Characterization of Weyl functions in the class of operator-valued generalized Nevanlinna functions
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866910888852717568 |
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| author | Borogovac, Muhamed |
| author_facet | Borogovac, Muhamed |
| contents | 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 $Π$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_06931 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Characterization of Weyl functions in the class of operator-valued generalized Nevanlinna functions Borogovac, Muhamed Functional Analysis Classical Analysis and ODEs (2020) 34B20 47A50 47A06 47B56 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 $Π$. |
| title | Characterization of Weyl functions in the class of operator-valued generalized Nevanlinna functions |
| topic | Functional Analysis Classical Analysis and ODEs (2020) 34B20 47A50 47A06 47B56 |
| url | https://arxiv.org/abs/2102.06931 |