Distribution semigroups and abstract Cauchy problems

Kunstmann, Peer

doi:10.1090/S0002-9947-99-02035-8

Distribution semigroups and abstract Cauchy problems
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by Peer Christian Kunstmann PDF

Trans. Amer. Math. Soc. 351 (1999), 837-856 Request permission

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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