Desirable Decompositions of Generalized Nevanlinna Functions

For a given generalized Nevanlinna function \(Q\in N_{\kappa }\left( H \right)\), we study decompositions that satisfy: \(Q=Q_{1}+Q_{2}\); \(Q_{i}{\in N}_{\kappa_{i}}\left( H \right)\), and \(\kappa_{1}+\kappa_{2}=\kappa \), \(0\le \kappa_{i}\), which we call desirable decompositions. In this paper,...

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Bibliographic Details
Published in:arXiv.org 2015-02
Main Author: Borogovac, Muhamed
Format: Article
Language:English
Subjects:
Decomposition
Economic models
Operators (mathematics)
Representations
Singularities
Online Access:Get full text
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Summary:For a given generalized Nevanlinna function \(Q\in N_{\kappa }\left( H \right)\), we study decompositions that satisfy: \(Q=Q_{1}+Q_{2}\); \(Q_{i}{\in N}_{\kappa_{i}}\left( H \right)\), and \(\kappa_{1}+\kappa_{2}=\kappa \), \(0\le \kappa_{i}\), which we call desirable decompositions. In this paper, some sufficient conditions for such decompositions of \(Q\) are given. One of the main results is a new operator representation of \(\hat{Q}\left(z\right):=-{Q(z)}^{-1}\) if \(Q\left( z \right):=\Gamma_{0}^{+}\left( A-z\right)^{-1}\Gamma_{0}\), where \(A\) is a bounded self-adjoint operator in a Pontryagin space. The new representation is used to get an interesting desirable decomposition of \(\hat{Q}\) and to obtain some information about singularities of \(\hat{Q}\).
ISSN:2331-8422