Representation formulas for integrated semigroups and sine families (Q1207294)

The investigations are concerned with integrated semigroups \((T_ t)\) [cf. \textit{W. Arendt}, Isr. J. Math. 59, 327-352 (1987; Zbl 0637.44001)] and operator families \((S_ t)\) fulfilling the sine-functional equation [\textit{W. Arendt} and \textit{H. Kellermann}, Pitman Res. Notes Math., Ser. 190, 21-51 (1989; Zbl 0675.45017)]. The aim of the paper under review is to prove representations of integrated semigroups as limits of resolvents of the infinitesimal generators, e.g. (Theorem 2.1 (2)) \[ T_ t=\lim_ n \int_ 0^ t \left( I-{s\over n} A\right)^{-n} ds. \] In \S2 and 3 sine-families are considered. There is an auxiliary Banach space \(X\times E\) such that a sine family \((S_ t)\) on \(X\) generates an integrated semigroup \((T_ t)\) on \(X\times E\). Therefore (\S5) the approximation formulas of \S1 can be applied in order to obtain approximation formulas for sine-families in terms of the resolvents of their generator.