Root functions of a meromorphic matrix function and applications

Abstract

A practical method is presented for determining root and pole cancellation functions of a matrix function $Q(z)$ meromorphic on the extended complex plane $\bar{\mathbb{C}}:=\mathbb{C}\cup \left\{\infty \right\}$. This method is applied to solve a nonlinear system of $n\in \mathbb{N}$ differential equations of order $l\in \mathbb{N}$ with $n$ unknown functions $u_i\left(t\right)$, where $i=1,\dots ,n$. For a function $Q\in {\mathcal{N}}_{\hat{I}}º(\mathcal{H}),\hat{I}º\in \mathbb{N}\cup \{0\}$, posesing a pole at infinity of order $m\in \mathbb{N}$, the following factorization is establish \[ Q(z)=(z-β)^{m}\tilde{Q}(z), \, z\in \mathcal{D}(Q), \] where ${\hat{I}}^2\in \mathbb{R}$ is a regular point of $Q$, and $\tilde{Q}\in {\mathcal{N}}_{\hat{I}{º}^{\prime }}(\mathcal{H})$ is holomotphic at $\infty$. Unlike the Krein-Langer representation of $Q$, which involves a linear relation $A$, this representation employs a bounded operator $\tilde{A}$ in the Krein-Langer representation of $\tilde{Q}$. The operator $\tilde{A}$ and the relation $A$ have identical spectra, except at ${\hat{I}}^2$ and $\infty$. We demonstrate how to obtain this representation for a given meromorphic function $Q\in {\mathcal{N}}_{\hat{I}}{º}^{n\times n}$ using the root functions developed in this work.

20 pages