On Operator Representations of Locally Definitizable Functions

Abstract

Let Ω be some domain in $\bar C$ symmetric with respect to the real axis and such that Ω ∩ $\bar R$ ≠ ø and the intersections of Ω with the upper and lower open half-planes are simply connected. We study the class of piecewise meromorphic R-symmetric operator functions G in Ω \ $\bar R$ such that for any subdomain Ω′ of Ω with $\overline {\Omega '}$ ⊂ Ω, G restricted to Ω′ can be written as a sum of a definitizable and a (in Ω′) holomorphic operator function. As in the case of a definitizable operator function, for such a function G we define intervals δ ⊂ R∩Ω of positive and negative type as well as some “local” inner products associated with intervals δ ⊂ R ∩ Ω.

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.