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Unconditional ideals in Banach spaces

100%

Godefroy G., Kalton N. J., Saphar P. D.

Studia Mathematica

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1993

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tom 104

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nr 1

13-59

EN

We show that a Banach space with separable dual can be renormed to satisfy hereditarily an "almost" optimal uniform smoothness condition. The optimal condition occurs when the canonical decomposition $X*** = X^{⊥} ⊕ X*$ is unconditional. Motivated by this result, we define a subspace X of a Banach space Y to be an h-ideal (resp. a u-ideal) if there is an hermitian projection P (resp. a projection P with ∥I-2P∥ = 1) on Y* with kernel $X^{⊥}$. We undertake a general study of h-ideals and u-ideals. For example we show that if a separable Banach space X is an h-ideal in X** then X has the complex form of Pełczyński's property (u) with constant one and the Baire-one functions Ba(X) in X** are complemented by an hermitian projection; the converse holds under a compatibility condition which is shown to be necessary. We relate these ideas to the more familiar notion of an M-ideal, and to Banach lattices. We further investigate when, for a separable Banach space X, the ideal of compact operators K(X) is a u-ideal or an h-ideal in ℒ(X) or K(X)**. For example, we show that K(X) is an h-ideal in K(X)** if and only if X has the "unconditional compact approximation property" and X is an M-ideal in X**.

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On general Franklin systems

77%

Gevorkyan G., Kamont A.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS 1. Introduction.....................................................................................5 1.1. Notation......................................................................................7 2. Definition and properties of general Franklin systems..................10 2.1. Piecewise linear functions.........................................................10 2.2. Franklin functions.....................................................................11 2.3. Sequences of partitions and Franklin functions........................13 2.3.1. Regularity of sequences of partitions...................................14 2.4. Sequences of partitions and general Haar systems.................16 2.5. Technical lemmas.....................................................................17 3. Franklin series in $L^p$, 1 < p < ∞...............................................21 4. Franklin series in $L^p$, 0 < p ≤ 1, and $H^p$, 1/2 < p ≤ 1..........27 5. The necessity of strong regularity in $H^p$, 1/2 < p ≤ 1...............42 6. Haar and Franklin series with identical coefficients......................46 7. Characterization of the spaces BMO and Lip(α), 0 < α < 1...........51 References.......................................................................................58

3

Unconditional convergence in lattice groups with respect to ideals

75%

Boccuto A., Dimitriou X., Papanastassiou N.

Commentationes Mathematicae

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2010

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tom 50

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nr 2

EN

We deal with unconditional convergence of series and some special classes of subsets of $\mathbb{N}$.

Ograniczanie wyników

1 Commentationes Mathematicae

1 Studia Mathematica

1 Boccuto A.

1 Dimitriou X.

1 Gevorkyan G.

1 Godefroy G.

1 Kalton N. J.

1 Kamont A.

1 Papanastassiou N.

1 Saphar P. D.

1 2010

1 1998

1 1993

Unconditional ideals in Banach spaces

On general Franklin systems

Unconditional convergence in lattice groups with respect to ideals