Representation formulas for C-semigroups (Q1206792)

A Korovkin-type approximation theorem for unbounded functions provides an unified way of deriving representation formulas for \(C_ 0\)-semigroups and cosine operator functions [see the first author, J. Approximation Theory 28, 238-259 (1980; Zbl 0452.41020) and the first author, \textit{C. S. Lee} and \textit{W. L. Chiou}, Aequationes Math. 31, 64-75 (1986; Zbl 0647.47048)]. In this note, it is applied again to derive representation formulas for \(C\)-semigroups. A family \(\{S(t); t\geq 0\}\) of bounded linear operators on a Banach space \(X\) is called a \(C\)-semigroup if it is strongly continuous on \([0, \infty)\) and satisfies \(S(s) S(t)= S(s+ t) C\), \(s\), \(t\geq 0\), and \(S(0)= C\), where \(C\) is a bounded injection. Among the seven formulas are the following two extensions of Kendall's and Hille's formulas: \[ \begin{aligned} S(t) x & = \lim_{n\to \infty} [(1- t) I+ tC^{-1} S(\textstyle{{1\over n}})]^ n Cx, \quad 0\leq t\leq 1;\\ S(t) x & = \lim_{n\to\infty} \exp[nt(C^{-1} S(\textstyle{{1\over n}})- I)] Cx, \quad t\geq 0.\end{aligned} \] {}.