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

1. Introduction

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

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

It is a compact convex subset of $\mathbb{C}$ (Toeplitz–Hausdorff theorem; see, e.g., [1,2]). A central open problem due to Crouzeix asks whether $W\left(A\right)$ is a 2-spectral set for A, i.e.,

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

(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 harmless normalization conventions, a recurring tool in the convex-domain approach of Delyon–Delyon and Crouzeix is the operator-valued boundary kernel

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

(1.2)

defined for a bounded convex domain $\Omega \subset \mathbb{C}$ with $C^1$ boundary containing $spec\left(A\right)$, where $n_{\Omega }\left(\sigma \right)$ is 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 $W\left(A\right)\subset \Omega$, 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 $\partial W\left(A\right)$. In applications and numerical implementations of boundary-integral calculi, one often approximates $W\left(A\right)$ by $C^1$ convex supersets ${\Omega }_{\epsilon }\downarrow W\left(A\right)$. 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 ${\sigma }_0\in \partial W\left(A\right)\setminus spec\left(A\right)$), the smallest eigenvalue of $P_{{\Omega }_{\epsilon }}(\sigma ,A)$ must deteriorate at boundary points $\sigma \in \partial {\Omega }_{\epsilon }$ approaching $\partial W\left(A\right)$ 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 $\Omega ⊋W\left(A\right)$ [4,7,8,9]. Here we analyze the complementary limiting regime in which $\Omega$shrinks to $W\left(A\right)$, 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 $\delta (\sigma ,n)=Re\left(\overline{n}\sigma \right)-{\lambda }_{max}(Re\left(\overline{n}A\right))$, 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 ${\Omega }_{\epsilon }\downarrow W\left(A\right)$, if ${\sigma }_{\epsilon }\in \partial {\Omega }_{\epsilon }$ approaches a non-spectral boundary point ${\sigma }_0\in \partial W\left(A\right)\setminus spec\left(A\right)$ with convergent outward normals $n_{{\Omega }_{\epsilon }}\left({\sigma }_{\epsilon }\right)\to n$, then ${\lambda }_{min}\left(P_{{\Omega }_{\epsilon }}({\sigma }_{\epsilon },A)\right)\to 0$ and the limiting min-eigenvector directions lie in $({\sigma }_{0}I-A)\mathcal{M}\left(n\right)$, where $\mathcal{M}\left(n\right)$ is the maximal eigenspace of $H\left(n\right)=Re\left(\overline{n}A\right)$.
• We establish two-sided bounds for ${\lambda }_{min}\left(P_{\Omega }(\sigma ,A)\right)$ 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 $W\left(A\right)+\epsilon \mathbb{D}$ (Proposition 2).
• Under a spectral-isolation hypothesis for ${\lambda }_{max}\left(H\left(n\right)\right)$, we obtain convergence of the entire near-kernel invariant subspace (spectral projector) along the exhaustion (Proposition 3).
• We analyze the contrasting spectral-support regime ${\sigma }_0\in spec\left(A\right)\cap \partial W\left(A\right)$ for normal matrices via an explicit eigenvalue formula for $P_{\Omega }(\sigma ,A)$, showing that degeneracy may fail at a spectral support point unless the supporting face contains multiple eigenvalues (Proposition 4 and Examples 1–2).

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, subspace convergence, and explicit examples, followed by a brief discussion of open problems.

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

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

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

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

Remark 1 

(Spectrum is contained in the numerical range). One has $spec\left(A\right)\subset W\left(A\right)$. Indeed, if $Ax=\lambda x$ with $∥x∥=1$, then $x^{*}Ax=\lambda \in W\left(A\right)$. Consequently, $W\left(A\right)\subset \Omega$ implies $spec\left(A\right)\subset \Omega$ 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

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

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

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

denote the corresponding maximal eigenspace.

Lemma 1 

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

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

(2.2)

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

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

Proof. 

