Root functions of a meromorphic matrix function and applications
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866912661713715200 |
|---|---|
| author | Borogovac, Muhamed |
| author_facet | Borogovac, Muhamed |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_06812 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Root functions of a meromorphic matrix function and applications Borogovac, Muhamed Complex Variables Functional Analysis (2020) 47A56 30D30 34M04 47B50 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. |
| title | Root functions of a meromorphic matrix function and applications |
| topic | Complex Variables Functional Analysis (2020) 47A56 30D30 34M04 47B50 |
| url | https://arxiv.org/abs/2505.06812 |