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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Distribution semigroups and
abstract Cauchy problems

Author: Peer Christian Kunstmann
Journal: Trans. Amer. Math. Soc. 351 (1999), 837-856
MSC (1991): Primary 47D03, 34G10, 47A10, 46F10
MathSciNet review: 1443882
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Abstract: We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator $A$ in a Banach space $E$ the following assertions are equivalent: (a) $A$ generates a distribution semigroup; (b) the convolution operator $\delta'\otimes I-\delta\otimes A$ has a fundamental solution in ${\mathcal D}'(L(E,D))$ where $D$ denotes the domain of $A$ supplied with the graph norm and $I$ denotes the inclusion $D\to E$; (c) $A$ generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup.

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Additional Information

Peer Christian Kunstmann
Affiliation: Mathematisches Seminar der Christian-Albrechts-Universität zu Kiel, Ludewig- Meyn-Straße 4, D-24098 Kiel, Germany
Address at time of publication: Mathematisches Institut I der Universität Karlsruhe, Englerstraße 2, D-76128 Karlsruhe, Germany

Received by editor(s): October 17, 1995
Received by editor(s) in revised form: February 6, 1997
Article copyright: © Copyright 1999 American Mathematical Society