Desirable Decompositions of Generalized Nevanlinna Functions

For a given generalized Nevanlinna function $Q\in N_{κ}\left( H \right)$, we study decompositions that satisfy: $Q=Q_{1}+Q_{2}$; $Q_{i}{\in N}_{κ_{i}}\left( H \right)$, and $κ_{1}+κ_{2}=κ$, $0\le κ_{i}$, 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 $Q\left( z \right):=Γ_{0}^{+}\left( A-z\right)^{-1}Γ_{0}$, 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 $\hat{Q}$.

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