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

1. Introduction

Let $A\in {\mathbb{C}}^{d\times d}$ and define its numerical range by

$$W(A):=\{x^{\ast }Ax:x\in {\mathbb{C}}^d,∥x∥=1\}.$$

The Toeplitz–Hausdorff theorem asserts that $W(A)$ is a compact convex subset of $\mathbb{C}$ (see, e.g., [1,2]). Crouzeix conjectured that $W(A)$ is a 2-spectral set for A, i.e.,

$$∥p(A)∥\le 2\underset{z\in W(A)}{max}\left|p(z)\right|\mathrm{for}\mathrm{every}\mathrm{polynomial}p.$$

(1.1)

See [3,4] for the formulation and [5] for the best known universal constant $1+\sqrt{2}$.

• Background and relation to the convex-domain functional calculus. Up to normalization conventions, a central tool in the convex-domain approach of Delyon–Delyon and Crouzeix is the operator-valued boundary kernel

  $$P_{\Omega }(\sigma ,A):=Re\left(n_{\Omega }(\sigma ){(\sigma I-A)}^{-1}\right),\sigma \in \partial \Omega ,$$

  (1.2)

  defined for a bounded convex domain $\Omega \subset \mathbb{C}$ with $C^1$ boundary and $spec(A)\subset \Omega$. Here $n_{\Omega }(\sigma )$ denotes the outward unit normal at $\sigma$. This kernel appears in double-layer potential representations and boundary integral operators used to obtain functional calculus bounds on convex domains [4,5,6,7,8]. For convex $\Omega$ with $W(A)\subset \Omega$, positivity/coercivity of $\sigma \mapsto P_{\Omega }(\sigma ,A)$ on $\partial \Omega$ encodes strict separation of supporting half-planes and serves as a key structural input in such estimates [7,8,9].

• Motivation: loss of coercivity near $\partial W(A)$. In applications and numerical implementations of boundary-integral calculi, one often approximates $W(A)$ by $C^1$ convex supersets ${\Omega }_{\epsilon }\downarrow W(A)$. It is therefore natural to ask whether coercivity of the pointwise kernel $P_{{\Omega }_{\epsilon }}(\sigma ,A)$ can remain uniform as $\epsilon \to 0$. The results below show that this is impossible in general: even when the resolvent stays bounded (i.e. at non-spectral boundary points ${\sigma }_0\in \partial W(A)\setminus spec(A)$), the smallest eigenvalue of $P_{{\Omega }_{\epsilon }}(\sigma ,A)$ must collapse to 0 at boundary points $\sigma \in \partial {\Omega }_{\epsilon }$ approaching $\partial W(A)$ in a fixed supporting direction.

• What is new in this paper. The existing convex-domain literature primarily exploits positivity of (1.2) for fixed domains $\Omega ⊋W(A)$ [4,7,8,9]. Here we analyze the complementary limiting regime in which $\Omega$shrinks to $W(A)$, and we make explicit the resulting loss of coercivity of the pointwise kernel. The analysis is driven by a congruence identity and by a scalar support gap $\delta (\sigma ,n)=Re(\overline{n}\sigma )-{\lambda }_{max}(Re(\overline{n}A))$, which admits a support-function interpretation in standard convex-geometry terminology.

• We prove a qualitative degeneracy theorem (Theorem 1): along any $C^1$ convex exhaustion ${\Omega }_{\epsilon }\downarrow W(A)$, if ${\sigma }_{\epsilon }\in \partial {\Omega }_{\epsilon }$ approaches a non-spectral boundary point ${\sigma }_0\in \partial W(A)\setminus spec(A)$ with convergent outward normals $n_{{\Omega }_{\epsilon }}({\sigma }_{\epsilon })\to n$, then ${\lambda }_{min}(P_{{\Omega }_{\epsilon }}({\sigma }_{\epsilon },A))\to 0$ and the limiting min-eigenvector directions lie in $({\sigma }_{0}I-A)\mathcal{M}(n)$, where $\mathcal{M}(n)$ is the maximal eigenspace of $H(n)=Re(\overline{n}A)$.
• We establish two-sided bounds for $P(\sigma ,n)$ in terms of the support gap $\delta (\sigma ,n)$, yielding a linear degeneracy rate under bounded-resolvent hypotheses (Lemma 3 and Corollary 3), and compute $\delta$ explicitly for standard outer offsets $W(A)+\epsilon \mathbb{D}$ (Proposition 2).
• Under a spectral-isolation hypothesis for ${\lambda }_{max}(H(n))$, we quantify the entire collapsing eigenvalue cluster: exactly $m=dim\mathcal{M}(n)$ eigenvalues collapse linearly with an explicit slope spectrum given by the eigenvalues of a computable $m\times m$ Gram matrix $G{(n,{\sigma }_0)}^{-1}$ (Proposition 7), while the remaining $d-m$ eigenvalues stay uniformly bounded away from 0 (Proposition 8). This yields a rigorous “face detector” based on counting eigenvalues below a threshold proportional to $\epsilon$ (Corollary 5). The same slope spectrum is shown to be intrinsic under arbitrary $C^1$ convex exhaustions after normalization by the support gap (Proposition 9).
• We analyze the contrasting spectral-support regime ${\sigma }_0\in spec(A)\cap \partial W(A)$. For general matrices we obtain a three-scale splitting under non-tangential offsets: an exact $1/\epsilon$ blow-up on $Ker({\sigma }_{0}I-A)$, an $O(\epsilon )$ collapsing cluster on $\mathcal{M}(n)\ominus Ker({\sigma }_{0}I-A)$ with an explicit slope spectrum, and an $O(1)$ bulk separated from 0 (Proposition 12). For normal matrices we recover a simple degeneracy dichotomy at spectral support points in terms of whether the supporting face contains multiple eigenvalues (Proposition 11 and Corollary 7).
• We include reproducible numerical experiments (Python) validating the predicted slopes, splittings, and direction-dependent sensitivity profiles (Section 4.9).

• Organization.Section 2 fixes notation and recalls support-function identities. Section 3 introduces $P_{\Omega }(\sigma ,A)$, proves the key congruence identity, and establishes quantitative support-gap bounds together with a geometric interpretation of $\delta$. Section 4 contains the degeneracy theorem, quantitative corollaries, slope spectra and two-/three-scale spectral splittings (including the spectral-support regime), subspace convergence, explicit examples, and reproducible numerical tests. It concludes with a brief discussion of remaining open questions.

2. Preliminaries

We use the standard notation for disks:

$$\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\},\overline{\mathbb{D}}:=\{z\in \mathbb{C}:|z|\le 1\}.$$

Throughout, $A\in {\mathbb{C}}^{d\times d}$ is fixed. For vectors $x\in {\mathbb{C}}^d$ we use $∥x∥:={(x^{\ast }x)}^{1/2}$. For matrices $B\in {\mathbb{C}}^{d\times d}$ we use the induced operator norm $∥B∥:={sup}_{∥x∥=1}∥Bx∥$. We write $B^{\ast }$ for the conjugate transpose and $Re(B):=(B+B^{\ast })/2$.

For a Hermitian matrix B, we write its eigenvalues in nondecreasing order as

$${\lambda }_1^{\uparrow }(B)\le \dots \le {\lambda }_d^{\uparrow }(B),$$

and in nonincreasing order as ${\lambda }_1^{\downarrow }(B)\ge \dots \ge {\lambda }_d^{\downarrow }(B)$. In particular, ${\lambda }_{min}(B)={\lambda }_1^{\uparrow }(B)$ and ${\lambda }_{max}(B)={\lambda }_1^{\downarrow }(B)$.

Remark 1

(Spectrum is contained in the numerical range). One has $spec(A)\subset W(A)$. Indeed, if $Ax=\lambda x$ with $∥x∥=1$, then $x^{\ast }Ax=\lambda \in W(A)$. Consequently, $W(A)\subset \Omega$ implies $spec(A)\subset \Omega$ for any open set $\Omega \subset \mathbb{C}$.

2.1. Support Functions and the Hermitian Pencil

For unimodular $\omega \in \mathbb{C}$ (i.e. $\left|\omega \right|=1$), define the Hermitian matrix

$$H(\omega ):=Re(\overline{\omega }A)=\frac{1}{2}(\overline{\omega }A+\omega A^{\ast }).$$

(2.1)

We will later write $n\in \mathbb{C}$ (with $\left|n\right|=1$) for outward unit normals on $\partial \Omega$; in the support-function identities below and throughout, such an n simply plays the role of the unimodular direction $\omega$.

Let ${\lambda }_{max}(H(\omega ))$ denote its largest eigenvalue and let

$$\mathcal{M}(\omega ):=Ker\left({\lambda }_{max}(H(\omega ))I-H(\omega )\right)$$

denote the corresponding maximal eigenspace.

Lemma 1

(Support function of the numerical range). For every unimodular $\omega \in \mathbb{C}$,

$$\underset{z\in W(A)}{max}Re(\overline{\omega }z)={\lambda }_{max}(H(\omega )).$$

(2.2)

Moreover, if $x\in {\mathbb{C}}^d$ is a unit eigenvector of $H(\omega )$ associated with ${\lambda }_{max}(H(\omega ))$, then $x^{\ast }Ax\in \partial W(A)$ and

$$Re\left(\overline{\omega }{x}^{\ast }Ax\right)={\lambda }_{max}(H(\omega )).$$

Proof. 

For $∥x∥=1$,

$$Re\left(\overline{\omega }{x}^{\ast }Ax\right)=Re\left(x^{\ast }(\overline{\omega }A)x\right)=x^{\ast }Re(\overline{\omega }A)x=x^{\ast }H(\omega )x.$$

Taking the maximum over $∥x∥=1$ yields (2.2) by Rayleigh–Ritz (see, e.g., [10]). If x is a maximizing unit vector, then $x^{\ast }Ax\in W(A)$ attains the support functional in direction $\omega$, hence lies on $\partial W(A)$ and satisfies the stated identity.  □

2.2. Convex Domains with $C^1$ Boundary and Normals

We identify $\mathbb{C}$ with ${\mathbb{R}}^2$ in the usual way. Let $\Omega \subset \mathbb{C}$ be a bounded open convex set with $C^1$ boundary. Then for each $\sigma \in \partial \Omega$ there is a unique outward unit normal vector. This $C^1$ assumption is used only to guarantee that the outward unit normal $n_{\Omega }(\sigma )$ exists and is unique at every boundary point, ensuring that $P_{\Omega }(\sigma ,A)$ is well-defined; no higher regularity (e.g. curvature bounds) is used. We represent the normal as a unimodular complex number $n_{\Omega }(\sigma )\in \mathbb{C}$ with $|n_{\Omega }(\sigma )|=1$ so that the supporting half-plane at $\sigma$ is

$${\Pi }_{\Omega }(\sigma )=\left\{z\in \mathbb{C}:Re\left(\overline{n_{\Omega }(\sigma )}(z-\sigma )\right)\le 0\right\}.$$

(2.3)

Equivalently, by convexity one has $\overline{\Omega }\subseteq {\Pi }_{\Omega }(\sigma )$ and $\Omega \subset \{z\in \mathbb{C}:Re(\overline{n_{\Omega }(\sigma )}(z-\sigma ))<0\}$. Under the identification $\mathbb{C}\simeq {\mathbb{R}}^2$, the functional $z\mapsto Re(\overline{n}z)$ is the Euclidean inner product with the unit vector corresponding to n.

Definition 1

($C^1$ convex exhaustion). A family ${\left\{{\Omega }_{\epsilon }\right\}}_{\epsilon >0}$ is called a$C^1$ convex exhaustionof a compact convex set $K\subset \mathbb{C}$ if:

(i) 

each ${\Omega }_{\epsilon }\subset \mathbb{C}$ is a bounded open convex set with $C^1$ boundary;

(ii) 

${\Omega }_{{\epsilon }^{\prime }}\subset {\Omega }_{\epsilon }$ for $0<{\epsilon }^{\prime }<\epsilon$;

(iii) 

$K\subset {\Omega }_{\epsilon }$ for all $\epsilon >0$;

(iv) 

${\bigcap }_{\epsilon >0}\overline{{\Omega }_{\epsilon }}=K$.

Remark 2

(Subsequence selection for convergent normals). Let ${\epsilon }_k\downarrow 0$ and ${\sigma }_k\in \partial {\Omega }_{{\epsilon }_k}$ be any sequence. Since each outward normal $n_k:=n_{{\Omega }_{{\epsilon }_k}}({\sigma }_k)$ is unimodular, the sequence $\left\{n_k\right\}\subset \{z\in \mathbb{C}:|z|=1\}$ lies in a compact set. Hence there is always a subsequence (not relabeled) such that $n_k\to n$ for some unimodular n. In particular, the normal convergence hypothesis in Theorem 1 can always be arranged by passing to a subsequence.

3. The Operator-Valued Poisson Kernel

Let $\Omega \subset \mathbb{C}$ be a bounded open convex set with $C^1$ boundary and assume $spec(A)\subset \Omega$. Then ${(\sigma I-A)}^{-1}$ exists for all $\sigma \in \partial \Omega$.

Definition 2

(Operator-valued Poisson kernel). For $\sigma \in \partial \Omega$, define

$$P_{\Omega }(\sigma ,A):=Re\left(n_{\Omega }(\sigma ){(\sigma I-A)}^{-1}\right).$$

(3.1)

3.1. A Congruence Identity

Lemma 2

(Congruence identity). Let $\sigma \notin spec(A)$ and let $n\in \mathbb{C}$ be unimodular. Then

$${(\sigma I-A)}^{\ast }Re\left(n{(\sigma I-A)}^{-1}\right)(\sigma I-A)=Re\left(\overline{n}(\sigma I-A)\right)=Re(\overline{n}\sigma )I-Re(\overline{n}A).$$

(3.2)

Proof. 

Write $R:={(\sigma I-A)}^{-1}$. Then $R(\sigma I-A)=I$ and ${(\sigma I-A)}^{\ast }{R}^{\ast }=I$. Using $Re(X)=\frac{1}{2}(X+X^{\ast })$,

$$\begin{array}{cc}\hfill {(\sigma I-A)}^{\ast }Re(nR)(\sigma I-A)& =\frac{1}{2}\left({(\sigma I-A)}^{\ast }(nR)(\sigma I-A)+{(\sigma I-A)}^{\ast }(\overline{n}{R}^{\ast })(\sigma I-A)\right)\hfill \\ & =Re\left(\overline{n}(\sigma I-A)\right).\hfill \end{array}$$

Expanding gives (3.2).  □

3.2. Support-Gap Bounds

For unimodular $n\in \mathbb{C}$ define the support gap

$$\delta (\sigma ,n):=Re(\overline{n}\sigma )-{\lambda }_{max}(H(n)),H(n)=Re(\overline{n}A).$$

Lemma 3

(Support-gap characterization and quantitative bounds). Let $A\in {\mathbb{C}}^{d\times d}$, let $\sigma \notin spec(A)$, and let $n\in \mathbb{C}$ be unimodular. Set

$$P(\sigma ,n):=Re\left(n{(\sigma I-A)}^{-1}\right),\alpha :=Re(\overline{n}\sigma ),\delta :=\alpha -{\lambda }_{max}(H(n)).$$

(This notation emphasizes dependence on the prescribed direction n; when $n=n_{\Omega }(\sigma )$ one has $P(\sigma ,n)=P_{\Omega }(\sigma ,A)$.) Then:

(a) 

$P(\sigma ,n)⪰0$ if and only if $\delta \ge 0$, and $P(\sigma ,n)\succ 0$ if and only if $\delta >0$.

(b) 

If $\delta =0$, then $P(\sigma ,n)$ is singular and

$$Ker(P(\sigma ,n))=(\sigma I-A)\mathcal{M}(n),\mathcal{M}(n)=Ker({\lambda }_{max}(H(n))I-H(n)).$$

(c) 

If $\delta >0$, then

$${\displaystyle \frac{\delta }{{\parallel \sigma I-A\parallel }^2}}\le {\lambda }_{min}\left(P(\sigma ,n)\right)\le \delta {\parallel {(\sigma I-A)}^{-1}\parallel }^2.$$

(3.3)

Proof. 