For $∥x∥=1$,

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

Taking the maximum over $∥x∥=1$ yields (2.2) by Rayleigh–Ritz. If x is a maximizing unit vector, then $x^{*}Ax\in W\left(A\right)$ attains the support functional in direction $\omega$, hence lies on $\partial W\left(A\right)$ 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 $n_{\Omega }\left(\sigma \right)$ 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 $n_{\Omega }\left(\sigma \right)\in \mathbb{C}$ with $|n_{\Omega }\left(\sigma \right)|=1$ so that the supporting half-plane at $\sigma$ is

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

(2.3)

Equivalently, by convexity one has $\overline{\Omega }\subseteq {\Pi }_{\Omega }\left(\sigma \right)$ and $\Omega \subset \{z\in \mathbb{C}:Re\left(\overline{n_{\Omega }\left(\sigma \right)}(z-\sigma )\right)<0\}$. Under the identification $\mathbb{C}\simeq {\mathbb{R}}^2$, the functional $z\mapsto Re\left(\overline{n}z\right)$ 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 $n_k:=n_{{\Omega }_{{\epsilon }_k}}\left({\sigma }_k\right)$ 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 $spec\left(A\right)\subset \Omega$. Then ${(\sigma I-A)}^{-1}$ exists for all $\sigma \in \partial \Omega$.

Definition 2 

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

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

(3.1)

3.1. A Congruence Identity

Lemma 2 

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

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

(3.2)

Proof. 

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

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

Expanding gives (3.2).    □

3.2. Support-Gap Bounds

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

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

Lemma 3 

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

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

(This notation emphasizes dependence on the prescribed direction n; when $n=n_{\Omega }\left(\sigma \right)$ 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

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

(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,

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

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 ${\lambda }_{min}\left(Q\right)=\alpha -{\lambda }_{max}\left(H\left(n\right)\right)=\delta$, this proves (a).

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

$$x^{*}Px=y^{*}Qy,$$

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

If $\delta >0$, then $Q\succ 0$ and $P=B^{-*}Q{B}^{-1}\succ 0$. 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

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

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

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

   □

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 ${\lambda }_{min}\left(P(\sigma ,n)\right)$ on the scalar support gap $\delta (\sigma ,n)$.

3.3. Strict positivity when $W\left(A\right)\subset \Omega$

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 $n=n_{\Omega }\left(\sigma \right)$ be the outward unit normal. Then

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

Proof. 

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

Proposition 1 

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

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

Proof. 

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

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

so $\delta (\sigma ,n)=\alpha -{\lambda }_{max}\left(H\left(n\right)\right)>0$. 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

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

If $\Omega \subset \mathbb{C}$ is a bounded open convex set with $C^1$ boundary and $\sigma \in \partial \Omega$ has outward normal $n=n_{\Omega }\left(\sigma \right)$, then necessarily $Re\left(\overline{n}\sigma \right)=h_{\overline{\Omega }}\left(n\right)$, 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 $n:=n_{\Omega }\left(\sigma \right)$. Then

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

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

Proof. 

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

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

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

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

Consequently, since $\sigma \in \partial {\Omega }_{\epsilon }$ implies $\sigma \notin W\left(A\right)$ and hence $\sigma \notin spec\left(A\right)$ (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 $\left|w\right|\le \epsilon$,

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

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

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

so $h_{\overline{K_{\epsilon }}}\left(n\right)\ge h_K\left(n\right)+\epsilon$. 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

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

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

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

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

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

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

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

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. $K+t\overline{\mathbb{D}}=\{u\in \mathbb{C}:(u,K)\le t\}$. 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 $h_M\left(n\right)\le h_N\left(n\right)$ 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 $Re\left(\overline{n}x\right)>{max}_{z\in N}Re\left(\overline{n}z\right)=h_N\left(n\right)$, hence $h_M\left(n\right)\ge Re\left(\overline{n}x\right)>h_N\left(n\right)$.)

