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 $Q\in N_{\kappa }^{n \times n}$ 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 $\Gamma: \mathbb{C}^{n}\rightarrow \mathcal{K}$ that represent the function $Q$ in a Krein-Langer type representation. We demonstrate the main results through examples of linear systems of ODEs.

Given Krein and Hilbert spaces $\left( \mathcal{K},[.,.] \right)$ and $\left( \mathcal{H}, \left( .,. \right) \right)$, respectively, the concept of the boundary triple $\Pi =(\mathcal{H}, \Gamma _{0}, \Gamma_{1})$ is generalized through the abstract Green's identity for the isometric relation $\Gamma$ 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 $\dom\, \Gamma$ and $\ran\, \Gamma$. This also means that we do not assume the existence of a closed symmetric linear relation $S$ such that $\dom\, \Gamma=S^{+}$, 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 $\Gamma : \mathcal{K}^{2} \rightarrow\mathcal{H}^{2}$ and the self-adjoint main transformation $\tilde{A}$ of $\Gamma$ 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 $\Gamma : \mathcal{K}^{2} \rightarrow\mathcal{H}^{2}$, 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 $\Gamma$ using the Green's boundary model approach.

: Let Q ∈ N κ ( H ) be a regular function, minimally represented by a self-adjoint linear relation A in the Pontryagin space ( K , [ ., . ]) of negative index κ and let ˆ Q := − Q − 1 be represented by a linear relation ˆ A . We give a necessary and sufficient condition for Q to be a Weyl function associated with S = A ∩ ˆ A and A . We also study a class of functions Q ∈ N κ ( H ) that have boundedly invertible derivative at infinity Q ′ ( ∞ ) := lim z →∞ zQ ( z ); we give relation matrices of A , ˆ A and S + in terms of S . We prove that every such function Q is a Weyl function associated with S = A | I − P and A . In examples we show how to apply the main results. For instance, for a given regular function Q ∈ N κ ( H ) with boundedly invertible Q ′ ( ∞ ) represented by A , we find the symmetric relation S so that Q is the Weyl function associated with ( S, A ). Then we find the corresponding boundary triple Π = ( H , Γ 0 , Γ 1 ). In another example, we apply main results to find linear relations ˆ R , S , ˆ A , S + which are associated with a given regular function Q ∈ N κ ( H ) represented by a given operator A .

Abstract: Let S be a symmetric linear relation in the Pontryagin space (K, [., .]) of index κ, let Π = (H,Γ0,Γ1) be an ordinary boundary triple for the relation S, and let Q be the Weyl function corresponding to S and Π. By means of a version of the Krein formula in Pontryagin space, we prove  = kerΓ1, where  is the representing relation of Q̂ := −Q−1 ∈ Nκ(H). For regular function Q ∈ Nκ(H), with representing relation A, we find symmetric relation S, {0} ⊆ S ( A, and the boundary triple Π such that Q is the Weyl function corresponding to S and Π. Then we assume that the derivative at infinity Q ′ (∞) := lim z→∞ zQ(z) is a boundedly invertible operator. That enables us to express

Let S be a symmetric linear relation in the Pontyagin space (K, [., .]) and let Π = (H,Γ0,Γ1) be the corresponding boundary triple. We prove that the corresponding Weyl function Q satisfies Q ∈ Nκ(H). Conversely, for regular Q ∈ Nκ(H), we find linear relation S ( A, where A is representing self-adjoint linear relation of Q, and we prove that Q is the Weyl function of the relation S. We also prove  = kerΓ1, where  is the representing relation of the Q̂ := −Q−1. In addition, if we assume that the derivative at infinity Q ′ (∞) := lim z→∞ zQ(z) is a boundedly invertible operator then we are able to decompose A,  and S in terms of S, i.e. we express relation matrices of A,  and S in terms of S, which is a bounded operator in this case.

We provide the necessary and sufficient conditions for a generalized Nevanlinna function $Q$ ($Q\in N_{\kappa }\left( \mathcal{H} \right)$) to be a Weyl function (also known as a Weyl-Titchmarch function). We also investigate an important subclass of $N_{\kappa }(\mathcal{H})$, 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 $Q\left( z \right)=\tilde{\Gamma}^{+}\left( A-z \right)^{-1}\tilde{\Gamma},z\in \rho \left( A \right)$, 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:=\tilde{\Gamma} \left( \tilde{\Gamma}^{+} \tilde{\Gamma} \right)^{-1} \tilde{\Gamma}^{+} $. 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 $Q\in N_{\kappa }\left( \mathcal{H} \right)$ and proceed to determine the closed symmetric linear relation $S$ and the boundary triple $\Pi$ so that $Q$ becomes the Weyl function associated with $\Pi$. 2020 Mathematics Subject Classification. 34B20, 47B50, 47A06, 47A56

Abstract: Let S be a symmetric linear relation in the Pontyagin space (K, [., .]) of index κ, let Π = (H,Γ0,Γ1) be an ordinary boundary triple for the relation S, and let Q be the Weyl function corresponding to S and Π. By means of a version of the Krein formula in Pontryagin space, we prove  = kerΓ1, where  is the representing relation of Q̂ := −Q−1 ∈ Nκ(H). For regular function Q ∈ Nκ(H), with representing relation A, we find symmetric relation S, {0} ⊆ S ( A, and the boundary triple Π such that Q is the Weyl function corresponding to S and Π. Then we assume that the derivative at infinity Q ′ (∞) := lim z→∞ zQ(z) is a boundedly invertible operator. That enables us to express

UDC 517.9We present necessary and sufficient conditions for the reducibility of a self-adjoint linear relation in a Krein space. Then a generalized Nevanlinna function Q represented by a self-adjoint linear relation A in a Pontryagin space is decomposed by means of the reducing subspaces of A . The sum of two functions Q i ∈ N κ i ( ℋ ) , i = 1,2 , minimally represented by the triplets ( 𝒦 i , A i , Γ i ) is also studied. For this purpose, we create a model ( 𝒦 ˜ , A ˜ , Γ ˜ ) to represent Q : = Q 1 + Q 2 in terms of ( 𝒦 i , A i , Γ i ) . By using this model, necessary and sufficient conditions for κ = κ 1 + κ 2 are proved in the analytic form. Finally, we explain how degenerate Jordan chains of the representing relation A affect the reducing subspaces of A and the decomposition of the corresponding function Q .

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 $\mathcal{N}_{\kappa} (\mathcal{H})$, where $\kappa$ 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 $\kappa$ negative squares. A focus is on the functions $Q \in \mathcal{N}_{\kappa} (\mathcal{H})$ 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 $\hat{Q}=\hat{Q}_{1}+\hat{Q}_{2},\, \, \hat{Q}_{i}\in \mathcal{N}_{\kappa_{i}}\left( \mathcal{H} \right)$ that satisfies $\kappa_{1}+\kappa_{2}=\kappa $. That decomposition enables us to study properties of both functions, $Q$ and $\hat{Q}$, by studying the simple components $\hat{Q}_{1}$ and $\hat{Q}_{2}$.