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THE COMPLEX INVERSION FORMULA REVISITED

Part of: Groups and semigroups of linear operators, their generalizations and applications General theory of linear operators Differential equations in abstract spaces Abstract harmonic analysis Integral transforms, operational calculus

Published online by Cambridge University Press:  01 February 2008

MARKUS HAASE
MARKUS HAASE*
Affiliation:
Delft Institute of Applied Analysis, TU Delft, PO Box 5031, 2600 GA Delft, The Netherlands (email: m.h.a.haase@tudelft.nl)
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Abstract

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We give a simplified proof of the complex inversion formula for semigroups and, more generally, solution families for scalar-type Volterra equations, including the stronger versions on unconditional martingale differences (UMD) spaces. Our approach is based on (elementary) Fourier analysis.

Keywords

C0 semigroupintegrated semigroupUMD spacecomplex inversionLaplace transformVolterra equation

MSC classification

Secondary: 34G10: Linear equations 43A50: Convergence of Fourier series and of inverse transforms 44A10: Laplace transform 47A60: Functional calculus 47D03: Groups and semigroups of linear operators 47D06: One-parameter semigroups and linear evolution equations 47D09: Operator sine and cosine functions and higher-order Cauchy problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 84 , Issue 1 , February 2008 , pp. 73 - 83
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Arendt, W., Batty, C. J. K., Hieber, M. and Neubrander, F., Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, 96 (Basel, Birkhäuser, 2001), pp. xi, 523.Google Scholar
[2]Burkholder, D. L., ‘Martingales and singluar integrals in Banach spaces’, in: Handbook of the geometry of Banach spaces, Vol. I (North-Holland, Amsterdam, 2001), pp. 233–269.CrossRefGoogle Scholar
[3]Cioranescu, I. and Lizama, C., ‘On the inversion of the Laplace transform for resolvent families in UMD spaces’, Arch. Math. (Basel) (2) 81 (2003), 182–192.CrossRefGoogle Scholar
[4]Driouich, A. and El-Mennaoui, O., ‘On the inverse Laplace transform for C 0-semigroups in UMD-spaces’, Arch. Math. (Basel) (1) 72 (1999), 56–63.Google Scholar
[5]Engel, K.-J. and Nagel, R., One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, 194 (Springer, Berlin, 2000), pp. xxi, 586.Google Scholar
[6]Haase, M., The functional calculus for sectorial operators, Operator Theory: Advances and Applications, 169 (Birkhäuser-Verlag, Basel, 2006).Google Scholar
[7]Haase, M., ‘Semigroup theory via functional calculus’, Preprint, 2006.Google Scholar
[8]Prüss, J., Evolutionary integral equations and applications, Monographs in Mathematics, 87 (Birkhäuser Verlag, Basel, 1993), pp. xxvi, 366.Google Scholar