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

Let $AinC^{dtimes d}$ and let $W(A)$ denote its numerical range. For a bounded convex domain $OmegasubsetC$ with $C^1$ boundary containing $spec(A)$, consider the operator-valued boundary kernel [ P_{Omega}(sigma,A);:=;Real!Bigl(n_{Omega}(sigma),(sigmaId-A)^{-1}Bigr), qquad sigmainpartialOmega, ] where $n_{Omega}(sigma)$ is the outward unit normal at $sigma$. For convex $Omega$ with $W(A)subsetOmega$ this kernel is strictly positive definite on $partialOmega$ and underlies boundary-integral functional calculi on convex domains. We analyze the opposite limiting regime $Omegadownarrow W(A)$. Along any $C^1$ convex exhaustion $Omega_varepsilondownarrow W(A)$, if $sigma_varepsiloninpartialOmega_varepsilon$ approaches $sigma_0inpartial W(A)$ with convergent outward normals and $sigma_0notinspec(A)$, 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 $(sigma_0Id-A)mathcal M(n)$ 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 $W(A)+varepsilonmathbb{D}$. 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.