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