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481Re: [harmonic] Singular values?

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  • Stephen Montgomery-Smith
    Jan 20, 2011
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      Zohreh Rahbani wrote:
      >
      >
      > Dear all,
      >
      > I'm dealing with a problem and I would be very thankful if someone could
      > help me.
      >
      > The following inequality is well known for singular values of a compact
      > operators in Hilbert spaces:
      >
      > s_(n+k-1)(T+L) <= s_(n)(T)+ s_(k)(L)
      >
      > Is there any similar inequality for operators in a Banach spaces?
      >
      > Any comment is very wellcome.

      There is a rather surprisingly large literature on generalizations of
      the concept of singular values to operators on Banach spaces. Here are
      a couple of books on the subject. Once you get away from Hilbert
      spaces, the subject becomes somewhat more difficult. I haven't looked
      at this stuff for about 20 years, so I am definitely not the expert.

      Pietsch, A.
      Eigenvalues and $s$s-numbers.
      Mathematik und ihre Anwendungen in Physik und Technik [Mathematics and
      its Applications in Physics and Technology], 43. Akademische
      Verlagsgesellschaft Geest & Portig K.-G., Leipzig, 1987. 360 pp. ISBN:
      3-321-00012-1

      Pietsch, Albrecht(DDR-FSU)
      Eigenvalues and $s$s-numbers.
      Cambridge Studies in Advanced Mathematics, 13. Cambridge University
      Press, Cambridge, 1987. 360 pp. ISBN: 0-521-32532-3

      Tomczak-Jaegermann, Nicole(3-AB)
      Banach-Mazur distances and finite-dimensional operator ideals.
      Pitman Monographs and Surveys in Pure and Applied Mathematics, 38.
      Longman Scientific & Technical, Harlow; copublished in the United States
      with John Wiley & Sons, Inc., New York, 1989. xii+395 pp. ISBN:
      0-582-01374-7
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