Powers of Generators and Taylor Expansions of Integrated Semigroups of Operators

Let A be the generator of an n-times integrated semigroup T (·) and let r ∈ N. We first prove the equivalence of Riemann, Peano, and Taylor operators, which are three different expressions of the r-th power of A1, 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 (·)x for x ∈ D(Ar−1 1 ), via the (n + r)-th Taylor expansion of T (·) in terms of A1 , k = 0, . . . , r− 1.