Let $B:=\sigma I-A$ and $P:=P(\sigma ,n)$. By Lemma 2,

$$B^{\ast }PB=Re(\overline{n}B)=\alpha I-Re(\overline{n}A)=\alpha I-H(n)=:Q.$$

Since B is invertible, congruence by B preserves (semi)definiteness, so $P⪰0\iff Q⪰0$ and $P\succ 0\iff Q\succ 0$. As Q is Hermitian with ${\lambda }_{min}(Q)=\alpha -{\lambda }_{max}(H(n))=\delta$, this proves (a).

If $\delta =0$, then $Q⪰0$ is singular with $Ker(Q)=\mathcal{M}(n)$, and $P⪰0$ by (a). For $P⪰0$, $x\in Ker(P)\iff x^{\ast }Px=0$. Writing $x=By$,

$$x^{\ast }Px=y^{\ast }Qy,$$

so $x\in Ker(P)\iff y\in Ker(Q)=\mathcal{M}(n)$, proving (b).

If $\delta >0$, then $Q\succ 0$ and $P=B^{-\ast }Q{B}^{-1}\succ 0$. For $\parallel x\parallel =1$ and $y=B^{-1}x$, one has $x=By$ and hence $\parallel y\parallel \ge 1/\parallel B\parallel$; thus

$$x^{\ast }Px=y^{\ast }Qy\ge {\lambda }_{min}{(Q)\parallel y\parallel }^2={\delta \parallel y\parallel }^2\ge \delta /{\parallel B\parallel }^2,$$

giving the lower bound in (3.3). For the upper bound, take y a unit eigenvector of Q for ${\lambda }_{min}(Q)=\delta$ and set $x=By/\parallel By\parallel$; then

$$x^{\ast }Px={\displaystyle \frac{y^{\ast }Qy}{{\parallel By\parallel }^2}}={\displaystyle \frac{\delta }{{\parallel By\parallel }^2}}\le \delta {\parallel B^{-1}\parallel }^2.$$

 □

Remark 3

(Connection with the convex-domain Poisson kernel literature). Up to normalization conventions, $P_{\Omega }(\sigma ,A)$ is the operator-valued boundary kernel appearing in the Carl Neumann double-layer potential framework for convex domains; see, e.g., [7,8,9]. Lemma 3 isolates the dependence of ${\lambda }_{min}(P(\sigma ,n))$ on the scalar support gap $\delta (\sigma ,n)$.

3.3. Strict Positivity when $W(A)\subset \Omega$

Lemma 4

(Strict separation at a supporting line). Let $\Omega \subset \mathbb{C}$ be a bounded open convex set with $C^1$ boundary and let $K\subset \Omega$ be compact. Fix $\sigma \in \partial \Omega$ and let $n=n_{\Omega }(\sigma )$ be the outward unit normal. Then

$$\underset{z\in K}{max}Re(\overline{n}z)<Re(\overline{n}\sigma ).$$

Proof. 

By (2.3), $\Omega \subset \{z:Re(\overline{n}(z-\sigma ))<0\}$, hence $K\subset \{z:Re(\overline{n}(z-\sigma ))<0\}$. The continuous function $z\mapsto Re(\overline{n}(z-\sigma ))$ attains its maximum on compact K, and this maximum is strictly negative. Rearranging yields the claim.  □

Proposition 1

(Positivity of the Poisson kernel). Assume $W(A)\subset \Omega$. Then for every $\sigma \in \partial \Omega$,

$$P_{\Omega }(\sigma ,A)\succ 0.$$

Proof. 

Fix $\sigma \in \partial \Omega$ and set $n:=n_{\Omega }(\sigma )$ and $\alpha :=Re(\overline{n}\sigma )$. By Lemma 4 with $K=W(A)$ and Lemma 1,

$${\lambda }_{max}(H(n))=\underset{z\in W(A)}{max}Re(\overline{n}z)<\alpha ,$$

so $\delta (\sigma ,n)=\alpha -{\lambda }_{max}(H(n))>0$. Now apply Lemma 3 (a).  □

3.4. Geometric Meaning of the Support Gap and Offset Exhaustions

For a compact convex set $K\subset \mathbb{C}$ and unimodular $n\in \mathbb{C}$, define its support function

$$h_K(n):=\underset{z\in K}{max}Re(\overline{n}z).$$

If $\Omega \subset \mathbb{C}$ is a bounded open convex set with $C^1$ boundary and $\sigma \in \partial \Omega$ has outward normal $n=n_{\Omega }(\sigma )$, then necessarily $Re(\overline{n}\sigma )=h_{\overline{\Omega }}(n)$, i.e. the boundary point lies on the supporting line in direction n.

Lemma 5

(Support gap as a support-function difference). Let $\Omega \subset \mathbb{C}$ be a bounded open convex set with $C^1$ boundary and $\sigma \in \partial \Omega$. Let $n:=n_{\Omega }(\sigma )$. Then

$$\delta (\sigma ,n)=Re(\overline{n}\sigma )-{\lambda }_{max}(H(n))=h_{\overline{\Omega }}(n)-h_{W(A)}(n).$$

In particular, $\delta (\sigma ,n)$ measures the separation between the supporting line of $\overline{\Omega }$ in direction n and the corresponding supporting line of $W(A)$.

Proof. 

Since n is the outward unit normal at $\sigma \in \partial \Omega$, the supporting half-plane characterization implies $Re(\overline{n}z)\le Re(\overline{n}\sigma )$ for all $z\in \overline{\Omega }$, hence $h_{\overline{\Omega }}(n)=Re(\overline{n}\sigma )$. By Lemma 1, $h_{W(A)}(n)={\lambda }_{max}(H(n))$. Combining gives the claim.  □

Proposition 2

(Outer offsets: $\delta$ is explicit). Let $K\subset \mathbb{C}$ be compact and convex and fix $\epsilon >0$. Define the outer offset (outer parallel set)

$$K_{\epsilon }:=K+\epsilon \mathbb{D}=\{z+w:z\in K,w\in \mathbb{C},|w|<\epsilon \}.$$

Then for every unimodular $n\in \mathbb{C}$,

$$h_{\overline{K_{\epsilon }}}(n)=h_K(n)+\epsilon .$$

In particular, taking $K=W(A)$ and ${\Omega }_{\epsilon }:=W(A)+\epsilon \mathbb{D}$, for any boundary point $\sigma \in \partial {\Omega }_{\epsilon }$ with outward normal $n=n_{{\Omega }_{\epsilon }}(\sigma )$ (whenever defined) one has

$$\delta (\sigma ,n)=\epsilon .$$

Consequently, since $\sigma \in \partial {\Omega }_{\epsilon }$ implies $\sigma \notin W(A)$ and hence $\sigma \notin spec(A)$ (Remark 1), Lemma 3 (c) yields

$${\displaystyle \frac{\epsilon }{{\parallel \sigma I-A\parallel }^2}}\le {\lambda }_{min}\left(P_{{\Omega }_{\epsilon }}(\sigma ,A)\right)\le \epsilon {\parallel {(\sigma I-A)}^{-1}\parallel }^2.$$

Proof. 

Fix unimodular n. For any $z\in K$ and $w\in \mathbb{C}$ with $\left|w\right|\le \epsilon$,

$$Re(\overline{n}(z+w))=Re(\overline{n}z)+Re(\overline{n}w)\le h_K(n)+\left|w\right|\le h_K(n)+\epsilon ,$$

so $h_{\overline{K_{\epsilon }}}(n)\le h_K(n)+\epsilon$. On the other hand, choosing $z_{★}\in K$ with $Re(\overline{n}{z}_{★})=h_K(n)$ and $w_{★}=\epsilon n$ gives $|w_{★}|=\epsilon$ and

$$Re(\overline{n}(z_{★}+w_{★}))=h_K(n)+\epsilon ,$$

so $h_{\overline{K_{\epsilon }}}(n)\ge h_K(n)+\epsilon$. This proves the support-function identity and hence the displayed formula for $\delta$ follows from Lemma 5.

The final eigenvalue bounds are an immediate substitution of $\delta =\epsilon$ into (3.3).  □

Remark 4

(Smoothness versus offsets). If K has flat faces, then $\partial (K+\epsilon \mathbb{D})$ is typically only $C^{1,1}$ (curvature may jump at transitions between translated faces and rounded arcs). Proposition 2 is therefore best viewed as a geometric model illustrating how the support gap scales with the outer distance parameter ε. For the purposes of Definition 1, one may replace $K+\epsilon \mathbb{D}$ by any convex domain with $C^1$ boundary whose support function differs from $h_K$ by a quantity comparable to ε; the same interpretation of δ then applies. For example, one may take Minkowski sums with a fixed smooth strictly convex unit ball (instead of $\mathbb{D}$) or smooth the support function to obtain a genuine $C^1$ (indeed smooth) convex exhaustion with the same first-order support-gap scaling.

3.5. Hausdorff Distance and Support-Function Control of the Support Gap

For a nonempty compact set $K\subset \mathbb{C}$ and $z\in \mathbb{C}$, write

$$dist(z,K):=\underset{w\in K}{inf}|z-w|.$$

For nonempty compact sets $K,L\subset \mathbb{C}$, define the (Euclidean) Hausdorff distance

$$d_{\mathrm{H}}(K,L):=max\left\{\underset{z\in K}{sup}dist(z,L),\underset{w\in L}{sup}dist(w,K)\right\}.$$

Lemma 6

(Hausdorff distance via support functions). Let $K,L\subset \mathbb{C}$ be nonempty compact convex sets and let $\overline{\mathbb{D}}:=\{z\in \mathbb{C}:|z|\le 1\}$. Then

$$d_{\mathrm{H}}(K,L)=\underset{\left|n\right|=1}{sup}\left|h_K(n)-h_L(n)\right|.$$

If moreover $K\subseteq L$, then $h_K(n)\le h_L(n)$ for all $\left|n\right|=1$ and hence

$$d_{\mathrm{H}}(L,K)=\underset{\left|n\right|=1}{sup}\left(h_L(n)-h_K(n)\right).$$

Proof. 

For $t\ge 0$ and a nonempty compact set K, the Minkowski sum

$$K+t\overline{\mathbb{D}}=\{z+w:z\in K,|w|\le t\}$$

is the closed t-neighborhood of K, i.e. $K+t\overline{\mathbb{D}}=\{u\in \mathbb{C}:dist(u,K)\le t\}$. Consequently,

$$d_{\mathrm{H}}(K,L)=inf\left\{t\ge 0:K\subseteq L+t\overline{\mathbb{D}}\mathrm{and}L\subseteq K+t\overline{\mathbb{D}}\right\}.$$

For compact convex sets $M,N\subset \mathbb{C}$ one has $M\subseteq N$ if and only if $h_M(n)\le h_N(n)$ for all $\left|n\right|=1$. (Indeed, the forward direction is immediate; conversely, if $x\in M\setminus N$, a separating supporting line for the convex compact set N yields a unimodular n with $Re(\overline{n}x)>{max}_{z\in N}Re(\overline{n}z)=h_N(n)$, hence $h_M(n)\ge Re(\overline{n}x)>h_N(n)$.)

Moreover, support functions add under Minkowski sums, and $h_{t\overline{\mathbb{D}}}(n)=t$ for $\left|n\right|=1$; hence

$$h_{K+t\overline{\mathbb{D}}}(n)=h_K(n)+t.$$

Therefore, $K\subseteq L+t\overline{\mathbb{D}}$ is equivalent to $h_K(n)\le h_L(n)+t$ for all $\left|n\right|=1$, and similarly $L\subseteq K+t\overline{\mathbb{D}}$ is equivalent to $h_L(n)\le h_K(n)+t$ for all $\left|n\right|=1$. Thus $d_{\mathrm{H}}(K,L)$ is the smallest t such that $|h_K(n)-h_L(n)|\le t$ for all $\left|n\right|=1$, i.e.

$$d_{\mathrm{H}}(K,L)=\underset{\left|n\right|=1}{sup}|h_K(n)-h_L(n)|.$$

If $K\subseteq L$, then $h_K\le h_L$, so the absolute value may be dropped, giving the second identity.  □

Corollary 1

(Support gap bounded by the Hausdorff approximation error). Assume $W(A)\subset \Omega$, and set

$$\Delta (\Omega ):=d_{\mathrm{H}}(\overline{\Omega },W(A))=\underset{\left|n\right|=1}{sup}\left(h_{\overline{\Omega }}(n)-h_{W(A)}(n)\right).$$

Then for every $\sigma \in \partial \Omega$ with outward normal $n=n_{\Omega }(\sigma )$,

$$\delta (\sigma ,n)=Re(\overline{n}\sigma )-{\lambda }_{max}(H(n))=h_{\overline{\Omega }}(n)-h_{W(A)}(n)\le \Delta (\Omega ).$$

Consequently, since $\sigma \notin spec(A)$ for $\sigma \in \partial \Omega$, Lemma 3 (c) yields

$${\lambda }_{min}\left(P_{\Omega }(\sigma ,A)\right){\le \delta (\sigma ,n)\parallel (\sigma I-A)}^{-1}{\parallel }^2\le \Delta (\Omega ){\parallel {(\sigma I-A)}^{-1}\parallel }^2.$$

Moreover, there exists ${\sigma }_{★}\in \partial \Omega$ such that

$$\delta ({\sigma }_{★},n_{\Omega }({\sigma }_{★}))=\Delta (\Omega ),$$

and for this point one has the two-sided estimate

$${\displaystyle \frac{\Delta (\Omega )}{\parallel {\sigma }_{★}{I-A\parallel }^2}}\le {\lambda }_{min}\left(P_{\Omega }({\sigma }_{★},A)\right)\le \Delta (\Omega ){\parallel {({\sigma }_{★}I-A)}^{-1}\parallel }^2.$$

Proof. 

The identity $\delta (\sigma ,n)=h_{\overline{\Omega }}(n)-h_{W(A)}(n)$ is Lemma 5, and the bound $\delta (\sigma ,n)\le \Delta (\Omega )$ follows from the definition of $\Delta (\Omega )$. The eigenvalue bounds are then immediate from Lemma 3 (c).

Finally, the function $n\mapsto h_{\overline{\Omega }}(n)-h_{W(A)}(n)$ is continuous on the unit circle, so it attains its maximum at some unimodular $n_{★}$. Choose ${\sigma }_{★}\in \overline{\Omega }$ such that $Re(\overline{n_{★}}{\sigma }_{★})=h_{\overline{\Omega }}(n_{★})$; then ${\sigma }_{★}\in \partial \Omega$ and the supporting line $\{z:Re(\overline{n_{★}}z)=Re(\overline{n_{★}}{\sigma }_{★})\}$ is a supporting line for $\overline{\Omega }$ at ${\sigma }_{★}$. Since $\partial \Omega$ is $C^1$, the outward unit normal at ${\sigma }_{★}$ is uniquely defined and equals $n_{★}$, and hence

$$\delta ({\sigma }_{★},n_{\Omega }({\sigma }_{★}))=h_{\overline{\Omega }}(n_{★})-h_{W(A)}(n_{★})=\Delta (\Omega ).$$

 □

4. Degeneracy Along a $C^1$ Convex Exhaustion

4.1. Qualitative Degeneracy and Limiting Kernel Directions

Theorem 1

(Degeneracy of the operator-valued Poisson kernel). Let $A\in {\mathbb{C}}^{d\times d}$ and let ${\left\{{\Omega }_{\epsilon }\right\}}_{\epsilon >0}$ be a $C^1$ convex exhaustion of $W(A)$ (Definition 1). For $\sigma \in \partial {\Omega }_{\epsilon }$, set

$$P_{{\Omega }_{\epsilon }}(\sigma ,A):=Re\left(n_{{\Omega }_{\epsilon }}(\sigma ){(\sigma I-A)}^{-1}\right).$$

Fix any sequence ${\epsilon }_k\downarrow 0$ and points ${\sigma }_k\in \partial {\Omega }_{{\epsilon }_k}$ such that

$${\sigma }_k\to {\sigma }_0\in \partial W(A),n_k:=n_{{\Omega }_{{\epsilon }_k}}({\sigma }_k)\to n\in \mathbb{C},\left|n\right|=1.$$

(After passing to a subsequence, the convergence $n_k\to n$ is automatic; see Remark 2.) Assume ${\sigma }_0\notin spec(A)$. Let $H(n)=Re(\overline{n}A)$ and $\mathcal{M}(n)=Ker({\lambda }_{max}(H(n))I-H(n))$.

