On operator representations of locally definitizable functions

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 $G$.