Root functions of a meromorphic matrix function and applications
Muhamed Borogovac
TL;DR
The paper develops a practical framework for root and pole cancellation analysis of meromorphic matrix functions $Q(z)$ on the extended complex plane, enabling a factorization $Q(z)=(z-\beta)^m\tilde{Q}(z)$ with $\tilde{Q}$ holomorphic at $\infty$ and a bounded operator $\tilde{A}$ that replaces the Krein–Langer relation. It introduces a constructive diagonalization $\tilde{D}(z)=S(z)Q(z)T(z)$ to extract canonical root and pole functions from zeros/poles of $\tilde{d}_i(z)$, and demonstrates how these canonical objects determine geometric and partial multiplicities. The framework is applied to nonlinear differential systems by translating them into a meromorphic matrix problem whose solutions are obtained from zeros of $Q$ and corresponding eigenvectors. An operator realization in Pontryagin spaces is provided, relating $Q$ to bounded self-adjoint operators and clarifying how spectral data transform under the $(z-\beta)^m$ factorization. Overall, the work extends root/pole analysis beyond matrix polynomials, with implications for spectral theory and differential equations.
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,\, \mathellipsis ,\,n $. For a function $Q\in \mathcal{N}_κ(\mathcal{H}) ,\, κ\in \mathbb{N} \cup \lbrace 0 \rbrace$, 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 $β\in \mathbb{R}$ is a regular point of $Q$, and $\tilde{Q}\in \mathcal{N}_{κ'}(\mathcal{H})$ 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 $β$ and $\infty$. We demonstrate how to obtain this representation for a given meromorphic function $Q\in \mathcal{N}_κ^{n \times n}$ using the root functions developed in this work.