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

Let $AinC^{dtimes d}$ 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 $P_Omega(sigma,A)=Real!bigl(n_Omega(sigma)(sigmaId-A)^{-1}bigr)$ on $partialOmega$, which is positive definite whenever $W(A)subsetOmega$.We study the loss of pointwise coercivity as $Omegadownarrow W(A)$. Along any $C^1$ convex exhaustion $Omega_varepsilondownarrow W(A)$, if boundary data $(sigma_varepsilon,n_{Omega_varepsilon}(sigma_varepsilon))$ converge to a supporting pair $(sigma_0,n)$ with $sigma_0inpartial W(A)setminusspec(A)$, then $lambda_{min}(P_{Omega_varepsilon}(sigma_varepsilon,A))to 0$ and the near-kernel aligns with $(sigma_0Id-A)mathcal M(n)$, 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 $sigma_0inspec(A)cappartial W(A)$ 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.