(p, q)-summing sequences
by
Oscar Blasco
Universidad de Valencia
Coauthors: José Luis Arregui (Universidad de Zaragoza)

Let X and Y be Banach spaces, and let p, q >= 1. A sequence (u_j)_{j in N} of operators in L(X, Y) is (p, q)-summing if there exists a constant C > 0 such that, for any finite collection of vectors x₁, x₂, ... x_n in X, it holds that

æ è n å j=1  ||uj xj||p ö ø 1/p   <= C sup ì í î æ è n å j=1  |x*xj|q ö ø 1/q    ; x* in BX* ü ý þ ,

(with the obvious changes for p=\infty or q=\infty).

A particular and very interesting case is Y=K which leads to the following sequence space.

For any Banach space X, we define the space l_{\pip, q}(X) as the set of all sequences (x_j) in X such that, for some constant C > 0, it holds that

æ è n å j=1  |x*j xj|p ö ø 1/p   <= C sup ì í î æ è n å j=1  |x*jx|q ö ø 1/q    ; x in BX ü ý þ

for any finite collection of vectors x^*₁, ..., x^*_n in X^*.

Our main objective is to relate these spaces to some classical aspects of the theory of geometry of Banach spaces and operator ideals.

For instance, it is shown that l_{\pi2, 1}(X) = l_\infty(X) if and only if l₂(X) subset or equal l_\pi1(X), if and only if X^* has Orlicz property. It is also seen that (x_j) in l_{\pi1, q}(X) if and only if l_q --> X, given by e_j --> x_j, is an integral operator, but p-integral operators cannot be (in general) identified with l_{\pip, q}(X). Besides, we point out an interesting statement, equivalent to Grothendieck's theorem, in terms of these spaces.

(T)

Date received: April 5, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caei-71.