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 positive definite on $\partial\Omega$ and underlies boundary-integral functional calculi and spectral-set bounds in the sense of Delyon--Delyon and Crouzeix.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 a non-spectral boundary point $\sigma_0\in\partial W(A)\setminus\spec(A)$ with convergent outward normals $n_{\Omega_\varepsilon}(\sigma_\varepsilon)\to n$, then $\lambda_{\min}(P_{\Omega_\varepsilon}(\sigma_\varepsilon,A))\to 0$ and the associated min-eigenvector directions 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 the support-gap scalar $\delta(\sigma,n)=\Real(\overline{n}\,\sigma)-\lambda_{\max}(H(n))$, yielding a linear degeneracy rate under bounded-resolvent hypotheses and an explicit rate for outer offsets $W(A)+\varepsilon\mathbb{D}$. Under a spectral-isolation hypothesis for $\lambda_{\max}(H(n))$, we characterize the entire collapsing eigenvalue cluster under non-tangential offsets: exactly $m=\dim\mathcal M(n)$ eigenvalues decay as $O(\varepsilon)$ with a computable slope spectrum given by the eigenvalues of an explicit Gram matrix $G(n,\sigma_0)^{-1}$, while the remaining eigenvalues stay uniformly bounded away from $0$. This yields a rigorous face detector based on counting small eigenvalues, and the rescaled cluster is intrinsic under arbitrary $C^1$ convex exhaustions after normalization by $\delta$.At spectral support points $\sigma_0\in\spec(A)\cap\partial W(A)$ we obtain a three-scale picture for nonnormal matrices: an exact $1/\varepsilon$ blow-up on $\Ker(\sigma_0\Id-A)$, an $O(\varepsilon)$ collapsing cluster on $\mathcal M(n)\ominus\Ker(\sigma_0\Id-A)$ with an explicit slope spectrum, and an $O(1)$ bulk separated from $0$. For normal matrices we compute the spectrum of $P_\Omega(\sigma,A)$ explicitly, recovering a simple dichotomy at spectral support points in terms of whether the supporting face contains multiple eigenvalues. Finally, we include reproducible numerical experiments (Python) validating the predicted slopes and splittings.