Norm closed operator ideals in Lorentz sequence spaces

Kamińska, Anna; Popov, Alexey I.; Spinu, Eugeniu; Tcaciuc, Adi; Troitsky, Vladimir G.

doi:10.1016/j.jmaa.2011.11.034

Cited by 13 publications

(17 citation statements)

References 18 publications

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“…We have that

S \circ P \circ I_{d, g}

preserves a copy of

d (w, p)

. (vii)

\Rightarrow

(iii): By [, Theorem 5.5], the sequence

{false((T \circ I_{d, g}) ({boldd}_{n}) false)}_{n = 1}^{\infty}

does not converge to zero in norm. This means that we can find a subsequence

{false(T ({boldg}_{n_{k}}) false)}_{k = 1}^{\infty}

that is semi‐normalized.…”

Section: Nonexistence Of Symmetric Basis In G(wp)mentioning

confidence: 97%

“…(vii) ⇒ (iii): By [8,Theorem 5.5], the sequence ((T • I d,g )(d n )) ∞ n=1 does not converge to zero in norm. This means that we can find a subsequence (T (g n k )) ∞ k=1 that is semi-normalized.…”

Section: Nonexistence Of Symmetric Basis In G(w P)mentioning

confidence: 99%

“…In particular, the set of all

X

‐strictly singular operators — those for which the restrictions to subspaces isomorphic to

X

are never bounded below — acting on a complementably homogeneous Banach space

X

forms the unique maximal ideal in

L (X)

whenever it is closed under addition. By an argument in [, Corollary 5.2] and the second paragraph of the proof of [, Theorem 5.3], it follows that the set of

g (w, p)

‐strictly singular operators forms the unique maximal ideal in

L (g (w, p))

.…”

Section: Introduction Notation and Terminologymentioning

confidence: 99%

See 2 more Smart Citations

Garling sequence spaces

Albiac

Ansorena

Wallis

2018

J. London Math. Soc.

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The aim of this paper is to introduce and investigate a new class of separable Banach spaces modeled after an example of Garling from 1968. For each 1⩽p<∞ and each nonincreasing weight w∈c0∖ℓ1, we exhibit an ℓp‐saturated, complementably homogeneous, and uniformly subprojective Banach space g(w,p). We also show that g(w,p) admits a unique subsymmetric basis despite the fact that for a wide class of weights it does not admit a symmetric basis. This provides the first known examples of Banach spaces where those two properties coexist.

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“…We have that

S \circ P \circ I_{d, g}

preserves a copy of

d (w, p)

. (vii)

\Rightarrow

(iii): By [, Theorem 5.5], the sequence

{false((T \circ I_{d, g}) ({boldd}_{n}) false)}_{n = 1}^{\infty}

does not converge to zero in norm. This means that we can find a subsequence

{false(T ({boldg}_{n_{k}}) false)}_{k = 1}^{\infty}

that is semi‐normalized.…”

Section: Nonexistence Of Symmetric Basis In G(wp)mentioning

confidence: 97%

Section: Nonexistence Of Symmetric Basis In G(w P)mentioning

confidence: 99%

“…In particular, the set of all

X

‐strictly singular operators — those for which the restrictions to subspaces isomorphic to

X

are never bounded below — acting on a complementably homogeneous Banach space

X

forms the unique maximal ideal in

L (X)

whenever it is closed under addition. By an argument in [, Corollary 5.2] and the second paragraph of the proof of [, Theorem 5.3], it follows that the set of

g (w, p)

‐strictly singular operators forms the unique maximal ideal in

L (g (w, p))

.…”

Section: Introduction Notation and Terminologymentioning

confidence: 99%

See 1 more Smart Citation

Garling sequence spaces

Albiac

Ansorena

Wallis

2018

J. London Math. Soc.

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“…Also the spaces E = ℓ ∞ and E = J p (1 < p < ∞) (where J p is the p-th James space), enjoy the property that WK(E) is the largest non-trivial closed ideal in L(E) (see [24, p. 253 [12], [21], [7] and [20]).…”

Section: Ideals In the Spacementioning

confidence: 99%

The Composition Operation on Spaces of Holomorphic Mappings

Acosta

Galindo

Moraes

2020

The Quarterly Journal of Mathematics

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We discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of the entire mappings that are bounded on bounded sets the composition turns to be even holomorphic. In such space we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.

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“…In the classical period, the only Banach spaces E for which the lattice of closed ideals in L(E) are fully understood are Hilbert space [2,4,11] and the sequence spaces c 0 and ℓ p , 1 ≤ p < ∞ [3]. In the past decade, starting with [7], there has been a resurgence of interest in the problem and new results have been obtained, see, e.g., [5,6,8,9,10,12,13]. One should also mention the recent breakthrough example AH by Argyros and Haydon [1], where L(AH) is well understood because of a very different reason.…”

mentioning

confidence: 99%

Ideals of operators on $(\oplus \ell ^\infty (n))_{\ell ^1}$

Leung

2015

Proc. Amer. Math. Soc.

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Abstract. The unique maximal ideal in the Banach algebra L(E), E = (⊕ℓ ∞ (n)) ℓ 1 , is identified. The proof relies on techniques developed by Laustsen, Loy and Read [7] and a dichotomy result for operators mapping into L 1 due to Laustsen, Odell, Schlumprecht and Zsák [8].Given a Banach space E, it is a natural problem to try to understand the ideal structure of the Banach algebra L(E). In the classical period, the only Banach spaces E for which the lattice of closed ideals in L(E) are fully understood are Hilbert space [2,4,11] and the sequence spaces c 0 and ℓ p , 1 ≤ p < ∞ [3]. In the past decade, starting with [7], there has been a resurgence of interest in the problem and new results have been obtained, see, e.g., [5,6,8,9,10,12,13]. One should also mention the recent breakthrough example AH by Argyros and Haydon [1], where L(AH) is well understood because of a very different reason. In this note, we make a small contribution to the program by identifying the unique maximal ideal in the algebra L(E), where E = (⊕ℓ ∞ (n)) ℓ 1 . Our method is slightly more general in the sense that it works also for the space E = (⊕ℓ 2 (n)) ℓ 1 , which gives an alternate proof of the main result of [9]. Let E n be finite dimensional Banach spaces and let E = (⊕E n ) ℓ 1 . Given a subset I of N, denote by E I the subspace (⊕ n∈I E n ) ℓ 1 . The natural embedding of E I into E is denoted by J I and the natural projection from E onto E I is denoted by P I . If I = {n}, then we write for short J n and P n respectively. Let T be a bounded linear operator on E, then T has a natural matrix representation (T mn ), where T mn = P m T J n : E n → E m . Denote the nth column in this representation by T n . Thus T n = T J n :Let X, Y, Z be Banach spaces. If T : X → Y is a bounded linear operator and ε > 0, defineFor Z = c 0 , respectively ℓ 1 , we write Fac In the sequel, T will always denote a bounded linear operator on E.

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Norm closed operator ideals in Lorentz sequence spaces

Cited by 13 publications

References 18 publications

Garling sequence spaces

Garling sequence spaces

The Composition Operation on Spaces of Holomorphic Mappings

Ideals of operators on $(\oplus \ell ^\infty (n))_{\ell ^1}$

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