On Operator Representations of Locally Definitizable Functions

Abstract

Let Ω be some domain in C symmetric with respect to the real axis and such that Ω ∩ R = ∅ 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 \ R such that for any subdomain Ω ′ of Ω with Ω ′ ⊂ 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 ∆ ⊂ R∩Ω of positive and negative type as well as some "local" inner products associated with intervals ∆ ⊂ 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.

Locally Definitizable Operators: The Local Structure of the Spectrum

“…Later, in a series of papers, P. Jonas studied these operators and introduced the notion of locally definitizable operators, cf. [52,53,55,59,60]. This class of operators will be of particular interest in the following, hence we recall here the definition of locally definitizable operators or, more precisely, operators definitizable over some subset of C. …”

Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

“…In [40] Jonas has given a corresponding characterization for so-called definitizable functions. For the particular case of generalized Nevanlinna functions, this result reads as follows.…”

“…Special attention was paid to the case when A is definitizable, roughly speaking this means that there exists a polynomial p such that p.A/ is nonnegative in the Krein space. The corresponding so-called definitizable functions have been characterized analytically and are studied in, e.g., [39,40]. Generalized Nevanlinna functions do belong to this class.…”

Boundary Value Problems with Local Generalized Nevanlinna Functions in the Boundary Condition

“…Here we consider a local variant of generalized Nevanlinna functions. We recall the definition of the class of local generalized Nevanlinna functions, which is a subclass of the class of the so-called locally definitizable functions (see [17]). …”