Abstract
LetA be a selfadjoint operator in a Hilbert space ℌ. Its rank one perturbations A+τ(·,ω)ω, ℝ, are studied when ω belongs to the scale space ℌ−2 associated with ℌ+2=domA and (·,·) is the corresponding duality. IfA is nonnegative and ω belongs to the scale space ℌ−1, Gesztesy and Simon [4] prove that the spectral measures ofA(τ), ℝ, converge weakly to the spectral measure of the limiting perturbationA(∞). In factA(∞) can be identified as a Friedrichs extension. Further results for nonnegative operatorsA were obtained by Kiselev and Simon [14] by allowing ω∈ℌ−2, Our purpose is to show that most results of Gesztesy, Kiselev, and Simon are valid for rank one perturbations of selfadjoint operators, which are not necessarily semibounded. We use the fact that rank one perturbations constitute selfadjoint extensions of an associated symmetric operator. The use of so-calledQ-functions [6, 8] facilitates the descriptions. In the special case that ω belongs to the scale space ℌ−1 associated with ℌ+2=dom |A|1/2 the limiting perturbationA(∞) is shown to be the generalized Friedrichs extension [5].
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References
S. Albeverio and P. Kurasov, “Rank one perturbations of not semibounded operators”, Integral Equations Operator Theory, 27 (1997), 379–400.
E. A. Coddington and R. C. Gilbert, “Generalized resolvents of ordinary differential operators”, Trans. Amer. Math. Soc., 93 (1959), 216–241.
E. A. Coddington and H.S.V. de Snoo, “Positive selfadjoint extensions of positive symmetric subspaces”, Math. Z., 159 (1978), 203–214.
F. Gesztesy and B. Simon, “Rank one perturbations at infinite coupling”, J. Functional Analysis, 128 (1995), 245–252.
S. Hassi, M. Kaltenbäck, and H.S.V. de Snoo, “Triplets of Hilbert spaces and Friedrichs extensions associated with the subclass N1 of Nevanlinna functions”, J. Operator Theory, 37 (1997), 155–181.
S. Hassi, H. Langer, and H.S.V. de Snoo, “Selfadjoint extensions for a class of symmetric operators with defect numbers (1,1)”, 15th OT Conference Proceedings, (1995), 115–145.
S. Hassi and H.S.V. de Snoo, “One-dimensional graph perturbations of selfadjoint relations”, Ann. Acad. Sci. Fenn. 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).
S. Hassi, H.S.V. de Snoo, and A.D.I. Willemsma, “Smooth rank one perturbations”, Proc. Amer. Math. Soc., (to appear).
P. Jonas and H. Langer, “Some questions in the perturbation theory ofJ-nonnegative operators in Kreîn spaces,”, Math. Nachr., 114 (1983), 205–226.
P. Jonas and H. Langer, “Selfadjoint extensions of a closed linear relation of defect one in a Kreîn space”, Operator Theory: Adv. Appl., 80 (1995), 176–205.
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).
I.S. Kac and M.G. Kreîn, “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. Functional Analysis, 130 (1995), 345–356.
M.G. Kreîn and V.A. Yavryan, “Spectral shift functions that arise in perturbations of a positive operator”, J. Operator Theory, 6 (1981), 155–191 (Russian).
P. Kurasov, “Rank one perturbations of nonsemibounded operators”, Research report 1995-15, Luleå University of Technology, Luleå, Sweden, 1995.
M. Reed and B. Simon,Methods of modern mathematical physics, I: Functional Analysis, Revised and enlarged edition, Academic Press, San Diego, 1980.
M. Reed and B. Simon,Methods of modern mathematical physics, II: Fourier Analysis, Self-Adjointness, Academic Press, San Diego, 1975.
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.
K. Yosida,Functional Analysis Third edition, Springer-Verlag, Berlin-Heidelberg-New York, 1971.
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Hassi, S., De Snoo, H. On rank one perturbations of selfadjoint operators. Integr equ oper theory 29, 288–300 (1997). https://doi.org/10.1007/BF01320702
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DOI: https://doi.org/10.1007/BF01320702