Abstract
Linear differential equations in Banach spaces are systematically treated with the help of Laplace transforms. The central tool is an “integrated version” of Widder’s theorem (characterizing 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 method presented here can be applied to operators whose domains are not dense.
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Arendt, W. Vector-valued laplace transforms and cauchy problems. Israel J. Math. 59, 327–352 (1987). https://doi.org/10.1007/BF02774144
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DOI: https://doi.org/10.1007/BF02774144