On an integral transform by R. S. Phillips

• Autorzy: Sten Bjon
• Języki publikacji: EN

Abstrakty

EN

The properties of a transformation $$ f \mapsto \tilde f_h $$ by R.S. Phillips, which transforms an exponentially bounded C 0-semigroup of operators T(t) to a Yosida approximation depending on h, are studied. The set of exponentially bounded, continuous functions f: [0, ∞[→ E with values in a sequentially complete L c-embedded space E is closed under the transformation. It is shown that $$ (\tilde f_h )\widetilde{_k } = \tilde f_{h + k} $$ for certain complex h and k, and that $$ f(t) = \lim _{h \to 0^ + } \tilde f_h (t) $$, where the limit is uniform in t on compact subsets of the positive real line. If f is Hölder-continuous at 0, then the limit is uniform on compact subsets of the non-negative real line. Inversion formulas for this transformation as well as for the Laplace transformation are derived. Transforms of certain semigroups of non-linear operators on a subset X of an L c-embedded space are studied through the C 0-semigroups, which they define by duality on a space of functions on X.

Słowa kluczowe

EN

One-parameter semigroups; Yosida approximation; Laplace transform

Kategorie tematyczne

• 47D06: One-parameter semigroups and linear evolution equations
• 44A10: Laplace transform
• 44A15: Special transforms (Legendre, Hilbert, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 1
• Strony: 98-113

Daty

wydano

2010-02-01

online

2010-02-02

Twórcy

autor

Sten Bjon

• Åbo Akademi University

Bibliografia

• [1] Beattie R., Butzmann H.-P., Convergence structures and applications to functional analysis, Kluwer Academic Publishers, Dordrecht, 2002
• [2] Binz E., Keller H.H., Funktionenräume in der Kategorie der Limesräume, Ann. Acad. Sci. Fenn. Ser. A I, 1966, 383, 1–21
• [3] Bjon S., Lindström M., A general approach to infinite-dimensional holomorphy, Monatsh. Math., 1986, 101, 11–26 http://dx.doi.org/10.1007/BF01326843
• [4] Bjon S., Einbettbarkeit in den Bidualraum und Darstellbarkeit als projektiver Limes in Kategorien von Limesvektorräumen, Math. Nachr., 1979, 97, 103–116 http://dx.doi.org/10.1002/mana.19790930108
• [5] Bjon S., On an exponential law for spaces of holomorphic mappings, Math. Nachr., 1987, 131, 201–204 http://dx.doi.org/10.1002/mana.19871310118
• [6] Bjon S., Differentiation under the integral sign and holomorphy, Math. Scand., 1987, 60, 77–95
• [7] Bjon S., One-parameter semigroups of operators on L c-embedded spaces, Semigroup Forum, 1994, 49, 369–380 http://dx.doi.org/10.1007/BF02573497
• [8] Bobrowski A., On the Yosida approximation and the Widder-Arendt representation theorem, Studia Math., 1997, 124, 281–290
• [9] Bobrowski A., Inversion of the Laplace transform and generation of Abel summable semigroups, J. Funct. Anal., 2001, 186, 1–24 http://dx.doi.org/10.1006/jfan.2001.3782
• [10] Frölicher A., Bucher W., Calculus in vector spaces without norm, Lecture Notes in Mathematics, 30, Springer-Verlag, Berlin, 1966
• [11] Hille E., Phillips R.S., Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., Vol. XXXI, Amer. Math. Soc., Providence, 1957
• [12] Ladyzhenskaya O., Attractors for semigroups and evolution equations, Cambridge University Press, Cambridge, 1991
• [13] Müller B., L c- und c-einbettbare Limesräume, Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser., 1978, 5, 509–526
• [14] Phillips R.S., An inversion formula for Laplace transforms and semi-groups of linear operators, Annals of Math., 1954, 59, 325–356 http://dx.doi.org/10.2307/1969697
• [15] Prudnikov A.P., Brychkov Yu.A., Marichev O.I., Integrals and series, Vol. 4: Direct Laplace transforms, Gordon and Breach Sci. Publ., New York, 1992

Identyfikatory

DOI

10.2478/s11533-009-0058-8