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Nonseparability of the quotient space cabv(∑,m;X)/L¹(m;X) for Banach spaces X without the Radon-Nikodym property

100%

Drewnowski L.

Studia Mathematica

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1993

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tom 104

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nr 2

125-132

EN

It is shown that if (S,∑,m) is an atomless finite measure space and X is a Banach space without the Radon-Nikodym property, then the quotient space cabv(∑,m;X)/L¹(m;X) is nonseparable.

2

The problem of complementability for some spaces of vector measures of bounded variation with values in Banach spaces containing copies of $c_{0}$

86%

Drewnowski L., Emmanuele G.

Studia Mathematica

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1993

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tom 104

|

nr 2

111-123

EN

Let (S, ∑, m) be any atomless finite measure space, and X any Banach space containing a copy of $c_0$. Then the Bochner space $L^1(m;X)$ is uncomplemented in ccabv(∑,m;X), the Banach space of all m-continuous vector measures that are of bounded variation and have a relatively compact range; and ccabv(∑,m;X) is uncomplemented in cabv(∑,m;X). It is conjectured that this should generalize to all Banach spaces X without the Radon-Nikodym property.

3

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Translations of functions iv vector Hardy classes on the unit disk

74%

Michalak A.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction...................................................................................................5 0. Preliminaries................................................................................................7 1. Fundamental properties of harmonic vector functions...............................13 2. Hardy spaces of vector functions...............................................................15 Relations between scalar and vector Hardy classes...................................15 The factorization theorem for $H^p(𝔻,X)$...................................................19 Nontangential limits of functions in $h^p(𝔻,X)$...........................................22 Properties of functions in $h^p(𝕋,X)$..........................................................27 3. Spaces $h^p(𝔻,X)$ and $M_p(𝕋,X)$..........................................................29 4. The sets of translates of harmonic functions..............................................33 5. Translations of functions from Hardy classes..............................................37 6. Translations of functions from Smirnov classes...........................................41 7. Translations of measures from $M_p(G,X)$................................................43 8. A criterion of uncomplementability of $L^p(λ_G,X)$ in $M_p(G,X)$.............53 9. Pettis integrability of the translation function for vector measures...............64 References...................................................................................................77

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Around Widder's characterization of the Laplace transform of an element of $L^{∞}(ℝ^{+})$

72%

Kisyński J.

Annales Polonici Mathematici

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2000

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tom 74

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nr 1

161-200

EN

Let ϰ be a positive, continuous, submultiplicative function on $ℝ^{+}$ such that $lim_{t→∞} e^{-ωt}t^{-α}ϰ(t) = a$ for some ω ∈ ℝ, α ∈ $\overline{ℝ^{+}}$ and $a ∈ ℝ^{+}$. For every λ ∈ (ω,∞) let $ϕ_{λ}(t) =e^{-λt}$ for $t ∈ ℝ^{+}$. Let $L^{1}_{ϰ}(ℝ^{+})$ be the space of functions Lebesgue integrable on $ℝ^{+}$ with weight $ϰ$, and let E be a Banach space. Consider the map $ϕ_{•}: (ω,∞) ∋ λ → ϕ_{λ} ∈ L_{ϰ}^{1}(ℝ^{+})$. Theorem 5.1 of the present paper characterizes the range of the linear map $T → Tϕ_{•}$ defined on $L(L_{ϰ}^{1}(ℝ^{+});E)$, generalizing a result established by B. Hennig and F. Neubrander for $ϰ(t)=e^{ωt}$. If ϰ ≡ 1 and E =ℝ then Theorem 5.1 reduces to D. V. Widder's characterization of the Laplace transform of a function in $L^{∞}(ℝ^{+})$. Some applications of Theorem 5.1 to the theory of one-parameter semigroups of operators are discussed. In particular a version of the Hille-Yosida generation theorem is deduced for $C_0$ semigroups $(S_t)_{t ∈ \overline{ℝ^{+}}}$ such that $sup_{t ∈ \overline{ℝ^{+}}} (ϰ(t))^{-1}∥ S_t∥ < ∞$.

Ograniczanie wyników

2 Studia Mathematica

1 Annales Polonici Mathematici

2 Drewnowski L.

1 Emmanuele G.

1 Kisyński J.

1 Michalak A.

1 2000

1 1996

2 1993

Nonseparability of the quotient space cabv(∑,m;X)/L¹(m;X) for Banach spaces X without the Radon-Nikodym property

The problem of complementability for some spaces of vector measures of bounded variation with values in Banach spaces containing copies of $c_{0}$

Translations of functions iv vector Hardy classes on the unit disk

Around Widder's characterization of the Laplace transform of an element of $L^{∞}(ℝ^{+})$