(p, q)-summing sequences

A sequence (xj ) in a Banach space X is (p, q)-summing if for any weakly q-summable sequence (x∗j ) in the dual space we get a p-summable sequence of scalars (x∗j (xj )). We consider the spaces formed by these sequences, relating them to the theory of (p, q)- summing operators. We give a characterization of the case p = 1 in terms of integral operators, and show how these spaces are relevant for a general question on Banach spaces and their duals, in connection with Grothendieck theorem.  2002 Elsevier Science (USA). All rights reserved.