Then:

(1) 

(Vanishing)${\lambda }_{min}\left(P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A)\right)\to 0$ as $k\to \infty$.

(2) 

(Limiting directions)If $u_k$ is any unit eigenvector of $P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A)$ for ${\lambda }_{min}\left(P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A)\right)$, then every accumulation point $u_0$ of $\left\{u_k\right\}$ satisfies

$$u_0\in ({\sigma }_{0}I-A)\mathcal{M}(n).$$

(3) 

(One-dimensional case)If $dim\mathcal{M}(n)=1$, then there exist phases ${\theta }_k\in \mathbb{R}$ such that

$$e^{i{\theta }_k}{u}_k⟶{\displaystyle \frac{({\sigma }_{0}I-A)v}{\parallel ({\sigma }_{0}I-A)v\parallel }}(k\to \infty ),$$

where v is any unit vector spanning $\mathcal{M}(n)$.

Proof. 

Set $B_k:={\sigma }_{k}I-A$ and $R_k:=B_k^{-1}$, and define

$$P_k:=Re(n_k{R}_k),{\alpha }_k:=Re(\overline{n_k}{\sigma }_k).$$

Define also $B_0:={\sigma }_{0}I-A$, $R_0:=B_0^{-1}$, $P_0:=Re(n{R}_0)$, ${\alpha }_0:=Re(\overline{n}{\sigma }_0)$.

• Step 1: Congruence identities. By Lemma 2,

  $$B_k^{\ast }{P}_k{B}_k={\alpha }_{k}I-H(n_k),B_0^{\ast }{P}_0{B}_0={\alpha }_{0}I-H(n),$$

  (4.1)

  where $H(n_k)=Re(\overline{n_k}A)$.

• Step 2: ${\alpha }_0={\lambda }_{max}(H(n))$. Since $n_k$ is the outward normal at ${\sigma }_k\in \partial {\Omega }_{{\epsilon }_k}$, the supporting half-plane property gives $Re(\overline{n_k}z)\le {\alpha }_k$ for all $z\in {\Omega }_{{\epsilon }_k}$ and hence for all $z\in W(A)$. Passing to the limit yields $Re(\overline{n}z)\le {\alpha }_0$ for all $z\in W(A)$. Because ${\sigma }_0\in W(A)$, equality holds at $z={\sigma }_0$, so ${\alpha }_0={max}_{z\in W(A)}Re(\overline{n}z)$. Lemma 1 now gives

  $${\alpha }_0={\lambda }_{max}(H(n)),Ker({\alpha }_{0}I-H(n))=\mathcal{M}(n),$$

  so ${\alpha }_{0}I-H(n)⪰0$ is singular.

• Step 3: $P_k\to P_0$ in operator norm. Since ${\sigma }_0\notin spec(A)$, $B_0$ is invertible. Write

  $$B_k=B_0+({\sigma }_k-{\sigma }_0)I=B_0(I+E_k),E_k:=({\sigma }_k-{\sigma }_0)R_0.$$

  Then $\parallel E_k\parallel \to 0$, so for large k, $I+E_k$ is invertible and

  $$R_k=B_k^{-1}={(I+E_k)}^{-1}{R}_0,\parallel R_k-R_0\parallel \to 0.$$

  Therefore,

  $$\parallel P_k-P_0\parallel =\parallel Re(n_k{R}_k-n{R}_0)\parallel \le |n_k-n|\parallel R_k\parallel +\parallel R_k-R_0\parallel \to 0.$$

• Step 4: ${\lambda }_{min}(P_0)=0$ and ${\lambda }_{min}(P_k)\to 0$. Since $P_k,P_0$ are Hermitian, Weyl’s inequality (see, e.g., [10]) yields

  $$\left|{\lambda }_{min}(P_k)-{\lambda }_{min}(P_0)\right|\le \parallel P_k-P_0\parallel \to 0,$$

  so ${\lambda }_{min}(P_k)\to {\lambda }_{min}(P_0)$. By (4.1) and Step 2,

  $$B_0^{\ast }{P}_0{B}_0={\alpha }_{0}I-H(n)⪰0\mathrm{is}\sin \mathrm{gular}.$$

  Since $B_0$ is invertible, $P_0⪰0$ is singular, hence ${\lambda }_{min}(P_0)=0$, proving (1). Moreover,

  $$Ker(P_0)=B_{0}Ker({\alpha }_{0}I-H(n))=({\sigma }_{0}I-A)\mathcal{M}(n)$$

  by Lemma 3 (b) (with $\delta =0$).

• Step 5: Limiting eigenvectors. Let $u_k$ be unit min-eigenvectors: $P_k{u}_k={\lambda }_{min}(P_k)u_k$. Along a convergent subsequence, $u_k\to u_0$. Then

  $$u_0^{\ast }{P}_0{u}_0=\underset{k\to \infty }{lim}{u}_k^{\ast }{P}_0{u}_k=\underset{k\to \infty }{lim}\left(u_k^{\ast }{P}_k{u}_k+u_k^{\ast }(P_0-P_k)u_k\right)=\underset{k\to \infty }{lim}\left({\lambda }_{min}(P_k)+o(1)\right)=0.$$

  Since $P_0⪰0$, this implies $u_0\in Ker(P_0)=({\sigma }_{0}I-A)\mathcal{M}(n)$, proving (2).

• Step 6: One-dimensional case. If $dim\mathcal{M}(n)=1$, then $dimKer(P_0)=1$, so the smallest eigenvalue of $P_0$ is simple. By the Davis–Kahan $sin\Theta$ theorem for invariant subspaces (see [11]), the corresponding one-dimensional eigenspaces of $P_k$ converge to $Ker(P_0)$ in gap metric, hence there exist phases ${\theta }_k$ such that $e^{i{\theta }_k}{u}_k\to u_{★}$, where $u_{★}$ spans $Ker(P_0)=({\sigma }_{0}I-A)\mathcal{M}(n)$. This gives (3).  □

Remark 5

(Why ${\sigma }_0\notin spec(A)$ is essential). The hypothesis ${\sigma }_0\notin spec(A)$ ensures that ${(\sigma I-A)}^{-1}$ remains bounded near ${\sigma }_0$, so $P_0$ is a finite Hermitian matrix. When ${\sigma }_0\in spec(A)$, the resolvent diverges and the behavior of ${\lambda }_{min}(P_{{\Omega }_{\epsilon }}({\sigma }_{\epsilon },A))$ depends on the spectral geometry; see Proposition 11 below and Section 4.11.

Corollary 2

(Global coercivity collapse along a $C^1$ convex exhaustion). Assume that A is not a scalar multiple of the identity (equivalently, $W(A)$ is not a singleton). Let ${\left\{{\Omega }_{\epsilon }\right\}}_{\epsilon >0}$ be a $C^1$ convex exhaustion of $W(A)$ and define the global coercivity constant

$$c(\epsilon ):=\underset{\sigma \in \partial {\Omega }_{\epsilon }}{inf}{\lambda }_{min}\left(P_{{\Omega }_{\epsilon }}(\sigma ,A)\right),P_{{\Omega }_{\epsilon }}(\sigma ,A)=Re\left(n_{{\Omega }_{\epsilon }}(\sigma ){(\sigma I-A)}^{-1}\right).$$

Then

$$\underset{\epsilon \downarrow 0}{lim\; inf}c(\epsilon )=0.$$

In particular, there do not exist ${\epsilon }_0>0$ and $c_0>0$ such that $P_{{\Omega }_{\epsilon }}(\sigma ,A)⪰c_{0}I$ for all $0<\epsilon <{\epsilon }_0$ and all $\sigma \in \partial {\Omega }_{\epsilon }$.

Proof. 

Since A is not scalar, the compact convex set $W(A)$ contains more than one point, hence $\partial W(A)$ is infinite, whereas $spec(A)$ is finite. Choose ${\sigma }_0\in \partial W(A)\setminus spec(A)$.

Fix any sequence ${\epsilon }_k\downarrow 0$. We claim that $dist({\sigma }_0,\partial {\Omega }_{{\epsilon }_k})\to 0$. Indeed, if not, then there exist $\delta >0$ and a subsequence (not relabeled) such that $dist({\sigma }_0,\partial {\Omega }_{{\epsilon }_k})\ge \delta$ for all k, hence the open ball $B({\sigma }_0,\delta )$ is contained in ${\Omega }_{{\epsilon }_k}$ for all k. Taking closures and intersecting over k yields $B({\sigma }_0,\delta )\subset {\bigcap }_k\overline{{\Omega }_{{\epsilon }_k}}=W(A)$, contradicting ${\sigma }_0\in \partial W(A)$.

Therefore we may choose ${\sigma }_k\in \partial {\Omega }_{{\epsilon }_k}$ with ${\sigma }_k\to {\sigma }_0$. By compactness of the unit circle, after passing to a subsequence we have $n_{{\Omega }_{{\epsilon }_k}}({\sigma }_k)\to n$ for some unimodular n. Theorem 1 then gives

$${\lambda }_{min}\left(P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A)\right)⟶0.$$

Since $c({\epsilon }_k)\le {\lambda }_{min}(P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A))$, it follows that ${lim\; inf}_{\epsilon \downarrow 0}c(\epsilon )=0$.  □

4.2. Quantitative Degeneracy Rate

Corollary 3

(Linear rate in terms of the support gap). In the setting of Theorem 1, define

$${\delta }_k:=Re(\overline{n_k}{\sigma }_k)-{\lambda }_{max}(H(n_k)),H(n_k)=Re(\overline{n_k}A).$$

Then ${\delta }_k>0$ for each k and ${\delta }_k\to 0$. Moreover, for all sufficiently large k,

$${\displaystyle \frac{{\delta }_k}{4\parallel {\sigma }_0{I-A\parallel }^2}}\le {\lambda }_{min}\left(P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A)\right)\le 4{\delta }_k{\parallel {({\sigma }_{0}I-A)}^{-1}\parallel }^2.$$

(4.2)

In particular, ${\lambda }_{min}(P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A))=\Theta ({\delta }_k)$.

Proof. 

Since $W(A)\subset {\Omega }_{{\epsilon }_k}$ and ${\sigma }_k\in \partial {\Omega }_{{\epsilon }_k}$ with normal $n_k$, Lemma 4 and Lemma 1 imply ${\lambda }_{max}(H(n_k))<Re(\overline{n_k}{\sigma }_k)$, so ${\delta }_k>0$.

As $n_k\to n$ and ${\sigma }_k\to {\sigma }_0$, $Re(\overline{n_k}{\sigma }_k)\to Re(\overline{n}{\sigma }_0)$. Also $H(n_k)\to H(n)$ in operator norm, hence ${\lambda }_{max}(H(n_k))\to {\lambda }_{max}(H(n))$. By Step 2 in the proof of Theorem 1, $Re(\overline{n}{\sigma }_0)={\lambda }_{max}(H(n))$, so ${\delta }_k\to 0$.

Set $B_k={\sigma }_{k}I-A$ and $B_0={\sigma }_{0}I-A$. Since $B_k\to B_0$ and $B_0$ is invertible, for large k one has $\parallel B_k\parallel \le 2\parallel B_0\parallel$ and $\parallel B_k^{-1}\parallel \le 2\parallel B_0^{-1}\parallel$. Applying Lemma 3 (c) to $(\sigma ,n)=({\sigma }_k,n_k)$ gives

$${\displaystyle \frac{{\delta }_k}{\parallel B_k{\parallel }^2}}\le {\lambda }_{min}\left(P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A)\right)\le {\delta }_k{\parallel B_k^{-1}\parallel }^2,$$

and the stated constants follow.  □

4.3. Sharpness and Refined Local/Global Bounds

Lemma 3 (c) bounds ${\lambda }_{min}(P(\sigma ,n))$ in terms of the scalar support gap $\delta (\sigma ,n)$ and the global operator norms $\parallel \sigma I-A\parallel$, ${\parallel (\sigma I-A)}^{-1}\parallel$. We first record that these norm-based bounds are optimal in the strongest possible sense, and then derive refinements that capture the correct constant in the degeneracy regime $\delta \downarrow 0$.

Proposition 3

(Optimality of the norm-based support-gap bounds). The constants in the two-sided inequality (3.3) are sharp and cannot be improved uniformly. More precisely, for $d=1$ both inequalities in (3.3) hold with equality.

Proof. 

Let $d=1$ and write $A=\left[a\right]$ with $a\in \mathbb{C}$. Then $B=\sigma -a$ is a nonzero scalar and

$$P(\sigma ,n)=Re\left({\displaystyle \frac{n}{\sigma -a}}\right)={\displaystyle \frac{Re(\overline{n}(\sigma -a))}{{|\sigma -a|}^2}}.$$

Moreover $\delta (\sigma ,n)=Re(\overline{n}\sigma )-Re(\overline{n}a)=Re(\overline{n}(\sigma -a))$, and $\parallel B\parallel =|\sigma -a|$, $\parallel B^{-1}\parallel =1/|\sigma -a|$. Therefore

$${\lambda }_{min}(P(\sigma ,n))={\displaystyle \frac{\delta (\sigma ,n)}{{\parallel B\parallel }^2}}=\delta (\sigma ,n){\parallel B^{-1}\parallel }^2,$$

so (3.3) is an equality in both directions. □

Proposition 4

(Generalized eigenvalue characterization). Let $\sigma \notin spec(A)$ and $\left|n\right|=1$. Set $B:=\sigma I-A$, $\alpha :=Re(\overline{n}\sigma )$, $H(n)=Re(\overline{n}A)$ and $Q:=\alpha I-H(n)$. Then

$${\lambda }_{min}(P(\sigma ,n))=\underset{y\ne 0}{min}{\displaystyle \frac{y^{\ast }Qy}{{\parallel By\parallel }^2}}={\lambda }_{min}(Q,B^{\ast }B),$$

(4.3)

where ${\lambda }_{min}(Q,B^{\ast }B)$ denotes the smallest generalized eigenvalue of the Hermitian definite pencil $(Q,B^{\ast }B)$.

Proof. 

By Lemma 2, $P(\sigma ,n)=B^{-\ast }Q{B}^{-1}$. For any $x\ne 0$ write $x=By$ (bijective since B is invertible). Then

$${\displaystyle \frac{x^{\ast }P(\sigma ,n)x}{{\parallel x\parallel }^2}}={\displaystyle \frac{y^{\ast }Qy}{{\parallel By\parallel }^2}}.$$

Taking the minimum over $x\ne 0$ is equivalent to taking the minimum over $y\ne 0$, which gives the first equality in (4.3); the second is the standard variational characterization of generalized eigenvalues.  □

Proposition 5

(Refined bounds via restriction to the maximal eigenspace). Let $\sigma \notin spec(A)$ and $\left|n\right|=1$, and define $B,Q$ as in Proposition 4. Let ${\lambda }_{max}(H(n))$ have maximal eigenspace $\mathcal{M}(n)$ and set

$$\delta :={\lambda }_{min}(Q)=\alpha -{\lambda }_{max}(H(n)),\beta (\sigma ,n):=∥B{|}_{\mathcal{M}(n)}∥=\underset{\begin{array}{c}y\in \mathcal{M}(n)\\ \parallel y\parallel =1\end{array}}{max}\parallel By\parallel .$$

Assume $\delta >0$ (equivalently $P(\sigma ,n)\succ 0$). Then:

(a) 

(Refined upper bound)

$${\lambda }_{min}(P(\sigma ,n))\le {\displaystyle \frac{\delta }{\beta {(\sigma ,n)}^2}}.$$

(4.4)

(b) 

(Asymptotically sharp two-sided bound)Assume in addition that ${\lambda }_{max}(H(n))$ is spectrally isolated with gap

$${\gamma }_H(n):={\lambda }_{max}(H(n))-{\lambda }_{m+1}^{\downarrow }(H(n))>0,m:=dim\mathcal{M}(n).$$

(4.5)

Then

$${\displaystyle \frac{\delta }{\beta {(\sigma ,n)}^2+\frac{{\parallel B\parallel }^2}{{\gamma }_H(n)}\delta }}\le {\lambda }_{min}(P(\sigma ,n))\le {\displaystyle \frac{\delta }{\beta {(\sigma ,n)}^2}}.$$

(4.6)

In particular, as $\delta \downarrow 0$ with B boundedly invertible, ${\lambda }_{min}(P(\sigma ,n))={\displaystyle \frac{\delta }{\beta {(\sigma ,n)}^2}}+O({\delta }^2)$.

