# Scalar Crouzeix inequality OA-1-H2-to-H3

## Setting

You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.

## Problem

H2: Let \(n\ge1\), let \(S\in M_n(\mathbb C)\) be invertible, and set
\[
B=S\operatorname{diag}(\beta_1,\ldots,\beta_n)S^{-1},\qquad
T=S\operatorname{diag}(\lambda_1,\ldots,\lambda_n)S^{-1},
\]
where the \(\beta_i\) are pairwise distinct and \(|\lambda_i|\le1\); the \(\lambda_i\) need not be distinct.
Suppose \(H:\{w\in\mathbb C:|w|<1\}\to M_n(\mathbb C)\) is analytic, \(H(0)=I\), \(\operatorname{Re}H(w)\succeq0\), and
\[
H(w)-(I-wT)^{-1}\in\operatorname{alg}(B^*)\qquad(|w|<1).
\]

Then \(\|T\|\le2\).

Here \(\operatorname{Re}X=(X+X^*)/2\), \(X\succeq0\) means Hermitian positive semidefinite, \(\operatorname{alg}(B^*)\) is the unital polynomial algebra generated by \(B^*\), and \(\|T\|\) is its Euclidean operator norm.

Prove H3: Let \(\Omega\subset\mathbb C\) be a bounded convex domain with \(C^2\) boundary, and let \(\gamma:[0,L]\to\partial\Omega\) traverse the boundary once counterclockwise at unit speed. Put \(\nu(s)=-\mathrm i\gamma'(s)\).
Let \(B\in M_n(\mathbb C)\) satisfy
\[
W(B)=\{x^*Bx:x\in\mathbb C^n,\ \|x\|_2=1\}\subset\Omega,
\]
and define
\[
R_s=(\gamma(s)I-B)^{-1},\qquad
D_B(s)=\frac1{2\pi}\left(\nu(s)R_s+\overline{\nu(s)}R_s^*\right).
\]
The map
\[
\Phi:C(\partial\Omega)\to M_n(\mathbb C),\qquad
\Phi(h)=\frac12\int_0^L h(\gamma(s))D_B(s)\,ds
\]
is positive and unital: \(h\ge0\) implies \(\Phi(h)\) is Hermitian positive semidefinite, and \(\Phi(1)=I\).

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.