Compact operators which are defined by<i>l</i><sub>p</sub>-spaces

Ain, Kati; Lillemets, Rauni; Oja, Eve

doi:10.2989/16073606.2012.696819

Cited by 23 publications

(30 citation statements)

References 15 publications

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“…(This result is due to [3,Proposition 3.11] where, to prove this result, the authors use a roundabout approach, first describing K dual p , and rely on Reinov's recent study [30] on operators with p-nuclear adjoints. An easy straightforward proof was independently proposed in [1] (see Remark 3.8 in [1]) and [24,Theorem 1].) It is well known (and easy to see, because 1 canonically embeds into 1 (B 1 )) that A sur ( 1 , X) = A( 1 , X) for any Banach operator ideal.…”

Section: Description Of C 0p (X) As a Chevet-saphar Tensor Productmentioning

confidence: 94%

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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: Description Of C 0p (X) As a Chevet-saphar Tensor Productmentioning

confidence: 94%

“…An operator T ∈ L(X, Y ) is p-compact if T (B X ) is a relatively p-compact subset of Y . A suitable formula for the Banach ideal norm in K p (X, Y ) was given by Delgado, Piñeiro, and Serrano [3] (see [1,Theorem 3.4 and Remark 3.7]) as follows:…”

Section: Description Of C 0p (X) As a Chevet-saphar Tensor Productmentioning

confidence: 99%

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

Oja

2012

Journal of Functional Analysis

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“…The notions of null sequences and compact sets were shown to be closely related from the nowadays classical result of Grothendieck which characterizes relatively compact sets as those contained in the absolutely convex hull of a norm null sequence of vectors of the space. In the recent years strong forms of compactness have been studied, see for instance , , , , , , , , , . Many of the results obtained can be revisited under the Carl–Stephani theory of

scriptA

‐compact sets and

scriptA

‐null sequences , where

scriptA

denotes an arbitrary operator ideal.…”

Section: Introductionmentioning

confidence: 99%

On null sequences for Banach operator ideals, trace duality and approximation properties

Lassalle

Turco

2017

Math. Nachr.

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Let scriptA be a Banach operator ideal and X be a Banach space. We undertake the study of the vector space of scriptA‐null sequences of Carl and Stephani on X, c0,Afalse(Xfalse), from a unified point of view after we introduce a norm which makes it a Banach space. To give accurate results we consider local versions of the different types of accessibility of Banach operator ideals. We show that in the most common situations, when scriptA is right‐accessible for (ℓ1;X), c0,Afalse(Xfalse) behaves much alike c0false(Xfalse). When this is the case we give a geometric tensor product representation of c0,Afalse(Xfalse). On the other hand, we show an example where the representation fails. Also, via a trace duality formula, we characterize the dual space of c0,Afalse(Xfalse). We apply our results to study some problems related with the KA‐approximation property giving a trace condition which is used to solve the remaining case (p=1) of a problem posed by Kim (2015). Namely, we prove that if a dual space has the K1‐approximation property then the space has the Ku1‐approximation property.

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“…(ii) For q > 0, a subset A of a Banach space E is a Bourgain-Reinov q-compact set (see [11,59,1]), in symbols A ∈ BR q (E), if there is a E-valued absolutely q-summable sequence (x n ) n such that A is contained in the closure of the absolutely convex hull of {x 1 , x 2 , . .…”

Section: Ideal Topologiesmentioning

confidence: 99%

“…Let C ⊆ BAN be given. Suppose that for every Banach space E it has been assigned a collection A(E) of bounded subsets of E containing the singletons, satisfying (1) and such that, for all n ∈ N and Banach spaces E 1 , . .…”

Section: Projective 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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Compact operators which are defined byl_p-spaces

Cited by 23 publications

References 15 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

On null sequences for Banach operator ideals, trace duality and approximation properties

Ideal topologies and corresponding approximation properties

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Compact operators which are defined bylp-spaces

Cited by 23 publications

References 15 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

On null sequences for Banach operator ideals, trace duality and approximation properties

Ideal topologies and corresponding approximation properties

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Product

Resources

About

Compact operators which are defined byl_p-spaces