Banach ideals of p-compact operators

Fourie, Jan; Swart, Johan

doi:10.1007/bf01170259

Cited by 35 publications

(30 citation statements)

References 10 publications

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“…The space

ℓ_{p}^{u} (X)

is defined as the (closed) subspace of

ℓ_{p}^{w} (X)

, formed by the

(x_{n}) \in ℓ_{p}^{w} (X)

satisfying

(x_{n}) = {trueprefixlim}_{N \to \infty} (x_{1}, \dots, x_{N}, 0, 0, \dots)

ℓ_{p}^{w} (X)

. The space

ℓ_{p}^{u} (X)

was introduced and thoroughly studied by Fourie and Swart in 1979. In particular, it follows from [, Theorem 1.4] that

Φ_{(x_{n})}

is compact whenever

(x_{n}) \in ℓ_{p}^{u} (X)

.…”

Section: Unconditionally and Weakly (Pr)‐null Sequencesmentioning

confidence: 99%

“…The space

ℓ_{p}^{u} (X)

was introduced and thoroughly studied by Fourie and Swart in 1979. In particular, it follows from [, Theorem 1.4] that

Φ_{(x_{n})}

is compact whenever

(x_{n}) \in ℓ_{p}^{u} (X)

. In fact,

Φ_{(x_{n})} : ℓ_{p^{*}} \to X

is compact if and only if

(x_{n}) \in ℓ_{p}^{u} (X)

(see [, Theorem 1.4] or, e.g., [, 8.2]).…”

Section: Unconditionally and Weakly (Pr)‐null Sequencesmentioning

confidence: 99%

“…Let

(x_{n}) \in ℓ_{p}^{u} (X)

. Then (see [, Lemma 1.2])

x_{n} = δ_{n} y_{n}

for some

(δ_{n}) \in c_{0}

and

(y_{n}) \in ℓ_{p}^{w} (X)

. Since, clearly,

Φ_{(x_{n})} = \sum_{n = 1}^{\infty} e_{n} \otimes x_{n} = \sum_{n = 1}^{\infty} δ_{n} e_{n} \otimes y_{n}

(where

e_{n} \in ℓ_{r}^{*}

are the unit vectors) and (as is well known and easy to verify)

(e_{n}) \in B_{ℓ_{r}^{w} (ℓ_{r}^{*})}

, we have, by the definition of

(t, u, v)

‐nuclear operators [, 18.1.1],

Φ_{(x_{n})} \in {scriptN}_{(\infty, p^{*}, r^{*})} (ℓ_{r}, X) .

Similarly, as in Section , we get that

{scriptU}_{(p, r)} = N_{(\infty, p^{*}, r^{*})}^{sur} .

This implies that

{scriptU}_{(p, r)} = K \circ {scriptU}_{(p}

…”

Section: Unconditionally and Weakly (Pr)‐null Sequencesmentioning

confidence: 99%

“…Recall (see [, Theorem 2.5] or, e.g., [, 18.3.2]) that

N_{(\infty, p, p^{*})}

coincides with the operator ideal K p of classical p ‐compact operators. Following Fourie and Swart or Pietsch [, 18.3.1 and 18.3.2], a linear operator

T : Y \to X

is called p ‐ compact , i.e.,

T \in {normalK}_{p} (Y, X)

, if there exist

A \in K (Y, ℓ_{p})

and

B \in K (ℓ_{p}, X)

such that

T = B A

. Remark (see and ) that

K_{p}

and K p are notably different as operator ideals.…”

Section: Unconditionally and Weakly (Pr)‐null Sequencesmentioning

confidence: 99%

See 3 more Smart Citations

On ‐null sequences and their relatives

Ain

Oja

2015

Mathematische Nachrichten

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Let 1 ≤ p < ∞ and 1 ≤ r ≤ p * , where p * is the conjugate index of p. We prove an omnibus theorem, which provides numerous equivalences for a sequence (x n ) in a Banach space X to be a ( p, r )-null sequence. One of them is that (x n ) is ( p, r )-null if and only if (x n ) is null and relatively ( p, r )-compact. This equivalence is known in the "limit" case when r = p * , the case of the p-null sequence and p-compactness. Our approach is more direct and easier than those applied for the proof of the latter result. We apply it also to characterize the unconditional and weak versions of ( p, r )-null sequences.

