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## Annales Polonici Mathematici

2000 | 74 | 1 | 161-200
Tytuł artykułu

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

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
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 $sup_{t ∈ \overline{ℝ^{+}}} (ϰ(t))^{-1}∥ S_t∥ < ∞$.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
161-200
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-07-20
Twórcy
autor
• Faculty of Electrical Engineering, Technical University of Lublin, Nadbystrzycka 38A, P.O. Box 189, 20-618 Lublin, Poland
Bibliografia
• [A] W. Arendt, Vector-valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327-352.
• [B] A. Bobrowski, On the Yosida approximation and the Widder-Arendt representation theorem, Studia Math. 124 (1997), 281-290.
• [C] W. Chojnacki, Multiplier algebras, Banach bundles, and one-parameter semigroups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 287-322.
• [D] E. B. Davies, One-Parameter Semigroups, Academic Press, 1980.
• [D-M;C] C. Dellacherie and P.-A. Meyer, Probabilities and Potential C. Potential Theory for Discrete and Continuous Semigroups, North-Holland Math. Stud. 151, North-Holland, 1988.
• [D-M;XII-XVI] C. Dellacherie and P.-A. Meyer, Probabilités et Potentiel, Chapitres XII à XVI. Théorie du Potentiel Associée à une Résolvante, Théorie des Processus de Markov, Hermann, Paris, 1987.
• [D-S;I] N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Interscience, New York, 1958.
• [D-S;II] N. Dunford and J. T. Schwartz, Linear Operators, Part II. Spectral Theory, Self Adjoint Operators in Hilbert Space, Interscience, New York, 1963.
• [D-U] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, RI, 1977.
• [E-K] S. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence, Wiley, 1986.
• [F-Y] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker, 1999.
• [G] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, 1985.
• [H] P. R. Halmos, Measure Theory, Springer, 1974
• [H-N] B. Hennig and F. Neubrander, On representations, inversions, and approximations of Laplace transforms in Banach spaces, Appl. Anal. 49 (1993), 151-170.
• [H-R;I] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Volume I, Structure of Topological Groups, Integration Theory, Group Representations, 2nd ed., Springer, 1979.
• [H-R;II] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Volume II, Structure and Analysis on Compact Groups, Analysis on Locally Compact Abelian Groups, 3rd printing, Springer, 1997.
• [H] E. Hille, Functional Analysis and Semi-groups, Colloq. Publ., Amer. Math. Soc., 1948.
• [H-P] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Colloq. Publ., Amer. Math. Soc., 1957.
• [Hi] F. Hirsch, Familles résolvantes, générateurs, cogénérateurs, potentiels, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 1, 89-210.
• [J] B. E. Johnson, Centralisers on certain topological algebras, J. London Math. Soc. 39 (1964), 603-614.
• [J-K-B] N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Vol. I, 2nd ed., Wiley, 1994.
• [K;I] T. Kato, Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, Proc. Japan Acad. 35 (1959), 467-468.
• [K;II] T. Kato, Perturbation Theory for Linear Operators, Springer, 1966.
• [Ki] J. Kisyński, The Widder spaces, representations of the convolution algebra $L^{1}(ℝ^{+})$, and one parameter semigroups of operators, preprint, Inst. Math., Polish Acad. Sci., 1998, 36 pp.
• [Kr] S. G. Krein, Linear Differential Equations in Banach Space, Transl. Math. Monographs 29, Amer. Math. Soc., Providence, RI, 1971.
• [deL-H-W-W] R. deLaubenfels, Z. Huang, S. Wang and Y. Wang, Laplace transforms of polynomially bounded vector-valued functions and semigroups of operators, Israel J. Math. 98 (1997), 189-207.
• [deL-J] R. deLaubenfels and M. Jazar, Functional calculi, regularized semigroups and integrated semigroups, Studia Math. 132 (1999), 151-172.
• [L] J. L. Lions, Les semigroupes distributions, Portugal. Math. 19 (1960), 141-164.
• [Pal] T. W. Palmer, Banach Algebras and the General Theory of *-Algebras. Volume I: Algebras and Banach Algebras, Encyclopedia Math. Appl. 49, Cambridge Univ. Press, 1994.
• [P] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 2nd printing, Springer, 1983.
• [Ph] R. S. Phillips, An inversion formula for Laplace transforms and semigroups of linear operators, Ann. of Math. 59 (1954), 325-356.
• [R-Y] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, 1991.
• [R;I] W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw- Hill, 1976.
• [R;II] W. Rudin, Real and Complex Analysis, 2nd ed., McGraw-Hill, 1974.
• [S-K] I. E. Segal and R. A. Kunze, Integrals and Operators, 2nd ed., Springer, 1987.
• [T] H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958), 887-919.
• [W;I] D. V. Widder, The Laplace Transform, Princeton Univ. Press, 1946.
• [W;II] D. V. Widder, An Introduction to Transform Theory, Academic Press, 1971.
• [Y;I] K. Yosida, On the differentiability and the representation of one-parameter semi-groups of linear operators, J. Math. Soc. Japan 1 (1948), 15-21.
• [Y;II] K. Yosida, Functional Analysis, 6th ed., Springer, 1980; reprint of the 1980 edition, Springer, 1995.
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Bibliografia
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