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 $\hat{Q}$.