The Numerical Range Is a 2-Spectral Set

For an \( n\times n \) complex matrix \( A \), let \( W(A)=\{x^*Ax : x\in\mathbb{C}^n,\ \lVert x\rVert_2=1\} \) be its numerical range. Here \( x^* \) denotes the conjugate transpose, \( \lVert\,\cdot\,\rVert_2 \) the Euclidean norm, and \( \lVert\,\cdot\,\rVert \) the induced operator norm. For every complex polynomial \( p \), we prove that \( \lVert p(A)\rVert\leq2\max_{z\in W(A)}\lvert p(z)\rvert \) and consequently that \( W(A) \) is a 2-spectral set for \( A \). This proves Crouzeix's conjecture, and the constant is optimal. The central ingredient is a positive-real completion theorem relative to an auxiliary eigenbasis: an adjoint-algebra constraint on the resolvent defect of a normalized matrix-valued Carathéodory function forces the underlying matrix to have norm at most 2. The classical positive double-layer calculus supplies such completions for \( f(B) \) whenever the auxiliary matrix \( B \) has simple spectrum; the eigenvalues of \( f(B) \) may repeat. Sampling the associated Herglotz kernel at half the conjugate diagonal entries and at the origin cancels the completion term and leaves a comparison of two weighted Gramians; a Stein identity yields the sharp estimate. Simple-spectrum approximation and convex outer approximation by domains with continuously differentiable boundary yield the general theorem.