Multilinear extensions of absolutely (p;q;r)-summing operators

Achour, Dahmane

doi:10.1007/s12215-011-0054-2

Cited by 16 publications

(13 citation statements)

References 29 publications

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“…So we can our results in the case of operators that are defined on reflexive spaces. As a consequence of the main Theorem in [4] and Proposition 5.1, we obtain the following result (see also [1,3]). …”

Section: 1supporting

confidence: 52%

Absolutely continuous multilinear operators

Dahia

Achour

Pérez

2013

Journal of Mathematical Analysis and Applications

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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.

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Section: 1supporting

confidence: 52%

Absolutely continuous multilinear operators

Dahia

Achour

Pérez

2013

Journal of Mathematical Analysis and Applications

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“…The following multilinear generalization of (p; q; r)-summing operators was recently introduced by D. Achour [1]: Definition 3.3. Let 0 < p, q 1 , .…”

Section: The First Multilinear and Polynomial Approaches To Summabilitymentioning

confidence: 99%

“…More precisely, the multilinear notions of mixing summing operators and absolutely (p; q; r)-summing multilinear operators were introduced following a different perspective (see [1,72]). The point is that these approaches do not carry out the essence of the respective linear concepts and this lack is clearly corroborated by the notions of coherence, compatibility and holomorphy types.…”

Section: Introduction and Historical Backgroundmentioning

confidence: 99%

Nonlinear absolutely summing operators revisited

Bernardino

Pellegrino

Seoane‐Sepúlveda

et al. 2015

Bull Braz Math Soc, New Series

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Abstract. In the last decades many authors have become interested in the study of multilinear and polynomial generalizations of families of operator ideals (such as, for instance, the ideal of absolutely summing operators). However, these generalizations must keep the essence of the given operator ideal and there seems not to be a universal method to achieve this. The main task of this paper is to discuss, study, and introduce multilinear and polynomial extensions of the aforementioned operator ideals taking into account the already existing methods of evaluating the adequacy of such generalizations. Besides this subject's intrinsic mathematical interest, the main motivation is our belief (based on facts that shall be presented) that some of the already existing approaches are not adequate.

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“…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].…”

Section: ])mentioning

confidence: 99%

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

Achour

Dahia

Rueda

et al. 2013

Linear and Multilinear Algebra

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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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Multilinear extensions of absolutely (p;q;r)-summing operators

Cited by 16 publications

References 29 publications

Absolutely continuous multilinear operators

Absolutely continuous multilinear operators

Nonlinear absolutely summing operators revisited

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

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