Wyniki wyszukiwania

1

Johnson's projection, Kalton's property (M*), and M-ideals of compact operators

100%

Nygaard O., Põldvere M.

Studia Mathematica

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2009

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tom 195

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nr 3

243-255

EN

Let X and Y be Banach spaces. We give a "non-separable" proof of the Kalton-Werner-Lima-Oja theorem that the subspace 𝒦(X,X) of compact operators forms an M-ideal in the space 𝓛(X,X) of all continuous linear operators from X to X if and only if X has Kalton's property (M*) and the metric compact approximation property. Our proof is a quick consequence of two main results. First, we describe how Johnson's projection P on 𝓛(X,Y)* applies to f ∈ 𝓛(X,Y)* when f is represented via a Borel (with respect to the relative weak* topology) measure on $\overline{B_{X**} ⊗ B_{Y*}}^{w*} ⊂ 𝓛(X,Y)*$: If Y* has the Radon-Nikodým property, then P "passes under the integral sign". Our basic theorem en route to this description-a structure theorem for Borel probability measures on $\overline{B_{X**} ⊗ B_{Y*}}^{w*}$-also yields a description of 𝒦(X,Y)* due to Feder and Saphar. Second, we show that property (M*) for X is equivalent to every functional in $\overline{B_{X**} ⊗ B_{X*}}^{w*}$ behaving as if 𝒦(X,X) were an M-ideal in 𝓛(X,X).

2

On the compact approximation property

81%

Lima V., Lima Å., Nygaard O.

Studia Mathematica

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2004

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tom 160

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nr 2

185-200

EN

We show that a Banach space X has the compact approximation property if and only if for every Banach space Y and every weakly compact operator T: Y → X, the space 𝔈 = {S ∘ T: S compact operator on X} is an ideal in 𝔉 = span(𝔈,T) if and only if for every Banach space Y and every weakly compact operator T: Y → X, there is a net $(S_γ)$ of compact operators on X such that $sup_{γ}||S_{γ}T|| ≤ ||T||$ and $S_{γ} → I_{X}$ in the strong operator topology. Similar results for dual spaces are also proved.

3

Diameter 2 properties and convexity

71%

Abrahamsen T., Hájek P., Nygaard O., Talponen J., Troyanski S.

Studia Mathematica

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2016

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tom 232

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nr 3

227-242

EN

We present an equivalent midpoint locally uniformly rotund (MLUR) renorming of C[0,1] with the diameter 2 property (D2P), i.e. every non-empty relatively weakly open subset of the unit ball has diameter 2. An example of an MLUR space with the D2P and with convex combinations of slices of arbitrarily small diameter is also given.

Ograniczanie wyników

3 Studia Mathematica

3 Nygaard O.

1 Abrahamsen T.

1 Hájek P.

1 Lima V.

1 Lima Å.

1 Põldvere M.

1 Talponen J.

1 Troyanski S.

1 2016

1 2009

1 2004

Johnson's projection, Kalton's property (M*), and M-ideals of compact operators

On the compact approximation property

Diameter 2 properties and convexity