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. We introduce the new ideal of strongly (p, σ)-continuous linear operators in order to study the adjoints of the (p, σ)-absolutely continuous linear operators. Starting from this ideal we build a new multi-ideal by using the composition method. We prove the corresponding Pietsch domination theorem and we present a representation of this multi-ideal by a tensor norm. A factorization theorem characterizing the corresponding multi-ideal -which is also new for the linear case-is given. When applied to the case of the Cohen strongly p-summing operators, this result gives also a new factorization theorem.

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