# On some Subclasses of Nevanlinna Functions

### S. Hassi

University of Helsinki, Finland### Jaap Top

University of Groningen, Netherlands

## Abstract

For any function $Q=Q(ℓ)$ belonging to the class $N$ of Nevanlinna functions, the function $Q_{τ}=Q_{τ}(ℓ)$ defined by $Q_{τ}(ℓ)=τQ(ℓ)+1Q(ℓ)–τ(lmQ(μ))_{2} $ belongs to $N$ for all values of $τ∈R⋃∞$. The class $N$ possesses subclasses $N_{0}⊂N_{1}$, each defined by some additional asymptotic conditions. If a function $Q$ belongs to such a subclass, then for all but one value of $τ∈R⋃∞$ the function $Q_{τ}$ belongs to the same subclass and the corresponding exceptional function can be characterized (cf. [4]). In this note we introduce two subclasses $N_{–2}⊂N_{–1}$ of $N_{0}$ 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 $N_{–2}$ and $N_{–1}$ to the classes $N_{0}$ and $N_{–1}$, 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.

## 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