Proof. 

By Proposition 4,

$${\lambda }_{min}(P(\sigma ,n))=\underset{y\ne 0}{min}{\displaystyle \frac{y^{\ast }Qy}{{\parallel By\parallel }^2}}.$$

(a) On $\mathcal{M}(n)$ one has $H(n)y={\lambda }_{max}(H(n))y$, hence $Qy=(\alpha -{\lambda }_{max}(H(n)))y=\delta y$. Choose $y\in \mathcal{M}(n)$ with $\parallel y\parallel =1$ attaining $\parallel By\parallel =\beta (\sigma ,n)$. Then

$${\lambda }_{min}(P(\sigma ,n))\le {\displaystyle \frac{y^{\ast }Qy}{{\parallel By\parallel }^2}}={\displaystyle \frac{\delta }{\beta {(\sigma ,n)}^2}},$$

which is (4.4).

(b) Decompose $y=y_M+y_{\perp }$ with $y_M\in \mathcal{M}(n)$ and $y_{\perp }\perp \mathcal{M}(n)$. Writing $Q=Q_0+\delta I$ with $Q_0:={\lambda }_{max}(H(n))I-H(n)⪰0$, one has $Q_0{y}_M=0$ and, by the gap hypothesis (4.5), $Q_0⪰{\gamma }_H(n)I$ on $\mathcal{M}{(n)}^{\perp }$. Hence

$$y^{\ast }Qy\ge \delta \parallel y_M{\parallel }^2+{\gamma }_H(n){\parallel y_{\perp }\parallel }^2.$$

Moreover, using $\parallel B{y}_M\parallel \le \beta (\sigma ,n)\parallel y_M\parallel$ and $\parallel B{y}_{\perp }\parallel \le \parallel B\parallel \parallel y_{\perp }\parallel$ gives

$$\parallel By\parallel \le \beta (\sigma ,n)\parallel y_M\parallel +\parallel B\parallel \parallel y_{\perp }\parallel .$$

By Cauchy–Schwarz, for $a=\parallel y_M\parallel$ and $b=\parallel y_{\perp }\parallel$,

$${\left(\beta (\sigma ,n)a+\parallel B\parallel b\right)}^2\le \left({\displaystyle \frac{\beta {(\sigma ,n)}^2}{\delta }}+{\displaystyle \frac{{\parallel B\parallel }^2}{{\gamma }_H(n)}}\right)\left(\delta a^2+{\gamma }_H(n)b^2\right),$$

so combining with the two displays above gives

$${\displaystyle \frac{y^{\ast }Qy}{{\parallel By\parallel }^2}}\ge {\displaystyle \frac{\delta }{\beta {(\sigma ,n)}^2+\frac{{\parallel B\parallel }^2}{{\gamma }_H(n)}\delta }}.$$

Taking the minimum over $y\ne 0$ yields the lower bound in (4.6); the upper bound is part (a). Dividing (4.6) by $\delta$ and letting $\delta \downarrow 0$ gives the stated first-order expansion.  □

Corollary 4

(Sharp first-order constant for outer offsets). Fix $\left|n\right|=1$ and let $z_0\in \partial W(A)$ be a support point in direction n, i.e. $Re(\overline{n}{z}_0)={\lambda }_{max}(H(n))$. Assume $z_0\notin spec(A)$ and ${\gamma }_H(n)>0$ as in (4.5). For $\epsilon >0$ set ${\sigma }_{\epsilon }:=z_0+\epsilon n$ and

$$P_{\epsilon }(n):=Re\left(n{({\sigma }_{\epsilon }I-A)}^{-1}\right).$$

Then

$$\underset{\epsilon \downarrow 0}{lim}{\displaystyle \frac{{\lambda }_{min}(P_{\epsilon }(n))}{\epsilon }}={∥(z_{0}I-A){|}_{\mathcal{M}(n)}∥}^{-2}.$$

(4.7)

In particular, ${\lambda }_{min}(P_{\epsilon }(n))=\epsilon \parallel (z_{0}I-A){{|}_{\mathcal{M}(n)}\parallel }^{-2}+O({\epsilon }^2)$.

Proof. 

For ${\sigma }_{\epsilon }=z_0+\epsilon n$ one has $\delta ({\sigma }_{\epsilon },n)=\epsilon$. Applying Proposition 5 (b) to $(\sigma ,n)=({\sigma }_{\epsilon },n)$ gives a two-sided bound of the form

$${\displaystyle \frac{\epsilon }{\beta {({\sigma }_{\epsilon },n)}^2+O(\epsilon )}}\le {\lambda }_{min}(P_{\epsilon }(n))\le {\displaystyle \frac{\epsilon }{\beta {({\sigma }_{\epsilon },n)}^2}}.$$

Since $z_0\notin spec(A)$, $B_{\epsilon }:={\sigma }_{\epsilon }I-A\to B_0:=z_{0}I-A$ in operator norm and $B_0$ is invertible. Hence $\beta ({\sigma }_{\epsilon },n)=\parallel B_{\epsilon }{|}_{\mathcal{M}(n)}\parallel \to \parallel B_0{|}_{\mathcal{M}(n)}\parallel$, and dividing by $\epsilon$ and letting $\epsilon \downarrow 0$ yields (4.7).  □

Proposition 6

(Global first-order slope for direction-sampled offsets). Assume that one can choose support pointscontinuously, i.e. there exists a continuous map $z_0:\{n\in \mathbb{C}:|n|=1\}\to \partial W(A)$ with $Re(\overline{n}{z}_0(n))={\lambda }_{max}(H(n))$ and such that:

(i) 

$z_0(n)\notin spec(A)$ (bounded resolvent on $\partial W(A)$), and

(ii) 

the top eigenspace $\mathcal{M}(n)$ is spectrally isolated with a uniform gap

$$\underset{\left|n\right|=1}{inf}{\gamma }_H(n)>0.$$

(4.8)

For each $\epsilon >0$ define the direction-sampled coercivity constant

$$\tilde{c}(\epsilon ):=\underset{\left|n\right|=1}{inf}{\lambda }_{min}\left(P_{\epsilon }(n)\right),P_{\epsilon }(n)=Re\left(n{\left((z_0(n)+\epsilon n)I-A\right)}^{-1}\right).$$

Set

$${\beta }_0(n):=∥(z_0(n)I-A){|}_{\mathcal{M}(n)}∥,{\beta }_{max}:=\underset{\left|n\right|=1}{sup}{\beta }_0(n).$$

Then

$$\underset{\epsilon \downarrow 0}{lim}{\displaystyle \frac{\tilde{c}(\epsilon )}{\epsilon }}={\displaystyle \frac{1}{{\beta }_{max}^2}}.$$

(4.9)

Proof. 

Under the uniform gap assumption (4.8), the spectral projector onto $\mathcal{M}(n)$ depends continuously on n. Since $n\mapsto z_0(n)$ is continuous by assumption, so is $B_0(n):=z_0(n)I-A$, and hence

$${\beta }_0(n)=∥B_0(n){|}_{\mathcal{M}(n)}∥=\parallel B_0(n)\Pi (n)\parallel$$

is continuous (where $\Pi (n)$ is the orthogonal projector onto $\mathcal{M}(n)$). In particular, ${\beta }_0$ attains its maximum ${\beta }_{max}$.

For $\epsilon >0$ set ${\sigma }_{\epsilon }(n):=z_0(n)+\epsilon n$ and $B_{\epsilon }(n):={\sigma }_{\epsilon }(n)I-A$. Since $z_0(·)$ is continuous on the compact unit circle and avoids $spec(A)$, there exists ${\epsilon }_0>0$ such that ${\sigma }_{\epsilon }(n)\notin spec(A)$ for all $\left|n\right|=1$ and all $0\le \epsilon \le {\epsilon }_0$. Proposition 5 (b) applied at $(\sigma ,n)=({\sigma }_{\epsilon }(n),n)$ (where $\delta ({\sigma }_{\epsilon }(n),n)=\epsilon$) gives, for all $0<\epsilon \le {\epsilon }_0$ and all $\left|n\right|=1$,

$${\displaystyle \frac{1}{{\beta }_{\epsilon }{(n)}^2+C\epsilon }}\le {\displaystyle \frac{{\lambda }_{min}(P_{\epsilon }(n))}{\epsilon }}\le {\displaystyle \frac{1}{{\beta }_{\epsilon }{(n)}^2}},{\beta }_{\epsilon }(n):=∥B_{\epsilon }(n){|}_{\mathcal{M}(n)}∥,$$

with a finite constant

$$C:=\underset{\begin{array}{c}\left|n\right|=1\\ 0\le \epsilon \le {\epsilon }_0\end{array}}{sup}{\displaystyle \frac{\parallel B_{\epsilon }{(n)\parallel }^2}{{\gamma }_H(n)}}<\infty$$

(using (4.8) and boundedness of $z_0$ on the unit circle). Since $(\epsilon ,n)\mapsto {\beta }_{\epsilon }(n)$ is continuous on $[0,{\epsilon }_0]\times \{n:|n|=1\}$, it is uniformly continuous, and hence ${\beta }_{\epsilon }(n)\to {\beta }_0(n)$ uniformly in n as $\epsilon \downarrow 0$. Moreover, ${\beta }_0(n)>0$ for all n, so by continuity on the compact unit circle there exists ${\beta }_{min}>0$ with ${\beta }_0(n)\ge {\beta }_{min}$ for all $\left|n\right|=1$, and therefore the same uniform convergence holds for the reciprocals. Therefore ${\lambda }_{min}(P_{\epsilon }(n))/\epsilon \to 1/{\beta }_0{(n)}^2$ uniformly in n, and taking infima over $\left|n\right|=1$ yields

$$\underset{\epsilon \downarrow 0}{lim}{\displaystyle \frac{\tilde{c}(\epsilon )}{\epsilon }}=\underset{\left|n\right|=1}{inf}{\displaystyle \frac{1}{{\beta }_0{(n)}^2}}={\displaystyle \frac{1}{{sup}_{\left|n\right|=1}{\beta }_0{(n)}^2}}={\displaystyle \frac{1}{{\beta }_{max}^2}},$$

which is (4.9).  □

4.4. Slope Spectra, Two-Scale Splitting, and Face Detection at Non-Spectral Support Points

The bounds and slope constants in Section 4.3 focus on ${\lambda }_{min}$. Under a spectral-gap hypothesis for the supporting pencil $H(n)$, one can quantify the entire cluster of eigenvalues that collapses to 0 as the support gap closes. This yields a two-scale spectral splitting (an $O(\epsilon )$ cluster plus an $O(1)$ bulk) and leads to a simple computational “face detector” based on counting small eigenvalues.

4.4.1. A Rigid Non-Tangential Offset Model

Fix a unimodular direction $n\in \mathbb{C}$ and assume that ${\lambda }_{max}(H(n))$ is isolated with multiplicity

$$m:=dim\mathcal{M}(n),{\gamma }_H:={\lambda }_{max}(H(n))-{\lambda }_{m+1}^{\downarrow }(H(n))>0(m<d),$$

(4.10)

where $\mathcal{M}(n)=Ker({\lambda }_{max}(H(n))I-H(n))$ is the maximal eigenspace of $H(n)=Re(\overline{n}A)$. Choose any unit vector $v\in \mathcal{M}(n)$ and define the associated numerical-range support point

$${\sigma }_0:=v^{\ast }Av\in \partial W(A),Re(\overline{n}{\sigma }_0)={\lambda }_{max}(H(n))$$

(4.11)

(Lemma 1). We impose the bounded-resolvent hypothesis

$${\sigma }_0\notin spec(A).$$

(4.12)

For $\epsilon >0$, define the outer offset point and the offset kernel

$${\sigma }_{\epsilon }:={\sigma }_0+\epsilon n,P_{\epsilon }:=P({\sigma }_{\epsilon },n)=Re\left(n{\left({\sigma }_{\epsilon }I-A\right)}^{-1}\right).$$

(4.13)

Let $V\in {\mathbb{C}}^{d\times m}$ have orthonormal columns spanning $\mathcal{M}(n)$ and set

$$B_0:={\sigma }_{0}I-A,G:=V^{\ast }{B}_0^{\ast }{B}_{0}V\in {\mathbb{C}}^{m\times m}.$$

(4.14)

Proposition 7

(Offset slope spectrum for the collapsing eigenvalue cluster). Under (4.10)–(4.12), $G\succ 0$ and

$$\underset{\epsilon \downarrow 0}{lim}{\displaystyle \frac{1}{\epsilon }}{\lambda }_j^{\uparrow }\left(P_{\epsilon }\right)={\lambda }_j^{\uparrow }\left(G^{-1}\right),j=1,\dots ,m.$$

(4.15)

Equivalently, if $0<g_1\le \dots \le g_m$ are the eigenvalues of G, then

$${\lambda }_j^{\uparrow }\left(P_{\epsilon }\right)={\displaystyle \frac{\epsilon }{g_{m+1-j}}}+o(\epsilon ),j=1,\dots ,m.$$

In particular, if $m=1$ and $\mathcal{M}(n)=span\left\{v\right\}$, then

$$\underset{\epsilon \downarrow 0}{lim}{\displaystyle \frac{1}{\epsilon }}{\lambda }_{min}\left(P_{\epsilon }\right)={\displaystyle \frac{1}{\parallel ({\sigma }_{0}I-A){v\parallel }^2}},$$

which recovers Corollary 4 for this rigid offset family.

Proof. 

Step 1: $G\succ 0$. Since ${\sigma }_0\notin spec(A)$, the matrix $B_0={\sigma }_{0}I-A$ is invertible. The $d\times m$ matrix V has full column rank (orthonormal columns), hence $B_{0}V$ has full column rank. Therefore the Gram matrix

$$G={(B_{0}V)}^{\ast }(B_{0}V)=V^{\ast }{B}_0^{\ast }{B}_{0}V$$

is Hermitian positive definite.

• Step 2: Congruence for the offset family. Set $B_{\epsilon }:={\sigma }_{\epsilon }I-A=B_0+\epsilon nI$ and

  $$Q_0:={\lambda }_{max}(H(n))I-H(n)⪰0.$$

  By Lemma 2 (applied at $(\sigma ,n)=({\sigma }_{\epsilon },n)$) and $Re(\overline{n}{\sigma }_{\epsilon })=Re(\overline{n}{\sigma }_0)+\epsilon ={\lambda }_{max}(H(n))+\epsilon$, we obtain

  $$B_{\epsilon }^{\ast }{P}_{\epsilon }{B}_{\epsilon }=Re(\overline{n}{\sigma }_{\epsilon })I-H(n)=Q_0+\epsilon I.$$

  (4.16)

  For sufficiently small $\epsilon$, $B_{\epsilon }$ remains invertible, hence $P_{\epsilon }\succ 0$.

