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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
Full Text:
##### References:
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