Green's Function Comparison Method

• Green's Function Comparison Method is a technique for comparing pluricomplex Green functions on matrix domains using holomorphic symmetrization.
• It establishes equality for cyclic matrices and strict inequality for derogatory ones through a rigorous analysis of nilpotence and Jordan decomposition.
• The method integrates pluripotential theory with spectral analysis to differentiate classical invariants like the Carathéodory distance and Green function in higher dimensions.

The Green's function comparison method is a technique for relating the pluricomplex Green functions on two distinct domains linked via a holomorphic mapping, with a particular focus on the spectral ball $\Omega_{\rm spec}$ and the symmetrized polydisk $G_n$ in several complex variables. Core results establish precise inequalities and characterizations for the Green functions on these domains, especially highlighting the distinction between cyclic (non-derogatory) and derogatory matrices and the role of nilpotence in the behavior of these functions. The method is rigorously developed in Thomas–Trao–Zwonek, "Green functions of the spectral ball and symmetrized polydisk" (Thomas et al., 2010).

1. Spectral Ball, Symmetrized Polydisk, and Symmetrization Map

Let $M_n$ denote the space of $n \times n$ complex matrices. The spectral ball is defined as

$\Omega_{\rm spec} = \{ A \in M_n : \rho(A) < 1 \},$

where $$\rho(A)=\max\{|\lambda|: \lambda \in\operatorname{sp}(A)\}$$ is the spectral radius of $A$.

Given the characteristic polynomial $$P_A(t)=\det(tI-A)=t^n - s_1(A) t^{n-1} + \cdots + (-1)^n s_n(A)$$, the symmetrization map

$$\pi: \Omega_{\rm spec}\to\mathbb{C}^n, \qquad \pi(A) = (s_1(A),...,s_n(A))$$

maps $A$ to the vector of its elementary symmetric functions. The image $G_n = \pi(\Omega_{\rm spec})\subset\mathbb{C}^n$ is called the symmetrized polydisk, which is a bounded, complete hyperbolic, and hyperconvex domain.

The pluricomplex Green function with pole at $p\in D\subset\mathbb{C}^m$ is

$$g_D(p,z) = \sup\{ u(z): u\in \operatorname{PSH}(D),\, u(w)\leq \log|w-p|+O(1) \},$$

with $g_\Omega(A,B)$ for $D=\Omega_{\rm spec}$ and $g_{G_n}(z,w)$ for $D=G_n$ (Thomas et al., 2010).

2. Main Comparison Theorem and Inequality

For any $A,B\in\Omega_{\rm spec}$, the Green's function on the spectral ball dominates the pullback via symmetrization: $$\boxed{ g_{\Omega_{\rm spec}}(A,B) \geq g_{G_n}\big(\pi(A),\pi(B)\big) }$$ If $A$ is cyclic (non-derogatory), i.e. it has a single Jordan block per eigenvalue, then equality holds for every $B$: $$A\ \mathrm{cyclic} \implies g_{\Omega}(A,B) = g_{G_n}\big(\pi(A),\pi(B)\big)$$ If $A$ is derogatory (it has a repeated eigenvalue and more than one Jordan block per eigenvalue), the inequality is always strict. The difference

$$\Delta(A,B) = g_\Omega(A,B) - g_{G_n}\big(\pi(A),\pi(B)\big)$$

is strictly positive and estimates for this gap depend on the orders of nilpotence in the Jordan decomposition of $A$ (Thomas et al., 2010).

3. Role of Derogatory Poles and Nilpotence

To formalize strictness, conjugate $A$ into Jordan form. For eigenvalue $\lambda$,

$n(\lambda) = \dim\ker(A-\lambda I),\quad m(\lambda) = \min\{k: (A-\lambda I)^k=0\, \text{on the %%%%21%%%%-block}\}$

$A$ is derogatory iff for some $\lambda$ one has $m(\lambda)<n(\lambda)$.

