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 \( ‖p(A)‖≤2maxz∈W(A)|p(z)| \). More generally, the same estimate holds for every function \( f \) holomorphic on a neighborhood of \( W(A) \), with \( p \) replaced by \( f \); consequently, \( W(A) \) is a 2-spectral set for \( A \). This proves Crouzeix's conjecture, and the constant is optimal. The central ingredient is a mass-parameterized positive-real completion theorem relative to an auxiliary eigenbasis: if a normalized matrix-valued Carathéodory function completes \( \frac{2}{\mathfrak m}(I-wT)^{-1} \) modulo the adjoint algebra, then \( \lVert T\rVert\leq\mathfrak m \) for \( \mathfrak m\geq2 \). The numerical-range double layer has mass two. 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 scaled conjugate diagonal entries and at the origin cancels the nonconstant completion term and leaves two ordered weighted Gramians; their first nonconstant terms yield the mass bound. Simple-spectrum approximation and canonical convex outer domains with supported oriented radial boundaries yield the general theorem. The mass formulation also provides a concrete scalar proof protocol for annular, multiply connected, operator-radius, and abstract spectral-constant problems. For a complex square matrix A, we prove the sharp polynomial Crouzeix inequality and its extension to every function f holomorphic near the numerical range of A. Consequently, the numerical range is a 2-spectral set, and the factor 2 is optimal. The proof uses a mass-parameterized positive-real completion theorem relative to an auxiliary eigenbasis. A matrix-valued Herglotz kernel is sampled so that the adjoint-algebra correction cancels, leaving ordered weighted Gramians that yield the sharp mass bound. Simple-spectrum approximation and convex outer domains give the general result. The mass formulation also supplies a reusable scalar protocol for sharp annular, operator-radius, and abstract spectral-constant problems.

Let $A\in\C^{d\times d}$ and let $W(A)$ denote its numerical range. In the convex-domain functional calculus of Delyon--Delyon and Crouzeix, a central role is played by the boundary kernel $P_\Omega(\sigma,A)=\Real\!\bigl(n_\Omega(\sigma)(\sigma\Id-A)^{-1}\bigr)$ on $\partial\Omega$, which is positive definite whenever $W(A)\subset\Omega$.We study the loss of pointwise coercivity as $\Omega\downarrow W(A)$. Along any $C^1$ convex exhaustion $\Omega_\varepsilon\downarrow W(A)$, if boundary data $(\sigma_\varepsilon,n_{\Omega_\varepsilon}(\sigma_\varepsilon))$ converge to a supporting pair $(\sigma_0,n)$ with $\sigma_0\in\partial W(A)\setminus\spec(A)$, then $\lambda_{\min}(P_{\Omega_\varepsilon}(\sigma_\varepsilon,A))\to 0$ and the near-kernel aligns with $(\sigma_0\Id-A)\mathcal M(n)$, where $\mathcal M(n)$ is the maximal eigenspace of $H(n)=\Real(\overline{n}A)$.Quantitatively, the collapse is governed by the support gap $\delta(\sigma,n)=\Real(\overline{n}\,\sigma)-\lambda_{\max}(H(n))$: under a spectral-gap hypothesis for $H(n)$ we obtain a full collapsing eigenvalue cluster with a computable slope spectrum given by an explicit Gram matrix, and show that these slopes are intrinsic after rescaling by $\delta$. This yields a rigorous face detector and explains a mechanism for ill-conditioning in boundary-integral discretizations as $\Omega$ approaches $W(A)$.At spectral support points $\sigma_0\in\spec(A)\cap\partial W(A)$ we obtain a three-scale splitting ($1/\varepsilon$ blow-up, $O(\varepsilon)$ cluster, and $O(1)$ bulk) under non-tangential offsets; for defective eigenvalues, higher-order blow-up related to Jordan structure may occur. In the normal case we give a complete description in terms of the supporting face. Numerical experiments validate the predicted slopes and splittings.

For a complex matrix \(A\), let \(\psi(A)\) be the norm of its polynomial functional calculus on the numerical range. We prove that \(\psi(A)=2\) forces the numerical range to be a nondegenerate closed disk. Equality in the sharp double-layer estimate gives a boundary-kernel identity for a finite Blaschke extremizer. A curved exposed arc turns this identity into a rational formula, and a resolvent argument identifies the full unit lemniscate with the numerical-range boundary. The associated tangent polynomial divides a Hermitian determinant. Its real roots correspond to the two support lines in a generic direction, forcing a conic; the points at infinity then make it a circle. In every fixed finite dimension at least two, we also prove that a proper convex domain has scalar domain constant two exactly when it is a disk. Compactness yields uniform near-equality rigidity up to translation and positive scaling, and a uniform gap for unbounded domains.

We study the open Euclidean-norm pseudospectra of \(A_n(a)=\operatorname{tridiag}(a,0,1)\), where \(0<a\gamma_n(a)\), where \(\gamma_n(a)\) is the maximum of \(s_{\min}(xI_n-A_n(a))\) over the real spectral interval. Every pseudospectral component is contractible. The thresholds decrease strictly with \(n\), so connectedness, once attained, persists at every larger order. The proof uses vertical monotonicity of the least singular value, a symmetric matrix pencil, and separate comparisons for noncentral and central spectral gaps. Explicit upper and lower bounds show that \(\gamma_n(a)\asymp_a{(n+1)a^{n/2}}\). Consequently, the first connected order equals \(2[\log(1/\varepsilon)+\log\log(1/\varepsilon)]/\log(1/a)+O_a(1)\) as \(\varepsilon\) tends to zero. We also give the connectedness thresholds at \(a=0\) and \(a=1\).