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Nevanlinna functions, perturbation formulas, and triplets of Hilbert spaces. (English) Zbl 0921.47019
Summary: Let $$S$$ be a closed symmetric operator with defect numbers $$(1,1)$$ in a Hilbert space $${\mathfrak H}$$ and let $$A$$ be a selfadjoint operator extension of $$S$$ in $${\mathfrak H}$$. Then $$S$$ is necessarily a graph restriction of $$A$$ and the selfadjoint extensions of $$S$$ can be considered as graph perturbations of $$A$$. 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 B. Simon [CRM Proc. Lect. Notes 8, 109-149 (1995; Zbl 0824.47019)]. 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 $${\mathfrak H}$$ the graph perturbations can be interpreted as compressions of rank one perturbations. We present two points of view: either the Hilbert space $${\mathfrak H}$$ 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 by interpolating the original triplet. We prove some invariance properties, which imply that such an interpolation is independent of the (nonexceptional) extension.

##### MSC:
 47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces 47A20 Dilations, extensions, compressions of linear operators 47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B25 Linear symmetric and selfadjoint operators (unbounded) 47A55 Perturbation theory of linear operators 47A57 Linear operator methods in interpolation, moment and extension problems
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