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Banach spaces of bounded Szlenk index

100%

Odell E. , Schlumprecht T. , Zsák A.

Studia Mathematica

2007

tom 183

nr 1

63-97

For a countable ordinal α we denote by $𝓒_{α}$ the class of separable, reflexive Banach spaces whose Szlenk index and the Szlenk index of their dual are bounded by α. We show that each $𝓒_{α}$ admits a separable, reflexive universal space. We also show that spaces in the class $𝓒_{ω^{α·ω}}$ embed into spaces of the same class with a basis. As a consequence we deduce that each $𝓒_{α}$ is analytic in the Effros-Borel structure of subspaces of C[0,1].

On norm closed ideals in $L(ℓ_{p},ℓ_{q})$

100%

Sari B. , Schlumprecht T. , Tomczak-Jaegermann N. , Troitsky V.

Studia Mathematica

2007

tom 179

nr 3

239-262

It is well known that the only proper non-trivial norm closed ideal in the algebra L(X) for $X = ℓ_{p}$ (1 ≤ p < ∞) or X = c₀ is the ideal of compact operators. The next natural question is to describe all closed ideals of $L(ℓ_{p}⊕ ℓ_{q})$ for 1 ≤ p,q < ∞, p ≠ q, or equivalently, the closed ideals in $L(ℓ_{p},ℓ_{q})$ for p < q. This paper shows that for 1 < p < 2 < q < ∞ there are at least four distinct proper closed ideals in $L(ℓ_{p},ℓ_{q})$, including one that has not been studied before. The proofs use various methods from Banach space theory.

Banach spaces of bounded Szlenk index II

100%

Freeman D. , Odell E. , Schlumprecht T. , Zsák A.

Fundamenta Mathematicae

2009

tom 205

nr 2

161-177

For every α < ω₁ we establish the existence of a separable Banach space whose Szlenk index is $ω^{αω+1}$ and which is universal for all separable Banach spaces whose Szlenk index does not exceed $ω^{αω}$. In order to prove that result we provide an intrinsic characterization of which Banach spaces embed into a space admitting an FDD with Tsirelson type upper estimates.