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

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

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

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

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 $W\left(A\right)\subset \Omega$, and set

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

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

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

Consequently, since $\sigma \notin spec\left(A\right)$ 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

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

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 $\delta (\sigma ,n)=h_{\overline{\Omega }}\left(n\right)-h_{W\left(A\right)}\left(n\right)$ 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 $n\mapsto h_{\overline{\Omega }}\left(n\right)-h_{W\left(A\right)}\left(n\right)$ is continuous on the unit circle, so it attains its maximum at some unimodular $n_{★}$. Choose ${\sigma }_{★}\in \overline{\Omega }$ such that $Re\left(\overline{n_{★}}{\sigma }_{★}\right)=h_{\overline{\Omega }}\left(n_{★}\right)$; then ${\sigma }_{★}\in \partial \Omega$ and the supporting line $\{z:Re\left(\overline{n_{★}}z\right)=Re\left(\overline{n_{★}}{\sigma }_{★}\right)\}$ 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

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

   □

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 $W\left(A\right)$ (Definition 1). For $\sigma \in \partial {\Omega }_{\epsilon }$, set

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

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

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

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

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

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

(3)

(One-dimensional case)If $dim\mathcal{M}\left(n\right)=1$, 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 $\mathcal{M}\left(n\right)$.

Proof. 

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

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

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

Step 1: Congruence identities. By Lemma 2,

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

(4.1)

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

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

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

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

Step 3: $P_k\to P_0$ in operator norm. Since ${\sigma }_0\notin spec\left(A\right)$, $B_0$ 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: ${\lambda }_{min}\left(P_0\right)=0$ and ${\lambda }_{min}\left(P_k\right)\to 0$. Since $P_k,P_0$ are Hermitian, Weyl’s inequality yields

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

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

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

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

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

by Lemma 3 (b) (with $\delta =0$).

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

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

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

Step 6: One-dimensional case. If $dim\mathcal{M}\left(n\right)=1$, then $dimKer\left(P_0\right)=1$, so the smallest eigenvalue of $P_0$ is simple. By the Davis–Kahan $sin\Theta$ theorem for invariant subspaces (see [10]), the corresponding one-dimensional eigenspaces of $P_k$ converge to $Ker\left(P_0\right)$ in gap metric, hence there exist phases ${\theta }_k$ such that $e^{i{\theta }_k}{u}_k\to u_{★}$, where $u_{★}$ spans $Ker\left(P_0\right)=({\sigma }_{0}I-A)\mathcal{M}\left(n\right)$. This gives (3).    □

Remark 5 

(Why ${\sigma }_0\notin spec\left(A\right)$ is essential). The hypothesis ${\sigma }_0\notin spec\left(A\right)$ ensures that ${(\sigma I-A)}^{-1}$ remains bounded near ${\sigma }_0$, so $P_0$ is a finite Hermitian matrix. When ${\sigma }_0\in spec\left(A\right)$, the resolvent diverges and the behavior of ${\lambda }_{min}\left(P_{{\Omega }_{\epsilon }}({\sigma }_{\epsilon },A)\right)$ depends on the spectral geometry; see Proposition 4 below and Section 4.7.

Corollary 2 

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

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

Then

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

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 $W\left(A\right)$ contains more than one point, hence $\partial W\left(A\right)$ is infinite, whereas $spec\left(A\right)$ is finite. Choose ${\sigma }_0\in \partial W\left(A\right)\setminus spec\left(A\right)$.

Fix any sequence ${\epsilon }_k\downarrow 0$. We claim that $({\sigma }_0,\partial {\Omega }_{{\epsilon }_k})\to 0$. Indeed, if not, then there exist $\delta >0$ and a subsequence (not relabeled) such that $({\sigma }_0,\partial {\Omega }_{{\epsilon }_k})\ge \delta$ 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 $B({\sigma }_0,\delta )\subset {\bigcap }_k\overline{{\Omega }_{{\epsilon }_k}}=W\left(A\right)$, contradicting ${\sigma }_0\in \partial W\left(A\right)$.