For a nilpotent matrix $V$, $\lambda=0$, $n(0)=n$, $m(0)=m\leq n-1$. For small perturbation $V+X$,

$$\log\rho(V+X)\leq \frac1m \log\|X\| + O(1) \implies g_{\Omega}(V,V+X)\geq m\log\|X\| + O(1)$$

Homogeneity of the elementary symmetric polynomials yields

$g_{G_n}(\pi(V),\pi(V+X))\leq n\log\|X\| + O(1)$

Therefore,

$\Delta(V,V+X)\geq (m-n)\log\|X\|+O(1)$

is strictly positive for small $X$ when $m<n$.

For general derogatory $A$ with repeated eigenvalue $\lambda_0$ and $m(\lambda_0)<n(\lambda_0)$, one applies a Möbius-type automorphism moving $\lambda_0$ to $0$ and repeats the estimation with the same order distinction, confirming strict inequality (Thomas et al., 2010).

4. Isospectral Fibres, Cyclicity, and the Proof Strategy

The Green function $g_\Omega$ is constant along fibers of the symmetrization map: $$g_\Omega(A,B) = g_\Omega(A,B') \quad \text{whenever}\quad \pi(B)=\pi(B')$$ This invariance under the isospectral equivalence is central. The composition $g_{G_n}(\pi(A),\pi(\cdot))$ is plurisubharmonic on $\Omega$, ensuring by the maximum property that it is always bounded above by $g_\Omega(A,\cdot)$, yielding the main inequality.

For $A$ cyclic, the differential $d\pi$ has maximal rank and $\pi$ is a submersion locally. A theorem of Jarnicki–Pflug shows that the pullback of the Green function along a submersion coincides with the Green function of the source, establishing equality for cyclic poles (Thomas et al., 2010).

In the derogatory case, one examines one-parameter families $B(t)=A+tX$ tangent to the isospectral fiber. The spectral radius of $B(t)$ grows as $|t|^{1/m(\lambda_0)}$, but under $\pi(B(t))$, the coordinate growth is $|t|^{1/n(\lambda_0)}$, so the difference in exponents after taking logarithms underscores the strict gap between $g_{\Omega}$ and $g_{G_n}$.

5. Infinitesimal Estimates and Metric Separation in $G_n$

For domains $D$, the following inequalities relate the (pseudo-)Carathéodory distance $c_D$, the Green function $g_D$, and the Lempert function $\ell_D$: $\log c_D(z,w) \leq g_D(z,w) \leq \log\ell_D(z,w)$ On the symmetrized polydisk, novel phenomena appear for $n\geq3$. At the origin $0\in G_n$ and in the “last coordinate” direction $e_{n-1}$,

$$A_{G_n}(0; e_{n-1})\geq Y_{G_n}(0; e_{n-1})\geq \frac{n-1}{n}$$

where $Y_D$ is the infinitesimal Azukawa metric and $A_D$ the Carathéodory–Reiffen metric.

For $n\geq3$,

$Y_{G_n}(0;e_{n-1}) < A_{G_n}(0;e_{n-1})$

Consequently, for small $|t|$,

$$c_{G_n}(0,\, t\, e_{n-1}) < \exp\big(g_{G_n}(0,t\,e_{n-1})\big)$$

Thus, the Green function and Carathéodory distance exhibit marked differences for the symmetrized polydisk in dimensions above two. This contrasts with the classical equivalence in the bidisk or the unit ball (Thomas et al., 2010).

6. Significance and Mathematical Context

The Green's function comparison method rigorously quantifies the distinction between the spectral ball and its symmetrized image, particularly through pluripotential-theoretic invariants and their extremal properties on matrix domains. The method demonstrates the central role of cyclicity, nilpotence, and isospectral geometry in the pluricomplex Green function's behavior and exposes deeper structure in fundamental complex-analytic metrics, revealing, for higher dimensions, non-trivial separations between classical objects such as the Carathéodory distance and the pluricomplex Green function (Thomas et al., 2010).