Inverse of generalized Nevanlinna function that is holomorphic at infinity

Abstract

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 \(\hat{Q}_{2}\).