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 $n_{{\Omega }_{{\epsilon }_k}}\left({\sigma }_k\right)\to n$ for some unimodular n. Theorem 1 then gives

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

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

4.2. Quantitative Degeneracy Rate

Corollary 3 

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

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

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

Proof. 

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

As $n_k\to n$ and ${\sigma }_k\to {\sigma }_0$, $Re\left(\overline{n_k}{\sigma }_k\right)\to Re\left(\overline{n}{\sigma }_0\right)$. Also $H\left(n_k\right)\to H\left(n\right)$ in operator norm, hence ${\lambda }_{max}\left(H\left(n_k\right)\right)\to {\lambda }_{max}\left(H\left(n\right)\right)$. By Step 2 in the proof of Theorem 1, $Re\left(\overline{n}{\sigma }_0\right)={\lambda }_{max}\left(H\left(n\right)\right)$, 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. Convergence of the Near-Kernel Subspace

Proposition 3 

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

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

(4.3)

Let

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

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

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

(4.4)

and consequently, for all sufficiently large k,

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

(4.5)

Proof. 

By Lemma 2 and Step 2 of Theorem 1,

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

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

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

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

Taking $j=m+1$ gives (4.4).

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 ${\lambda }_{m+1}^{\uparrow }\left(P_0\right)>0$, the Davis–Kahan $sin\Theta$ theorem for invariant subspaces [10] yields (4.5), and hence $\parallel {\Pi }_k-{\Pi }_0\parallel \to 0$.    □

4.4. The Spectral-Support Regime for Normal Matrices

Proposition 4 

(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 $\sigma \notin spec\left(A\right)$ 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

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

(4.6)

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

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

Let ${\sigma }_k\notin spec\left(A\right)$ 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

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

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

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

(4.7)

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

(iii)

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

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

(4.8)

Then for every $j\notin J_0$,

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

and $p_j({\sigma }_0,n)=0$ if and only if ${\lambda }_j$ lies on the same supporting line $\{z:Re\left(\overline{n}z\right)=Re\left(\overline{n}{\sigma }_0\right)\}$. If moreover (4.7) 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 $\{z:Re\left(\overline{n}z\right)=Re\left(\overline{n}{\sigma }_0\right)\}$.

Proof. 

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

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

Therefore,

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

which proves (4.6). 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

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

and (4.7) implies $p_{k,j}\ge c/|{\sigma }_k-{\sigma }_0|\to +\infty$.

Finally, (4.8) implies $Re\left(\overline{n}({\sigma }_0-{\lambda }_j)\right)\ge 0$ for all j, giving the nonnegativity (and the characterization of equality) in (iii). If additionally (4.7) 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.    □

Example 1 

(Nondegeneracy at a spectral support point). Let $A=\mathrm{diag}(0,1)$, so $W\left(A\right)=[0,1]$. 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 ${\lambda }_{min}\left(P(\sigma ,n)\right)={\displaystyle \frac{1}{1+\epsilon }}\to 1$ 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 ${\lambda }_{min}\left(P(\sigma ,n)\right)={\displaystyle \frac{\epsilon }{{\epsilon }^2+1}}\sim \epsilon \downarrow 0$. Here the supporting functional $Re\left(z\right)$ is maximized by more than one eigenvalue, and degeneracy persists at ${\sigma }_0=1\in spec\left(A\right)$.

4.5. 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 $W\left(A\right)=\{z\in \mathbb{C}:|z|\le \frac{1}{2}\}$. 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 $n_{{\Omega }_r}\left(\sigma \right)=e^{it}$ 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 ${\lambda }_{\pm }\left(r\right)={\displaystyle \frac{2r\pm 1}{2r^2}}$. 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 $n=e^{it}$,

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

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

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

in agreement with Theorem 1.

