Compactness in L<sup>1</sup>of a vector measure

Calabuig, J. M.; Lajara, Sebastián; Rodríguez, José; Pérez, Enrique Alfonso Sánchez

doi:10.4064/sm225-3-6

Cited by 6 publications

(17 citation statements)

References 31 publications

(30 reference statements)

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“…In the proof of part (i) of the following result we will use that the set

Γ \subseteq B_{L^{1} {(m)}^{*}}

(defined in ) is

w^{*}

‐thick, see [, Lemma 3.2(ii)]. Theorem Let

(f_{n})

be a sequence in

L^{1} false(m false)

.…”

Section: Resultsmentioning

confidence: 99%

“…

\underset{μ (A) \to 0}{trueprefixlim} \underset{n \in N}{trueprefixsup} {∥ f_{n} χ_{A} ∥}_{L^{1} false(m false)} = 0,

(cf. [, Lemma 3.2 and Proposition 3.3]). It follows that

(f_{n})

is relatively weakly compact in

L^{1} false(m false)

(see e.g.…”

Section: Preliminariesmentioning

confidence: 98%

“…It is defined by declaring that a net

(f_{α})

L^{1} false(m false)

τ_{m}

‐convergent to

f \in L^{1} false(m false)

if and only if

false∥ T_{g} false(f_{α} - f false) ∥_{X} \to 0

for every

g \in L^{\infty} false(m false)

. This topology and its compact sets have been studied in .…”

Section: Preliminariesmentioning

confidence: 99%

“…Example below). We will use the following characterization, which can be found in [, Proposition 3.5]: Fact

m (Σ)

is relatively norm compact if and only if

B_{L^{\infty} (m)}

τ_{m}

‐compact as a subset of

L^{1} false(m false)

.…”

Section: Preliminariesmentioning

confidence: 99%

“…It is known that

Γ = {normalΓ}_{B_{L^{\infty} (m)}}

is a James boundary of

L^{1} false(m false)

whenever

m (Σ)

is relatively norm compact (see [, Proposition 4.38], cf. [, Lemma 3.3] or [, Corollary 4.2]). Corollary below improves that result.…”

Section: Preliminariesmentioning

confidence: 99%

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Summability in of a vector measure

Calabuig

Rodríguez

Pérez

2016

Mathematische Nachrichten

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We show a picture of the relations among different types of summability of series in the space L1false(mfalse) of integrable functions with respect to a vector measure m of relatively norm compact range. In order to do that, we study the class of the so‐called m‐1‐summing operators. We give several applications regarding the existence of copies of c0 in L1false(mfalse), as well as on m‐1‐summing operators which are weakly compact, Asplund or weakly precompact.

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“…In the proof of part (i) of the following result we will use that the set

Γ \subseteq B_{L^{1} {(m)}^{*}}

(defined in ) is

w^{*}

‐thick, see [, Lemma 3.2(ii)]. Theorem Let

(f_{n})

be a sequence in

L^{1} false(m false)

.…”

Section: Resultsmentioning

confidence: 99%

“…

\underset{μ (A) \to 0}{trueprefixlim} \underset{n \in N}{trueprefixsup} {∥ f_{n} χ_{A} ∥}_{L^{1} false(m false)} = 0,

(cf. [, Lemma 3.2 and Proposition 3.3]). It follows that

(f_{n})

is relatively weakly compact in

L^{1} false(m false)

(see e.g.…”

Section: Preliminariesmentioning

confidence: 98%

“…It is defined by declaring that a net

(f_{α})

L^{1} false(m false)

τ_{m}

‐convergent to

f \in L^{1} false(m false)

if and only if

false∥ T_{g} false(f_{α} - f false) ∥_{X} \to 0

for every

g \in L^{\infty} false(m false)

. This topology and its compact sets have been studied in .…”

Section: Preliminariesmentioning

confidence: 99%

“…Example below). We will use the following characterization, which can be found in [, Proposition 3.5]: Fact

m (Σ)

is relatively norm compact if and only if

B_{L^{\infty} (m)}

τ_{m}

‐compact as a subset of

L^{1} false(m false)

.…”

Section: Preliminariesmentioning

confidence: 99%

“…It is known that

Γ = {normalΓ}_{B_{L^{\infty} (m)}}

is a James boundary of

L^{1} false(m false)

whenever

m (Σ)

is relatively norm compact (see [, Proposition 4.38], cf. [, Lemma 3.3] or [, Corollary 4.2]). Corollary below improves that result.…”

Section: Preliminariesmentioning

confidence: 99%

See 3 more Smart Citations

Summability in of a vector measure

Calabuig

Rodríguez

Pérez

2016

Mathematische Nachrichten

Self Cite

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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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