Extension of eigenvalue and eigenprojection to its complex neighbourhood for constantly hyperbolic operator

Question

I am reading a paper named The block structure condition for symmetric hyperbolic systems written by G. Metivier. There is a statement really has bothered me for a long time.

Consider the following matrix $$ L(\tau,\eta,\xi)=\tau I_N+\xi A+\eta B. $$ with $I_N$ the identity matrix, constant matrices $A,B \in \mathbb{R}^{N\times N},$ and real parameters $\tau,\xi,\eta.$ When $(\xi,\eta) \in \mathbb{R}^2\backslash\{(0,0)\},$ there exist fixed positive integers $J$ and $\alpha_i$ for $i=1,\cdots,J,$ and real eigenvalue functions $\lambda_i(\xi,\eta) \in C^\infty$ for $i=1,\cdots,J$ such that $$ \det L(\tau,\eta,\xi)=\prod \limits_{i=1}^J (\tau - \lambda_i(\xi,\eta))^{\alpha_i}, \quad \lambda_1(\xi,\eta)<\lambda_2(\xi,\eta)<\cdots<\lambda_J(\xi,\eta),\qquad \qquad (1). $$ Here $J,\alpha_i$ are independent on $(\xi,\eta).$

Now we assume $A$ is an invertible matrix. We introduce a new real parameter $\gamma \geq 0$ and use the notation $$ M(\tau,\gamma,\eta)=A^{-1}((\tau-i\gamma)I_N+\eta B). $$ Now we want to analyze the eigenvalue of the matrix $M(\tau,\gamma,\eta)$. We introduce the following function $$ \Delta(\tau,\gamma,\eta,\xi)=\det (\xi I_N+M(\tau,\gamma,\eta)). $$ Here $\xi$ could be a complex function.

We have an important observation that when $\gamma>0$ and $\xi \in \mathbb{R},$ the function $$ \Delta(\tau,\gamma,\eta,\xi)=\det (\xi I_N+M(\tau,\gamma,\eta))=\det A^{-1} \cdot \det(\xi I_N+(\tau-i\gamma)I_N+\eta B) $$ does not vanish, otherwise the complex number $\tau-i\gamma$ will be a root of $\det L(\tau,\gamma,\eta),$ which contradicts the fact that the polynomial $\det L(\tau,\gamma,\eta)$ only has real roots in $\tau.$

Now we consider a fixed point $(\tau_0,0,\eta_0)$ (that is, $\gamma_0=0$.) For $\xi_0 \in \mathbb{R}$ so that $\Delta(\tau_0,0,\eta_0,\xi_0)=0,$ and we know $(\xi_0,\eta_0)\neq (0,0).$ By (1), there is a unique eigenvalue function $\lambda_j(\xi,\eta)$ so that $\tau_0=\lambda_j(\xi_0,\eta_0).$ Then the author stated that the function $\lambda_j(\xi,\eta)$ can be extended to a complex neighbourhood of $\xi_0$ due to the constant multiplicity property (1). Moreover, there is a $C^\infty$ matrix valued function $\Pi(\tau,\eta,\xi)$ defined in a neighbourhood of $(\eta_0,\xi_0)$ in $\mathbb{R}\times \mathbb{C}$, holomorphic in $\xi,$ such that $\Pi$ is a projector of rank $\alpha_j$ (recall that $\alpha_j$ is the algebraic multiplicity of $\lambda_j$), and $$ \ker (\xi I_N+M(\tau,\gamma,\eta))=\Pi(\eta,\xi)\mathbb{C}^N, \quad \text{when} \ \tau-i\gamma+\lambda_j(\xi,\eta)=0. $$ The author simply said the reason is also due to the constant multiplicity.

I am confused about the statement that $\lambda_j(\xi,\eta)$ can be locally smoothly extended to complex variable with respect to $\xi,$ why?

