Pietsch's factorization theorem for dominated polynomials

Botelho, Geraldo; Pellegrino, Daniel; Rueda, Pilar

doi:10.1016/j.jfa.2006.10.001

Cited by 23 publications

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“…We study projections and injections between projective tensor products spaces or spaces of polynomials and we show that the example of a polynomial constructed in [4], that is neither p-dominated nor compact, can be identified with the projection map of the symmetric tensor product onto the space. Also we give a characterization of the weak and quasi approximation properties on symmetric tensor products.Since Ryan [17] proved that the projective symmetric k-fold tensor product of a Banach space E is a predual of the space of continuous k-homogeneous polynomials on E; the relationship between both spaces has been deeply studied and shown to be a powerful tool in infinite dimensional holomorphy (see [10,9,6]). This duality between polynomials and linear operators on symmetric tensor product spaces is generalized to the vector-valued case by means of the process known as linearization, that arose first in [17].…”

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confidence: 99%

“…Since Ryan [17] proved that the projective symmetric k-fold tensor product of a Banach space E is a predual of the space of continuous k-homogeneous polynomials on E; the relationship between both spaces has been deeply studied and shown to be a powerful tool in infinite dimensional holomorphy (see [10,9,6]). This duality between polynomials and linear operators on symmetric tensor product spaces is generalized to the vector-valued case by means of the process known as linearization, that arose first in [17].…”

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confidence: 99%

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On distinguished polynomials and their projections

Çalışkan¹,

Rueda²

2012

Ann. Acad. Sci. Fenn. Math.

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Abstract. We study projections and injections between projective tensor products spaces or spaces of polynomials and we show that the example of a polynomial constructed in [4], that is neither p-dominated nor compact, can be identified with the projection map of the symmetric tensor product onto the space. Also we give a characterization of the weak and quasi approximation properties on symmetric tensor products.Since Ryan [17] proved that the projective symmetric k-fold tensor product of a Banach space E is a predual of the space of continuous k-homogeneous polynomials on E; the relationship between both spaces has been deeply studied and shown to be a powerful tool in infinite dimensional holomorphy (see [10,9,6]). This duality between polynomials and linear operators on symmetric tensor product spaces is generalized to the vector-valued case by means of the process known as linearization, that arose first in [17]. The linearization process establishes an isomorphism from the space of vector valued continuous k-homogeneous polynomials P( k E; F ) and the space of continuous linear operators defined on the projective symmetric tensor product⊗ k,s π E into F . Aron and Schottenloher [1] proved that the space P( k E) of continuous k-homogeneous polynomials defined on a Banach space E is complemented in P( n E) whenever k < n. Later, Blasco [3] got the vector valued case by proving that the completed projective symmetric k-fold tensor product⊗ k,s π E is a complemented subspace of⊗ n,s π E for k < n. In this note we undertake a detailed study of the behavior of the main distinguished polynomials and linear mappings involved in the linearization process and their interplay when the space of linear operators is identified with a complemented subspace of homogeneous polynomials. We show two applications. On the one hand we apply our study to the weak approximation property and the quasi approximation property on projective symmetric tensor products. The results we obtain are related to the results of [2], especially [2, Theorem 4.2]. On the other hand, we show that the example of a polynomial P γ constructed in [4], that is neither p-dominated nor compact, and weakly compact only when the underlying space is reflexive, can be identified with the projection map of the symmetric tensor product onto the space.

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confidence: 99%

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confidence: 99%

On distinguished polynomials and their projections

Çalışkan¹,

Rueda²

2012

Ann. Acad. Sci. Fenn. Math.

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“…Brought to you by | MIT Libraries Authenticated Download Date | 5/12/18 2:08 AM Similarly to the Pietsch domination theorem, there is a Pietsch factorization theorem for dominated operators see [4]. To prove a factorization theorem for l r -dominated operators we mention the following elementary proposition.…”

Section: Finally We Havementioning

confidence: 98%

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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“…Botelho [4] (see also [17]) proved the existence of a p-dominated polynomial that is not weakly compact. This example was used in [10] to prove that p-dominated m-homogeneous polynomials could not be expected to factor through a subspace of an L p -space, and it justified the introduction of a new norm in order to obtain a factorization theorem for all pdominated polynomials. The fact that we keep the L p (μ)-norm on the subspace S guarantees that the m-homogeneous polynomial j m p : i X (X) → S is weakly compact.…”

Section: Proof Consider X Imentioning

confidence: 99%

“…However, the search of a canonical prototype of a p-dominated polynomial through which any p-dominated polynomial could factor in the spirit of the linear theorem for absolutely summing operators has turned out difficult and tricky. The factorization of p-dominated polynomials requires new techniques, which have been recently developed and can be found in [10] and [14]. These results allow us to prove that the factorization of p-dominated polynomials needs to consider spaces based on L p (μ) for a Pietsch measure μ but endowed with a different norm in a way that they do not coincide with the L p (μ)-norm.…”

Section: Introductionmentioning

confidence: 99%