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“…The space

ℓ_{p}^{u} (X)

is defined as the (closed) subspace of

ℓ_{p}^{w} (X)

, formed by the

(x_{n}) \in ℓ_{p}^{w} (X)

satisfying

(x_{n}) = {trueprefixlim}_{N \to \infty} (x_{1}, \dots, x_{N}, 0, 0, \dots)

ℓ_{p}^{w} (X)

. The space

ℓ_{p}^{u} (X)

was introduced and thoroughly studied by Fourie and Swart in 1979. In particular, it follows from [, Theorem 1.4] that

Φ_{(x_{n})}

is compact whenever

(x_{n}) \in ℓ_{p}^{u} (X)

.…”

Section: Unconditionally and Weakly (Pr)‐null Sequencesmentioning

confidence: 99%

“…The space

ℓ_{p}^{u} (X)

was introduced and thoroughly studied by Fourie and Swart in 1979. In particular, it follows from [, Theorem 1.4] that

Φ_{(x_{n})}

is compact whenever

(x_{n}) \in ℓ_{p}^{u} (X)

. In fact,

Φ_{(x_{n})} : ℓ_{p^{*}} \to X

is compact if and only if

(x_{n}) \in ℓ_{p}^{u} (X)

(see [, Theorem 1.4] or, e.g., [, 8.2]).…”

Section: Unconditionally and Weakly (Pr)‐null Sequencesmentioning

confidence: 99%

“…Let

(x_{n}) \in ℓ_{p}^{u} (X)

. Then (see [, Lemma 1.2])

x_{n} = δ_{n} y_{n}

for some

(δ_{n}) \in c_{0}

and

(y_{n}) \in ℓ_{p}^{w} (X)

. Since, clearly,

Φ_{(x_{n})} = \sum_{n = 1}^{\infty} e_{n} \otimes x_{n} = \sum_{n = 1}^{\infty} δ_{n} e_{n} \otimes y_{n}

(where

e_{n} \in ℓ_{r}^{*}

are the unit vectors) and (as is well known and easy to verify)

(e_{n}) \in B_{ℓ_{r}^{w} (ℓ_{r}^{*})}

, we have, by the definition of

(t, u, v)

‐nuclear operators [, 18.1.1],

Φ_{(x_{n})} \in {scriptN}_{(\infty, p^{*}, r^{*})} (ℓ_{r}, X) .

Similarly, as in Section , we get that

{scriptU}_{(p, r)} = N_{(\infty, p^{*}, r^{*})}^{sur} .

This implies that

{scriptU}_{(p, r)} = K \circ {scriptU}_{(p}

…”

Section: Unconditionally and Weakly (Pr)‐null Sequencesmentioning

confidence: 99%

“…Recall (see [, Theorem 2.5] or, e.g., [, 18.3.2]) that

N_{(\infty, p, p^{*})}

coincides with the operator ideal K p of classical p ‐compact operators. Following Fourie and Swart or Pietsch [, 18.3.1 and 18.3.2], a linear operator

T : Y \to X

is called p ‐ compact , i.e.,

T \in {normalK}_{p} (Y, X)

, if there exist

A \in K (Y, ℓ_{p})

and

B \in K (ℓ_{p}, X)

such that

T = B A

. Remark (see and ) that

K_{p}

and K p are notably different as operator ideals.…”

Section: Unconditionally and Weakly (Pr)‐null Sequencesmentioning

confidence: 99%

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On ‐null sequences and their relatives

Ain

Oja

2015

Mathematische Nachrichten

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“…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%

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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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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Banach ideals of p-compact operators

Cited by 35 publications

References 10 publications

On ‐null sequences and their relatives

On ‐null sequences and their relatives

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

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

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