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Laplace transforms and integrated, regularized semigroups in locally convex spaces. (English) Zbl 0999.47030
Summary: In this paper, we consider vector-valued Laplace transforms, \(r\)-times \((r\in[0,\infty))\) integrated semigroups and regularized semigroups in the context of sequentially complete locally convex spaces. Our theorems develop the corresponding results in [W. Arendt, Isr. J. Math. 59, No. 3, 327-352 (1987; Zbl 0637.44001); M. Hieber, Forum Math. 3, No. 6, 595-612 (1991; Zbl 0766.47013)] including the well-known integrated version of the classical Widder representation theorem of Laplace transforms for functions taking values in Banach spaces. Moreover, we study a class of differential operators on certain function spaces. Optimal conditions, making them the generators of integrated or regularized semigroups, are obtained. Finally, we show some applications to abstract Cauchy problems.

MSC:
47D62 Integrated semigroups
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