A class of weakly compact sets in Lebesgue–Bochner spaces

Rodríguez, José Antonio Piqueras

doi:10.1016/j.topol.2017.02.075

Cited by 4 publications

(3 citation statements)

References 23 publications

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“…Our next result strengthens the conclusion of the Bourgain-Maurey-Pisier theorem (i.e., Theorem 2.11 with ε = 0) for Banach spaces having property KM w and multi-functions which are "measurable" in a certain sense. The proof is similar to that of [36,Proposition 2.8].…”

Section: Unconditional Sumsmentioning

confidence: 66%

$\varepsilon$-weakly precompact sets in Banach spaces

Rodríguez¹

2021

Preprint

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A bounded subset M of a Banach space X is said to be ε-weakly precompact, for a given ε ≥ 0, if every sequenceIn this paper we discuss several aspects of ε-weakly precompact sets. On the one hand, we give quantitative versions of the following known results: (a) the absolutely convex hull of a weakly precompact set is weakly precompact (Stegall), and (b) for any probability measure µ, the set of all Bochner µ-integrable functions taking values in a weakly precompact subset of X is weakly precompact in L 1 (µ, X) (Bourgain, Maurey, Pisier). On the other hand, we introduce a relative of a Banach space property considered by Kampoukos and Mercourakis when studying subspaces of strongly weakly compactly generated spaces. We say that a Banach space X has property KM w if there is a family {M n,p : n, p ∈ N} of subsets of X such that: (i) M n,p is 1 p -weakly precompact for all n, p ∈ N, and (ii) for each weakly precompact set C ⊆ X and for each p ∈ N there is n ∈ N such that C ⊆ M n,p . All subspaces of strongly weakly precompactly generated spaces have property KM w . Among other things, we study the three-space problem and the stability under unconditional sums of property KM w .

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Section: Unconditional Sumsmentioning

confidence: 66%

$\varepsilon$-weakly precompact sets in Banach spaces

Rodríguez¹

2021

Preprint

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“…In general, the projective tensor product of two SWCG spaces is not SWCG, [22,Example 2.11]. It is an open problem whether the Lebesgue-Bochner space L 1 ([0, 1], X) (which is isometrically isomorphic to L 1 [0, 1] ⊗ π X) is SWCG whenever X is SWCG, see [19] and the references therein. In the other direction, it is known that ℓ 2 ⊗ π ℓ 2 is SWCG, [22,Example 2.3(c)].…”

Section: Introductionmentioning

confidence: 99%

On weak compactness in projective tensor products

Rodríguez¹

2022

Preprint

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We study the property of being strongly weakly compactly generated (and some relatives) in projective tensor products of Banach spaces. Our main result is as follows. Let 1 < p, q < ∞ be such that 1/p + 1/q ≥ 1. Let X (resp., Y ) be a Banach space with a countable unconditional finite-dimensional Schauder decomposition having a disjoint lower p-estimate (resp., q-estimate). If X and Y are strongly weakly compactly generated, then so is its projective tensor product X ⊗ π Y . 2020 Mathematics Subject Classification. 46B28, 46B50.

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“…Any δS-set of L 1 (ν, X) is relatively weakly compact, while the converse is not true in general. For more information on these sets, see [37] and the references therein. Concerning positive results, we show that L 1 (ν, X) has property (µ s δS ) whenever X is a subspace of a S 2 WCG space (Theorem 5.8).…”

Section: Introductionmentioning

confidence: 99%

Cesàro Convergent Sequences in the Mackey Topology

Rodríguez

2019

Mediterr. J. Math.

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A Banach space X is said to have property (µ s ) if every weak *null sequence in X * admits a subsequence such that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. This is stronger than the so-called property (K) of Kwapień. We prove that property (µ s ) holds for every subspace of a Banach space which is strongly generated by an operator with Banach-Saks adjoint (e.g. a strongly super weakly compactly generated space). The stability of property (µ s ) under ℓ p -sums is discussed. For a family A of relatively weakly compact subsets of X, we consider the weaker property (µ s A ) which only requires uniform convergence on the elements of A, and we give some applications to Banach lattices and Lebesgue-Bochner spaces. We show that every Banach lattice with order continuous norm and weak unit has property (µ s A ) for the family of all L-weakly compact sets. This sharpens a result of de Pagter, Dodds and Sukochev. On the other hand, we prove that L 1 (ν, X) (for a finite measure ν) has property (µ s A ) for the family of all δS-sets whenever X is a subspace of a strongly super weakly compactly generated space.2010 Mathematics Subject Classification. Primary: 46B50. Secondary: 47B07.

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A class of weakly compact sets in Lebesgue–Bochner spaces

Cited by 4 publications

References 23 publications

$\varepsilon$-weakly precompact sets in Banach spaces

$\varepsilon$-weakly precompact sets in Banach spaces

On weak compactness in projective tensor products

Cesàro Convergent Sequences in the Mackey Topology

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