Extension of vector-valued integral polynomials

Carando, Daniel; Lassalle, Silvia

doi:10.1016/j.jmaa.2004.10.020

Cited by 12 publications

(5 citation statements)

References 20 publications

(26 reference statements)

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“…εs E, F ) isometrically [6,22]. An application of Lemma 1.7 shows that the sequence of symmetric injective tensor norms verifies the conditions of the previous proposition with constants C = 1 and D = e. Also, we obtain the same constants for compatibility.…”

Section: Relation With Tensor Normssupporting

confidence: 67%

Coherent sequences of polynomial ideals on Banach spaces

Carando

Dimant

Muro

2009

Mathematische Nachrichten

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Abstract. This article deals with the relationship between an operator ideal and its natural polynomial extensions. We define the concept of coherent sequence of polynomial ideals and also the notion of compatibility between polynomial and operator ideals. We study the stability of these properties for maximal and minimal hulls, adjoint and composition ideals. We also relate these concepts with conditions on the underlying tensor norms.

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Section: Relation With Tensor Normssupporting

confidence: 67%

Coherent sequences of polynomial ideals on Banach spaces

Carando

Dimant

Muro

2009

Mathematische Nachrichten

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“…different Aron-Berner extensions. Some multi-ideals are known to be Aron-Berner stable (for example, integral mappings [12,Proposition 8]) and some are known not to be (for example, weakly sequentially continuous mappings [26, Example 1.9]).…”

Section: Aron-berner Extensionsmentioning

confidence: 99%

Uniform approximation on ideals of multilinear mappings

Botelho¹,

Galindo²,

Pellegrini³

2010

MATH. SCAND.

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For each ideal of multilinear mappings M we explicitly construct a corresponding ideal a M such that multilinear forms in a M are exactly those which can be approximated, in the uniform norm, by multilinear forms in M. This construction is then applied to finite type, compact, weakly compact and absolutely summing multilinear mappings. It is also proved that the correspondence M → a M is Aron-Berner stability preserving.

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“…In the Banach space setting, Grothendieck-integral bilinear mappings are always extendible [5,Proposition 7]. Let us see that an analogous contention holds in the operator space framework.…”

Section: Example 62 a Multiplicatively Bounded Bilinear Mapping Whimentioning

confidence: 94%

Bilinear ideals in operator spaces

Dimant

Fernández-Unzueta

2015

Journal of Mathematical Analysis and Applications

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Abstract. We introduce a concept of bilinear ideal of jointly completely bounded mappings between operator spaces. In particular, we study the bilinear ideals N of completely nuclear, I of completely integral, E of completely extendible bilinear mappings, MB multiplicatively bounded and its symmetrization SMB. We prove some basic properties of them, one of which is the fact that I is naturally identified with the ideal of (linear) completely integral mappings on the injective operator space tensor product. Introduction and PreliminariesLet V, W and X be operator spaces. If we consider the underlying vector space structure, the relationshold through the two natural linear isomorphisms ν, ρ. In order for ν and ρ to induce natural morphisms in the operator space category, it is necessary to have appropriately defined an operator space tensor norm on V ⊗ W and specific classes of linear and bilinear mappings. This is the case, for instance, of the so called projective operator space tensor norm · ∧ , the completely bounded maps and the jointly completely bounded bilinear mappings, where ν and ρ induce the following completely bounded isometric isomorphisms:There are many possible ways to provide V ⊗ W with an operator space tensor norm and, of course, to define classes of mappings. Several authors, inspired by the success that the study of the relations between tensor products and mappings has had in the Banach space setting, have systematically study some analogous relations for operator spaces. This is the case, for instance, of the completely nuclear and completely integral linear mappings (see [7, Section III]).In this paper we follow this approach as well, but with the attention focused on the relations involving ν, the isomorphism in (1) which concerns bilinear mappings. In Section 2 we introduce the notion of an ideal of completely bounded bilinear mappings and study its general properties. In Section 3 we define the ideals of completely nuclear and completely integral bilinear mappings. The main result proved here is that the ideal of completely integral bilinear mappings is naturally identified with the ideal of completely integral linear mappings on the injective operator space tensor product, that is I(V × W, X) ∼ = L I (V ∨ ⊗ W, X) (see Theorem 3.8). This implies that, 2010 Mathematics Subject Classification. 47L25,47L22, 46M05.

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Extension of vector-valued integral polynomials

Cited by 12 publications

References 20 publications

Coherent sequences of polynomial ideals on Banach spaces

Coherent sequences of polynomial ideals on Banach spaces

Uniform approximation on ideals of multilinear mappings

Bilinear ideals in operator spaces

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