Results on the numerical range, exterior coordinates and Faber polynomials: the Toeplitz–Hausdorff theorem, outer approximations, the extended Riemann map, the positive kernel, the coefficient estimate, the resolvent and the theorem, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• The complete Crouzeix inequality (OpenAI) — Theorem A. For all n, m, every A in M_n(C) and every polynomial P with coefficients in M_m(C), ||P[A]|| <= 2 max over W(A) of ||P(z)||, where P[A] = sum A^k (x) B_k; the constant 2 is sharp. Proved in the last lesson.
• Toeplitz–Hausdorff theorem — Theorem 1.1. The numerical range W(A) = {u^*Au : ||u|| = 1} of a complex square matrix is convex; it is compact and contains the spectrum.
• Outer approximation by sublevel sets (OpenAI) — Lemma 2.1. For K compact convex and 0 < delta < 1, Omega_delta = {Phi_delta < 1} is a bounded convex domain containing K with closure {Phi_delta <= 1}, every point of which lies within delta(1 + diam K)/(1 - delta) of K.
• Analytic polar boundary — Proposition 2.3. Omega_delta is star-shaped about each of its points with a polar radius function that extends holomorphically to a strip; it is a smooth convex domain.
• The Riemann map across an analytic polar boundary — Theorem 3.5. If U is a bounded domain, star-shaped about 0 with analytic polar boundary, the Riemann map g of the unit disc onto U with g(0) = 0 extends to an injective holomorphic map on a disc of radius R_0 > 1, with nonvanishing derivative, mapping the unit circle onto the boundary of U.
• Exterior coordinate of a smooth convex domain — Theorem 3.7. There are 0 < r_0 < 1, c > 0 and an injective h(lambda) = c lambda + kappa(1/lambda) on |lambda| > r_0, kappa holomorphic on |u| < 1/r_0, mapping |lambda| > 1 onto the exterior and the unit circle onto the boundary, with Re(conj(q(lambda))(h(lambda) - z)) >= 0 for |lambda| = 1 and z in the closed domain, q = lambda h'.
• Faber polynomials — Proposition 4.1. q(lambda)/(h(lambda) - z) = sum b_k(z) lambda^(-k) defines polynomials b_k of degree k with leading coefficient c^(-k), given by an explicit recursion; they form a basis of the polynomials.
• Separately holomorphic functions on a bidisc — Lemma 4.2. A continuous function on a bidisc that is holomorphic in each variable is a double power series, with Cauchy estimates for its coefficients.
• Mixed coefficients (OpenAI) — Theorem 4.3. s(lambda, mu) = q(lambda)/(h(lambda) - h(mu)) - lambda/(lambda - mu) = sum s_kj lambda^(-k) mu^(-j) with |s_kj| <= C_rho rho^(k+j) for r_0 < rho < 1, and b_k(h(mu)) = mu^k + sum_j s_kj mu^(-j).
• The positive kernel (OpenAI) — Theorem 4.4. K = 1 + s + conj(s) is continuous and nonnegative on the torus, and both of its marginals equal one.
• The coefficient estimate (OpenAI) — Proposition 4.6. For G = sum b_k C_k with C_k in M_m(C), sum ||C_k||_2^2 <= the integral over the circle of ||G(h(mu))||_2^2 <= ||C_0||_2^2 + 2 sum_{k >= 1} ||C_k||_2^2.
• The resolvent and its positive Hermitian part (OpenAI) — Theorem 5.1. If W(A) lies in the domain, R(lambda) = q(lambda)(h(lambda)I - A)^(-1) = sum b_k(A) lambda^(-k) on the unit circle, with ||b_k(A)|| <= C theta^k for some theta < 1, and R + R^* is positive semidefinite there.
• The bound on a smooth convex domain (OpenAI) — Theorem B. If W(A) lies in a smooth convex domain Omega and F has coefficients in M_m(C) with ||F(z)|| <= 1 on the boundary of Omega, then ||F[A]|| <= 2.

Reading order

Read the lessons in order. The first lesson is elementary; the second and third are the geometric and complex-analytic preparation and can be read on their own; the fourth uses the coordinate of the third, and the last uses all of them.