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 positive definite on $partialOmega$ and underlies boundary-integral functional calculi and spectral-set bounds in the sense of Delyon–Delyon and Crouzeix.We analyze the opposite limiting regime $Omegadownarrow W(A)$. Along any $C^1$ convex exhaustion $Omega_varepsilondownarrow W(A)$, if $sigma_varepsiloninpartialOmega_varepsilon$ approaches a non-spectral boundary point $sigma_0inpartial W(A)setminusspec(A)$ 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 $(sigma_0Id-A)mathcal M(n)$ determined by the maximal eigenspace of $H(n)=Real(overline{n}A)$.Quantitatively, we obtain two-sided bounds in terms of 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 $W(A)+varepsilonmathbb{D}$. Under a spectral-isolation hypothesis for $lambda_{max}(H(n))$, we characterize the entire collapsing eigenvalue cluster under non-tangential offsets: exactly $m=dimmathcal M(n)$ 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 $sigma_0inspec(A)cappartial W(A)$ we obtain a three-scale picture for nonnormal matrices: an exact $1/varepsilon$ blow-up on $Ker(sigma_0Id-A)$, an $O(varepsilon)$ collapsing cluster on $mathcal M(n)ominusKer(sigma_0Id-A)$ 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.