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On bounded approximation properties of Banach spaces

100%
Oja E.
Banach Center Publications
|
2010
|
tom 91
|
nr 1
219-231
EN
This survey features some recent developments concerning the bounded approximation property in Banach spaces. As a central theme, we discuss the weak bounded approximation property and the approximation property which is bounded for a Banach operator ideal. We also include an overview around the related long-standing open problem: Is the approximation property of a dual Banach space always metric?
2
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Uniform factorization for compact sets of weakly compact operators

64%
Mikkor K., Oja E.
Studia Mathematica
|
2006
|
tom 174
|
nr 1
85-97
EN
We prove uniform factorization results that describe the factorization of compact sets of compact and weakly compact operators via Hölder continuous homeomorphisms having Lipschitz continuous inverses. This yields, in particular, quantitative strengthenings of results of Graves and Ruess on the factorization through $ℓ_{p}$-spaces and of Aron, Lindström, Ruess, and Ryan on the factorization through universal spaces of Figiel and Johnson. Our method is based on the isometric version of the Davis-Figiel-Johnson-Pełczyński factorization construction due to Lima, Nygaard, and Oja.
3
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On subspaces of Banach spaces where every functional has a unique norm-preserving extension

64%
Oja E., Põldvere M.
Studia Mathematica
|
1995-1996
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tom 117
|
nr 3
289-306
EN
Let X be a Banach space and Y a closed subspace. We obtain simple geometric characterizations of Phelps' property U for Y in X (that every continuous linear functional g ∈ Y* has a unique norm-preserving extension f ∈ X*), which do not use the dual space X*. This enables us to give an intrinsic geometric characterization of preduals of strictly convex spaces close to the Beauzamy-Maurey-Lima-Uttersrud criterion of smoothness. This also enables us to prove that the U-property of the subspace K(E,F) of compact operators from a Banach space E to a Banach space F in the corresponding space L(E,F) of all operators implies the U-property for F in F** whenever F is isomorphic to a quotient space of E.
4
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Ideals of finite rank operators, intersection properties of balls, and the approximation property

64%
Lima Å., Oja E.
Studia Mathematica
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1999
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tom 133
|
nr 2
175-186
EN
We characterize the approximation property of Banach spaces and their dual spaces by the position of finite rank operators in the space of compact operators. In particular, we show that a Banach space E has the approximation property if and only if for all closed subspaces F of $c_0$, the space ℱ(F,E) of finite rank operators from F to E has the n-intersection property in the corresponding space K(F,E) of compact operators for all n, or equivalently, ℱ(F,E) is an ideal in K(F,E).
5
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Some duality results on bounded approximation properties of pairs

64%
Oja E., Treialt S.
Studia Mathematica
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2013
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tom 217
|
nr 1
79-94
EN
The main result is as follows. Let X be a Banach space and let Y be a closed subspace of X. Assume that the pair $(X*,Y^{⊥})$ has the λ-bounded approximation property. Then there exists a net $(S_{α})$ of finite-rank operators on X such that $S_{α}(Y) ⊂ Y$ and $||S_{α}|| ≤ λ$ for all α, and $(S_{α})$ and $(S*_{α})$ converge pointwise to the identity operators on X and X*, respectively. This means that the pair (X,Y) has the λ-bounded duality approximation property.
6
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On the norm of a projection onto the space of compact operators

64%
Lippus J., Oja E.
Studia Mathematica
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2007
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tom 182
|
nr 3
263-272
EN
Let X and Y be Banach spaces and let 𝓐(X,Y) be a closed subspace of 𝓛(X,Y), the Banach space of bounded linear operators from X to Y, containing the subspace 𝒦(X,Y) of compact operators. We prove that if Y has the metric compact approximation property and a certain geometric property M*(a,B,c), where a,c ≥ 0 and B is a compact set of scalars (Kalton's property (M*) = M*(1, {-1}, 1)), and if 𝓐(X,Y) ≠ 𝒦(X,Y), then there is no projection from 𝓐(X,Y) onto 𝒦(X,Y) with norm less than max|B| + c. Since, for given λ with 1 < λ < 2, every Y with separable dual can be equivalently renormed to satisfy M*(a,B,c) with max|B| + c = λ, this implies and improves a theorem due to Saphar.
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