  Let $\left(\mathcal{H},\left(.,.\right)\right)$ be a Hilbert space and let
$\mathcal{L}\left(\mathcal{H}\right)$ be the linear space of bounded operators
in $\mathcal{H}$. In this paper, we deal with $\mathcal{L}(\mathcal{H})$-valued
function $Q$ that belongs to the generalized Nevanlinna class
$\mathcal{N}_{\kappa} (\mathcal{H})$, where $\kappa$ is a non-negative integer.
It is the class of functions meromorphic on $C \backslash R$, such that
$Q(z)^{*}=Q(\bar{z})$ and the kernel $\mathcal{N}_{Q}\left( z,w
\right):=\frac{Q\left( z \right)-{Q\left( w \right)}^{\ast }}{z-\bar{w}}$ has
$\kappa$ negative squares. A focus is on the functions $Q \in
\mathcal{N}_{\kappa} (\mathcal{H})$ which are holomorphic at $ \infty$. A new
operator representation of the inverse function $\hat{Q}\left( z
\right):=-{Q\left( z \right)}^{-1}$ is obtained under the condition that the
derivative at infinity $Q^{'}\left( \infty\right):=\lim\limits_{z\to
\infty}{zQ(z)}$ is boundedly invertible operator. It turns out that $\hat{Q}$
is the sum $\hat{Q}=\hat{Q}_{1}+\hat{Q}_{2},\, \, \hat{Q}_{i}\in
\mathcal{N}_{\kappa_{i}}\left( \mathcal{H} \right)$ that satisfies
$\kappa_{1}+\kappa_{2}=\kappa $. That decomposition enables us to study
properties of both functions, $Q$ and $\hat{Q}$, by studying the simple
components $\hat{Q}_{1}$ and $\hat{Q}_{2}$.