4.6. Numerical Experiments

This section provides numerical illustrations of: (i) the linear degeneracy predicted by Corollary 3 (and, in offset form, Proposition 2), (ii) the global coercivity collapse of Corollary 2, and (iii) the contrasting behavior at spectral support points for normal matrices (Proposition 4 and Examples 1–2).

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

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

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

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

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

We evaluate the pointwise kernel

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

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

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

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 ${\sigma }_{\epsilon }\left(n\right)=z_0\left(n\right)+\epsilon n$ as above. Figure 2 shows ${\lambda }_{min}\left(P_{\epsilon }\left(n\right)\right)$ 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 $P_{\epsilon }\left(n\right)$ to the predicted limiting subspace $(z_0\left(n\right)I-A)\mathcal{M}\left(n\right)$, quantified by $\parallel (I-\Pi )u_{\epsilon }\parallel$ where $\Pi$ is the orthogonal projector onto $(z_0\left(n\right)I-A)\mathcal{M}\left(n\right)$ (consistent with Theorem 1).

Experiment 3: approximate global coercivity collapse. For the same A we approximate the global coercivity constant

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

by sampling a fine grid of directions $\left\{n_j\right\}$ and using the offset model ${\sigma }_{\epsilon }\left(n_j\right)=z_0\left(n_j\right)+\epsilon n_j$. Figure 4 plots the sampled minimum ${min}_j{\lambda }_{min}\left(P_{\epsilon }\left(n_j\right)\right)$ versus $\epsilon$, illustrating the collapse asserted by Corollary 2. (Here the offset model has $\Delta \left({\Omega }_{\epsilon }\right)=\epsilon$, so Corollary 1 also predicts that uniform coercivity cannot persist as $\epsilon \downarrow 0$.)

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 5 shows that the former remains bounded away from 0 as $\epsilon \downarrow 0$, while the latter degenerates linearly, consistent with Proposition 4.

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.

4.7. Discussion and Open Problems: the Nonnormal Spectral-Support Regime

Remark 6 

(Beyond the normal case at spectral support points). Theorem 1 treats the bounded-resolvent regime ${\sigma }_0\notin spec\left(A\right)$, while Proposition 4 gives a complete description of what can happen at a spectral support point ${\sigma }_0\in spec\left(A\right)\cap \partial W\left(A\right)$ fornormalmatrices.

Fornonnormalmatrices, the regime ${\sigma }_0\in spec\left(A\right)\cap \partial W\left(A\right)$ appears substantially more delicate: the resolvent ${(\sigma I-A)}^{-1}$ typically diverges as $\sigma \to {\sigma }_0$, and the interplay between (i) the approach geometry $\sigma \to {\sigma }_0$ along $\partial {\Omega }_{\epsilon }$, (ii) the supporting direction n, and (iii) the Jordan/pseudospectral behavior of A near ${\sigma }_0$ can produce several qualitatively different limits for ${\lambda }_{min}\left(P_{\Omega }(\sigma ,A)\right)$.

Natural questions suggested by the present results include:

• Can one classify (or even bound) the possible asymptotic behavior of ${\lambda }_{min}\left(P_{{\Omega }_{\epsilon }}({\sigma }_{\epsilon },A)\right)$ as ${\sigma }_{\epsilon }\to {\sigma }_0\in spec\left(A\right)\cap \partial W\left(A\right)$, in terms of the local spectral data of A (e.g. Jordan structure) and the support direction n?
• In analogy with Proposition 4, is there a purely geometric/spectral criterion characterizing when degeneracy must occur at a spectral support point for a general (possibly defective) A?
• How do these boundary effects interact with quantitative constants in boundary-integral functional calculi and with conditioning of numerical schemes based on $C^1$ domain approximations ${\Omega }_{\epsilon }\downarrow W\left(A\right)$?

We leave these questions for future work.