• Step 3: Work with the inverse. From (4.16),

  $$P_{\epsilon }^{-1}=B_{\epsilon }{(Q_0+\epsilon I)}^{-1}{B}_{\epsilon }^{\ast }.$$

  (4.17)

  Let $\Pi :=V{V}^{\ast }$ be the orthogonal projector onto $\mathcal{M}(n)=Ker(Q_0)$. On $\mathcal{M}(n)$ one has ${(Q_0+\epsilon I)}^{-1}=(1/\epsilon )I$. On $\mathcal{M}{(n)}^{\perp }$ the gap assumption gives $Q_0⪰{\gamma }_H(I-\Pi )$ and hence

  $$\parallel {(Q_0+\epsilon I)}^{-1}(I-\Pi )\parallel \le {\displaystyle \frac{1}{{\gamma }_H}}.$$

  Thus

  $$\epsilon {(Q_0+\epsilon I)}^{-1}=\Pi +\epsilon R_{\epsilon },R_{\epsilon }:={(Q_0+\epsilon I)}^{-1}(I-\Pi ),\parallel R_{\epsilon }\parallel \le {\gamma }_H^{-1}.$$

  Multiplying (4.17) by $\epsilon$ yields

  $$\epsilon P_{\epsilon }^{-1}=B_{\epsilon }\Pi B_{\epsilon }^{\ast }+\epsilon B_{\epsilon }{R}_{\epsilon }{B}_{\epsilon }^{\ast }.$$

  (4.18)

  Since $B_{\epsilon }\to B_0$ and $\left\{R_{\epsilon }\right\}$ is uniformly bounded, we conclude that

  $$\epsilon P_{\epsilon }^{-1}⟶S:=B_0\Pi B_0^{\ast }(\epsilon \downarrow 0)$$

  (4.19)

  in operator norm.

• Step 4: Identify the nonzero spectrum of the limit. Since $S=B_{0}V{V}^{\ast }{B}_0^{\ast }=(B_{0}V){(B_{0}V)}^{\ast }$, $S⪰0$ has rank m. Its nonzero eigenvalues coincide with the eigenvalues of

  $${(B_{0}V)}^{\ast }(B_{0}V)=V^{\ast }{B}_0^{\ast }{B}_{0}V=G.$$

  Writing eigenvalues in nondecreasing order,

  $${\lambda }_{d-m+i}^{\uparrow }(S)={\lambda }_i^{\uparrow }(G),i=1,\dots ,m,{\lambda }_j^{\uparrow }(S)=0(j\le d-m).$$

• Step 5: Pass to eigenvalues and invert. By Weyl’s inequality and (4.19),

  $${\lambda }_{d-m+i}^{\uparrow }\left(\epsilon P_{\epsilon }^{-1}\right)\to {\lambda }_i^{\uparrow }(G),i=1,\dots ,m.$$

  Since $P_{\epsilon }\succ 0$, the eigenvalues of $P_{\epsilon }$ and $P_{\epsilon }^{-1}$ are reciprocal and reversed:

  $${\lambda }_j^{\uparrow }\left(P_{\epsilon }\right)={\displaystyle \frac{1}{{\lambda }_{d+1-j}^{\uparrow }\left(P_{\epsilon }^{-1}\right)}}.$$

  Hence for $j=1,\dots ,m$,

  $${\displaystyle \frac{1}{\epsilon }}{\lambda }_j^{\uparrow }\left(P_{\epsilon }\right)={\displaystyle \frac{1}{{\lambda }_{d+1-j}^{\uparrow }\left(\epsilon P_{\epsilon }^{-1}\right)}}⟶{\displaystyle \frac{1}{{\lambda }_{d+1-j}^{\uparrow }(S)}}.$$

  Now $d+1-j=d-m+(m+1-j)$, so ${\lambda }_{d+1-j}^{\uparrow }(S)={\lambda }_{m+1-j}^{\uparrow }(G)$ and therefore

  $$\underset{\epsilon \downarrow 0}{lim}{\displaystyle \frac{1}{\epsilon }}{\lambda }_j^{\uparrow }(P_{\epsilon })={\displaystyle \frac{1}{{\lambda }_{m+1-j}^{\uparrow }(G)}}={\lambda }_j^{\uparrow }(G^{-1}),$$

  which is (4.15). The case $m=1$ follows from $G=\parallel B_0{v\parallel }^2$.  □

Remark 6

(Basis invariance). Although G is defined using a particular orthonormal basis V of $\mathcal{M}(n)$, its eigenvalues (and hence the slope spectrum ${\lambda }_j^{\uparrow }(G^{-1})$) depend only on the subspace $\mathcal{M}(n)$: if $\tilde{V}=VU$ for unitary $U\in {\mathbb{C}}^{m\times m}$, then $\tilde{G}=U^{\ast }GU$ has the same spectrum.

Proposition 8

(Two-scale spectral splitting and a uniform $O(1)$ lower bound). Assume the hypotheses of Proposition 7. Let ${\mu }_1\le \dots \le {\mu }_m$ be the eigenvalues of $G^{-1}$ (the slope spectrum). Then:

(i) 

${\lambda }_j^{\uparrow }(P_{\epsilon })={\mu }_j\epsilon +o(\epsilon )$ for $j=1,\dots ,m$.

(ii) 

The remaining eigenvalues stay bounded away from 0: there exist ${\epsilon }_0>0$ and $c_{★}>0$ such that

$${\lambda }_{m+1}^{\uparrow }(P_{\epsilon })\ge c_{★}\mathrm{for}\mathrm{all}0<\epsilon \le {\epsilon }_0.$$

A concrete bound is

$$c_{★}:={\displaystyle \frac{{\gamma }_H}{2\parallel B_0{\parallel }^2}},{\epsilon }_0:=min\left\{{\displaystyle \frac{1}{2\parallel B_0^{-1}\parallel }},{\displaystyle \frac{{\gamma }_H}{4\parallel B_0{\parallel }^2{\parallel B_0^{-1}\parallel }^2}}\right\}.$$

(4.20)

Proof. 

Part (i) is exactly Proposition 7.

For (ii), define the $\epsilon =0$ kernel

$$P_0:=P({\sigma }_0,n)=Re\left(n{({\sigma }_{0}I-A)}^{-1}\right),$$

which is well-defined since ${\sigma }_0\notin spec(A)$. By Lemma 2 at $(\sigma ,n)=({\sigma }_0,n)$ and the support identity $Re(\overline{n}{\sigma }_0)={\lambda }_{max}(H(n))$,

$$B_0^{\ast }{P}_0{B}_0={\lambda }_{max}(H(n))I-H(n)=Q_0⪰0.$$

The eigenvalues of $Q_0$ are 0 (multiplicity m) and at least ${\gamma }_H$ on $\mathcal{M}{(n)}^{\perp }$ by the gap assumption. By the Courant–Fischer min–max principle (see, e.g., [10]) with the change of variables $x=B_{0}y$,

$${\lambda }_{m+1}^{\uparrow }(P_0)=\underset{dimS=m+1}{min}\underset{\begin{array}{c}x\in S\\ \parallel x\parallel =1\end{array}}{max}{x}^{\ast }{P}_{0}x=\underset{dimS=m+1}{min}\underset{\begin{array}{c}y\in B_0^{-1}S\\ y\ne 0\end{array}}{max}{\displaystyle \frac{y^{\ast }{Q}_{0}y}{\parallel B_0{y\parallel }^2}}\ge {\displaystyle \frac{1}{\parallel B_0{\parallel }^2}}{\lambda }_{m+1}^{\uparrow }(Q_0)={\displaystyle \frac{{\gamma }_H}{\parallel B_0{\parallel }^2}}.$$

Now compare $P_{\epsilon }$ to $P_0$ in operator norm. For $\epsilon \le 1/(2\parallel B_0^{-1}\parallel )$, the Neumann series gives $B_{\epsilon }^{-1}={(B_0+\epsilon nI)}^{-1}$ well-defined and

$$\parallel B_{\epsilon }^{-1}\parallel \le 2\parallel B_0^{-1}\parallel ,\parallel B_{\epsilon }^{-1}-B_0^{-1}\parallel =\parallel B_{\epsilon }^{-1}(B_0-B_{\epsilon })B_0^{-1}\parallel \le 2\epsilon \parallel B_0^{-1}{\parallel }^2.$$

Therefore

$$\parallel P_{\epsilon }-P_0\parallel =\parallel Re(n(B_{\epsilon }^{-1}-B_0^{-1}))\parallel \le \parallel B_{\epsilon }^{-1}-B_0^{-1}\parallel \le 2\epsilon \parallel B_0^{-1}{\parallel }^2.$$

Weyl’s inequality gives

$${\lambda }_{m+1}^{\uparrow }(P_{\epsilon })\ge {\lambda }_{m+1}^{\uparrow }(P_0)-\parallel P_{\epsilon }-P_0\parallel \ge {\displaystyle \frac{{\gamma }_H}{\parallel B_0{\parallel }^2}}-2\epsilon {\parallel B_0^{-1}\parallel }^2.$$

If $\epsilon \le {\gamma }_H/(4\parallel B_0{\parallel }^2\parallel B_0^{-1}{\parallel }^2)$, then the last term is $\le {\gamma }_H/(2\parallel B_0{\parallel }^2)$, and we obtain

$${\lambda }_{m+1}^{\uparrow }(P_{\epsilon })\ge {\displaystyle \frac{{\gamma }_H}{2\parallel B_0{\parallel }^2}}=c_{★},$$

with the explicit choices (4.20).  □

Corollary 5

(A rigorous “face detector” threshold). Assume the hypotheses of Proposition 7. Let ${\mu }_{max}:={\mu }_m={\lambda }_{max}(G^{-1})$. Fix any $\tau >{\mu }_{max}$. Then there exists ${\epsilon }_{\tau }>0$ such that for all $0<\epsilon \le {\epsilon }_{\tau }$,

$$\#\left\{j:{\lambda }_j^{\uparrow }(P_{\epsilon })\le \tau \epsilon \right\}=m.$$

Proof. 

By Proposition 7, for each $j\le m$, ${\lambda }_j^{\uparrow }(P_{\epsilon })/\epsilon \to {\mu }_j\le {\mu }_{max}<\tau$, so for sufficiently small $\epsilon$ one has ${\lambda }_j^{\uparrow }(P_{\epsilon })\le \tau \epsilon$ for all $j\le m$. By Proposition 8 (ii), ${\lambda }_{m+1}^{\uparrow }(P_{\epsilon })\ge c_{★}>0$ for small $\epsilon$, so ${\lambda }_{m+1}^{\uparrow }(P_{\epsilon })>\tau \epsilon$ for all sufficiently small $\epsilon$. This implies the stated count.  □

4.4.2. Intrinsic Rescaled Clusters Under General $C^1$ Convex Exhaustions

The offset analysis above uses the rigid non-tangential approach ${\sigma }_{\epsilon }={\sigma }_0+\epsilon n$. Along a general $C^1$ convex exhaustion ${\Omega }_{\epsilon }\downarrow W(A)$, the natural small parameter is the support gap $\delta =Re(\overline{n_{\Omega }(\sigma )}\sigma )-{\lambda }_{max}(H(n_{\Omega }(\sigma )))$. After rescaling by $\delta$, the slope spectrum is intrinsic (independent of the exhaustion).

Proposition 9

(Intrinsic rescaled collapsing cluster under arbitrary $C^1$ exhaustions). Let $\left\{{\Omega }_{\epsilon }\right\}$ be a $C^1$ convex exhaustion of $W(A)$. Let ${\epsilon }_k\downarrow 0$ and let ${\sigma }_k\in \partial {\Omega }_{{\epsilon }_k}$ satisfy

$${\sigma }_k\to {\sigma }_0\in \partial W(A)\setminus spec(A),n_k:=n_{{\Omega }_{{\epsilon }_k}}({\sigma }_k)\to n,|n_k|=|n|=1.$$

Assume that ${\lambda }_{max}(H(n))$ is isolated with multiplicity

$$m:=dim\mathcal{M}(n),\mathcal{M}(n)=Ker({\lambda }_{max}(H(n))I-H(n)),$$

and gap

$${\gamma }_H:={\lambda }_{max}(H(n))-{\lambda }_{m+1}^{\downarrow }(H(n))>0.$$

Define the support gap

$${\delta }_k:=Re(\overline{n_k}{\sigma }_k)-{\lambda }_{max}(H(n_k))\ge 0,$$

and assume ${\delta }_k>0$ for all sufficiently large k.

Let $V\in {\mathbb{C}}^{d\times m}$ have orthonormal columns spanning $\mathcal{M}(n)$ and set

$$B_0:={\sigma }_{0}I-A,G:=V^{\ast }{B}_0^{\ast }{B}_{0}V.$$

Then $G\succ 0$ and for each $j=1,\dots ,m$,

$$\underset{k\to \infty }{lim}{\displaystyle \frac{1}{{\delta }_k}}{\lambda }_j^{\uparrow }\left(P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A)\right)={\lambda }_j^{\uparrow }\left(G^{-1}\right).$$

(4.21)

Moreover, the remaining eigenvalues stay uniformly bounded away from 0: there exist $c_{★}>0$ and $k_0$ such that

$${\lambda }_{m+1}^{\uparrow }\left(P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A)\right)\ge c_{★}\mathrm{for}\mathrm{all}k\ge k_0.$$

Proof. 

Set $P_k:=P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A)$. Since ${\sigma }_0\notin spec(A)$ and $spec(A)$ is finite, $dist({\sigma }_0,spec(A))>0$ and thus ${\sigma }_k\notin spec(A)$ for all sufficiently large k. In particular, $B_k:={\sigma }_{k}I-A$ is invertible for large k.

• Step 1: Positivity of G. As in Proposition 7, $B_0$ is invertible and V has full column rank, hence $B_{0}V$ has full column rank and $G={(B_{0}V)}^{\ast }(B_{0}V)\succ 0$.

• Step 2: Congruence at $({\sigma }_k,n_k)$ and a useful bound ${\delta }_k\to 0$. Because ${\Omega }_{{\epsilon }_k}$ is $C^1$ convex and $n_k$ is the outward unit normal at ${\sigma }_k\in \partial {\Omega }_{{\epsilon }_k}$,

  $$P_k=Re\left(n_k{({\sigma }_{k}I-A)}^{-1}\right)=:P({\sigma }_k,n_k).$$

  Let $H_k:=H(n_k)=Re(\overline{n_k}A)$ and define

  $$Q_k:={\lambda }_{max}(H_k)I-H_k⪰0.$$

  Applying Lemma 2 with $(\sigma ,n)=({\sigma }_k,n_k)$ yields

  $$B_k^{\ast }{P}_k{B}_k=Re(\overline{n_k}{\sigma }_k)I-H_k=Q_k+{\delta }_{k}I.$$

  (4.22)

  Since ${\sigma }_k\to {\sigma }_0\in W(A)$, we have the estimate

  $$0\le {\delta }_k=Re(\overline{n_k}{\sigma }_k)-\underset{z\in W(A)}{max}Re(\overline{n_k}z)\le Re(\overline{n_k}{\sigma }_k)-Re(\overline{n_k}{\sigma }_0)\le |{\sigma }_k-{\sigma }_0|,$$

  (4.23)

  hence ${\delta }_k\to 0$.

• Step 3: Uniform gap persistence and convergence of spectral projectors. Since $n_k\to n$ and $H(·)$ is continuous, $H_k\to H:=H(n)$ in operator norm. By Weyl’s inequality for Hermitian matrices,

  $$|{\lambda }_j(H_k)-{\lambda }_j(H)|\le \parallel H_k-H\parallel (j=1,\dots ,d),$$

  so for all sufficiently large k the top eigenvalue cluster of $H_k$ has the same multiplicity m and a uniform gap

  $${\gamma }_k:={\lambda }_{max}(H_k)-{\lambda }_{m+1}^{\downarrow }(H_k)\ge {\gamma }_H/2.$$

  (4.24)

  Let ${\Pi }_k$ be the orthogonal projector onto $\mathcal{M}(n_k)=Ker(Q_k)$ and $\Pi :=V{V}^{\ast }$ the orthogonal projector onto $\mathcal{M}(n)$. Standard Riesz projector arguments (cf. [10]) yield $\parallel {\Pi }_k-\Pi \parallel \to 0$.

