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 $Q(z{)}^{\ast }=Q(\bar{z})$ 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 $Q^{{}^{\prime }}\left(\infty \right):=\underset{z\to \infty }{\lim }zQ(z)$ 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 ${\hat{Q}}_2$.

19 pages

Cited by in corpus (2)

• Reducibility of self-adjoint linear relations and application to generalized Nevanlinna functions
• Characterization of Weyl functions in the class of operator-valued generalized Nevanlinna functions