Inner and outer inequalities with applications to approximation properties

Oja, Eve

doi:10.1090/s0002-9947-2011-05241-4

Cited by 20 publications

(21 citation statements)

References 40 publications

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“…A remarkable fact, essentially due to Grothendieck [9] (see, e.g., [5, p. 47] or [32, Proposition 8.1]), is that A and B are Banach operator ideals. (Using terminology from [32], A is the ideal of the α -integral operators (where α denotes the dual norm of α) and B is the ideal of α t -integral operators, called "the dual operator ideal of α" in [18] (see [18,Remark 1.5]). ) We shall use the following known result which is essentially contained, e.g., in [32, proof of Theorem 8.4].…”

Section: Lemma 22 Let X and Y Be Banach Spaces And Let α Be A Tensomentioning

confidence: 99%

Grothendieckʼs nuclear operator theorem revisited with an application to p-null sequences

Oja

2012

Journal of Functional Analysis

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Let X and Y be Banach spaces and let α be a tensor norm. The principal result is the following theorem. If either X * * * or Y has the approximation property, then each α-nuclear operator T : X * → Y such that T * (Y * ) ⊂ X can be approximated in the α-nuclear norm by finite-rank operators of type X ⊗ Y . In the special case of (Grothendieck) nuclear operators, the theorem provides a strengthening for the classical theorem on the nuclearity of operators with a nuclear adjoint. The hypotheses about the approximation property are essential. The main application yields an affirmative answer to [C. Piñeiro, J.M. Delgado, p-Convergent sequences and Banach spaces in which p-compact sets are q-compact, Proc. Amer. Math. Soc. 139 (2011) 957-967]: for p 1, a sequence (x n ) ⊂ X is p-null if and only if lim x n = 0 and (x n ) is relatively p-compact in X.

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Section: Lemma 22 Let X and Y Be Banach Spaces And Let α Be A Tensomentioning

confidence: 99%

Grothendieckʼs nuclear operator theorem revisited with an application to p-null sequences

Oja

2012

Journal of Functional Analysis

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“…Since X has the weak BAP, there exists an extension operator ∈ X ⊗ X * w * ⊂ L (X * , X * * * ) = (X * ⊗ π X * * ) * (see [13, Propositions 2.1, 2.3, and 2.5] and [20,Corollary 3.18…”

Section: Proof Let T ∈ a (Y Z * )mentioning

confidence: 99%

Bounded approximation properties in terms of $C[0,1]$

Lima¹,

Lima²,

Oja³

2012

MATH. SCAND.

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Let X be a Banach space and let I be the Banach operator ideal of integral operators. We prove that X has the λ-bounded approximation property (λ-BAP) if and only if for every operator T ∈ I (X, C[0, 1] * ) there exists a net (S α ) of finite-rank operators on X such that S α → I X pointwise and lim supWe also prove that replacing I by the ideal N of nuclear operators yields a condition which is equivalent to the weak λ-BAP.

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“…Reinforced by the fact that there are Banach spaces which lack the approximation property (the first example given by Enflo [13]), important variants of this property have emerged and were intensively studied, see [4,11,18,21] and references therein. In particular, there is a recent inclination to study approximation properties related to (Banach) operator ideals, as it can be seen for instance in [1,5,7,9,14,16,17,19,22,23,29,30].…”

Section: Introductionmentioning

confidence: 99%

The Banach ideal of A-compact operators and related approximation properties

Lassalle

Turco

2013

Journal of Functional Analysis

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Abstract. We use the notion of A-compact sets (determined by an operator ideal A), introduced by Carl and Stephani (1984), to show that many known results of certain approximation properties and several ideals of compact operators can be systematically studied under this framework. For Banach operator ideals A, we introduce a way to measure the size of A-compact sets and use it to give a norm on K A , the ideal of A-compact operators. Then, we study two types of approximation properties determined by A-compact sets. We focus our attention on an approximation property which makes use of the norm defined on K A . This notion fits the definition of the A-approximation property, recently introduced by Oja (2012), with K A instead of A. We exemplify the power of the Carl-Stephani theory and the geometric structure introduced here by appealing to some recent developments on p-compactness.

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Inner and outer inequalities with applications to approximation properties

Cited by 20 publications

References 40 publications

Grothendieckʼs nuclear operator theorem revisited with an application to p-null sequences

Grothendieckʼs nuclear operator theorem revisited with an application to p-null sequences

Bounded approximation properties in terms of $C[0,1]$

The Banach ideal of A-compact operators and related approximation properties

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