In 1983, A. Pietsch [20] presented his “designs of a theory” for ideals of multilinear forms. This work provided a general framework from which different lines of investigation developed. Some ideals of multilinear forms appeared as the multilinear natural extension of ideals of linear operators (e.g., nuclear and integral multilinear forms). However, it is not always clear what the multilinear...

for allx ∈X and all T ∈ . Since the appearance of Grothendieck-Pietsch’s domination theorem for p-summing operators, there is a great interest in finding out the structure of uniformly dominated sets. We will denote by p(μ) the set of all operators T ∈ Πp(X,Y) satisfying (1.1) for all x ∈ X. It is easy to prove that p(μ) is absolutely convex, closed, and bounded (for the p-summing norm). In [4]...

w,q = supφ∈BX́ ( ∑k j=1 | φ(xj) | ) 1 q . This is a natural generalization of the concept of (p; q)-summing operators and in the last years has been studied by several authors. The infimum of the L > 0 for which the inequality holds defines a norm ‖.‖as(p;q) for the case p ≥ 1 or a p-norm for the case p < 1 on the space of (p; q)-summing homogeneous polynomials. The space of all m-homogeneous (p...

In this paper we study absolutely summing mappings on Banach spaces by exploring the cotype of their domains and ranges. It is proved that every n-linear mapping from L∞-spaces into K is (2; 2, ..., 2,∞)-summing and also shown that every n-linear mapping from L∞-spaces into F is (q; 2, ..., 2)-summing whenever F has cotype q. We also give new examples of analytic summing mappings and polynomial...