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Nevanlinna Functions, Perturbation Formulas, and Triplets of Hilbert Spaces
Author(s) -
Hassi Seppo,
de Snoo Henk
Publication year - 1998
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19981950107
Subject(s) - mathematics , hilbert space , bounded function , pure mathematics , graph , perturbation (astronomy) , rank (graph theory) , operator (biology) , extension (predicate logic) , unitary operator , mathematical analysis , discrete mathematics , combinatorics , biochemistry , chemistry , physics , quantum mechanics , repressor , computer science , transcription factor , gene , programming language
Let S be a closed symmetric operator with defect numbers (1,1) in a Hilbert space and let A be a selfadjoint operator extension of S in . 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, 5 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 No of Nevanlinna functions. In this paper we show that by going beyond the Hilbert space the graph perturbations can be interpreted as compressions of rank one perturbations. We present two points of view: either the Hilbert space 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.