Root functions of a meromorphic matrix function and applications

A practical method is presented for determining root and pole cancellation functions of a matrix function $Q(z)$ meromorphic on the extended complex plane $\bar{\mathbb{C}}:=\mathbb{C}\cup \left\{\mathrm{\infty }\right\}$. 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 $Q\in {\mathcal{N}}_{\kappa }(\mathcal{H}),\kappa \in \mathbb{N}\cup \{0\}$, posesing a pole at infinity of order $m\in \mathbb{N}$, the following factorization is establish $$Q(z)=(z-\beta {)}^m\tilde{Q}(z),z\in \mathcal{D}(Q),$$ where $\beta \in \mathbb{R}$ is a regular point of $Q$, and $\tilde{Q}\in {\mathcal{N}}_{{\kappa }^{\mathrm{\prime }}}(\mathcal{H})$ is holomotphic at $\mathrm{\infty }$. 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 $\beta$ and $\mathrm{\infty }$. We demonstrate how to obtain this representation for a given meromorphic function $Q\in {\mathcal{N}}_{\kappa }^{n\times n}$ using the root functions developed in this work.