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Main Title: On operator representations of locally definitizable functions
Author(s): Jonas, Peter
Type: Research Paper
Abstract: Let $\Omega$ be some domain in $\overline{{\bf C}}$ symmetric with respect to the real axis and such that $\Omega \cap \overline{{\bf R}} \neq \emptyset$ and the intersections of $\Omega$ with the upper and lower open half-planes are simply connected. We study the class of piecewise meromorphic ${\bf R}$-symmetric operator functions $G$ in $\Omega \setminus \overline{{\bf R}}$ such that for any subdomain $\Omega'$ of $\Omega$ with $\overline{\Omega'} \subset \Omega$, $G$ restricted to $\Omega'$ can be written as a sum of a definitizable and a (in $\Omega'$) holomorphic operator function. As in the case of a definitizable operator function, for such a function $G$ we define intervals $\Delta \subset {\bf R} \cap \Omega$ of positive and negative type as well as some ``local'' inner products associated with intervals $\Delta \subset {\bf R} \cap \Omega$. Representations of $G$ with the help of linear operators and relations are studied, and it is proved that there is a representing locally definitizable selfadjoint relation $A$ in a Krein space which locally exactly reflects the sign properties of $G$: The ranks of positivity and negativity of the spectral subspaces of $A$ coincide with the numbers of positive and negative squares of the "local'' inner products corresponding to $G$.
Subject(s): definitizable operator functions
generalized Nevanlinna functions
selfadjoint and unitary operators in Krein spaces
locally definitizable operators
spectral points of positive and negative type
Issue Date: 1-Sep-2005
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 47B50 Operators on spaces with an indefinite metric
47A56 Functions whose values are linear operators
47A60 Functional calculus
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2005, 20
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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