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

Kisyński, Jan

Around Widder’s characterization of the Laplace transform of an element of

L^{\infty} (ℝ^{+})

Kisyński, Jan

Annales Polonici Mathematici, Tome 75 (2000), p. 161-200 / Harvested from The Polish Digital Mathematics Library

Access to full text
Full (PDF)

Résumé

Let ϰ be a positive, continuous, submultiplicative function on $ℝ^{+}$ such that $l i m_{t \to \infty} e^{- ω t} t^{- α} ϰ (t) = a$ for some ω ∈ ℝ, α ∈ $\bar{ℝ^{+}}$ and $a \in ℝ^{+}$ . For every λ ∈ (ω,∞) let $ϕ_{λ} (t) = e^{- λ t}$ for $t \in ℝ^{+}$ . 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 $ϕ_{•} : (ω, \infty) ∋ λ \to ϕ_{λ} \in L_{ϰ}^{1} (ℝ^{+})$ . Theorem 5.1 of the present paper characterizes the range of the linear map $T \to T ϕ_{•}$ 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 $L^{\infty} (ℝ^{+})$ . 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 ${(S_{t})}_{t \in \bar{ℝ^{+}}}$ such that $s u p_{t \in \bar{ℝ^{+}}} {(ϰ (t))}^{- 1} ∥ S_{t} ∥ < \infty$ .

Publié le : 2000-01-01

Zbl 0964.44001

EUDML-ID : urn:eudml:doc:208364

@article{bwmeta1.element.bwnjournal-article-apmv74z1p161bwm,
     author = {Kisy\'nski, Jan},
     title = {Around Widder's characterization of the Laplace transform of an element of $L^{$\infty$}($\mathbb{R}$^{+})$
            },
     journal = {Annales Polonici Mathematici},
     volume = {75},
     year = {2000},
     pages = {161-200},
     zbl = {0964.44001},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv74z1p161bwm}
}

Kisyński, Jan. Around Widder’s characterization of the Laplace transform of an element of $L^{∞}(ℝ^{+})$
            . Annales Polonici Mathematici, Tome 75 (2000) pp. 161-200. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv74z1p161bwm/

Bibliographie

[000] [A] W. Arendt, Vector-valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), 327-352. | Zbl 0637.44001

[001] [B] A. Bobrowski, On the Yosida approximation and the Widder-Arendt representation theorem, Studia Math. 124 (1997), 281-290. | Zbl 0876.44001

[002] [C] W. Chojnacki, Multiplier algebras, Banach bundles, and one-parameter semigroups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 287-322. | Zbl 0970.46032

[003] [D] E. B. Davies, One-Parameter Semigroups, Academic Press, 1980. | Zbl 0457.47030

[004] [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.

[005] [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.

[006] [D-S;I] N. Dunford and J. T. Schwartz, Linear Operators, Part I, General Theory, Interscience, New York, 1958.

[007] [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. | Zbl 0128.34803

[008] [D-U] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, RI, 1977.

[009] [E-K] S. N. Ethier and T. G. Kurtz, Markov Processes, Characterization and Convergence, Wiley, 1986.

[010] [F-Y] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker, 1999.

[011] [G] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, 1985. | Zbl 0592.47034

[012] [H] P. R. Halmos, Measure Theory, Springer, 1974

[013] [H-N] B. Hennig and F. Neubrander, On representations, inversions, and approximations of Laplace transforms in Banach spaces, Appl. Anal. 49 (1993), 151-170. | Zbl 0791.44002

[014] [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.

[015] [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.

[016] [H] E. Hille, Functional Analysis and Semi-groups, Colloq. Publ., Amer. Math. Soc., 1948. | Zbl 0033.06501

[017] [H-P] E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, Colloq. Publ., Amer. Math. Soc., 1957.

[018] [Hi] F. Hirsch, Familles résolvantes, générateurs, cogénérateurs, potentiels, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 1, 89-210. | Zbl 0219.31015

[019] [J] B. E. Johnson, Centralisers on certain topological algebras, J. London Math. Soc. 39 (1964), 603-614. | Zbl 0124.06902

[020] [J-K-B] N. L. Johnson, S. Kotz and N. Balakrishnan, Continuous Univariate Distributions, Vol. I, 2nd ed., Wiley, 1994.

[021] [K;I] T. Kato, Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, Proc. Japan Acad. 35 (1959), 467-468. | Zbl 0095.10502

[022] [K;II] T. Kato, Perturbation Theory for Linear Operators, Springer, 1966.

[023] [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.

[024] [Kr] S. G. Krein, Linear Differential Equations in Banach Space, Transl. Math. Monographs 29, Amer. Math. Soc., Providence, RI, 1971.

[025] [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. | Zbl 0896.47034

[026] [deL-J] R. deLaubenfels and M. Jazar, Functional calculi, regularized semigroups and integrated semigroups, Studia Math. 132 (1999), 151-172. | Zbl 0923.47011

[027] [L] J. L. Lions, Les semigroupes distributions, Portugal. Math. 19 (1960), 141-164. | Zbl 0103.09001

[028] [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.

[029] [P] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, 2nd printing, Springer, 1983. | Zbl 0516.47023

[030] [Ph] R. S. Phillips, An inversion formula for Laplace transforms and semigroups of linear operators, Ann. of Math. 59 (1954), 325-356. | Zbl 0059.10704

[031] [R-Y] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, 1991. | Zbl 0731.60002

[032] [R;I] W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw- Hill, 1976. | Zbl 0346.26002

[033] [R;II] W. Rudin, Real and Complex Analysis, 2nd ed., McGraw-Hill, 1974.

[034] [S-K] I. E. Segal and R. A. Kunze, Integrals and Operators, 2nd ed., Springer, 1987.

[035] [T] H. F. Trotter, Approximation of semi-groups of operators, Pacific J. Math. 8 (1958), 887-919. | Zbl 0099.10302

[036] [W;I] D. V. Widder, The Laplace Transform, Princeton Univ. Press, 1946. | Zbl 0060.24801

[037] [W;II] D. V. Widder, An Introduction to Transform Theory, Academic Press, 1971. | Zbl 0219.44001

[038] [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. | Zbl 0037.35302

[039] [Y;II] K. Yosida, Functional Analysis, 6th ed., Springer, 1980; reprint of the 1980 edition, Springer, 1995.