• Step 4: Work with the inverse and rescale by the support gap. From (4.22) and invertibility of $B_k$,

  $$P_k^{-1}=B_k{(Q_k+{\delta }_{k}I)}^{-1}{B}_k^{\ast }.$$

  (4.25)

  On $Ker(Q_k)=ran({\Pi }_k)$ one has ${(Q_k+{\delta }_{k}I)}^{-1}=(1/{\delta }_k)I$, so

  $${\delta }_k{(Q_k+{\delta }_{k}I)}^{-1}{\Pi }_k={\Pi }_k.$$

  On $ran(I-{\Pi }_k)$, the gap bound (4.24) implies $Q_k⪰{\gamma }_k(I-{\Pi }_k)$ and hence

  $$\parallel {(Q_k+{\delta }_{k}I)}^{-1}(I-{\Pi }_k)\parallel \le {\displaystyle \frac{1}{{\gamma }_k}}\le {\displaystyle \frac{2}{{\gamma }_H}}.$$

  Therefore

  $${\delta }_k{(Q_k+{\delta }_{k}I)}^{-1}={\Pi }_k+{\delta }_k{R}_k,R_k:={(Q_k+{\delta }_{k}I)}^{-1}(I-{\Pi }_k),\parallel R_k\parallel \le {\displaystyle \frac{2}{{\gamma }_H}}.$$

  Multiplying (4.25) by ${\delta }_k$ gives

  $${\delta }_k{P}_k^{-1}=B_k{\Pi }_k{B}_k^{\ast }+{\delta }_k{B}_k{R}_k{B}_k^{\ast }.$$

  (4.26)

• Step 5: Take the limit $k\to \infty$ and identify the spectrum. Since $B_k\to B_0$ and ${\Pi }_k\to \Pi$ in operator norm, we have

  $${\delta }_k{P}_k^{-1}⟶S:=B_0\Pi B_0^{\ast }\mathrm{in}\mathrm{operator}\mathrm{norm}.$$

  As in Proposition 7, $S=(B_{0}V){(B_{0}V)}^{\ast }$ has rank m and its nonzero eigenvalues coincide with those of $G={(B_{0}V)}^{\ast }(B_{0}V)$.

• Step 6: Invert the corresponding eigenvalues. By Weyl’s inequality and the reciprocity of eigenvalues of $P_k$ and $P_k^{-1}$, one obtains (4.21) exactly as in the offset proof. The uniform lower bound for ${\lambda }_{m+1}^{\uparrow }(P_k)$ follows by the Courant–Fischer principle and the uniform gap bound ${\gamma }_k\ge {\gamma }_H/2$, together with boundedness of $\parallel B_k\parallel$ for large k.  □

Corollary 6

(Exhaustion-invariant face-detector threshold). Assume the hypotheses of Proposition 9 and let ${\mu }_{max}:={\lambda }_{max}(G^{-1})$. Fix any $\tau >{\mu }_{max}$. Then there exists $k_{\tau }$ such that for all $k\ge k_{\tau }$,

$$\#\left\{j:{\lambda }_j^{\uparrow }\left(P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A)\right)\le \tau {\delta }_k\right\}=m.$$

Proof. 

By (4.21), for each $j\le m$, ${\lambda }_j^{\uparrow }(P_k)/{\delta }_k\to {\lambda }_j^{\uparrow }(G^{-1})\le {\mu }_{max}<\tau$, hence ${\lambda }_j^{\uparrow }(P_k)\le \tau {\delta }_k$ for all sufficiently large k. The uniform bound in Proposition 9 gives ${\lambda }_{m+1}^{\uparrow }(P_k)\ge c_{★}>0$ for large k. Since ${\delta }_k\to 0$, eventually $\tau {\delta }_k<c_{★}$, hence ${\lambda }_{m+1}^{\uparrow }(P_k)>\tau {\delta }_k$ for large k. This yields the count.  □

Remark 7

(Geometric meaning of the support gap). For a convex exhaustion ${\Omega }_{\epsilon }\downarrow W(A)$, the scalar

$${\delta }_k=Re(\overline{n_k}{\sigma }_k)-{\lambda }_{max}(H(n_k))$$

is the support-function mismatch between ${\Omega }_{{\epsilon }_k}$ and $W(A)$ in direction $n_k$. Support functions and their stability properties are classical in convex geometry; see, e.g., [12].

4.5. Convergence of the near-Kernel Subspace

Proposition 10

(Convergence of the near-kernel spectral projector). Assume the setting of Theorem 1 and set $m:=dim\mathcal{M}(n)$. Assume that ${\lambda }_{max}(H(n))$ is isolated with multiplicity m, i.e.

$${\gamma }_H:={\lambda }_{max}(H(n))-{\lambda }_{m+1}^{\downarrow }(H(n))>0.$$

(4.27)

Let

$$P_0:=Re\left(n{({\sigma }_{0}I-A)}^{-1}\right),{\mathcal{K}}_0:=Ker(P_0)=({\sigma }_{0}I-A)\mathcal{M}(n),{\Pi }_0:{\mathbb{C}}^d\to {\mathcal{K}}_0$$

be the orthogonal projector onto ${\mathcal{K}}_0$.

For each k, let $P_k:=P_{{\Omega }_{{\epsilon }_k}}({\sigma }_k,A)$ and let ${\Pi }_k$ be the orthogonal projector onto the direct sum of the eigenspaces of $P_k$ corresponding to its m smallest eigenvalues. Then $\parallel {\Pi }_k-{\Pi }_0\parallel \to 0$ as $k\to \infty$.

Moreover, writing $B_0={\sigma }_{0}I-A$, one has the explicit spectral-gap bound

$${\lambda }_{m+1}^{\uparrow }(P_0)\ge {\displaystyle \frac{{\gamma }_H}{\parallel B_0{\parallel }^2}},$$

(4.28)

and consequently, for all sufficiently large k,

$$\parallel {\Pi }_k-{\Pi }_0\parallel \le {\displaystyle \frac{2\parallel P_k-P_0\parallel }{{\lambda }_{m+1}^{\uparrow }(P_0)}}\le {\displaystyle \frac{2\parallel B_0{\parallel }^2}{{\gamma }_H}}\parallel P_k-P_0\parallel .$$

(4.29)

Proof. 

By Lemma 2 and Step 2 of Theorem 1,

$$B_0^{\ast }{P}_0{B}_0=Q_0:={\lambda }_{max}(H(n))I-H(n)⪰0.$$

The eigenvalues of $Q_0$ are 0 with multiplicity m and at least ${\gamma }_H$ on $\mathcal{M}{(n)}^{\perp }$, so ${\lambda }_{m+1}^{\uparrow }(Q_0)={\gamma }_H$.

Using the Courant–Fischer characterization with the change of variables $x=B_{0}y$, one obtains for every j

$${\lambda }_j^{\uparrow }(P_0)=\underset{dimS=j}{min}\underset{\begin{array}{c}x\in S\\ \parallel x\parallel =1\end{array}}{max}{x}^{\ast }{P}_{0}x=\underset{dimS=j}{min}\underset{\begin{array}{c}y\in B_0^{-1}S\\ y\ne 0\end{array}}{max}{\displaystyle \frac{y^{\ast }{Q}_{0}y}{\parallel B_0{y\parallel }^2}}\ge {\displaystyle \frac{1}{\parallel B_0{\parallel }^2}}{\lambda }_j^{\uparrow }(Q_0).$$

Taking $j=m+1$ gives (4.28).

Next, Theorem 1 gives $\parallel P_k-P_0\parallel \to 0$. Since $P_0$ has an isolated cluster of m eigenvalues at 0 separated by the gap ${\lambda }_{m+1}^{\uparrow }(P_0)>0$, the Davis–Kahan $sin\Theta$ theorem for invariant subspaces [11] yields (4.29), and hence $\parallel {\Pi }_k-{\Pi }_0\parallel \to 0$.  □

4.6. The Spectral-Support Regime for Normal Matrices

Proposition 11

(Normal matrices: explicit eigenvalues near a spectral support point). Let A be normal with eigenvalues ${\lambda }_1,\dots ,{\lambda }_d$ (listed with algebraic multiplicity). Fix $\sigma \notin spec(A)$ and unimodular $n\in \mathbb{C}$. Then

$$P(\sigma ,n):=Re\left(n{(\sigma I-A)}^{-1}\right)$$

is unitarily diagonalizable and its eigenvalues are the scalars

$$p_j(\sigma ,n):=Re\left({\displaystyle \frac{n}{\sigma -{\lambda }_j}}\right)={\displaystyle \frac{Re(\overline{n}(\sigma -{\lambda }_j))}{|\sigma -{\lambda }_j{|}^2}},j=1,\dots ,d.$$

(4.30)

Now fix ${\sigma }_0\in spec(A)$ and let

$$J_0:=\{j\in \{1,\dots ,d\}:{\lambda }_j={\sigma }_0\}.$$

Let ${\sigma }_k\notin spec(A)$ and unimodular $n_k$ satisfy

$${\sigma }_k\to {\sigma }_0,n_k\to n.$$

Write $p_{k,j}:=p_j({\sigma }_k,n_k)$. Then:

(i) 

For every $j\notin J_0$,

$$p_{k,j}⟶p_j({\sigma }_0,n)=Re\left({\displaystyle \frac{n}{{\sigma }_0-{\lambda }_j}}\right).$$

(ii) 

For every $j\in J_0$ one has theexactidentity

$$p_{k,j}=Re\left({\displaystyle \frac{n_k}{{\sigma }_k-{\sigma }_0}}\right)={\displaystyle \frac{Re(\overline{n_k}({\sigma }_k-{\sigma }_0))}{|{\sigma }_k-{\sigma }_0{|}^2}}.$$

In particular, if there exists $c>0$ such that

$$Re(\overline{n_k}({\sigma }_k-{\sigma }_0))\ge c|{\sigma }_k-{\sigma }_0|\mathrm{for}\mathrm{all}\mathrm{sufficiently}\mathrm{large}k,$$

(4.31)

then $p_{k,j}\to +\infty$ for every $j\in J_0$.

(iii) 

Assume in addition that n is a supporting direction for $W(A)=\mathrm{conv}\{{\lambda }_1,\dots ,{\lambda }_d\}$ at ${\sigma }_0$, i.e.

$$Re(\overline{n}{\lambda }_j)\le Re(\overline{n}{\sigma }_0),j=1,\dots ,d.$$

(4.32)

Then for every $j\notin J_0$,

$$p_j({\sigma }_0,n)={\displaystyle \frac{Re(\overline{n}({\sigma }_0-{\lambda }_j))}{|{\sigma }_0-{\lambda }_j{|}^2}}={\displaystyle \frac{Re(\overline{n}{\sigma }_0)-Re(\overline{n}{\lambda }_j)}{|{\sigma }_0-{\lambda }_j{|}^2}}\ge 0,$$

and $p_j({\sigma }_0,n)=0$ if and only if ${\lambda }_j$ lies on the same supporting line $\{z:Re(\overline{n}z)=Re(\overline{n}{\sigma }_0)\}$. If moreover (4.31) holds (so that all $p_{k,j}\to +\infty$ for $j\in J_0$), then

$${\lambda }_{min}\left(P({\sigma }_k,n_k)\right)⟶\underset{j\notin J_0}{min}{p}_j({\sigma }_0,n),$$

which is strictly positive if and only if no eigenvalue ${\lambda }_j\ne {\sigma }_0$ lies on the supporting line $\{z:Re(\overline{n}z)=Re(\overline{n}{\sigma }_0)\}$.

Proof. 

Since A is normal, $A=U\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_d)U^{\ast }$ for some unitary U, hence

$${(\sigma I-A)}^{-1}=U\mathrm{diag}\left({\displaystyle \frac{1}{\sigma -{\lambda }_1}},\dots ,{\displaystyle \frac{1}{\sigma -{\lambda }_d}}\right)U^{\ast }.$$

Therefore,

$$\begin{array}{cc}\hfill P(\sigma ,n)& =Re\left(n{(\sigma I-A)}^{-1}\right)\hfill \\ & =URe\left(\mathrm{diag}\left({\displaystyle \frac{n}{\sigma -{\lambda }_1}},\dots ,{\displaystyle \frac{n}{\sigma -{\lambda }_d}}\right)\right)U^{\ast }\hfill \\ & =U\mathrm{diag}\left(Re\left({\displaystyle \frac{n}{\sigma -{\lambda }_1}}\right),\dots ,Re\left({\displaystyle \frac{n}{\sigma -{\lambda }_d}}\right)\right)U^{\ast },\hfill \end{array}$$

which proves (4.30). The limit in (i) follows by continuity of the map $(\sigma ,n)\mapsto Re\left(n/(\sigma -{\lambda }_j)\right)$ when ${\sigma }_0\ne {\lambda }_j$. For (ii), if ${\lambda }_j={\sigma }_0$ then

$$Re\left({\displaystyle \frac{n_k}{{\sigma }_k-{\sigma }_0}}\right)=Re\left({\displaystyle \frac{n_k\overline{{\sigma }_k-{\sigma }_0}}{|{\sigma }_k-{\sigma }_0{|}^2}}\right)={\displaystyle \frac{Re(\overline{n_k}({\sigma }_k-{\sigma }_0))}{|{\sigma }_k-{\sigma }_0{|}^2}},$$

and (4.31) implies $p_{k,j}\ge c/|{\sigma }_k-{\sigma }_0|\to +\infty$.

Finally, (4.32) implies $Re(\overline{n}({\sigma }_0-{\lambda }_j))\ge 0$ for all j, giving the nonnegativity (and the characterization of equality) in (iii). If additionally (4.31) holds, then $p_{k,j}\to +\infty$ for all $j\in J_0$ while $p_{k,j}\to p_j({\sigma }_0,n)\in [0,\infty )$ for $j\notin J_0$, so for large k the minimum eigenvalue is attained among indices $j\notin J_0$, yielding the stated limit and positivity criterion.  □

Definition 3

(Poisson degeneracy exponent for normal matrices). Let A be normal and let ${\sigma }_0\in spec(A)\cap \partial W(A)$ be a spectral support point with supporting direction n, i.e. $\left|n\right|=1$ and $Re(\overline{n}{\sigma }_0)={\lambda }_{max}(H(n))$. For the non-tangential offset family ${\sigma }_{\epsilon }={\sigma }_0+\epsilon n$ ($\epsilon >0$), define thePoisson degeneracy exponent

$$\alpha ({\sigma }_0,n):=\underset{\epsilon \downarrow 0}{lim}{\displaystyle \frac{log{\lambda }_{min}\left(P({\sigma }_{\epsilon },n)\right)}{log\epsilon }},$$

whenever the limit exists (note that $log\epsilon \to -\infty$).

Corollary 7

(Normal matrices: a dichotomy for the exponent). Let A be normal with eigenvalues ${\left\{{\lambda }_j\right\}}_{j=1}^d$. Fix a spectral support point ${\sigma }_0\in spec(A)\cap \partial W(A)$ and a supporting direction n. Let $J_0:=\{j:Re(\overline{n}{\lambda }_j)={\lambda }_{max}(H(n))\}$ be the set of eigenvalues on the supporting line. Then:

(i) 

If $J_0=\left\{j_0\right\}$ and ${\lambda }_{j_0}={\sigma }_0$ (i.e. no other eigenvalue lies on the supporting line), then ${\lambda }_{min}(P({\sigma }_0+\epsilon n,n))\to c_0>0$ as $\epsilon \downarrow 0$ and $\alpha ({\sigma }_0,n)=0$.

(ii) 

If there exists $j\in J_0$ with ${\lambda }_j\ne {\sigma }_0$ (equivalently, the supporting face contains at least two eigenvalues), then

$${\lambda }_{min}\left(P({\sigma }_0+\epsilon n,n)\right)=C\epsilon +o(\epsilon ),C=\underset{\begin{array}{c}j\in J_0\\ {\lambda }_j\ne {\sigma }_0\end{array}}{min}{\displaystyle \frac{1}{|{\sigma }_0-{\lambda }_j{|}^2}},$$

and $\alpha ({\sigma }_0,n)=1$.

Proof. 

