Comparison of Non-Commutative 2- and $p$-Summing Operators from $B(l_2)$ into $OH$

Mezrag, Lahcène

doi:10.4171/zaa/1104

Cited by 5 publications

(7 citation statements)

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“…As in the linear case (see [15]), we have a domination theorem similar to the commutative case which is appeared in [12, p. 57], the domination theorem is contained in [19,Theorem 14]. A proof of the general case, including vector-valued operators, can be found in [14,Proposition 3.1].…”

Section: Basic Definitions and Propertiesmentioning

confidence: 88%

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: 88%

Characterization of lr-dominated m-linear operators

Mezrag¹

2010

Demonstratio Mathematica

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“…We recall that π l p (X, Y ) is the space of all l p -summing operators from an operator space X into a Banach space Y . For more information on this notion we refer the reader to [Mez02]. In fact, Theorem 2.3 and [Mez02, Theorem 2.3] give us the following characterization result.…”

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confidence: 98%

“…Using this notion we prove some properties of multilinear operators in the non-commutative case. Our motivation is that the adjoint of a strongly l p -summing m-linear operator is an l p -summing operator as studied in [Mez02]. This paper is organized as follows.…”

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confidence: 99%

“…, X m ; R)). We recall that π l p (X, Y ) (see [Mez02] for more details) is the space of all l p -summing operators from X into Y , where X is an operator space.…”

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confidence: 99%

“…We introduce the adjoint of a multilinear operator as in [PS05] and we show that a multilinear operator T is strongly l p -summing if and only if T * is l p -summing. The notion of l p -summing operators has been introduced in [Mez02].…”

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On strongly l_p-summing m-linear operators

Mezrag

2008

Colloq. Math.

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Abstract. We introduce and study a new concept of strongly l p -summing m-linear operators in the category of operator spaces. We give some characterizations of this notion such as the Pietsch domination theorem and we show that an m-linear operator is strongly l p -summing if and only if its adjoint is l p -summing.1. Introduction. The development of the theory of polynomials and multilinear operators can be divided into two periods. The first starts in the thirties of the last century, essentially motivated externally through holomorphic and differential functions on infinite-dimensional spaces. The second begins in the eighties, mainly due to Pietsch [Pie83] where the idea to generalize the theory of ideals to the multilinear setting appears. Motived by the importance of this theory, several authors have developed and studied many concepts relating to summability; see [Ale85, AM89, Dia03, Mat96, Mat03, MT99, Sch91] among so many others. In this note we introduce a new concept concerning summability of multilinear operators.The concept of strongly p-summing linear operators (1 ≤ p < ∞) was introduced by J. S. Cohen [Coh73] in order to obtain a characterization of the conjugates of absolutely p * -summing linear operators. In [AM07], we have generalized this concept to the multilinear case. It is natural to try to develop the same concept in the non-commutative case.In the present work, we introduce a new notion of summability for multilinear operators, which we call strongly l p -summing m-linear operators. Using this notion we prove some properties of multilinear operators in the non-commutative case. Our motivation is that the adjoint of a strongly l p -summing m-linear operator is an l p -summing operator as studied in [Mez02]. This paper is organized as follows.In Section 1, we recall some basic definitions and properties concerning the theory of operator spaces.

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Little Grothendieck's theorem for sublinear operators

Achour

Mezrag

2004

Journal of Mathematical Analysis and Applications

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Let SB(X, Y ) be the set of the bounded sublinear operators from a Banach space X into a Banach lattice Y . Consider π 2 (X, Y ) the set of 2-summing sublinear operators. We study in this paper a variation of Grothendieck's theorem in the sublinear operators case. We prove under some conditions that every operator in SB(C(K), H ) is in π 2 (C(K), H ) for any compact K and any Hilbert H . In the noncommutative case the problem is still open.

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Comparison of Non-Commutative 2- and $p$-Summing Operators from $B(l_2)$ into $OH$

Cited by 5 publications

References 6 publications

Characterization of lr-dominated m-linear operators

Characterization of lr-dominated m-linear operators

On strongly l_p-summing m-linear operators

Little Grothendieck's theorem for sublinear operators

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Comparison of Non-Commutative 2- and $p$-Summing Operators from $B(l_2)$ into $OH$

Cited by 5 publications

References 6 publications

Characterization of lr-dominated m-linear operators

Characterization of lr-dominated m-linear operators

On strongly lp-summing m-linear operators

Little Grothendieck's theorem for sublinear operators

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On strongly l_p-summing m-linear operators