Characterization of Weyl functions in the class of operator-valued generalized Nevanlinna functions
Muhamed Borogovac
TL;DR
The paper provides necessary and sufficient conditions for a operator-valued generalized Nevanlinna function $Q\in N_{\kappa}(\mathcal{H})$ to be a Weyl function, clarifying the roles of regularity and the strict part via Krein–Langer representations in Pontryagin spaces. It proves a bidirectional realization: a regular, strict $Q$ is the Weyl function of a simple closed symmetric operator $S$ in a Pontryagin space with an ordinary boundary triple, and conversely, every such Weyl function arises from such data with $A=\ker\Gamma_0$, $\hat{A}=\ker\Gamma_1$, and $S^+ = A\hat{+}\hat{A}$. For functions holomorphic at infinity with invertible $Q'(\infty)$, it develops a constructive inverse problem, giving explicit block realizations and showing $\mathcal{R}$-regular extensions $\hat{A}$, $A$, and $S^+$ of $S$. The results are demonstrated through detailed examples that recover the associated symmetric operator and boundary triple from a given Weyl function, highlighting practical realization techniques in indefinite inner product spaces.
Abstract
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 $Π$.