On some Subclasses of Nevanlinna Functions

  • S. Hassi

    University of Helsinki, Finland
  • Jaap Top

    University of Groningen, Netherlands


For any function Q=Q()Q = Q(\ell) belonging to the class N\mathbb N of Nevanlinna functions, the function Qτ=Qτ()Q_\tau = Q_{\tau}(\ell) defined by Qτ()=Q()τ(lmQ(μ))2τQ()+1Q_{\tau}(\ell) = \frac{Q (\ell) –\tau(lmQ(\mu))^2}{\tau Q(\ell)+1} belongs to N\mathbb N for all values of τR\tau \in \mathbb R \bigcup {\infty}. The class N\mathbb N possesses subclasses N0N1\mathbb N_0 \subset \mathbb N_1, each defined by some additional asymptotic conditions. If a function QQ belongs to such a subclass, then for all but one value of τR\tau \in \mathbb R \bigcup {\infty} the function QτQ_{\tau} belongs to the same subclass and the corresponding exceptional function can be characterized (cf. [4]). In this note we introduce two subclasses N2N1\mathbb N_{–2} \subset \mathbb N_{–1} of N0\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 N2\mathbb N_{–2} and N1\mathbb N_{–1} to the classes N0\mathbb N_0 and N1\mathbb N_{–1}, respectively, as studied in [4]. In operator-theoretic terms these notions have a translation in terms of QQ-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.

Cite this article

S. Hassi, Jaap Top, On some Subclasses of Nevanlinna Functions. Z. Anal. Anwend. 15 (1996), no. 1, pp. 45–55

DOI 10.4171/ZAA/687