Symmetric structures in Banach spaces

Johnson, William B.; Maurey, Bernard; Schechtman, Gideon; Tzafriri, Lior

doi:10.1090/memo/0217

Cited by 180 publications

(197 citation statements)

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“…This result means that Li(0,1) © L2(0,1) has, up to isomorphism, a unique structure as a Banach function space on [0,1]. Further results on uniqueness of structures in Banach function spaces can be found in [9].…”

Section: Anß#0mentioning

confidence: 87%

Decompositions of Banach lattices into direct sums

Casazza¹,

Kalton²

1987

Trans. Amer. Math. Soc.

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ABSTRACT. We consider the problem of decomposing a Banach lattice Z as a direct sum Z = X®Y where X and Y are complemented subspaces satisfying a condition of incomparability (e.g. every operator from Y to X is strictly singular). We treat both the atomic and nonatomic cases. In particular we answer a question of Wojtaszczyk by showing that L\ ®¿2 has unique structure as a nonatomic Banach lattice.One of the most important problems in the theory of Banach lattices, which is still open, is whether any complemented subspace of a Banach lattice must be linearly isomorphic to a Banach lattice. The main difficulty seems to lie in the fact that most of the criteria for a Banach space to be isomorphic to a lattice do not really distinguish between lattices and their complemented subspaces.We do not actually treat this question in the present paper but rather consider the situation Z = X © Y, where Z is a Banach lattice and X and Y two complemented subspaces which are assumed to satisfy different conditions that make them "distinct" in some or another sense. This line of research was initiated by P. Wojtaszczyk [28] (and also by I. S. Edelstein and P. Wojtaszczyk [3]) who proved that if Z has a normalized unconditional basis {zn}^=x (i.e. it is a separable atomic lattice) so that every linear operator from Y into X is compact then {zn}^Lx splits into two disjoint parts which are respectively equivalent to bases of X and Y. In particular, both X and Y have unconditional bases. The proof of this result is based on a fundamental theorem from [28 and 3], which is mentioned below as Theorem A. We give here a different proof which does not make use of Theorem A but instead is based on a simple "change of signs" result from [2], which is described below as Theorem B. We also consider the case when the compactness assumption above is replaced by the total incomparability of X and Y for which we prove a similar result provided X and Y have unconditional bases. Unfortunately, the most interesting case when every operator from Y into X is assumed to be strictly singular (which was raised as an open problem in [28]) remains unsolved. We conclude the section devoted to the atomic case with a simple theorem on block bases of a space with unconditional basis {zn}^Lx whose span is complemented. Such a block basis splits into two disjoint parts, the first equivalent to a subsequence of {zn}^7x

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Section: Anß#0mentioning

confidence: 87%

Decompositions of Banach lattices into direct sums

Casazza¹,

Kalton²

1987

Trans. Amer. Math. Soc.

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“…Therefore our paper serves two purposes. On one hand it decides the natural question, which Lorentz spaces embed into L and on the other hand it gives some examples of spaces that show up naturally in functional analysis and that are symmetric subspaces of L though they are not Orlicz spaces.In particular we show that the Lorentz spaces Lp '" embed into L if and only if I < q < p <2 or p = q = 2 .We would like to mention that infinite-dimensional versions of the Orlicz spaces that we use in our standard embedding were already used in [6] for some other purpose. This type of space was first studied by Rosenthal [12].…”

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confidence: 85%

“…We would like to mention that infinite-dimensional versions of the Orlicz spaces that we use in our standard embedding were already used in [6] for some other purpose. This type of space was first studied by Rosenthal [12].…”

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confidence: 99%

Lorentz spaces that are isomorphic to subspaces of $L\sp 1$

Schütt¹

1989

Trans. Amer. Math. Soc.

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Abstract. We show which Lorentz spaces are isomorphic to subspaces of Lx and which are not.The class of subspaces of L1 is known to be quite big. Even if one considers only symmetric subspaces of L1, i.e. subspaces that have symmetric basis or that are isomorphic to a rearrangement invariant function space, one finds that the class is quite rich.It can be shown that those spaces are isomorphic to averages of Orlicz spaces [3,11]. It was first believed that those spaces are in fact just Orlicz spaces [1]. Examples were given that this is not the case [3]. These examples have an abstract nature. Therefore our paper serves two purposes. On one hand it decides the natural question, which Lorentz spaces embed into L and on the other hand it gives some examples of spaces that show up naturally in functional analysis and that are symmetric subspaces of L though they are not Orlicz spaces.In particular we show that the Lorentz spaces Lp '" embed into L if and only if I < q < p <2 or p = q = 2 .We would like to mention that infinite-dimensional versions of the Orlicz spaces that we use in our standard embedding were already used in [6] for some other purpose. This type of space was first studied by Rosenthal [12].In § 1 we state the theorem and derive some corollaries. Also we present some propositions from which the theorem follows. The propositions are proved in the following section. PreliminariesA basis {e¡}"=x of a normed space is C-symmetric if we have for all signs e, = ± 1, x,■ e R, and all permutations n Í2xiei í=i

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“…This terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870000882X [13] The Schur and (weak) Dunford-Pettis properties 263 yields by the Rosenthal £ r theorem that («;) contains a subsequence (M,J which is equivalent to the unit vector basis of t\. It is easy to see that («,,) is equivalent to the unit vector basis (e k ) of £ (i ,.…”

Section: A(mentioning

confidence: 99%

The Schur and (weak) Dunford-Pettis properties in Banach lattices

Kamińska¹,

Mastyło

2002

J. Aust. Math. Soc.

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We study the Schur and (weak) Dunford-Pettis properties in Banach lattices. We show that £i, c 0 and loo are the only Banach symmetric sequence spaces with the weak Dunford-Pettis property. We also characterize a large class of Banach lattices without the (weak) Dunford-Pettis property. In MusielakOrlicz sequence spaces we give some necessary and sufficient conditions for the Schur property, extending the Yamamuro result. We also present a number of results on the Schur property in weighted Orlicz sequence spaces, and, in particular, we find a complete characterization of this property for weights belonging to class A. We also present examples of weighted Orlicz spaces with the Schur property which are not S£\ -spaces. Finally, as an application of the results in sequence spaces, we provide a description of the weak Dunford-Pettis and the positive Schur properties in Orlicz spaces over an infinite non-atomic measure space.2000 Mathematics subject classification: primary 46B20,46E30,46B42,46B45.

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Symmetric structures in Banach spaces

Cited by 180 publications

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Decompositions of Banach lattices into direct sums

Decompositions of Banach lattices into direct sums

Lorentz spaces that are isomorphic to subspaces of $L\sp 1$

The Schur and (weak) Dunford-Pettis properties in Banach lattices

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