I also not sure about the existence of the projection matrix $\Pi(\eta,\xi).$ Here is my thought. After the eigenvalue $\lambda_j(\xi,\eta)$ extended to complex neighbourhood of $\xi_0,$ then there exists a smooth function $e(\tau,\gamma,\eta,\xi)$ defined in the neighbourhood of $(\tau_0,0,\eta_0,\xi_0)$ in $\mathbb{R}\times \mathbb{R}_+ \times \mathbb{R}\times \mathbb{C}$, so that $$ \Delta(\tau,\gamma,\eta,\xi)=e(\tau,\gamma,\eta,\xi) (\tau-i\gamma+\lambda_j(\xi,\eta))^{\alpha_j}, \quad \quad (2), $$ where $e(\tau_0,0,\eta_0,\xi_0)\neq 0.$ Thus in the neighbourhood of $(\tau_0,0,\eta_0,\xi_0),$ the kernel space $ \ker (\xi I_N+M(\tau,\gamma,\eta))=\ker ((\tau-i\gamma)I_N+\xi A+\eta B) $ can be seen as the eigenspace of $-(\xi A+\eta B)$ where the eigenvalue $\tau-i\gamma$ satisfies $\tau-i\gamma+\lambda_j(\xi,\eta)=0.$ I hope to use Dunford-Taylor calculus to construct the projection matrix. Since in the neighbourhood of $(\tau_0,0),$ when $\tau-i\gamma+\lambda_j(\xi,\eta)=0,$ we know from (2) that $\tau-i\gamma$ is always a $\alpha_j$-multiple root. Select a circle $\mathcal{C}$ around $\tau_0$ in the complex plane, such that when $(\xi,\eta)$ vary in a neighbourhood of $(\xi_0,\eta_0),$ there is a $\alpha_j$-multiple root $\tau-i\gamma$ inside the circle. Then the generalized eigenspace projector of eigenvalue $\tau-i\gamma$ is $$ \Pi(\xi,\eta)=\frac{1}{2\pi i}\int_{\mathcal{C}} (zI_N+\xi A+\eta B)^{-1} dz. $$ The fact that the rank of $\Pi$ is $\alpha_j$ comes from the dimension of generalized eigenspace is the algebraic multiplicity $\alpha_j$. Am I right?

Answer 1

Denote $P(\tau,\eta,\xi):=\det L(\tau,\eta,\xi)\in{\mathbb R}[\tau,\eta,\xi]$. This is a homogeneous polynomial of degree $N$, whose leading term in $\tau$ is just $\tau^N$.

Let us fix $j$, so that $\lambda_j(\xi,\eta)$ is a root of $P(\cdot,\eta,\xi)$ of constant multiplicity $\alpha_j$, at least when $(\xi,\eta)\ne0$ is real. Thus $P=(\tau-\lambda_j)^{\alpha_j}R(\tau,\xi,\eta)$ where $R$ is a polynomial in $\tau$ and is analytic in the real variables $\xi,\eta$, and $R(\lambda_j(\xi,\eta),\xi,\eta)$ does not vanish. Consider $$Q=\frac{\partial^{\alpha_j-1}P}{\partial\tau^{\alpha_j-1}}\in{\mathbb R}[\tau,\eta,\xi].$$ We have easily $Q=(\tau-\lambda_j)(\alpha_j!+(\tau-\lambda_j)R_1)$ for some other polynomial/analytical function $R_1$. Therefore $\lambda_j(\xi,\eta)$ is a simple root of $Q(\cdot,\eta,\xi)$.

Now, by the analytical version of the IFT, there exist a unique analytical function $f(\xi,\eta)$ over a complex neighbourhood $\cal U$ of $(\xi_0,\eta_0)$, which extends $\lambda_j$ as a root of $Q(\cdot,\eta,\xi)$. Then the analytical function $(\xi,\eta)\mapsto P(f(\xi,\eta),\eta,\xi)$ vanishes over ${\mathbb R}^2\cap\cal U$, and therefore vanishes identically. The same is true (same argument) when replacing $P$ by its $\tau$-derivatives up to order $\alpha_j-1$. At last, one has $$\frac{\partial^{\alpha_j}P}{\partial\tau^{\alpha_j}}(\lambda_j(\xi,\tau),\tau,\xi)\ne0,$$ a property that extends locally. Therefore, choosing $\cal U$ small enough, $f(\xi,\tau)$ is a root of $P(\cdot,\tau,\xi)$ of order exactly $\alpha_j$.