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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 | References | Similar Articles | Additional Information

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. Akhiezer, The classical moment problem and some related questions in analysis, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965. MR 0184042
  • 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. Earl A. Coddington and Hendrik S. V. de Snoo, Positive selfadjoint extensions of positive symmetric subspaces, Math. Z. 159 (1978), no. 3, 203–214. MR 0500265
  • 4. William F. Donoghue Jr., Monotone matrix functions and analytic continuation, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 207. MR 0486556
  • 5. F. Gesztesy and B. Simon, Rank-one perturbations at infinite coupling, J. Funct. Anal. 128 (1995), no. 1, 245–252. MR 1317717, 10.1006/jfan.1995.1030
  • 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, Z. Anal. Anwendungen 15 (1996), no. 1, 45–55. MR 1376588, 10.4171/ZAA/687
  • 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. Funct. Anal. 130 (1995), no. 2, 345–356. MR 1335385, 10.1006/jfan.1995.1074
  • 15. Martin Jurchescu, Riemann surfaces and holomorphic mappings, Acad. R. P. Rom\cflexıne Stud. Cerc. Mat. 12 (1961), 575–590 (Romanian, with Russian and English summaries). MR 0131542
  • 16. Barry Simon, Spectral analysis of rank one perturbations and applications, Mathematical quantum theory. II. Schrödinger operators (Vancouver, BC, 1993) CRM Proc. Lecture Notes, vol. 8, Amer. Math. Soc., Providence, RI, 1995, pp. 109–149. MR 1332038
  • 17. A. V. Shtraus, Generalized resolvents of nondensely defined bounded symmetric operators, Functional analysis, No. 27 (Russian), Ul′yanovsk. Gos. Ped. Inst., Ul′yanovsk, 1987, pp. 187–196 (Russian). MR 1129513

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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