Nevanlinna functions, perturbation formulas, and triplets of Hilbert spaces

Let S be a closed symmetric operator with defect numbers (1, 1) in a Hilbert space h and let A be a selfadjoint operator extension of S in h. Then S is necessarily a graph restriction of A and the selfadjoint extensions of S can be considered as graph perturbations of A, cf. [8]. Only when S is not densely defined and, in particular, when S is bounded, S is given by a domain restriction of A and the graph perturbations reduce to rank one perturbations in the sense of [23]. This happens precisely when the Q-function of S and A belongs to the subclass N-0 of Nevanlinna functions. In this paper we show that by going beyond the Hilbert space h the graph perturbations can be interpreted as compressions of rank one perturbations. We present two points of view: either the Hilbert space h is given a one-dimensional extension, or the use of Hilbert space triplets associated with A is invoked. If the Q-function of S and A belongs to the subclass N-1 of Nevanlinna functions, then it is convenient to describe the selfadjoint extensions of S including its generalized Friedrichs extension (see [6]) by interpolating the original triplet, cf. [5]. For the case when A is semibounded, see also [4]. We prove some invariance properties, which imply that such an interpolation is independent of the (nonexceptional) extension.