Please use this identifier to cite or link to this item: http://dx.doi.org/10.14279/depositonce-14331
 Main Title: On operator representations of locally definitizable functions Author(s): Jonas, Peter Type: Research Paper URI: https://depositonce.tu-berlin.de/handle/11303/15558http://dx.doi.org/10.14279/depositonce-14331 License: http://rightsstatements.org/vocab/InC/1.0/ 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 functionsgeneralized Nevanlinna functionsselfadjoint and unitary operators in Krein spaceslocally definitizable operatorsspectral 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 metric47A56 Functions whose values are linear operators47A60 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