Dominated polynomials on infinite dimensional spaces

Botelho, Geraldo; Pellegrino, Daniel; Rueda, Pilar

doi:10.1090/s0002-9939-09-10002-3

Cited by 11 publications

(9 citation statements)

References 36 publications

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“…The notion of p-dominated multilinear mappings and homogeneous polynomials between Banach spaces plays an important role in the nonlinear theory of absolutely summing operators. It was introduced by Pietsch [17] and has been investigated by several authors since then (see, e.g., [5,6] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Dominated Bilinear Forms and 2-homogeneous Polynomials

Botelho¹,

Pellegrino²,

Rueda³

2010

Publ. Res. Inst. Math. Sci.

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The main goal of this note is to establish a connection between the cotype of the Banach space X and the parameters r for which every 2-homogeneous polynomial on X is rdominated. Let cot X be the infimum of the cotypes assumed by X and (cot X) * be its conjugate. The main result of this note asserts that if cot X > 2, then for every 1 ≤ r < (cot X)* there exists a non-r-dominated 2-homogeneous polynomial on X.2010 Mathematics Subject Classification: 46G25, 46B20, 46B28. Keywords: r-dominated multilinear form, r-dominated homogeneous polynomial, absolutely (p; q)-summing mapping, cotype. §1. IntroductionThe notion of p-dominated multilinear mappings and homogeneous polynomials between Banach spaces plays an important role in the nonlinear theory of absolutely summing operators. It was introduced by Pietsch [17] and has been investigated by several authors since then (see, e.g., [5,6] and references therein). Let X be a Banach space and m be a positive integer. A continuous m-are weakly r-summable in X. In a similar way, a scalarvalued m-homogeneous polynomial P on X is r-dominated if (P (x j )) ∞ j=1 ∈ r/m whenever (x j ) ∞ j=1 is weakly r-summable in X.

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Section: Introductionmentioning

confidence: 99%

Dominated Bilinear Forms and 2-homogeneous Polynomials

Botelho¹,

Pellegrino²,

Rueda³

2010

Publ. Res. Inst. Math. Sci.

Self Cite

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“…Since the proof of Pietsch's composition theorem is based on the Grothendieck-Pietsch domination theorem, it seems natural to work in the multilinear case with the class for which there is a domination theorem. One such a class is the class of dominated operators, see [4,8,10,12,13]. In this paper we prove that, as in the linear case, the class of dominated operators has some general splitting theorem and, as a consequence, we deduce some possible extensions of Pietsch's composition theorem to multilinear settings.…”

Section: Introduction and Notationmentioning

confidence: 81%

“…We would like to express our gratitude to the referee for his/her carefully reading of the manuscript, many valuable comments, suggestions, and showing us the references [2][3][4]10] which have improved the final version of the paper.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Multilinear variants of Pietsch's composition theorem

Popa

2010

Journal of Mathematical Analysis and Applications

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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%

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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Dominated polynomials on infinite dimensional spaces

Abstract: Abstract. The aim of this paper is to prove a stronger version of a conjecture posed earlier on the existence of nondominated scalar-valued m-homogeneous polynomials, m ≥ 3, on arbitrary infinite dimensional Banach spaces.

Cited by 11 publications

References 36 publications

Dominated Bilinear Forms and 2-homogeneous Polynomials

Dominated Bilinear Forms and 2-homogeneous Polynomials

Multilinear variants of Pietsch's composition theorem

Characterization of lr-dominated m-linear operators

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