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• ## Singular values?

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• Jan 19, 2011
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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.