Degeneracy of the Operator-Valued Poisson Kernel Near the Numerical Range Boundary

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.