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. For a bounded convex domain $\Omega\subset\C$ with $C^1$ boundary containing $\spec(A)$, consider the operator-valued boundary kernel \[ P_{\Omega}(\sigma,A)\;:=\;\Real\!\Bigl(n_{\Omega}(\sigma)\,(\sigma\Id-A)^{-1}\Bigr), \qquad \sigma\in\partial\Omega, \] where $n_{\Omega}(\sigma)$ is the outward unit normal at $\sigma$. For convex $\Omega$ with $W(A)\subset\Omega$ this kernel is strictly positive definite on $\partial\Omega$ and underlies boundary-integral functional calculi on convex domains. We analyze the opposite limiting regime $\Omega\downarrow W(A)$. Along any $C^1$ convex exhaustion $\Omega_\varepsilon\downarrow W(A)$, if $\sigma_\varepsilon\in\partial\Omega_\varepsilon$ approaches $\sigma_0\in\partial W(A)$ with convergent outward normals and $\sigma_0\notin\spec(A)$, then $\lambda_{\min}(P_{\Omega_\varepsilon}(\sigma_\varepsilon,A))\to 0$ and the corresponding min-eigenvectors converge (up to subsequences and phases) to the canonical subspace $(\sigma_0\Id-A)\mathcal M(n)$ determined by the maximal eigenspace of $H(n)=\Real(\overline{n}A)$. Quantitatively, we obtain two-sided bounds in terms of an explicit support-gap scalar, yielding a linear degeneracy rate under bounded-resolvent hypotheses and an explicit rate for outer offsets $W(A)+\varepsilon\mathbb{D}$. For normal matrices we compute the eigenvalues of $P_{\Omega}(\sigma,A)$ explicitly, showing that degeneracy may fail at spectral support points unless the supporting face contains multiple eigenvalues.