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

Abstract

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 $Q^{\prime }\left(\infty \right):=\underset{z\to \infty }{\lim }zQ(z)$. 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 $S:=A_{|(I-P)\mathcal{K}}$, 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 $\Pi$.

23 pages, Accepted in Sarajevo Journal of Mathematics

References in corpus (2)

• Inverse of generalized Nevanlinna function that is holomorphic at infinity
• Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions