Compactness arguments for spaces of $p$-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces

Pérez, Enrique Alfonso Sánchez

doi:10.1215/ijm/1258138159

Cited by 40 publications

(52 citation statements)

References 9 publications

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“…By (14) and (15), to establish that E q (μ) is a μ-K.f.s. it only remains to prove that E q (μ) is complete.…”

Section: The Function Normmentioning

confidence: 99%

The dual space of L p of a vector measure

Galaz-Fontes

2010

Positivity

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For a vector measure ν having values in a real or complex Banach space and p ∈ [1, ∞), we consider L p (ν) and L p w (ν), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E q (μ) and show it is σ -order continuous when p > 1. In this case, for the associate spaces we prove that L p (ν) × = E q (μ) and E q (μ) × = L p w (ν). It follows that L p (ν) * * = L p w (ν). We also show that L 1 (ν) × may be equal or not to E ∞ (μ).

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“…By (14) and (15), to establish that E q (μ) is a μ-K.f.s. it only remains to prove that E q (μ) is complete.…”

Section: The Function Normmentioning

confidence: 99%

The dual space of L p of a vector measure

Galaz-Fontes

2010

Positivity

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“…If 1 < p < ∞, the space L p (m) of p-integrable functions with respect to m is the set of (classes of) functions that satisfy that |f | p ∈ L 1 (m). It is also a Banach function space over each Rybakov control measure for m; see [12], [4] or [11,Proposition 3. 28] for the definition and main results regarding these spaces.…”

Section: Example 43mentioning

confidence: 99%

Ordered representations of spaces of integrable functions

Fernández¹,

Mayoral²,

Naranjo³

et al. 2008

Positivity

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Let X be a Banach space, (Ω, Σ) a measurable space and let m : Σ → X be a (countably additive) vector measure. Consider the corresponding space of integrable functions L 1 (m). In this paper we analyze the set of (countably additive) vector measures n satisfying that L 1 (n) = L 1 (m). In order to do this we define a (quasi) order relation on this set to obtain under adequate requirements the simplest representation of the space L 1 (m) associated to downward directed subsets of the set of all the representations. Mathematics Subject Classification (2000). Primary 46E30, 46G10; Secondary 47B65, 47H07.

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“…Let f i and f j two elements of the λ-orthonormal sequence. Since f = lim n n i=1 α i f i ∈ L 2 (λ f ) then the properties of L 2 (λ) (see [19]) give…”

Section: Commutative Sequences Of Functions and Thementioning

confidence: 99%

“…In fact, L 2 (λ) is a Köthe function space. However, we do not use the properties of these spaces in this paper, that can be found by the interested reader in [19] and [5]. Following [21], we will use the following definition.…”

Section: Introductionmentioning

confidence: 99%

Commutative Sequences of Integrable Functions and Best Approximation With Respect to the Weighted Vector Measure Distance

Raffi

Pérez

2005

Integr. equ. oper. theory

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Let λ be a countably additive vector measure with values in a separable real Hilbert space H. We define and study a pseudo metric on a Banach lattice of integrable functions related to λ that we call a λ-weighted distance. We compute the best approximation with respect to this distance to elements of the function space by the use of sequences with special geometric properties. The requirements on the sequence of functions are given in terms of a commutation relation between these functions that involves integration with respect to λ. We also compare the approximation that is obtained in this way with the corresponding projection on a particular Hilbert space. Mathematics Subject Classification (2000). 46G10, 46E30, 46C05.

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Compactness arguments for spaces of $p$-integrable functions with respect to a vector measure and factorization of operators through Lebesgue-Bochner spaces

Cited by 40 publications

References 9 publications

The dual space of L p of a vector measure

The dual space of L p of a vector measure

Ordered representations of spaces of integrable functions

Commutative Sequences of Integrable Functions and Best Approximation With Respect to the Weighted Vector Measure Distance

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