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.