# Scalar Crouzeix inequality — 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.