Desirable Decompositions of Generalized Nevanlinna Functions

Abstract

For a given generalized Nevanlinna function $Q\in N_{\kappa }\left(H\right)$, we study decompositions that satisfy: $Q=Q_1+Q_2$; $Q_i{\in N}_{{\kappa }_i}\left(H\right)$, and ${\kappa }_1+{\kappa }_2=\kappa$, $0\le {\kappa }_i$, which we call desirable decompositions. In this paper, some sufficient conditions for such decompositions of $Q$ are given. One of the main results is a new operator representation of $\hat{Q}\left(z\right):=-{Q(z)}^{-1}$ if $Q\left(z\right):={\Gamma }_0^{+}{\left(A-z\right)}^{-1}{\Gamma }_0$, where $A$ is a bounded self-adjoint operator in a Pontryagin space. The new representation is used to get an interesting desirable decomposition of $\hat{Q}$ and to obtain some information about singularities of $\hat{Q}$.

16 pages