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 $Q$ ($Q\in N_{\kappa }\left( \mathcal{H} \right)$) 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 $\Pi$.