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Vector-valued Laplace transforms and Cauchy problems. (English) Zbl 0637.44001
The author symmetrically treats linear differential equations in Banach spaces with the help of Laplace transforms. The central tool used is an “integrated version” of Widder’s theorem (characterising Laplace transforms of bounded functions). It holds in any Banach space, whereas the vector-valued version of Widder’s theorem itself holds if and only if the Banach space has the Radon - Nikodým property. The Hille-Yosida theorem and other generation theorems are immediate consequences. The technique presented in the paper can be applied to operators whose domains are not dense.
Reviewer: S.D.Bajpai

##### MSC:
 44A10 Laplace transform 34G10 Linear ODE in abstract spaces 47D03 (Semi)groups of linear operators
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