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**wrote:*<zrahbani@...>*

From: Zohreh Rahbani <zrahbani@...>

Subject: [harmonic] Singular values?

To: harmonicanalysis@yahoogroups.com

Date: Wednesday, 19 January, 2011, 18:43Dear 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