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On operators from separable reflexive spaces with asymptotic structure

100%
Zheng B.
Studia Mathematica
|
2008
|
tom 185
|
nr 1
87-98
EN
Let 1 < q < p < ∞ and q ≤ r ≤ p. Let X be a reflexive Banach space satisfying a lower-$ℓ_{q}$-tree estimate and let T be a bounded linear operator from X which satisfies an upper-$ℓ_{p}$-tree estimate. Then T factors through a subspace of $(∑ Fₙ)_{ℓ_{r}}$, where (Fₙ) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an $(ℓ_{p}, ℓ_{q})$ FDD. Similarly, let 1 < q < r < p < ∞ and let X be a separable reflexive Banach space satisfying an asymptotic lower-$ℓ_{q}$-tree estimate. Let T be a bounded linear operator from X which satisfies an asymptotic upper-$ℓ_{p}$-tree estimate. Then T factors through a subspace of $(∑ Gₙ)_{ℓ_{r}}$, where (Gₙ) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an asymptotic $(ℓ_{p},ℓ_{q})$ FDD.
2
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On operators which factor through $l_{p}$ or c₀

100%
Zheng B.
Studia Mathematica
|
2006
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tom 176
|
nr 2
177-190
EN
Let 1 < p < ∞. Let X be a subspace of a space Z with a shrinking F.D.D. (Eₙ) which satisfies a block lower-p estimate. Then any bounded linear operator T from X which satisfies an upper-(C,p)-tree estimate factors through a subspace of $(∑Fₙ)_{l_{p}}$, where (Fₙ) is a blocking of (Eₙ). In particular, we prove that an operator from $L_{p}$ (2 < p < ∞) satisfies an upper-(C,p)-tree estimate if and only if it factors through $l_{p}$. This gives an answer to a question of W. B. Johnson. We also prove that if X is a Banach space with X* separable and T is an operator from X which satisfies an upper-(C,∞)-estimate, then T factors through a subspace of c₀.
3
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Corrigendum to "Commutators on $(∑ℓ_{q})_{p}$" (Studia Math. 206 (2011), 175-190)

51%
Chen D., Johnson W., Zheng B.
Studia Mathematica
|
2014
|
tom 223
|
nr 2
187-191
EN
We give a corrected proof of Theorem 2.10 in our paper "Commutators on $(∑ℓ_{q})_{p}$" [Studia Math. 206 (2011), 175-190] for the case 1 < q < p < ∞. The case when 1 = q < p < ∞ remains open. As a consequence, the Main Theorem and Corollary 2.17 in that paper are only valid for 1 < p,q < ∞.
4
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Commutators on $(∑ ℓ_{q})_{p}$

51%
Chen D., Johnson W., Zheng B.
Studia Mathematica
|
2011
|
tom 206
|
nr 2
175-190
EN
Let T be a bounded linear operator on $X = (∑ ℓ_{q})_{p}$ with 1 ≤ q < ∞ and 1 < p < ∞. Then T is a commutator if and only if for all non-zero λ ∈ ℂ, the operator T - λI is not X-strictly singular.
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