Around Widder’s characterization of the Laplace transform of an element of L ∞ ( ℝ + )

Jan Kisyński

Annales Polonici Mathematici (2000)

  • Volume: 74, Issue: 1, page 161-200
  • ISSN: 0066-2216

Abstract

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Let ϰ be a positive, continuous, submultiplicative function on ℝ + such that l i m t → ∞ e - ω t t - α ϰ ( t ) = a for some ω ∈ ℝ, α ∈ ℝ + ¯ and a ∈ ℝ + . For every λ ∈ (ω,∞) let ϕ λ ( t ) = e - λ t for t ∈ ℝ + . Let L ϰ 1 ( 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 → T ϕ • defined on L ( L ϰ 1 ( ℝ + ) ; E ) , generalizing a result established by B. Hennig and F. Neubrander for ϰ ( t ) = e ω t . 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 ∈ ℝ + ¯ such that s u p t ∈ ℝ + ¯ ( ϰ ( t ) ) - 1 ∥ S t ∥ < ∞ .

How to cite

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Kisyński, Jan. "Around Widder’s characterization of the Laplace transform of an element of $L^{∞}(ℝ^{+})$." Annales Polonici Mathematici 74.1 (2000): 161-200. <http://eudml.org/doc/208364>.

@article{Kisyński2000,
abstract = {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 $sup_\{t ∈ \overline\{ℝ^\{+\}\}\} (ϰ(t))^\{-1\}∥ S_t∥ < ∞$.},
author = {Kisyński, Jan},
journal = {Annales Polonici Mathematici},
keywords = {operators from $L_\{ϰ\}^\{1\}(ℝ^\{+\})$ into a Banach space; complete monotonicity and positivity with respect to a cone; one-parameter semigroups of operators; vector measures; Gelfand space; Radon-Nikodym property; representations of the convolution algebra $L_\{ϰ\}^\{1\}(ℝ^\{+\})$; pseudoresolvents and their generators; real inversion formulas for the Laplace transform; characterization of Laplace-transforms; Banach space; real inversion formulas; semigroups of operators; complete monotonicity},
language = {eng},
number = {1},
pages = {161-200},
title = {Around Widder’s characterization of the Laplace transform of an element of $L^\{∞\}(ℝ^\{+\})$},
url = {http://eudml.org/doc/208364},
volume = {74},
year = {2000},
}

TY - JOUR
AU - Kisyński, Jan
TI - Around Widder’s characterization of the Laplace transform of an element of $L^{∞}(ℝ^{+})$
JO - Annales Polonici Mathematici
PY - 2000
VL - 74
IS - 1
SP - 161
EP - 200
AB - 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 $sup_{t ∈ \overline{ℝ^{+}}} (ϰ(t))^{-1}∥ S_t∥ < ∞$.
LA - eng
KW - operators from $L_{ϰ}^{1}(ℝ^{+})$ into a Banach space; complete monotonicity and positivity with respect to a cone; one-parameter semigroups of operators; vector measures; Gelfand space; Radon-Nikodym property; representations of the convolution algebra $L_{ϰ}^{1}(ℝ^{+})$; pseudoresolvents and their generators; real inversion formulas for the Laplace transform; characterization of Laplace-transforms; Banach space; real inversion formulas; semigroups of operators; complete monotonicity
UR - http://eudml.org/doc/208364
ER -

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