Factorization theorems for some new classes of multilinear operators

Abstract

Two new classes of summing multilinear operators, factorable (q, p)-summing operators and (r; p, q)-summing operators are studied. These classes are described in terms of factorization. It is shown that operators in the first (resp., the second) class admit the factorization through the injective tensor product of Banach spaces (resp., through some Banach lattices). Applications in different contexts related to Grothendieck Theorem and Fourier integral bilinear operators are shown. Motivated by Pisier's Theorem on factorization of (q, 1)-summing operators from C(K)-spaces through Lorentz spaces Lq,1 on some probability Borel measure spaces, we prove two variants of Pisier's Theorem for bilinear operators on the product of C(K)-spaces. We also prove bilinear versions of Mityagin-Pe lczyński and Kislyakov Theorems.

Integral representation of product factorable bilinear operators and summability of bilinear maps on C(K)-spaces

“…In this section we will investigate summability conditions of bilinear maps defined on the Cartesian product of CpKq-spaces. Although pq, 1q-summing bilinear operators on these products have recently been analyzed in [23] (see also [24] for the polynomial version), the results presented here and the type of summing inequalities we consider differ from those studied there. However, some related arguments and additional explanations about the general framework of summability of multilinear operators can be found in these papers.…”

“…This notion is inspired by the definition of factorable pp, qq-summing multilinear operators given in [23].…”

Pietsch’s variants of s-numbers for multilinear operators

“…Pietsch's paper is the origin of studies of several authors on some classes of ideals of multilinear operators. We refer to the study of such ideals to [2,11,12] and also to the survey paper [16] and references therein.…”