Absolutely continuous multilinear operators

Abstract

Abstract. We introduce the new class of the absolutely (p; p1, ..., pm; σ)-continuous multilinear operators, that is defined using a summability property that provides the multilinear version of the absolutely (p, σ)-continuous operators. We give an analogue of Pietsch's Domination Theorem and a multilinear version of the associated Factorization Theorem that holds for absolutely (p, σ)-continuous operators, obtaining in this way a rich factorization theory. We present also a tensor norm which represents this multiideal by trace duality. As an application, we show that absolutely (p; p1, ..., pm; σ)-continuous multilinear operators are compact under some requirements. Applications to factorization of linear maps on Banach function spaces through interpolation spaces are also given.

Abstract extrapolation theorems for absolutely summing nonlinear operators

“…A series of recent works [6][7][8][28][29][30] on Pietsch Domination-Factorization Theorems have shown that the domination theorem actually needs almost no linear structure, and a quite general version is in fact valid (see Theorem 2.2 below). This general approach recovers several previous Pietsch-type domination theorems (see [8]) and also rapidly found applications in different contexts (see [1,11,13]). …”

Domination spaces and factorization of linear and multilinear summing operators

“…Then Theorem 3.7 when applied with these elements gives the factorization result that can be found in [5, Theorem 2.1.20] (see also [1] and [4]). …”

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

“…Using classical methods it can be shown that g p,σ is a norm on X 1 ⊗ · · · ⊗ X m ⊗ X with the metric mapping property (see the proof of Proposition 4.1 and 4.2 in [10])…”

“…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].…”