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THE COMPLEX INVERSION FORMULA REVISITED

Published online by Cambridge University Press:  01 February 2008

MARKUS HAASE*
Affiliation:
Delft Institute of Applied Analysis, TU Delft, PO Box 5031, 2600 GA Delft, The Netherlands (email: [email protected])
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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.

Type
Research Article
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. 233269.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), 182192.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), 5663.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