POWERS OF GENERATORS AND TAYLOR EXPANSIONS OF INTEGRATED SEMIGROUPS OF OPERATORS

Abstract

Let $A$ be the generator of an $n$-times integrated semigroup $T(\cdot)$ and let $r \in \mathbb{N}$. We first prove the equivalence of Riemann, Peano, and Taylor operators, which are three different expressions of the $r$-th power of $A_1$, the part of $A$ in the closure of the domain $D(A)$ of $A$. Then we discuss optimal and non-optimal rates of approximation of $T(\cdot)x$ for $x \in D(A^{r−1}_{1})$, via the $(n+r)$-th Taylor expansion of $T(\cdot)$ in terms of $A^k_1$, $k = 0, \ldots, r−1$.