By Proposition 11, for $\sigma ={\sigma }_0+\epsilon n$ the eigenvalues of $P(\sigma ,n)$ are the scalars

$$p_j(\sigma ,n)={\displaystyle \frac{Re(\overline{n}(\sigma -{\lambda }_j))}{|\sigma -{\lambda }_j{|}^2}}.$$

If $J_0=\left\{j_0\right\}$ with ${\lambda }_{j_0}={\sigma }_0$, then for all $j\ne j_0$ one has $Re(\overline{n}({\sigma }_0-{\lambda }_j))>0$, hence $p_j({\sigma }_0,n)>0$. Continuity gives ${min}_{j\ne j_0}{p}_j({\sigma }_0+\epsilon n,n)\to {min}_{j\ne j_0}{p}_j({\sigma }_0,n)=:c_0>0$, while $p_{j_0}({\sigma }_0+\epsilon n,n)=1/\epsilon \to \infty$. Thus ${\lambda }_{min}(P({\sigma }_0+\epsilon n,n))\to c_0$ and $\alpha ({\sigma }_0,n)=0$.

If there exists $j\in J_0$ with ${\lambda }_j\ne {\sigma }_0$, then $Re(\overline{n}({\sigma }_0-{\lambda }_j))=0$ and

$$p_j({\sigma }_0+\epsilon n,n)={\displaystyle \frac{\epsilon }{|{\sigma }_0-{\lambda }_j{+\epsilon n|}^2}}={\displaystyle \frac{\epsilon }{|{\sigma }_0-{\lambda }_j{|}^2}}+o(\epsilon ).$$

All $j\notin J_0$ give $p_j({\sigma }_0,n)>0$, hence contribute $O(1)$ values as $\epsilon \downarrow 0$. Therefore the minimum is attained among $j\in J_0$ with ${\lambda }_j\ne {\sigma }_0$, yielding the stated C and $\alpha ({\sigma }_0,n)=1$.  □

Example 1

(Nondegeneracy at a spectral support point). Let $A=\mathrm{diag}(0,1)$, so $W(A)=[0,1]$. Take $\sigma =1+\epsilon$ with $\epsilon >0$ and $n=1$. Then

$$P(\sigma ,n)=Re\left({(\sigma I-A)}^{-1}\right)=\mathrm{diag}\left({\displaystyle \frac{1}{1+\epsilon }},{\displaystyle \frac{1}{\epsilon }}\right),$$

so ${\lambda }_{min}(P(\sigma ,n))={\displaystyle \frac{1}{1+\epsilon }}\to 1$ as $\epsilon \downarrow 0$. Thus the smallest eigenvalue doesnotdegenerate when the limiting support point is spectral and unique on the support face.

Example 2

(Degeneracy at a spectral point with a flat support face). Let $A=\mathrm{diag}(1,1+i)$ and take $\sigma =1+\epsilon$, $n=1$. Then

$$P(\sigma ,n)=\mathrm{diag}\left({\displaystyle \frac{1}{\epsilon }},Re\left({\displaystyle \frac{1}{\epsilon -i}}\right)\right)=\mathrm{diag}\left({\displaystyle \frac{1}{\epsilon }},{\displaystyle \frac{\epsilon }{{\epsilon }^2+1}}\right),$$

so ${\lambda }_{min}(P(\sigma ,n))={\displaystyle \frac{\epsilon }{{\epsilon }^2+1}}\sim \epsilon \downarrow 0$. Here the supporting functional $Re(z)$ is maximized by more than one eigenvalue, and degeneracy persists at ${\sigma }_0=1\in spec(A)$.

4.7. Spectral Support Points for Nonnormal Matrices: A Three-Scale Splitting

We now turn to the spectral-support regime, where the boundary point is itself an eigenvalue:

$${\sigma }_0\in spec(A)\cap \partial W(A).$$

In contrast to the non-spectral case, $P(\sigma ,n)$ may develop both a collapsing cluster and an exact blow-up as $\sigma \to {\sigma }_0$ along non-tangential offsets.

Fix a supporting direction n with $\left|n\right|=1$ and

$$Re(\overline{n}{\sigma }_0)={\lambda }_{max}(H(n)),H(n)=Re(\overline{n}A),$$

(4.33)

and assume the same spectral-isolation hypothesis (4.10) for ${\lambda }_{max}(H(n))$ with multiplicity $m=dim\mathcal{M}(n)$ and gap ${\gamma }_H>0$. For $\epsilon >0$ define ${\sigma }_{\epsilon }={\sigma }_0+\epsilon n$ and $P_{\epsilon }=P({\sigma }_{\epsilon },n)$ as in (4.13).

Let

$$E:=Ker({\sigma }_{0}I-A),r:=dimE\ge 1.$$

Lemma 7

(Spectral eigenspace inclusion). Under (4.33), one has $E\subseteq \mathcal{M}(n)$.

Proof. 

Let $0\ne v\in E$, so $Av={\sigma }_{0}v$. Then

$$v^{\ast }H(n)v=Re(\overline{n}{v}^{\ast }Av)=Re(\overline{n}{\sigma }_0){\parallel v\parallel }^2={\lambda }_{max}(H(n)){\parallel v\parallel }^2.$$

Since ${\lambda }_{max}(H(n))$ is the maximal Rayleigh quotient of $H(n)$, this forces $v\in \mathcal{M}(n)$.  □

Set $F:=\mathcal{M}(n)\ominus E$, so $dimF=m-r$ (possibly 0), and choose matrices with orthonormal columns $V_E\in {\mathbb{C}}^{d\times r}$ spanning E and $V_F\in {\mathbb{C}}^{d\times (m-r)}$ spanning F. Define

$$B_0:={\sigma }_{0}I-A,G_F:=V_F^{\ast }{B}_0^{\ast }{B}_0{V}_F\in {\mathbb{C}}^{(m-r)\times (m-r)}.$$

(4.34)

Proposition 12

(Three-scale splitting at a spectral support point). Assume (4.33) and the gap hypothesis (4.10) for $H(n)$. For $\epsilon >0$ sufficiently small, $P_{\epsilon }\succ 0$ and:

(i) 

Exact blow-up on the geometric eigenspace. For every $v\in E$,

$$P_{\epsilon }v={\displaystyle \frac{1}{\epsilon }}v.$$

(4.35)

In particular, $P_{\epsilon }$ has an eigenvalue $1/\epsilon$ with multiplicity at least r.

(ii) 

$O(\epsilon )$ collapsing cluster on $\mathcal{M}(n)\ominus E$. If $m>r$, then $G_F\succ 0$ and

$$\underset{\epsilon \downarrow 0}{lim}{\displaystyle \frac{1}{\epsilon }}{\lambda }_j^{\uparrow }(P_{\epsilon })={\lambda }_j^{\uparrow }(G_F^{-1}),j=1,\dots ,m-r.$$

(4.36)

Equivalently, the $m-r$ smallest eigenvalues collapse linearly with slopes given by the eigenvalues of $G_F^{-1}$.

(iii) 

$O(1)$ bulk separated from 0.There exist ${\epsilon }_0>0$ and $c_{★}>0$ such that

$${\lambda }_{m-r+1}^{\uparrow }(P_{\epsilon })\ge c_{★}\mathrm{for}\mathrm{all}0<\epsilon \le {\epsilon }_0.$$

Proof. 

(i). If $v\in E$, then $({\sigma }_{\epsilon }I-A)v=({\sigma }_0+\epsilon n-{\sigma }_0)v=\epsilon nv$ and hence ${({\sigma }_{\epsilon }I-A)}^{-1}v=(1/(\epsilon n))v$. Therefore

$$P_{\epsilon }v=Re\left(n·{\displaystyle \frac{1}{\epsilon n}}v\right)={\displaystyle \frac{1}{\epsilon }}v,$$

which is (4.35).

(ii) Assume $m>r$. Since $F\cap Ker(B_0)=\left\{0\right\}$, the restriction of $B_0$ to F is injective and $B_0{V}_F$ has full column rank, hence $G_F\succ 0$.

Let $\Pi :=V_E{V}_E^{\ast }+V_F{V}_F^{\ast }$ be the orthogonal projector onto $\mathcal{M}(n)=E\oplus F$. The congruence identity (Lemma 2) at $(\sigma ,n)=({\sigma }_{\epsilon },n)$ yields

$$B_{\epsilon }^{\ast }{P}_{\epsilon }{B}_{\epsilon }=Re(\overline{n}{\sigma }_{\epsilon })I-H(n)=({\lambda }_{max}(H(n))I-H(n))+\epsilon I=Q_0+\epsilon I,$$

where $B_{\epsilon }={\sigma }_{\epsilon }I-A$ and $Q_0⪰0$ has kernel $\mathcal{M}(n)$. Exactly as in (4.18)–(4.19), one obtains the operator-norm limit

$$\epsilon P_{\epsilon }^{-1}⟶S:=B_0\Pi B_0^{\ast }(\epsilon \downarrow 0).$$

Since $B_0$ annihilates E and is injective on F, $rank(S)=m-r$ and the nonzero eigenvalues of S coincide with the eigenvalues of

$${(B_0{V}_F)}^{\ast }(B_0{V}_F)=V_F^{\ast }{B}_0^{\ast }{B}_0{V}_F=G_F.$$

The eigenvalue convergence (4.36) then follows by the same inversion/reversal argument as in Proposition 7.

(iii) The lower bound is obtained by the Courant–Fischer principle exactly as in Proposition 8 (ii), using that the multiplicity of the kernel of $Q_0$ is m and the gap is ${\gamma }_H>0$ on $\mathcal{M}{(n)}^{\perp }$.  □

Remark 8

(Defective eigenvalues and higher-order blow-up). Proposition 12 isolates the contribution of thegeometriceigenspace $E=Ker({\sigma }_{0}I-A)$, producing an exact $1/\epsilon$ blow-up along non-tangential offsets. If ${\sigma }_0$ isdefective(nontrivial Jordan chains), generalized eigenvectors can lead to stronger growth (typically $1/{\epsilon }^p$ along a Jordan block of length p), and the interaction between Jordan structure and the Hermitian support pencil is a pseudospectral phenomenon; see, e.g., [13].

4.8. A Fully Explicit $2\times 2$ Example: A Nilpotent Jordan Block

Example 3

(Exact Poisson kernel and exact degeneracy rate for a disk exhaustion). Let

$$A=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right).$$

Then $W(A)=\{z\in \mathbb{C}:|z|\le \frac{1}{2}\}$. For $r>\frac{1}{2}$, let ${\Omega }_r:=\{z\in \mathbb{C}:|z|<r\}$ and choose $\sigma =r{e}^{it}\in \partial {\Omega }_r$. The outward normal at σ is $n_{{\Omega }_r}(\sigma )=e^{it}$ and

$${(\sigma I-A)}^{-1}=\left(\begin{array}{cc}{\displaystyle \frac{1}{\sigma }}& {\displaystyle \frac{1}{{\sigma }^2}}\\ 0& {\displaystyle \frac{1}{\sigma }}\end{array}\right).$$

Hence

$$P_{{\Omega }_r}(\sigma ,A)=Re\left(e^{it}{(\sigma I-A)}^{-1}\right)=\left(\begin{array}{cc}{\displaystyle \frac{1}{r}}& {\displaystyle \frac{e^{-it}}{2r^2}}\\ {\displaystyle \frac{e^{it}}{2r^2}}& {\displaystyle \frac{1}{r}}\end{array}\right),$$

whose eigenvalues are ${\lambda }_{\pm }(r)={\displaystyle \frac{2r\pm 1}{2r^2}}$. In particular,

$${\lambda }_{min}\left(P_{{\Omega }_r}(\sigma ,A)\right)={\displaystyle \frac{2r-1}{2r^2}}={\displaystyle \frac{r-\frac{1}{2}}{r^2}},$$

so the degeneracy islinearas $r\downarrow \frac{1}{2}$.

Moreover, a min-eigenvector is $u(r,t)\propto {(-e^{-it},1)}^{\top }$ (independent of r). For the support direction $n=e^{it}$,

$$H(n)=Re(\overline{n}A)={\displaystyle \frac{1}{2}}\left(\begin{array}{cc}0& e^{-it}\\ e^{it}& 0\end{array}\right),\mathcal{M}(n)=span\left\{{(e^{-it},1)}^{\top }\right\}.$$

At ${\sigma }_0=\frac{1}{2}{e}^{it}\in \partial W(A)$,

$$({\sigma }_{0}I-A){(e^{-it},1)}^{\top }=\frac{1}{2}{(-1,e^{it})}^{\top }\propto {(-e^{-it},1)}^{\top },$$

in agreement with Theorem 1.

4.9. Numerical Experiments

This section provides numerical illustrations of: (i) the linear degeneracy predicted by Corollary 3 (and, in offset form, Proposition 2), (ii) local sharpness and improved bounds (Proposition 5 and Corollary 4), (iii) global coercivity collapse and its first-order slope under direction sampling (Corollary 2 and Proposition 6), and (iv) the contrasting behavior at spectral support points for normal matrices (Proposition 11 and Examples 1–2).

• Sampling model for an “outer offset” exhaustion. Fix a unimodular direction $n\in \mathbb{C}$. Let $H(n)=Re(\overline{n}A)$ and let $v\in \mathcal{M}(n)$ be a unit vector in the maximal eigenspace of $H(n)$ (Lemma 1). The corresponding numerical-range support point is

  $$z_0(n):=v^{\ast }Av\in \partial W(A),Re(\overline{n}{z}_0(n))={\lambda }_{max}(H(n)).$$

  For $\epsilon >0$ we define the offset boundary point

  $${\sigma }_{\epsilon }(n):=z_0(n)+\epsilon n.$$

  Then $Re(\overline{n}{\sigma }_{\epsilon }(n))={\lambda }_{max}(H(n))+\epsilon$, so the support gap equals $\delta ({\sigma }_{\epsilon }(n),n)=\epsilon$ (cf. Proposition 2). Moreover, ${\sigma }_{\epsilon }(n)\notin W(A)$, hence ${\sigma }_{\epsilon }(n)\notin spec(A)$ (because $spec(A)\subset W(A)$; Remark 1), so the resolvent is well-defined.

We evaluate the pointwise kernel

$$P_{\epsilon }(n):=Re\left(n{\left({\sigma }_{\epsilon }(n)I-A\right)}^{-1}\right),$$

and track ${\lambda }_{min}(P_{\epsilon }(n))$ as $\epsilon \downarrow 0$. In the generic (bounded-resolvent) regime $z_0(n)\notin spec(A)$, Corollary 3 predicts the linear scaling ${\lambda }_{min}(P_{\epsilon }(n))=\Theta (\epsilon )$ and convergence of min-eigenvectors to $(z_0(n)I-A)\mathcal{M}(n)$ (Theorem 1).

• Experiment 1: exact linear rate for the nilpotent Jordan block. We revisit Example 3 with $A=(\begin{array}{c}\hfill \begin{array}{cc}0& 1\\ 0& 0\end{array}\end{array})$ and the disk exhaustion ${\Omega }_r=\{z:|z|<r\}$, $r>\frac{1}{2}$. Writing $\epsilon =r-\frac{1}{2}$, one has the exact formula ${\lambda }_{min}(P_{{\Omega }_r}(\sigma ,A))={\displaystyle \frac{\epsilon }{r^2}}$, hence linear degeneracy as $\epsilon \downarrow 0$. Figure 1 compares the computed smallest eigenvalue to the exact expression.

Figure 1. Nilpotent Jordan block (Example 3): log–log plot of ${\lambda }_{min}$ versus $\epsilon =r-\frac{1}{2}$, illustrating the predicted linear scaling.

Figure 1. Nilpotent Jordan block (Example 3): log–log plot of ${\lambda }_{min}$ versus $\epsilon =r-\frac{1}{2}$, illustrating the predicted linear scaling.

