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Re: [harmonic] Singular values?

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  • venku naidu
    The same inequality holds good in a normed linear space, in the case of approximation numbers also. It simply depends on the fact that the sum of two finite
    Message 1 of 4 , Jan 22, 2011
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      The same inequality holds good in a normed linear space, in the case of approximation numbers also.

      It simply depends on the fact that the sum of two finite rank operators, one of rank less than or equal to k-1 and other less than or equal to n-1, will have rank less than or equal to n+k-2.

      You can refer a book called 'operator ideals' by Piestch, A.




      --- On Wed, 19/1/11, Zohreh Rahbani <zrahbani@...> wrote:

      From: Zohreh Rahbani <zrahbani@...>
      Subject: [harmonic] Singular values?
      To: harmonicanalysis@yahoogroups.com
      Date: Wednesday, 19 January, 2011, 18:43

       

      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.


      From: lakhmau <lakhmau@...>
      To: harmonicanalysis@yahoogroups.com
      Sent: Wed, January 12, 2011 6:05:31 PM
      Subject: [harmonic] a problem with harmonic functions.

       

      Dear all,

      I'm stock on a problem and I would be very thankful if someone had an idea. I believe I reduced the problem to a basic one, so anyone can get it. (I'm not a harmonic analyst...)

      Let f_n be a sequence of nonnegative functions defined on the unit ball of R^2. We assume

      *) each f_n is 1-Lipschitz.
      *) each f_n is harmonic on {x : f_n(x) > 0} (where it is positive)
      *) f_n converges uniformly to a function f.
      *) f_n(0) > 0 for all integer n, but f(0) = 0.
      *) f is differentiable at 0 and Df(0) = 0

      Is is true that Df_n(0) converges to 0 ?

      The main difficulty here is that each f_n is positive in a neighborhood of 0, which is not necessarily uniform in k.

      Thank for your answers,
      Lakhdar



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