×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Albeverio, Rank One Perturbations of Not Semibounded Operators, Integral Equations Operator Theory 27 pp 379– (1997) · Zbl 0901.47004
[2] Dijksma, Lectures on Operator Theory and its Applications, Fields Institute Monographs 3, in: Operator Theory and Ordinary Differential Operators (1995)
[3] Fleige , A. 1995
[4] Gesztesy, Rank One Perturbations at Infinite Coupling, J. Functional Analysis 128 pp 245– (1995) · Zbl 0828.47009
[5] Hassi, Triplets of Hilbert Spaces and Friedrichs Extensions Associated with the Subclass N1 of Nevanlinna Functions, J. Operator Theory 37 pp 155– (1997) · Zbl 0891.47013
[6] Hassi , S. Langer , H. de Snoo , H. S. V. 1995 115 145
[7] Hassi, On Some Subclasses of Nevanlinna Functions, Zeitschrift für Analysis und ihre Anwendungen 15 pp 45– (1996) · Zbl 0844.47014 · doi:10.4171/ZAA/687
[8] Hassi, One-Dimensional Graph Perturbations of Selfadjoint Relations, Ann. Acad. Sci. Fenn. A. I. Math. 22 pp 123– (1997) · Zbl 0894.47021
[9] Jonas, Some Questions in the Perturbation Theory of J - Nonnegative Operators in KreTn Spaces, Math. Nachr. 114 pp 205– (1983)
[10] Jonas, Selfadjoint Extensions of a Closed Linear Relation of Defect One in a Kreîn Space, Operator Theory: Adv. Appl. 80 pp 176– (1995)
[11] Kac, On Integral Representations of Analytic Functions Mapping the Upper Half-Plane onto a Part of Itself, Uspehi Mat. Nauk 11 pp 139– (1956)
[12] Kac, Discrete and Continuous Boundary Problems (1968)
[13] Amer. Math. Soc. Transl. (2) 103 pp 1– (1974) · Zbl 0291.34016 · doi:10.1090/trans2/103/01
[14] Kreîn, Spectral Shift Functions that Arise in Perturbations of a Positive Operator, J. Operator Theory 6 pp 155– (1981)
[15] Kurasov, Distribution Theory for Discontinuous Test Functions and Differential Operators with Generalized Coefficients, J. Math. Anal. Appl. 201 pp 297– (1996) · Zbl 0878.46030
[16] Langer, On Generalized Resolvents and Q-functions of Symmetric Linear Relations (Subspaces) in-Hilbert Space, Pacific J. Math. 72 pp 135– (1977) · Zbl 0335.47014 · doi:10.2140/pjm.1977.72.135
[17] Langer, L - Resolvent Matrices of Symmetric Linear Relations with Equal Defect Numbers; Applications to Canonical Differential Equations, Integral Equations Operator Theory 5 pp 208– (1982)
[18] McIntosh, Bilinear Forms in Hilbert Space, J. Math. Mech. 19 pp 1027– (1970) · Zbl 0204.15704
[19] McIntosh, Hermitian Bilinear Forms Which Are not Semibounded, Bull. Amer. Math. Soc. 76 pp 732– (1970) · Zbl 0198.16403
[20] McIntosh, On the Comparability of A1/2 and A1/2, Proc. Amer. Math. Soc. 32 pp 430– (1972)
[21] Shmul’yan, Extended Resolvents and Extended Spectral Functions of a Symmetric Operator, Mat. Sb. 84 pp 440– (1971)
[22] Shondin, Quantum - Mechanical Models in Rn Associated with Extensions of the Energy Operator in Pontryagin Space, Theor. Math. Phys. 74 pp 220– (1988) · Zbl 0685.46047
[23] Shondin , Yu. G. 1991
[24] Simon, Proceedings on Mathematical Quantum Theory II: Schrödinger Operators, CRM Proceedings and Lecture Notes 8 (1995)
[25] Tsekanovskiî, The Theory of Bi - Extensions of Operators on Rigged Hilbert Spaces. Unbounded Operator Colligations and Characteristic Functions, Uspekhi Mat. Nauk 32 pp 5– (1977)
[26] Russian Math. Surveys 32 pp 5– (1977)
[27] Vonhoff , R. 1995
[28] Winkler, The Inverse Spectral Problem for Canonical Systems, Integr. Equat. Oper. Th. 22 pp 360– (1995) · Zbl 0843.34031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.