• Experiment 2: generic nonnormal matrix—linear degeneracy and eigenvector convergence. We generate a fixed random complex matrix $A\in {\mathbb{C}}^{5\times 5}$ (seeded for reproducibility), fix one direction $n=e^{i\theta }$, and form ${\sigma }_{\epsilon }(n)=z_0(n)+\epsilon n$ as above. Figure 2 shows ${\lambda }_{min}(P_{\epsilon }(n))$ against $\epsilon$ on a log–log scale, together with a reference $\propto \epsilon$ line; the observed slope is $\approx 1$ on the plotted range. Figure 3 tracks the distance of a min-eigenvector $u_{\epsilon }$ of $P_{\epsilon }(n)$ to the predicted limiting subspace $(z_0(n)I-A)\mathcal{M}(n)$, quantified by $\parallel (I-\Pi )u_{\epsilon }\parallel$ where $\Pi$ is the orthogonal projector onto $(z_0(n)I-A)\mathcal{M}(n)$ (consistent with Theorem 1).

Figure 2. Random nonnormal $A\in {\mathbb{C}}^{5\times 5}$ (fixed seed) and fixed direction n: ${\lambda }_{min}(P_{\epsilon }(n))$ scales linearly with $\epsilon$ (Corollary 3).

Figure 2. Random nonnormal $A\in {\mathbb{C}}^{5\times 5}$ (fixed seed) and fixed direction n: ${\lambda }_{min}(P_{\epsilon }(n))$ scales linearly with $\epsilon$ (Corollary 3).

Figure 3. Same setup as Figure 2: the direction of a min-eigenvector $u_{\epsilon }$ converges to $(z_0(n)I-A)\mathcal{M}(n)$ as $\epsilon \downarrow 0$ (Theorem 1). The plotted “gap” is $\parallel (I-\Pi )u_{\epsilon }\parallel$.

Figure 3. Same setup as Figure 2: the direction of a min-eigenvector $u_{\epsilon }$ converges to $(z_0(n)I-A)\mathcal{M}(n)$ as $\epsilon \downarrow 0$ (Theorem 1). The plotted “gap” is $\parallel (I-\Pi )u_{\epsilon }\parallel$.

• Experiment 2b: local sharpness of norm-based versus refined bounds. We compare the norm-based bounds from Lemma 3 (c) with the refined bounds of Proposition 5. For the Jordan block, Figure 4 shows that the refined bounds track the exact eigenvalue closely and are asymptotically sharp, while the norm-based upper bound is typically much looser. For the random nonnormal matrix, Figure 5 shows the same qualitative behavior.

Figure 4. Nilpotent Jordan block: comparison of ${\lambda }_{min}(P)$ with the norm-based bounds from Lemma 3 (c) and the refined bounds from Proposition 5.

Figure 4. Nilpotent Jordan block: comparison of ${\lambda }_{min}(P)$ with the norm-based bounds from Lemma 3 (c) and the refined bounds from Proposition 5.

Figure 5. Random nonnormal $A\in {\mathbb{C}}^{5\times 5}$, fixed direction n: comparison of ${\lambda }_{min}(P_{\epsilon }(n))$ with the norm-based and refined bounds. The refined bounds recover the correct first-order constant as $\epsilon \downarrow 0$.

Figure 5. Random nonnormal $A\in {\mathbb{C}}^{5\times 5}$, fixed direction n: comparison of ${\lambda }_{min}(P_{\epsilon }(n))$ with the norm-based and refined bounds. The refined bounds recover the correct first-order constant as $\epsilon \downarrow 0$.

• Experiment 2c: full slope spectrum at a flat face ($m=2$). Proposition 7 predicts a two-dimensional $O(\epsilon )$ collapsing cluster when the supporting pencil $H(n)$ has a two-dimensional maximal eigenspace and the chosen support point ${\sigma }_0$ lies in the interior of the corresponding face. To isolate this mechanism in a fully explicit setting, we take a normal diagonal matrix

  $$A=\mathrm{diag}(1,1+2i,0,-1),n=1,$$

  so that $H(n)=Re(A)$ has ${\lambda }_{max}=1$ with multiplicity $m=2$ and $\mathcal{M}(n)=span\{e_1,e_2\}$. With $v=(e_1+e_2)/\sqrt{2}$, the associated support point is ${\sigma }_0=v^{\ast }Av=1+i\notin spec(A)$, lying in the relative interior of the face joining 1 and $1+2i$. In this basis, $B_0={\sigma }_{0}I-A$ restricts to $\mathrm{diag}(i,-i)$ on $\mathcal{M}(n)$, hence $G=I_2$ and the predicted slope spectrum is $\{1,1\}$. Numerically we observe ${\lambda }_1^{\uparrow }(P({\sigma }_0+\epsilon n,n))/\epsilon \to 1$ and ${\lambda }_2^{\uparrow }(P({\sigma }_0+\epsilon n,n))/\epsilon \to 1$ as $\epsilon \downarrow 0$, and the face-detector count in Corollary 5 stabilizes at $m=2$.

Figure 6. Experiment 2c: two collapsing eigenvalues and their rescaled slopes in the $m=2$ flat-face normal example.

Figure 6. Experiment 2c: two collapsing eigenvalues and their rescaled slopes in the $m=2$ flat-face normal example.

• Experiment 3: approximate global coercivity collapse. For the same random nonnormal matrix A as in Experiment 2, we approximate the global coercivity constant

  $$c(\epsilon )=\underset{\sigma \in \partial {\Omega }_{\epsilon }}{inf}{\lambda }_{min}(P_{{\Omega }_{\epsilon }}(\sigma ,A))$$

  by sampling a fine grid of directions $\left\{n_j\right\}$ and using the offset model ${\sigma }_{\epsilon }(n_j)=z_0(n_j)+\epsilon n_j$. Figure 7 plots the sampled minimum ${min}_j{\lambda }_{min}(P_{\epsilon }(n_j))$ versus $\epsilon$, illustrating the collapse asserted by Corollary 2. (Here the offset model has $\Delta ({\Omega }_{\epsilon })=\epsilon$, so Corollary 1 also predicts that uniform coercivity cannot persist as $\epsilon \downarrow 0$.)

Figure 7. Approximate global coercivity constant $c(\epsilon )$ computed by sampling directions and using the offset model ${\sigma }_{\epsilon }(n)=z_0(n)+\epsilon n$. The sampled minimum tends to 0 as $\epsilon \downarrow 0$ (Corollary 2).

Figure 7. Approximate global coercivity constant $c(\epsilon )$ computed by sampling directions and using the offset model ${\sigma }_{\epsilon }(n)=z_0(n)+\epsilon n$. The sampled minimum tends to 0 as $\epsilon \downarrow 0$ (Corollary 2).

• Experiment 3b: local slope profile across directions. For the same random nonnormal matrix and the offset model, Corollary 4 predicts the local first-order constant ${\lambda }_{min}(P_{\epsilon }(n))/\epsilon \to 1/{\beta }_0{(n)}^2$ as $\epsilon \downarrow 0$. Figure 8 compares the empirically observed slope ${\lambda }_{min}(P_{\epsilon }(n))/\epsilon$ at a small fixed $\epsilon$ to the prediction $1/{\beta }_0{(n)}^2$ across directions $n=e^{i\theta }$, while Figure 9 plots the relative error.

Figure 8. Random nonnormal matrix: direction-wise comparison of the empirical slope ${\lambda }_{min}(P_{\epsilon }(n))/\epsilon$ with the predicted limit $1/{\beta }_0{(n)}^2$ (Corollary 4).

Figure 8. Random nonnormal matrix: direction-wise comparison of the empirical slope ${\lambda }_{min}(P_{\epsilon }(n))/\epsilon$ with the predicted limit $1/{\beta }_0{(n)}^2$ (Corollary 4).

Figure 9. Random nonnormal matrix: relative error of the prediction in Figure 8. The error decreases as the test value of $\epsilon$ is reduced.

Figure 9. Random nonnormal matrix: relative error of the prediction in Figure 8. The error decreases as the test value of $\epsilon$ is reduced.

• Experiment 3c: global slope constant and the ${\beta }_0$ profile. Proposition 6 predicts that the direction-sampled coercivity constant satisfies $\tilde{c}(\epsilon )/\epsilon \to 1/{max}_{\theta }{\beta }_0{(\theta )}^2$. Figure 10 plots the profile of ${\beta }_0{(\theta )}^2$, and Figure 11 compares the sampled ratio $\tilde{c}(\epsilon )/\epsilon$ to the predicted limit.

Figure 10. Random nonnormal matrix: profile of ${\beta }_0{(\theta )}^2=\parallel (z_0(\theta )I-A){{|}_{\mathcal{M}(e^{i\theta })}\parallel }^2$. The global slope constant is set by the maximum of this profile (Proposition 6).

Figure 10. Random nonnormal matrix: profile of ${\beta }_0{(\theta )}^2=\parallel (z_0(\theta )I-A){{|}_{\mathcal{M}(e^{i\theta })}\parallel }^2$. The global slope constant is set by the maximum of this profile (Proposition 6).

Figure 11. Random nonnormal matrix: the sampled ratio $\tilde{c}(\epsilon )/\epsilon$ versus $\epsilon$, together with the predicted limit $1/{max}_{\theta }{\beta }_0{(\theta )}^2$ (Proposition 6).

Figure 11. Random nonnormal matrix: the sampled ratio $\tilde{c}(\epsilon )/\epsilon$ versus $\epsilon$, together with the predicted limit $1/{max}_{\theta }{\beta }_0{(\theta )}^2$ (Proposition 6).

• Experiment 4: normal matrices at spectral support points. We reproduce the contrasting behavior in Examples 1–2 by evaluating $P(1+\epsilon ,1)=Re\left({((1+\epsilon )I-A)}^{-1}\right)$ for two diagonal (hence normal) matrices: $A=\mathrm{diag}(0,1)$ and $A=\mathrm{diag}(1,1+i)$. Figure 12 shows that the former remains bounded away from 0 as $\epsilon \downarrow 0$, while the latter degenerates linearly, consistent with Proposition 11.

Figure 12. Normal matrices at a spectral support point: $A=\mathrm{diag}(0,1)$ remains bounded away from 0, whereas $A=\mathrm{diag}(1,1+i)$ degenerates linearly, consistent with Proposition 11.

Figure 12. Normal matrices at a spectral support point: $A=\mathrm{diag}(0,1)$ remains bounded away from 0, whereas $A=\mathrm{diag}(1,1+i)$ degenerates linearly, consistent with Proposition 11.

• Experiment 4b: three-scale splitting at spectral support points (nonnormal). Proposition 12 predicts a three-scale structure at ${\sigma }_0\in spec(A)\cap \partial W(A)$ under non-tangential offsets ${\sigma }_{\epsilon }={\sigma }_0+\epsilon n$: an exact $1/\epsilon$ blow-up on the geometric eigenspace $E=Ker({\sigma }_{0}I-A)$, an $O(\epsilon )$ collapsing cluster of dimension $m-r$ on $\mathcal{M}(n)\ominus E$, and an $O(1)$ bulk bounded away from 0. We test this on three small nonnormal examples with $n=1$ and ${\sigma }_0=1$:

  (SS1)

  $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]$, for which $m=r=1$ (blow-up only; no $O(\epsilon )$ cluster);

  (SS2)

  $A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& i\\ 0& i& 0\end{array}\right]$, for which $m=2$ and $r=1$ (one collapsing slope);

  (SS3)

  $A=H+iK$ with $H=\mathrm{diag}(1,1,1,0,0)$ and $K_{24}=K_{42}=1$, $K_{35}=K_{53}=2$, for which $m=3$ and $r=1$ (two collapsing slopes).

In each case we observe the predicted behavior: the largest eigenvalue scales as $1/\epsilon$, the smallest $m-r$ eigenvalues scale linearly in $\epsilon$ with slopes given by $G_F^{-1}$, and the next eigenvalue remains separated from 0 uniformly in $\epsilon$.

• Reproducibility. All figures are generated by the accompanying scripts poisson_utils.py and run_numerical_experiments.py, which require only NumPy and Matplotlib and save PDF figures into a figs/ folder.

Figure 13. Experiment 4b: three-scale splitting at spectral support points (SS0–SS2). Each panel shows the $1/\epsilon$ blow-up eigenvalue(s), the collapsing $O(\epsilon )$ cluster, and the $O(1)$ bulk.

Figure 13. Experiment 4b: three-scale splitting at spectral support points (SS0–SS2). Each panel shows the $1/\epsilon$ blow-up eigenvalue(s), the collapsing $O(\epsilon )$ cluster, and the $O(1)$ bulk.

4.10. A Curvature Surrogate Question at Smooth Exposed Points

The direction-dependent slope data in Corollary 4 and Proposition 7 suggest that the rescaled Poisson degeneracy rate can be viewed as a quantitative “stiffness” of the numerical range boundary in a given direction.

Let $n=e^{i\theta }$ and consider the support function

$$h(\theta ):=\underset{z\in W(A)}{max}Re(e^{-i\theta }z)={\lambda }_{max}\left(H(e^{i\theta })\right),$$

where $H(e^{i\theta })=Re(e^{-i\theta }A)$. At directions where the maximal eigenvalue is simple, choose the corresponding unit eigenvector $v(\theta )$ and set the exposed support point

$${\sigma }_0(\theta ):=v{(\theta )}^{\ast }Av(\theta )\in \partial W(A).$$

For the non-tangential offset family ${\sigma }_{\epsilon }(\theta )={\sigma }_0(\theta )+\epsilon e^{i\theta }$, define $P_{\epsilon }(\theta )=P({\sigma }_{\epsilon }(\theta ),e^{i\theta })$. When $dim\mathcal{M}(e^{i\theta })=1$, Proposition 7 gives the explicit slope

$$s(\theta ):=\underset{\epsilon \downarrow 0}{lim}{\displaystyle \frac{1}{\epsilon }}{\lambda }_{min}\left(P_{\epsilon }(\theta )\right)={\displaystyle \frac{1}{\parallel ({\sigma }_0(\theta )I-A){v(\theta )\parallel }^2}}.$$

(4.37)

A natural question is how $s(\theta )$ compares to geometric curvature data of $\partial W(A)$. In convex geometry, the support function determines the boundary, and for $C^2$ strictly convex curves the radius of curvature can be expressed in terms of h and its second derivative (see, e.g., [12]). For numerical ranges, $\partial W(A)$ is an algebraic curve determined by the Kippenhahn polynomial [14], with corners and flat portions corresponding to eigenvalue multiplicities and supporting-face degeneracies.

Question. At smooth exposed boundary points where $dim\mathcal{M}(e^{i\theta })=1$, is the slope $s(\theta )$ in (4.37) comparable (in an appropriate quantitative sense) to the curvature (or radius of curvature) of $\partial W(A)$ at ${\sigma }_0(\theta )$? Numerically, plots of $s(\theta )$ (Experiment 3b) show sharp spikes near directions associated with nearly-flat faces, suggesting that $s(\theta )$ may serve as an easily computable curvature surrogate.

4.11. Discussion and Remaining Open Problems

Remark 9

(Beyond the geometric three-scale picture). The bounded-resolvent regime ${\sigma }_0\notin spec(A)$ is treated in Theorem 1 and quantified by the slope spectra in Section 4.4. At spectral support points ${\sigma }_0\in spec(A)\cap \partial W(A)$, Proposition 11 and Corollary 7 give a complete normal-matrix description, and Proposition 12 provides a three-scale splitting for general matrices under a gap hypothesis for the supporting pencil $H(n)$.

Several aspects of the nonnormal spectral-support regime remain open:

(i) 

Defective eigenvalues. When ${\sigma }_0$ has nontrivial Jordan chains, generalized eigenvectors may exhibit higher-order resolvent blow-up (typically $1/{\epsilon }^p$ for a length-p Jordan block). A systematic description of how Jordan structure and the support pencil interact to determine the full singular-value/eigenvalue profile of $P({\sigma }_0+\epsilon n,n)$ is closely related to pseudospectral growth; see [13].

(ii) 

Tangential approach geometry. Our slope spectra are derived for non-tangential offsets and for general $C^1$ exhaustions after normalization by the scalar support gap δ. Understanding when the $O(\delta )$ scaling persists under more tangential approach, or when higher-order scalings appear, is largely open.

(iii) 

Loss of spectral isolation. The explicit slope spectra rely on a gap separating ${\lambda }_{max}(H(n))$ from the rest of the spectrum. When this gap closes, the associated projectors become ill-conditioned and new multi-scale behaviors may appear.