividing by 2 a^2 > 0 gives gamma^2 <= 4, so gamma <= 2.

The expected enclosure claim holds. For any
nonempty compact convex K in C and any delta in (0, 1) there is a
bounded convex open Omega_delta with regular real-analytic Jordan
boundary, containing K, and

   sup_{z in closure(Omega_delta)} dist(z, K) <= delta (1 + diam K) / (1 - delta).

Choose a finite Euclidean delta-net D_delta of unit directions
containing 1, -1, i, -i. Let s_K(nu) = max_{z in K} Re(conj(nu) z),
and choose t_delta > 0 with |D_delta| e^{-t_delta delta} < 1. Define

   Phi_delta(z) = sum_{nu in D_delta} exp( t_delta ( Re(conj(nu) z) - s_K(nu) - delta ) ),

   Omega_delta = {z in C : Phi_delta(z) < 1}.

For z in K, Re(conj(nu) z) <= s_K(nu), so each summand is at most
e^{-t_delta delta} and K sits inside Omega_delta. If Phi_delta(z) <= 1,
every summand is at most one, so Re(conj(nu) z) <= s_K(nu) + delta for
each nu in D_delta. Taking nu = 1, -1, i, -i bounds closure(Omega_delta)
inside a fixed rectangle R.

The Hessian of Phi_delta is a positive combination of nu nu^T over nu
in D_delta; the four axis directions make it positive definite, so
Phi_delta is strictly convex and real analytic. Its gradient cannot
vanish on the level {Phi_delta = 1}: a critical point of a convex
function is a global minimum, while Phi_delta < 1 on K. The implicit
function theorem makes the level a regular real-analytic curve, and
radial projection from an interior point identifies it as a Jordan
curve. Hence Omega_delta is a bounded convex open domain with a regular
real-analytic Jordan boundary.

For the distance bound, take z in closure(Omega_delta) with
d = dist(z, K) > 0, let w be nearest to z in K, put nu = (z - w)/d,
choose mu in D_delta with |mu - nu| <= delta, and let v in K attain
s_K(mu). Minimality of w gives Re(conj(nu)(v - w)) <= 0. Since
Phi_delta(z) <= 1 forces Re(conj(mu) z) <= Re(conj(mu) v) + delta,

   (1 - delta) d <= Re(conj(mu)(z - w)) <= delta + delta |v - w| <= delta (1 + diam K).

Points of K have distance 0, so the bound covers all of closure(Omega_delta).

https://eudml.org/doc/168868
https://eudml.org/doc/168874

Take K = W(A), which is compact (image of the unit sphere under a
continuous map) and convex (Toeplitz-Hausdorff). Set Omega_j =
Omega_{delta_j} with delta_j = 1/(j+1). For each j, Omega_j contains
W(A), lies in the fixed rectangle R, and closure(Omega_j) converges to
W(A) in Hausdorff distance.

Fix an M_m(C)-valued polynomial P and set
M_j = max_{z in closure(Omega_j)} ||P(z)||. For unit x, y, the
function z -> y* P(z) x is scalar holomorphic, and
||P(z)|| = sup_{|x|=|y|=1} |y* P(z) x|, so the maximum over
closure(Omega_j) equals the maximum over the boundary. If M_j = 0 then
P vanishes on the open set Omega_j, so P = 0 and the inequality is
trivial. Otherwise, scaling P by 1/M_j makes ||P/M_j|| <= 1 on the
boundary of Omega_j, and the constant-two bound already proved for
analytic convex enclosures gives ||P[A]|| <= 2 M_j.

Because ||P|| is continuous on the fixed compact rectangle R, it is
uniformly continuous there. Containment and the convergence of
closure(Omega_j) to W(A) in Hausdorff distance then force
M_j -> max_{z in W(A)} ||P(z)||. Passing to the limit in
||P[A]|| <= 2 M_j,

   ||P[A]|| <= 2 max_{z in W(A)} ||P(z)||.

The constant 2 is optimal: A = [[0, 2], [0, 0]] with P(z) = z gives
P[A] = A, of norm 2, while max_{W(A)} |P(z)| = 1.

The exterior coordinate h admits a construction from classical
conformal mapping. Fix a bounded convex open Omega with regular
real-analytic Jordan boundary, choose z_* in Omega, and apply the
inversion z -> 1/(z - z_*) (with infinity mapped to 0). The spherical
exterior (C \ closure(Omega)) union {infinity} maps to a bounded
simply connected U containing 0: each ray from z_* meets the boundary
at one point by convexity, so the exterior portions become a
star-shaped domain about 0, and inversion preserves the regularity and
real-analyticity of the boundary.

The Riemann mapping theorem (Ahlfors, Chapter 6, Section 1.1) supplies
a conformal bijection g : D -> U with g(0) = 0. Caratheodory's
Jordan-domain theorem (Ahlfors, Chapter 6, Section 1.2) extends g to a
homeomorphism of closures.

Real-analyticity of the boundary makes Schwarz reflection applicable.
At each boundary point, the Schwarz reflection principle (Ahlfors,
Chapter 4 Section 6.5; Chapter 6 Sections 1.3-1.4) extends the
analytic arc of g holomorphically across the unit circle. Patching by the identity
theorem produces a holomorphic extension to a neighborhood of the
closed disk.

The extension stays injective on a slightly smaller neighborhood:
otherwise, two sequences of distinct points with matching images would
have subsequential limits in the closed disk that coincide by
continuity and injectivity, so eventually the points would lie in a
local injectivity neighborhood of a common limit, contradiction. By
compactness, g is univalent on {|w| < R_0} for some R_0 > 1, and

   h(lambda) = z_* + 1 / g(1/lambda)

is meromorphic on {|lambda| > r_0} union {infinity} with r_0 = 1/R_0,
has a simple pole at infinity, and maps {|lambda| > 1} conformally onto
C \ closure(Omega). Its Laurent expansion at infinity is
h(lambda) = c lambda + c_0 + O(1/lambda) with c = 1/g'(0) != 0.

Finally, q(lambda) = lambda h'(lambda) is nonzero on {|lambda| = 1}
and points along the outward normal: at lambda = e^{i theta} the
radial and angular derivatives of h are q and i q, perpendicular and
nonzero, and increasing r = |lambda| moves into the exterior of
Omega. Convexity of Omega then gives the supporting-line inequality
Re(conj(q)(h(lambda) - z)) >= 0 for every z in closure(Omega).

https://archive.org/details/complexanalysis0000ahlf
https://eudml.org/doc/168868
+command_not_found_handler:22> echo 'zsh: command not found: 77.'