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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

On Krein's example


Authors: Vadim Kostrykin and Konstantin A. Makarov
Journal: Proc. Amer. Math. Soc. 136 (2008), 2067-2071
MSC (2000): Primary 47B35; Secondary 47A55, 45P05
Published electronically: February 12, 2008
MathSciNet review: 2383512
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09141-7
PII: S 0002-9939(08)09141-7
Keywords: Spectral shift function, Hankel operators, absolutely continuous spectrum.
Received by editor(s): June 12, 2006
Received by editor(s) in revised form: February 26, 2007
Published electronically: February 12, 2008
Dedicated: Dedicated to Eduard Tsekanovskiĭ on the occasion of his 70th birthday
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 V. Kostrykin, K. A. Makarov