On Krein’s example
HTML articles powered by AMS MathViewer
- by Vadim Kostrykin and Konstantin A. Makarov PDF
- Proc. Amer. Math. Soc. 136 (2008), 2067-2071
Abstract:
In his 1953 paper [Matem. Sbornik 33 (1953), 597–626] Mark Krein presented an example of a symmetric rank one perturbation of a self-adjoint operator such that for all values of the spectral parameter in the interior of the spectrum, the difference of the corresponding spectral projections is not trace class. In the present note it is shown that in the case in question this difference has simple Lebesgue spectrum filling in the interval $[-1,1]$ and, therefore, the pair of the spectral projections is generic in the sense of Halmos but not Fredholm.References
- J. Avron, R. Seiler, and B. Simon, The index of a pair of projections, J. Funct. Anal. 120 (1994), no. 1, 220–237. MR 1262254, DOI 10.1006/jfan.1994.1031
- M. Sh. Birman and A. B. Pushnitski, Spectral shift function, amazing and multifaceted, Integral Equations Operator Theory 30 (1998), no. 2, 191–199. Dedicated to the memory of Mark Grigorievich Krein (1907–1989). MR 1607900, DOI 10.1007/BF01238218
- P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389. MR 251519, DOI 10.1090/S0002-9947-1969-0251519-5
- James S. Howland, Spectral theory of selfadjoint Hankel matrices, Michigan Math. J. 33 (1986), no. 2, 145–153. MR 837573, DOI 10.1307/mmj/1029003344
- James S. Howland, Spectral theory of operators of Hankel type. I, II, Indiana Univ. Math. J. 41 (1992), no. 2, 409–426, 427–434. MR 1183350, DOI 10.1512/iumj.1992.41.41022
- James S. Howland, Spectral theory of operators of Hankel type. I, II, Indiana Univ. Math. J. 41 (1992), no. 2, 409–426, 427–434. MR 1183350, DOI 10.1512/iumj.1992.41.41022
- M. G. Kreĭn, On the trace formula in perturbation theory, Mat. Sbornik N.S. 33(75) (1953), 597–626 (Russian). MR 0060742
- I. M. Lifšic, On a problem of the theory of perturbations connected with quantum statistics, Uspehi Matem. Nauk (N.S.) 7 (1952), no. 1(47), 171–180 (Russian). MR 0049490
- Vladimir V. Peller, Hankel operators and their applications, Springer Monographs in Mathematics, Springer-Verlag, New York, 2003. MR 1949210, DOI 10.1007/978-0-387-21681-2
- S. C. Power, Hankel operators on Hilbert space, Research Notes in Mathematics, vol. 64, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. MR 666699
- Marvin Rosenblum, On the Hilbert matrix. II, Proc. Amer. Math. Soc. 9 (1958), 581–585. MR 99599, DOI 10.1090/S0002-9939-1958-0099599-2
Additional Information
- Vadim Kostrykin
- Affiliation: Institut für Mathematik, Technische Universität Clausthal, Erzstraße 1, D-38678 Clausthal-Zellerfeld, Germany
- Email: kostrykin@math.tu-clausthal.de, kostrykin@t-online.de
- Konstantin A. Makarov
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- Email: makarov@math.missouri.edu
- Received by editor(s): June 12, 2006
- Received by editor(s) in revised form: February 26, 2007
- Published electronically: February 12, 2008
- Communicated by: Joseph A. Ball
- © Copyright 2006 V. Kostrykin, K. A. Makarov
- Journal: Proc. Amer. Math. Soc. 136 (2008), 2067-2071
- MSC (2000): Primary 47B35; Secondary 47A55, 45P05
- DOI: https://doi.org/10.1090/S0002-9939-08-09141-7
- MathSciNet review: 2383512
Dedicated: Dedicated to Eduard Tsekanovskiĭ on the occasion of his 70th birthday