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Smooth rank one perturbations
of selfadjoint operators

Authors: S. Hassi, H. S. V. de Snoo and A. D. I. Willemsma
Journal: Proc. Amer. Math. Soc. 126 (1998), 2663-2675
MSC (1991): Primary 47A55, 47A57, 47B25; Secondary 81Q15
MathSciNet review: 1451805
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Abstract: Let $A$ be a selfadjoint operator in a Hilbert space ${\mathfrak H}$ with inner product $[\cdot,\cdot]$. The rank one perturbations of $A$ have the form $A+\tau[\cdot,\omega] \omega$, $\tau \in {\mathbb R}$, for some element $\omega \in {\mathfrak H}$. In this paper we consider smooth perturbations, i.e. we consider $\omega \in \operatorname{dom}|A|^{k/2}$ for some $k \in {\mathbb N}\cup \{0\}$. Function-theoretic properties of their so-called $Q$-functions and operator-theoretic consequences will be studied.

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  • 1. N.I. Achieser, The classical moment problem and some related questions in analysis, Fizmatgiz, Moscow, 1961 (Russian) (English translation: Oliver and Boyd, Edinburgh, and Hafner, New York, 1965). MR 32:1518
  • 2. S. Albeverio and P. Kurasov, "Rank one perturbations of not semibounded operators", Integral Equations Operator Theory 27 (1997), 379-400. CMP 97:10
  • 3. E.A. Coddington and H.S.V. de Snoo, "Positive selfadjoint extensions of positive symmetric subspaces", Math. Z., 159 (1978), 203-214. MR 58:17936
  • 4. W.F. Donoghue, Monotone matrix functions and analytic continuation, Springer-Verlag, Berlin-Heidelberg-New York, 1974. MR 58:6279
  • 5. F. Gesztesy and B. Simon, "Rank one perturbations at infinite coupling", J. Functional Analysis, 128 (1995), 245-252. MR 95m:47014
  • 6. S. Hassi, M. Kaltenbäck, and H.S.V. de Snoo, "Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass ${\bf N}_1$ of Nevanlinna functions", J. Operator Theory, 37 (1997), 155-181. CMP 97:09
  • 7. S. Hassi, M. Kaltenbäck, and H.S.V. de Snoo, "A characterization of semibounded selfadjoint operators", Proc. Amer. Math. Soc. 125 (1997), 2681-2692. CMP 97:12
  • 8. S. Hassi, H. Langer, and H.S.V. de Snoo, "Selfadjoint extensions for a class of symmetric operators with defect numbers $(1,1)$", Topics in Operator Theory, Operator Algebras and Applications (Timisoara, 1994), Romanian Acad., Bucharest, 1995, pp. 115-145. CMP 97:04
  • 9. S. Hassi and H.S.V. de Snoo, "On some subclasses of Nevanlinna functions", Zeitschrift für Analysis und ihre Anwendungen, 15 (1996), 45-55. MR 96k:47044
  • 10. S. Hassi and H.S.V. de Snoo, "One-dimensional graph perturbations of selfadjoint relations", Ann. Acad. Sci. Fenn., Series A.I. Math., 22 (1997), 123-164. CMP 97:07
  • 11. S. Hassi and H.S.V. de Snoo, "Nevanlinna functions, perturbation formulas, and triplets of Hilbert spaces", Math. Nachr., (to appear).
  • 12. I.S. Kac, ''On integral representations of analytic functions mapping the upper half-plane onto a part of itself'', Uspehi Mat. Nauk, 11 (1956), no. 3 (69), 139-144 (Russian). MR 18:293C
  • 13. I.S. Kac and M.G. Kre[??]in, "$R$-functions-analytic functions mapping the upper halfplane into itself", Supplement I to the Russian edition of F.V. Atkinson, Discrete and continuous boundary problems, Mir, Moscow, 1968 (Russian) (English translation: Amer. Math. Soc. Transl., (2) 103 (1974), 1-18).
  • 14. A. Kiselev and B. Simon, "Rank one perturbations with infinitesimal coupling", J. Functional Analysis, 130 (1995), 345-356. MR 96e:47012
  • 15. E. Lukacs, Characteristic functions, Charles Griffin & Company Limited, London, 1960. MR 24:A1392
  • 16. B. Simon, "Spectral analysis of rank one perturbations and applications", in J. Feldman, R. Froese, and L.M. Rosen (editors), Proceedings on Mathematical Quantum Theory II: Schrödinger operators, CRM Proceedings and Lecture Notes, Vol. 8, Amer. Math. Soc., Providence, R.I., 1995. MR 97c:47008
  • 17. A.V. \v{S}traus, ``Generalized resolvents of bounded symmetric operators'', Funkts. Anal., 27 (1987), 187-196 (Russian). MR 93d:47050

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Additional Information

S. Hassi
Affiliation: Department of Statistics University of Helsinki PL 54, 00014 Helsinki Finland

H. S. V. de Snoo
Affiliation: Department of Mathematics University of Groningen Postbus 800, 9700 AV Groningen Nederland

Keywords: Rank one perturbation, moments, selfadjoint extension, $Q$-function, Friedrichs extension
Received by editor(s): December 26, 1996
Received by editor(s) in revised form: January 28, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

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