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a -times integrated semigroups and fractional derivation

a -times integrated semigroups and fractional derivation Abstract. By applying fractional integration and derivation to the vector-valued Laplace transform, a functional calculus for -times integrated semigroups is obtained. This functional calculus is related to smooth distribution semigroups. As application, fractional powers of its in®nitesimal generator are de®ned. 2000 Mathematics Subject Classi®cation: 47D62, 26A33. 0 Introduction One of the concepts used in the last ®fteen years to study the Abstract Cauchy Problem V b dutY x ` AutY x for t 0, dt b X u0Y x xY AXCXPX when A does not generate a C0 -semigroup of bounded operators in a Banach space X (x e X ), is the n-times integrated semigroup with n e N [32], [19]. These semigroups are related to other notions: C-semigroups [9], spectral distributions [4], holomorphic semigroups [2]. They have been used with C-semigroups [39], cosine functions [26], [15] or sine functions [23]. In 1991 Hieber [18] introduced -times integrated semigroups with e R . In an informal way, the ``formal solution'' of the Cauchy Problem is smoothed integrating it times in the fractional sense of Riemann-Liouville. Some works about these semigroups were published later on, see for instance [29], [8] (about spectral mapping theorem), [37] (in locally convex http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

a -times integrated semigroups and fractional derivation

Miana, Pedro J.
Forum Mathematicum , Volume 14 (1) – Jan 29, 2002
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Publisher
de Gruyter
Copyright
Copyright © 2002 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2002.004
Publisher site
See Article on Publisher Site

Abstract

Abstract. By applying fractional integration and derivation to the vector-valued Laplace transform, a functional calculus for -times integrated semigroups is obtained. This functional calculus is related to smooth distribution semigroups. As application, fractional powers of its in®nitesimal generator are de®ned. 2000 Mathematics Subject Classi®cation: 47D62, 26A33. 0 Introduction One of the concepts used in the last ®fteen years to study the Abstract Cauchy Problem V b dutY x ` AutY x for t 0, dt b X u0Y x xY AXCXPX when A does not generate a C0 -semigroup of bounded operators in a Banach space X (x e X ), is the n-times integrated semigroup with n e N [32], [19]. These semigroups are related to other notions: C-semigroups [9], spectral distributions [4], holomorphic semigroups [2]. They have been used with C-semigroups [39], cosine functions [26], [15] or sine functions [23]. In 1991 Hieber [18] introduced -times integrated semigroups with e R . In an informal way, the ``formal solution'' of the Cauchy Problem is smoothed integrating it times in the fractional sense of Riemann-Liouville. Some works about these semigroups were published later on, see for instance [29], [8] (about spectral mapping theorem), [37] (in locally convex

Journal

Forum Mathematicum – de Gruyter

Published: Jan 29, 2002

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