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Smooth rank one perturbations of selfadjoint operators
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by S. Hassi, H. S. V. de Snoo and A. D. I. Willemsma PDF
Proc. Amer. Math. Soc. 126 (1998), 2663-2675 Request permission


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.
  • N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR 0184042
  • S. Albeverio and P. Kurasov, "Rank one perturbations of not semibounded operators", Integral Equations Operator Theory 27 (1997), 379–400.
  • 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 500265, DOI 10.1007/BF01214571
  • William F. Donoghue Jr., Monotone matrix functions and analytic continuation, Die Grundlehren der mathematischen Wissenschaften, Band 207, Springer-Verlag, New York-Heidelberg, 1974. MR 0486556, DOI 10.1007/978-3-642-65755-9
  • F. Gesztesy and B. Simon, Rank-one perturbations at infinite coupling, J. Funct. Anal. 128 (1995), no. 1, 245–252. MR 1317717, DOI 10.1006/jfan.1995.1030
  • S. Hassi, M. Kaltenbäck, and H.S.V. de Snoo, "Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass $\textbf {N}_1$ of Nevanlinna functions", J. Operator Theory, 37 (1997), 155-181.
  • S. Hassi, M. Kaltenbäck, and H.S.V. de Snoo, "A characterization of semibounded selfadjoint operators", Proc. Amer. Math. Soc. 125 (1997), 2681–2692.
  • 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.
  • 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, DOI 10.4171/ZAA/687
  • 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.
  • S. Hassi and H.S.V. de Snoo, "Nevanlinna functions, perturbation formulas, and triplets of Hilbert spaces", Math. Nachr., (to appear).
  • Cahit Arf, Untersuchungen ĂĽber reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
  • I.S. Kac and M.G. KreÄ­n, "$R$-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).
  • A. Kiselev and B. Simon, Rank one perturbations with infinitesimal coupling, J. Funct. Anal. 130 (1995), no. 2, 345–356. MR 1335385, DOI 10.1006/jfan.1995.1074
  • Martin Jurchescu, Riemann surfaces and holomorphic mappings, Acad. R. P. RomĂ®ne. Stud. Cerc. Mat. 12 (1961), 575–590 (Romanian, with English and Russian summaries). MR 131542
  • 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, DOI 10.1090/crmp/008/04
  • 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
  • Email:
  • H. S. V. de Snoo
  • Affiliation: Department of Mathematics University of Groningen Postbus 800, 9700 AV Groningen Nederland
  • Email:
  • Received by editor(s): December 26, 1996
  • Received by editor(s) in revised form: January 28, 1997
  • Communicated by: Palle E. T. Jorgensen
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 2663-2675
  • MSC (1991): Primary 47A55, 47A57, 47B25; Secondary 81Q15
  • DOI:
  • MathSciNet review: 1451805