On ‐null sequences and their relatives

Ain, Kati; Oja, Eve

doi:10.1002/mana.201400300

Cited by 12 publications

(14 citation statements)

References 28 publications

(88 reference statements)

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“…When

ℓ_{p} false(X false)

is replaced with

ℓ_{p}^{u} false(X false)

the unconditionally p ‐null sequences of J. M. Kim are obtained. By [, Corollary 4.2], unconditionally p ‐null sequences and

K_{p^{'}}

‐null sequences coincide. Therefore, once these concepts are described under the framework of the metric Carl–Stephani theory, an application of Theorem (c) for

N^{p}

covers the result due to Oja [, Theorem 4.1] for p ‐null sequences,

c_{0, {scriptN}^{p}} false(X false) = c_{0} {\overset{\otimes}{true}}_{d_{p}} X

.…”

Section: Tensor Representation Of Scripta‐null Sequencesmentioning

confidence: 99%

“…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%

“…The link between these notions of nullity and null sequences given by operator ideals is as follows: p ‐null sequences correspond with

N^{p}

‐null sequences with

N^{p}

the ideal of right p ‐nuclear operators [, Proposition 1.4 and Remark 1.3]. Also, unconditionally p ‐null sequences coincides with

K_{p^{'}}

‐null sequences [, Corollary 4.2], where

K_{p^{'}}

is the ideal of the classical

p^{'}

‐compact operators of Fourie and Swart and

p^{'}

is the conjugate of p . Taking this into account, the results by Oja and Kim read as follows:

c_{0, {scriptN}^{p}} false(X false) = c_{0} {\overset{\otimes}{true}}_{d_{p}} X

and

c_{0, {frakturK}_{p^{'}}} false(X false) = c_{0} {\overset{\otimes}{true}}_{w_{p}} X

, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…Operator ideals, null sequences, compact sets, approximation properties M S C ( 2 0 1 0 ) Primary: 46B04, 46B45; Secondary: 46B28, 46B50 ′ -null sequences [3, Corollary 4.2], where ′ is the ideal of the classical ′ -compact operators of Fourie and Swart [11] and ′ is the conjugate of . Taking this into account, the results by Oja and Kim read as follows: 0, ( ) = 0⊗ and 0, ′ ( ) = 0⊗ , respectively.…”

Section: Introductionmentioning

confidence: 99%

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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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“…When

ℓ_{p} false(X false)

is replaced with

ℓ_{p}^{u} false(X false)

the unconditionally p ‐null sequences of J. M. Kim are obtained. By [, Corollary 4.2], unconditionally p ‐null sequences and

K_{p^{'}}

‐null sequences coincide. Therefore, once these concepts are described under the framework of the metric Carl–Stephani theory, an application of Theorem (c) for

N^{p}

covers the result due to Oja [, Theorem 4.1] for p ‐null sequences,

c_{0, {scriptN}^{p}} false(X false) = c_{0} {\overset{\otimes}{true}}_{d_{p}} X

.…”

Section: Tensor Representation Of Scripta‐null Sequencesmentioning

confidence: 99%

scriptA

‐compact sets and

scriptA

‐null sequences , where

scriptA

denotes an arbitrary operator ideal.…”

Section: Introductionmentioning

confidence: 99%

“…The link between these notions of nullity and null sequences given by operator ideals is as follows: p ‐null sequences correspond with

N^{p}

‐null sequences with

N^{p}

the ideal of right p ‐nuclear operators [, Proposition 1.4 and Remark 1.3]. Also, unconditionally p ‐null sequences coincides with

K_{p^{'}}

‐null sequences [, Corollary 4.2], where

K_{p^{'}}

is the ideal of the classical

p^{'}

‐compact operators of Fourie and Swart and

p^{'}

is the conjugate of p . Taking this into account, the results by Oja and Kim read as follows:

c_{0, {scriptN}^{p}} false(X false) = c_{0} {\overset{\otimes}{true}}_{d_{p}} X

and

c_{0, {frakturK}_{p^{'}}} false(X false) = c_{0} {\overset{\otimes}{true}}_{w_{p}} X

, respectively.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Lassalle

Turco

2017

Math. Nachr.

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“…The missing link c 0,N p (X) = c 0,P dual p (X) is provided by [2,Corollary 3.4]. Finally, to show (c) we appeal to [8, p. 73] implying that the classes of τ p -and τ N p -continuous functionals on L(X; Y ) coincide.…”

mentioning

confidence: 99%

Weaker relatives of the bounded approximation property for a Banach operator ideal

Lassalle

Oja

Turco

2016

Journal of Approximation Theory

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Abstract. Fixed a Banach operator ideal A, we introduce and investigate two new approximation properties, which are strictly weaker than the bounded approximation property (BAP) for A of Lima, Lima and Oja (2010). We call them the weak BAP for A and the local BAP for A, showing that the latter is in turn strictly weaker than the former. Under this framework, we address the question of approximation properties passing from dual spaces to underlying spaces. We relate the weak and local BAPs for A with approximation properties given by tensor norms and show that the Saphar BAP of order p is the weak BAP for the ideal of absolutely p

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On approximation properties related to unconditionally p‐compact operators and Sinha–Karn p‐compact operators

Wirzenius

2023

Mathematische Nachrichten

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We establish new results on the -approximation property for the Banach operator ideal  =  𝑢𝑝 of the unconditionally 𝑝-compact operators in the case of 1 ≤ 𝑝 < 2. As a consequence of our results, we provide a negative answer for the case 𝑝 = 1 of a problem posed by Kim. Namely, the  𝑢1 -approximation property implies neither the  1 -approximation property nor the (classical) approximation property; and the  1 -approximation property implies neither the  𝑢1 -approximation property nor the approximation property. Here,  𝑝 denotes the 𝑝-compact operators of Sinha and Karn for 𝑝 ≥ 1. We also show for all 2 < 𝑝, 𝑞 < ∞ that there is a closed subspace 𝑋 ⊂ 𝓁 𝑞 that fails the  𝑟 -approximation property for all 𝑟 ≥ 𝑝.

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On ‐null sequences and their relatives

Cited by 12 publications

References 28 publications

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

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

Weaker relatives of the bounded approximation property for a Banach operator ideal

On approximation properties related to unconditionally p‐compact operators and Sinha–Karn p‐compact operators

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