On multilinear generalizations of the concept of nuclear operators

Abstract

This paper introduces the class of Cohen p-nuclear m-linear operators between Banach spaces. A characterization in terms of Pietsch's domination theorem is proved. The interpretation in terms of factorization gives a factorization theorem similar to Kwapień's factorization theorem for dominated linear operators. Connections with the theory of absolutely summing m-linear operators are established. As a consequence of our results, we show that every Cohen p-nuclear (1 < p ≤ ∞) m-linear mapping on arbitrary Banach spaces is weakly compact.

Domination and Kwapień’s factorization theorems for positive Cohen p–nuclear m–linear operators

“…Proof. Suppose that T ∈ N m p (E 1 , ..., E m ; F) by Theorem 3.1 in [1] there exist Banach spaces X 1 , ..., X m such that…”

“…where D m p , Π p are the Banach spaces of Cohen strongly p−summing multilinear operators and p−summing linear operators, respectively and in [1,Theorem 2.5], they characterized this class by Piestch's domination theorem. In our work, we generalize this notion for positive operators and prove, among other results, the domination theorem for this new class of operators.…”

Coincidence Results for Summing Multilinear Mappings

“…, p n )-summing operators that shed light on the summability properties for operators. Let us introduce another variant of the notion of a summing multilinear operator (for a related notion we refer the reader to [2]).…”

“…be well defined (hence linear) and bounded. Hence CN (p;q) (E; F ) ⊆ Π (p;q) (E; F ) and, due to (1),…”

Factorization of strongly (p,σ)-continuous multilinear operators

“…In [19,20], inclusions between the class of Cohen strongly summing multilinear operators and other classes of operators were systematically analyzed. A related concept and a new generalizations of the concept of Cohen strongly summing multilinear operators have also been recently studied in [8,7,2,3]). For more details concerning the nonlinear theory of summing operators and recent developments and applications we refer to [1,10].…”