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Muhamed Borogovac

Publications and source records attributed to Muhamed Borogovac.

11 recordsLinked to original sources
arXiv2026

Construction of Pole Cancellation Functions at Ordinary Poles of Operator-Valued Functions

A pole of order $m \in \mathbb{N}$ at $β\in \mathbb{C}$ of a regular operator valued function $Q : \mathcal{D}(Q) \to \mathcal{L}(\mathcal{H})$ is investigated. We provide a characterization of pole cancellation functions $\boldsymbolψ(z)$ of $Q(z)$ of order $k \le m$ at $β$ in terms of the coefficients of the Laurent expansion of $Q$. This characterization yields practical and explicit constructions of pole cancellation functions $\boldsymbolψ(z)$. Moreover, it leads to an explicit formula for the associated functions $\boldsymbol{\hatφ}(z) := Q(z)\boldsymbolψ(z)$, which are root functions of order $k$ at the zero $β$ of $Q^{-1}$. The results are illustrated by an example.

math.CV↗
arXiv2025

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 $\bar{\mathbb{C}}:=\mathbb{C} \cup \left\{ \infty \right\}$. 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.

math.CV↗
arXiv2024

Green's boundary relation model in a Krein space

Given Krein and Hilbert spaces $\left( \mathcal{K},[.,.] \right)$ and $\left( \mathcal{H}, \left( .,. \right) \right)$, respectively, the concept of the boundary triple $Π=(\mathcal{H}, Γ_{0}, Γ_{1})$ is generalized through the abstract Green's identity for the isometric relation $Γ$ between Krein spaces $\left( \mathcal{K}^{2}, \left[ .,.\right]_{\mathcal{K}^{2}} \right) $ and $\left(\mathcal{H}^{2}, \left[ .,.\right]_{\mathcal{H}^{2}} \right) $ without any conditions on $\dom\, Γ$ and $\ran\, Γ$. This also means that we do not assume the existence of a closed symmetric linear relation $S$ such that $\dom\, Γ=S^{+}$, which is a standard assumptions in all previous research of boundary triples. The main properties of such a general Green's boundary model are proven. In the process, some useful properties of the isometric relation $V$ between two Krein spaces $X$ and $Y$ are proven. Additionally, surprising properties of the unitary relation $Γ: \mathcal{K}^{2} \rightarrow\mathcal{H}^{2}$ and the self-adjoint main transformation $\tilde{A}$ of $Γ$ are discovered. Then, two statements about generalized Nevanlinna families are generalized using this Green's boundary model. Furthermore, several previously known boundary triples involving a Hilbert space $\mathcal{K}$ and reduction operator $Γ: \mathcal{K}^{2} \rightarrow\mathcal{H}^{2}$, such as AB-generalized, B-generalized, ordinary, isometric, unitary, quasi-boundary, and S-generalized boundary triples, have been extended to a Krein space $\mathcal{K}$ and linear relation $Γ$ using the Green's boundary model approach.

math.FA↗
arXiv2024

Novel approach to root functions of matrix polynomials with applications in differential equations and meromorphic matrix functions

In the first part of the paper, we address an invertible matrix polynomial $L(z)$ and its inverse $\hat{L}(z) := -L(z)^{-1}$. We present a method for obtaining a canonical set of root functions and Jordan chains of $L(z)$ through elementary transformations of the matrix $L(z)$ alone. This method provides a new and simple approach to deriving a general solution of the system of ordinary linear differential equations $L\left(\frac{d}{dt}\right)u=0$ using only elementary transformations of the corresponding matrix polynomial $L(z)$. In the second part of the paper, given a matrix generalized Nevanlinna function $Q\in N_{κ}^{n \times n}$ and a canonical set of root functions of $\hat{Q}(z) := -Q(z)^{-1}$, we provide an algorithm to determine a specific Pontryagin space $(\mathcal{K}, [.,.])$, a specific self-adjoint operator $A:\mathcal{K}\rightarrow \mathcal{K}$ and an operator $Γ: \mathbb{C}^{n}\rightarrow \mathcal{K}$ that represent the function $Q$ in a Krein-Langer type representation. We demonstrate the main results through examples of linear systems of ODEs.

math.FA↗
arXiv2024

Characterization of Weyl functions in the class of operator-valued generalized Nevanlinna functions

We provide the necessary and sufficient conditions for a generalized Nevanlinna function $Q$ ($Q\in N_{κ}\left( \mathcal{H} \right)$) to be a Weyl function (also known as a Weyl-Titchmarch function). We also investigate an important subclass of $N_{κ}(\mathcal{H})$, the functions that have a boundedly invertible derivative at infinity $Q'\left( \infty \right):=\lim \limits_{z \to \infty}{zQ(z)}$. These functions are regular and have the operator representation $Q\left( z \right)=\tildeΓ^{+}\left( A-z \right)^{-1}\tildeΓ,z\in ρ\left( A \right)$, where $A$ is a bounded self-adjoint operator in a Pontryagin space $\mathcal{K}$. We prove that every such strict function $Q$ is a Weyl function associated with the symmetric operator $S:=A_{\vert (I-P)\mathcal{K}}$, where $P$ is the orthogonal projection, $P:=\tildeΓ \left( \tildeΓ^{+} \tildeΓ \right)^{-1} \tildeΓ^{+} $. Additionally, we provide the relation matrices of the adjoint relation $S^{+}$ of $S$, and of $\hat{A}$, where $\hat{A}$ is the representing relation of $\hat{Q}:=-Q^{-1}$. We illustrate our results through examples, wherein we begin with a given function $Q\in N_{κ}\left( \mathcal{H} \right)$ and proceed to determine the closed symmetric linear relation $S$ and the boundary triple $Π$ so that $Q$ becomes the Weyl function associated with $Π$.

