zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Vector-valued Laplace transforms and Cauchy problems. (English) Zbl 0637.44001
The author symmetrically treats linear differential equations in Banach spaces with the help of Laplace transforms. The central tool used is an “integrated version” of Widder’s theorem (characterising Laplace transforms of bounded functions). It holds in any Banach space, whereas the vector-valued version of Widder’s theorem itself holds if and only if the Banach space has the Radon - Nikodým property. The Hille-Yosida theorem and other generation theorems are immediate consequences. The technique presented in the paper can be applied to operators whose domains are not dense.
Reviewer: S.D.Bajpai

44A10Laplace transform
34G10Linear ODE in abstract spaces
47D03(Semi)groups of linear operators
Full Text: DOI
[1] W. Arendt,Resolvent positive operators, Porc. London Math. Soc., to appear. · Zbl 0617.47029
[2] E. B. Davies,One-parameter Semigroups, Academic Press, London, 1980. · Zbl 0457.47030
[3] E. B. Davies and M. M. H. Pang,The Cauchy problem and a generalization of the Hille-Yosida theorem, preprint, 1986.
[4] J. Chazarain,Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes, J. Functional Anal.7 (1971), 387--446. · Zbl 0211.12902 · doi:10.1016/0022-1236(71)90027-9
[5] G. Da Prato and E. Sinestrari,Differential operators with nondense domain and evolution equations, preprint, 1985.
[6] J. Diestel and J. J. Uhl,Vector Measures, Amer. Math. Soc., Providence, Rhode Island, 1977.
[7] H. O. Fattorini,The Cauchy Problem, Addison-Wesley, London, 1983. · Zbl 0493.34005
[8] H. O. Fattorini,Second Order Differential Equations in Banach Spaces, North-Holland, Amsterdam, 1985. · Zbl 0564.34063
[9] W. Feller,On the generation of unbounded semigroups of bounded linear operators, Ann. Math. (2)58 (1953), 166--174. · Zbl 0050.34201 · doi:10.2307/1969826
[10] J. A. Goldstein,Semigroups of Operators and Applications, Oxford University Press, New York, 1985. · Zbl 0592.47034
[11] E. Hille and R. S. Phillips,Functional Analysis and Semigroups, Amer. Math. Soc. Colloquium Publications, Vol. 31, Providence, R.I., 1957. · Zbl 0078.10004
[12] H. Kellermann,Integrated semigroups, Dissertation, Tübingen, 1986. · Zbl 0604.47025
[13] J. Kisyński,Semi-groups of operators and some of their applications to partial differential equations, inControl Theory and Topics in Functional Analysis, Vol. II, IAEA, Vienna, 1976.
[14] S. G. Krein and M. I. Khazan,Differential equations in a Banach space, J. Soviet Math.30 (1985), 2154--2239. · Zbl 0611.34059 · doi:10.1007/BF02105398
[15] J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces I, Springer-Verlag, Berlin, 1977. · Zbl 0362.46013
[16] J. L. Lions,Semi-groupes distributions, Portugalae Math.19 (1960), 141--164. · Zbl 0103.09001
[17] I. Miyadera,Generation of a strongly continuous semi-groups of operators, Tôhoku Math. J.2 (1952), 109--114. · Zbl 0048.09304 · doi:10.2748/tmj/1178245412
[18] I. Miyadera,On the representation theorem by the Laplace transformation of vector-valued functions, Tôhoku Math. J.8 (1956), 170--180. · Zbl 0073.08602 · doi:10.2748/tmj/1178244980
[19] I. Miyadera, S. Oharu and N. Okazawa,Generation theorems of linear operators, PRIMS, Kyoto Univ.8 (1973), 509--555. · Zbl 0262.47030 · doi:10.2977/prims/1195192960
[20] R. Nagel (ed.),One-parameter Semigroups of Positive Operators, Lecture Notes in Math.1184, Springer, Berlin, 1986. · Zbl 0585.47030
[21] F. Neubrander,Integrated semigroups and their applications to the abstract Cauchy problem, preprint, 1986. · Zbl 0589.34004
[22] A. Pazy,Semi-groups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983. · Zbl 0516.47023
[23] R. S. Phillips,Perturbation theory for semi-groups of linear operators, Trans. Amer. Math. Soc.74 (1953), 199--221. · Zbl 0053.08704 · doi:10.1090/S0002-9947-1953-0054167-3
[24] H. H. Schaefer,Banach Lattices and Positive Operators, Springer, Berlin, 1974. · Zbl 0296.47023
[25] M. Sova,Problèmes de Cauchy pour équations hyperboliques operationelles à coéfficients non-bornés, Ann. Scuola Norm. Sup. Pisa22 (1968), 67--100.
[26] Y. Sova,Problèmes de Cauchy paraboliques abstraits de classes supérieurs et les semigroupes distributions, Ricerche Mat.18 (1969), 215--238. · Zbl 0196.16301
[27] D. V. Widder,The inversion of the Laplace integral and the related moment problem, Trans. Amer. Math. Soc.36 (1934), 107--200. · Zbl 0008.30603 · doi:10.1090/S0002-9947-1934-1501737-7
[28] D. V. Widder,An Introduction to Transform Theory, Academic Press, New York, 1971. · Zbl 0219.44001
[29] K. Yosida,Functional Analysis, Springer, Berlin, 1978. · Zbl 0365.46001