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

Lassalle, Silvia; Oja, Eve; Turco, Pablo

doi:10.1016/j.jat.2016.01.005

Cited by 5 publications

(5 citation statements)

References 33 publications

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“…Clearly,

c_{0} false(X false) = c_{0, K} false(X false)

. In fact,

c_{0} false(X false) = c_{0, A} false(X false)

for every operator ideal

scriptA

such that

\bar{scriptF} \subset A

, as can be inferred from [, Proposition 2.4].…”

Section: Introductionmentioning

confidence: 91%

“…As usual,

τ_{c}

denotes de topology of uniform convergence on compact sets. There is a recent inclination to study approximation properties related to (Banach) operator ideals, as it can be seen, among others, in , , , , , , , , , , . Our interest lies in the study of the

scriptA

‐ approximation property (

scriptA

‐AP for short) which is defined taking into account the geometry of

scriptA

.…”

Section: On the Dual Of (Lfalse(boldx;boldyfalse)boldτsa) And Applicmentioning

confidence: 99%

“…Recall that X has the approximation property (AP for short) if

I d_{X}

can be uniformly approximated by finite rank operators on compact sets. Thus, the

K_{A}

‐AP, defined and studied in and (see also ), can be seen as the analogous to the AP where the system of compact sets is replaced with the system of

scriptA

‐compact sets and the uniform norm with

m_{A}

.…”

Section: Introductionmentioning

confidence: 99%

“…Clearly, 0 ( ) = 0, ( ). In fact, 0 ( ) = 0, ( ) for every operator ideal  such that  ⊂ , as can be inferred from [20,Proposition 2.4].…”

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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“…Clearly,

c_{0} false(X false) = c_{0, K} false(X false)

. In fact,

c_{0} false(X false) = c_{0, A} false(X false)

for every operator ideal

scriptA

such that

\bar{scriptF} \subset A

, as can be inferred from [, Proposition 2.4].…”

Section: Introductionmentioning

confidence: 91%

“…As usual,

τ_{c}

scriptA

‐ approximation property (

scriptA

‐AP for short) which is defined taking into account the geometry of

scriptA

.…”

Section: On the Dual Of (Lfalse(boldx;boldyfalse)boldτsa) And Applicmentioning

confidence: 99%

“…Recall that X has the approximation property (AP for short) if

I d_{X}

can be uniformly approximated by finite rank operators on compact sets. Thus, the

K_{A}

‐AP, defined and studied in and (see also ), can be seen as the analogous to the AP where the system of compact sets is replaced with the system of

scriptA

‐compact sets and the uniform norm with

m_{A}

.…”

Section: Introductionmentioning

confidence: 99%

“…Clearly, 0 ( ) = 0, ( ). In fact, 0 ( ) = 0, ( ) for every operator ideal  such that  ⊂ , as can be inferred from [20,Proposition 2.4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Lassalle

Turco

2017

Math. Nachr.

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“…and π(u) = α 1,1 (u). The tensor norms g p and w p were introduced and studied by Chevet and Saphar [12,13]; see [10,11,[14][15][16][17][18][19][20][21][22][23][24] and the references therein for the investigation on related topics.…”

Section: Introductionmentioning

confidence: 99%