math.FA↗
arXiv2020

Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions

Necessary and sufficient conditions for reducidibility of a self-adjoint linear relation in a Krein space are given. Then a generalized Nevanlinna function $Q$, represented by a self-adjoint linear relation $A$, is decomposed by means of the reducing subspaces of $A$. The sum of two functions $Q_{i}{\in N}_{κ_{i}}\left( \mathcal{H} \right),\thinspace i=1,\thinspace 2$, minimally represented by the triplets $\left( \mathcal{K}_{i},A_{i},Γ_{i} \right)$, is also studied. For that purpose, a model $( \tilde{\mathcal{K}},\tilde{A},\tilde{Γ} )$ to represent $Q:=Q_{1}+Q_{2}$ in terms of $\left( \mathcal{K}_{i},A_{i},Γ_{i} \right)$ is created. By means of that model, necessary and sufficient conditions for $κ=κ_{1}+κ_{2}$ are proven in analytic terms. At the end, it is explained how degenerate Jordan chains of the representing relation $A$ affect reducing subspaces of $A$ and decomposition of the corresponding function $Q$.

math.FA↗
arXiv2020

Inverse of generalized Nevanlinna function that is holomorphic at infinity

Let $\left(\mathcal{H},\left(.,.\right)\right)$ be a Hilbert space and let $\mathcal{L}\left(\mathcal{H}\right)$ be the linear space of bounded operators in $\mathcal{H}$. In this paper, we deal with $\mathcal{L}(\mathcal{H})$-valued function $Q$ that belongs to the generalized Nevanlinna class $\mathcal{N}_κ (\mathcal{H})$, where $κ$ is a non-negative integer. It is the class of functions meromorphic on $C \backslash R$, such that $Q(z)^{*}=Q(\bar{z})$ and the kernel $\mathcal{N}_{Q}\left( z,w \right):=\frac{Q\left( z \right)-{Q\left( w \right)}^{\ast }}{z-\bar{w}}$ has $κ$ negative squares. A focus is on the functions $Q \in \mathcal{N}_κ (\mathcal{H})$ which are holomorphic at $ \infty$. A new operator representation of the inverse function $\hat{Q}\left( z \right):=-{Q\left( z \right)}^{-1}$ is obtained under the condition that the derivative at infinity $Q^{'}\left( \infty\right):=\lim\limits_{z\to \infty}{zQ(z)}$ is boundedly invertible operator. It turns out that $\hat{Q}$ is the sum $\hat{Q}=\hat{Q}_{1}+\hat{Q}_{2},\, \, \hat{Q}_{i}\in \mathcal{N}_{κ_{i}}\left( \mathcal{H} \right)$ that satisfies $κ_{1}+κ_{2}=κ$. That decomposition enables us to study properties of both functions, $Q$ and $\hat{Q}$, by studying the simple components $\hat{Q}_{1}$ and $\hat{Q}_{2}$.

math.FA↗
arXiv2015

Two Applications of Brouwer's Fixed Point Theorem: in Insurance and in Biology Models

In the first part of the article, a new interesting system of difference equations is introduced. It is developed for re-rating purposes in general insurance. A nonlinear transformation $φ$ of a d-dimensional $(d \ge 2)$ Euclidean space is introduced that enables us to express the system in the form $f^{t+1}:=φ(f^t),\, t=0,\, 1,\, 2,\, \ldots $. Under typical actuarial assumptions, existence of solutions of that system is proven by means of Brouwer's fixed point theorem in normed spaces. In addition, conditions that guarantee uniqueness of a solution are given. The second, smaller part of the article is about Leslie-Gower's system of $d \ge 2$ difference equations. We focus on the system that satisfies conditions consistent with weak inter-specific competition. We prove existence and uniqueness of the equilibrium of the model under surprisingly simple and very general conditions. Even though the two parts of this article have applications in two different sciences, they are connected with similar mathematics, in particular by our use of Brouwer's Fixed point Theorem.

math.OC↗
arXiv2015

Desirable Decompositions of Generalized Nevanlinna Functions

For a given generalized Nevanlinna function $Q\in N_{κ}\left( H \right)$, we study decompositions that satisfy: $Q=Q_{1}+Q_{2}$; $Q_{i}{\in N}_{κ_{i}}\left( H \right)$, and $κ_{1}+κ_{2}=κ$, $0\le κ_{i}$, which we call desirable decompositions. In this paper, some sufficient conditions for such decompositions of $Q$ are given. One of the main results is a new operator representation of $\hat{Q}\left(z\right):=-{Q(z)}^{-1}$ if $Q\left( z \right):=Γ_{0}^{+}\left( A-z\right)^{-1}Γ_{0}$, where $A$ is a bounded self-adjoint operator in a Pontryagin space. The new representation is used to get an interesting desirable decomposition of $\hat{Q}$ and to obtain some information about singularities of $\hat{Q}$.

math.FA↗
arXiv2014

Characterizations of generalized poles by pole cancellation functions of higher order

In this paper the analytic characterization of generalized poles of operator valued generalized Nevanlinna functions (including the length of Jordan chains of the representing relation) is completed. In particular, given a Jordan chain of length $\ell$, we show that there exists a pole cancellation function of order at least $\ell$, and, moreover, this function is of surprisingly simple form.

math.FA↗