Ideals of Homogeneous Polynomials

Aron, Richard M.; Rueda, Pilar

doi:10.2977/prims/93

Cited by 25 publications

(42 citation statements)

References 10 publications

(18 reference statements)

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“…In Section 2 we study A-compact homogeneous polynomials. We show that this class fits in the theory of locally K A -bounded homogeneous polynomials studied in [4]. Also, we show that A-compact homogeneous polynomials are a composition ideal of polynomials (see definitions below).…”

Section: Introductionmentioning

confidence: 99%

“…Aron and Rueda [4] introduced the class of locally A-bounded polynomials by considering the theory of generating system of sets of Stephani. For a Banach operator ideal A, an n-…”

Section: A-compact Polynomialsmentioning

confidence: 99%

“…These Banach operator ideals are called surjective. Based on this, several authors had introduced and studied different classes of functions between Banach spaces (in particular polynomials and holomorphic mappings) somehow extending this property, see for instance [2,5,4,3,6,17,18,23].…”

Section: Introductionmentioning

confidence: 99%

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$${\mathcal A}$$ A -compact mappings

Turco

2015

RACSAM

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Abstract. For a fixed Banach operator ideal A, we use the notion of A-compact sets of Carl and Stephani to study A-compact polynomials and A-compact holomorphic mappings. Namely, those mappings g : X → Y such that every x ∈ X has a neighborhood V x such that g(V x ) is relatively A-compact. We show that the behavior of A-compact polynomials is determined by its behavior in any neighborhood of any point. We transfer some known properties of A-compact operators to A-compact polynomials. In order to study A-compact holomorphic functions, we appeal to the A-compact radius of convergence which allows us to characterize the functions in this class. Under certain hypothesis on the ideal A, we give examples showing that our characterization is sharp.

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Section: Introductionmentioning

confidence: 99%

“…Aron and Rueda [4] introduced the class of locally A-bounded polynomials by considering the theory of generating system of sets of Stephani. For a Banach operator ideal A, an n-…”

Section: A-compact Polynomialsmentioning

confidence: 99%

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$${\mathcal A}$$ A -compact mappings

Turco

2015

RACSAM

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“…It is clear that I-bounded sets are norm bounded and that singletons are I-bounded (indeed, this is obvious for x = 0, and for x = 0 just pick a funcional ϕ ∈ E ′ such that ϕ(x) = x and note that ϕ ⊗ x ∈ I(E; E) and ϕ ⊗ x (x/ x ) = x). By the ideal property of I it follows that bounded linear operators send I-bounded sets to I-bounded sets, so the topology τ C I of uniform convergence on I-bounded sets (cf., e.g., [2]) is an ideal topology by Proposition 2.4.…”

Section: Ideal Topologiesmentioning

confidence: 99%

Ideal topologies and corresponding approximation properties

Berrios¹,

Botelho²

2016

Ann. Acad. Sci. Fenn. Math.

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Abstract. We propose a unifying approach to numerous approximation properties in Banach spaces studied from the 1930s up to our days. To do so, we introduce the concept of ideal topology and say that a Banach space E has the (I, J , τ )-approximation property if E-valued operators belonging to the operator ideal I can be approximated, with respect to the ideal topology τ , by operators belonging to the operator ideal J . This concept recovers many classical/recent approximation properties as particular instances and several important known results are particular cases of more general results that are valid in this general framework.

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“…This multi-ideal has many good properties and extends almost all the ones that are satisfied by the ideals of absolutely p-summing and p-dominated multilinear operators, as inclusion theorems, Pietsch domination theorems, factorization theorems and tensor product representations. On the other hand, in the last ten years a considerable effort has been made in order to increase the knowledge on the polynomials that belong to some operator ideal (see for instance [2,3,4,12,22] and the references therein). The case of the p-dominated polynomials is particularly relevant and has been intensively studied ( [6,7,8,9,10]).…”

Section: Introductionmentioning

confidence: 99%

Factorization of absolutely continuous polynomials

Achour

Dahia

Rueda

et al. 2013

Journal of Mathematical Analysis and Applications

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Abstract. In this paper we study the ideal of dominated (p; σ)-continuous polynomials, that extend the nowadays well known ideal of p-dominated polynomials to the more general setting of the interpolated ideals of polynomials. We give the polynomial version of Pietsch's factorization Theorem for this new ideal. Although based in [11], our factorization theorem requires new techniques inspired in the theory of Banach lattices.

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Ideals of Homogeneous Polynomials

Abstract: Given a surjective ideal of operators, we undertake a new general procedure to construct an ideal of polynomials. The relation with the ideal of polynomials obtained by the well-known method of composition is established.

Cited by 25 publications

References 10 publications

$${\mathcal A}$$ A -compact mappings

$${\mathcal A}$$ A -compact mappings

Ideal topologies and corresponding approximation properties

Factorization of absolutely continuous polynomials

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