On Multilinear Mappings of Nuclear Type

Matos, Mário C.

doi:10.5209/rev_rema.1993.v6.n1.17846

Cited by 39 publications

(45 citation statements)

References 6 publications

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“…This terminology was introduced in the commutative case by Pietsch [19] for scalar valued mappings. The reader interested by previous work on this and related properties can consult [3,5,6,7,8,13,14,16,17].…”

Section: Basic Definitions and Propertiesmentioning

confidence: 99%

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Characterization of lr-dominated m-linear operators

Mezrag¹

2010

Demonstratio Mathematica

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Abstract. In this paper, we define and study a new concept of r-dominated multilinear operators in the category of operator spaces, which we call l r -dominated m-linear operators. We give some characterizations of this concept such as the theorem of factorization. IntroductionThe concept of r-dominated multilinear operators was mainly introduced at the beginning of the 1980s by Pietsch [19] where the idea of generalizing the theory of ideals of linear operators to the multilinear setting appears. Motivated by the importance of this theory, several authors have developed and studied many concepts relating to summability (we mention for example [13,14,16,18] among so many authors). Regarding this, it is natural to try to develop analogous in the noncommutative case. Hence, this paper deals with the equivalent of this concept in the theory of operator spaces; which will be called the "l r -dominated m-linear operators". Firstly, we will introduce the notion of l r -dominated m-linear operators and we prove an analogue to the Pietsch domination theorem. After that, we shall give the factorization theorem of such operators. We finish this paper by giving the finite dimensional version of our result. This paper is organized as follows.In the first section, we recall some basic definitions and properties concerning the theory of operator spaces.In the second section of the present paper, we introduce and study the notion of l r -dominated multilinear operators. We give the Pietsch domination theorem and related properties.2000 Mathematics Subject Classification: 46B28, 47H60, 46G25, 46B25. Key words and phrases: completely bounded operator, l r -dominated multilinear operator, operator space, Pietsch domination theorem. Brought to you by | MIT LibrariesAuthenticated

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Section: Basic Definitions and Propertiesmentioning

confidence: 99%

“…Motivated by the importance of this theory, several authors have developed and studied many concepts relating to summability (we mention for example [13,14,16,18] among so many authors). Regarding this, it is natural to try to develop analogous in the noncommutative case.…”

Section: Introductionmentioning

confidence: 99%

Characterization of lr-dominated m-linear operators

Mezrag¹

2010

Demonstratio Mathematica

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“…(1) (p 1 , ..., p m )-dominated multilinear operators (see [20] or [12]). Let 1 ≤ p, p 1 , ..., p m < ∞ with…”

Section: Let Us Consider Now the Quotient Mapmentioning

confidence: 99%

Domination spaces and factorization of linear and multilinear summing operators

Achour

Dahia

Rueda

et al. 2016

Quaestiones Mathematicae

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Abstract. It is well known that not every summability property for multilinear operators leads to a factorization theorem. In this paper we undertake a detailed study of factorization schemes for summing linear and nonlinear operators. Our aim is to integrate under the same theory a wide family of classes of mappings for which a Pietsch type factorization theorem holds. Our construction includes the cases of absolutely p-summing linear operators, (p, σ)-absolutely continuous linear operators, factorable strongly p-summing multilinear operators, (p1, . . . , pn)-dominated multilinear operators and dominated (p1, . . . , pn; σ)-continuous multilinear operators.

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“…The theory has been developed by several authors, and among the advances obtained thus far we mention: Pietsch-type domination/factorization theorems ( [11], Geiss [26], Pérez-García [38]), different types of absolutely summing multilinear mappings and polynomials (Achour and Mezrag [1], Carando and Dimant [15], Ç aliskan and Pellegrino [14], Dimant [23], Pellegrino and Souza [36], Pérez-García [38]), Grothendieck-type theorems (Bombal et al [4], Pérez-García and Villanueva [41]), coincidence/inclusion/composition theorems (Alencar and Matos [2], Botelho et al [8], Pérez-García [39,37], Popa [44]), connections with the geometry of Banach spaces ( [5], Floret and Matos [25], Meléndez and Tonge [33], [35], Pérez-García [40]), interplay with other multi-ideals and polynomial ideals (Botelho et al [7], [12], Cilia and Gutiérrez [17], Jarchow et al [27], Matos [29]), estimates for absolutely summing norms (Aron et al [3], [13], Choi et al [16], Defant and Sevilla-Peris [21], Zalduendo [45]), extensions of the theory to more general nonlinear mappings (Junek, Matos and Pellegrino [28], Matos [30,31], Matos and Pellegrino [32]). …”

Section: Introductionmentioning

confidence: 99%