On some Subclasses of Nevanlinna Functions

Abstract

For any function $Q=Q(\ell )$ belonging to the class $\mathbb{N}$ of Nevanlinna functions, the function $Q_{\tau }=Q_{\tau }(\ell )$ defined by $Q_{\tau }(\ell )=\frac{Q(\ell )–\tau (lmQ(\mu ){)}^2}{\tau Q(\ell )+1}$ belongs to $\mathbb{N}$ for all values of $\tau \in \mathbb{R}\bigcup \infty$. The class $\mathbb{N}$ possesses subclasses ${\mathbb{N}}_0\subset {\mathbb{N}}_1$, each defined by some additional asymptotic conditions. If a function $Q$ belongs to such a subclass, then for all but one value of $\tau \in \mathbb{R}\bigcup \infty$ the function $Q_{\tau }$ belongs to the same subclass and the corresponding exceptional function can be characterized (cf. [4]). In this note we introduce two subclasses ${\mathbb{N}}_{–2}\subset {\mathbb{N}}_{–1}$ of ${\mathbb{N}}_0$ which can be described in terms of the moments of the spectral measures in the associated integral representations. We characterize the corresponding exceptional function in a purely function-theoretic way by suitably estimating certain quadratic terms. The behaviour of the exceptional function connects the subclasses ${\mathbb{N}}_{–2}$ and ${\mathbb{N}}_{–1}$ to the classes ${\mathbb{N}}_0$ and ${\mathbb{N}}_{–1}$, respectively, as studied in [4]. In operator-theoretic terms these notions have a translation in terms of $Q$-functions of selfadjoint extensions of a symmetric operator with defect numbers (1, 1). In this sense the exceptional function has an interpretation in terms of a generalized Friedrichs extension of the symmetric operator.