Inhalt des Dokuments
Preprint 20-2005
On operator representations of locally definitizable functions
Author(s) :
Peter Jonas
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 20-2005
MSC 2000
- 47B50 Operators on spaces with an indefinite metric
-
47A56 Functions whose values are linear operators
-
47A60 Functional calculus
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 $G$.
Keywords :
Definitizable operator functions, generalized Nevanlinna functions, selfadjoint and unitary operators in Krein spaces, locally definitizable operators, spectral points of positive and negative type.