Cⁿ-Operational Calculus, Non-commutative Taylor Formula and Perturbation of Semigroups

Let A be a unital complex Banach algebra and a ∈ A. For an interval K, we consider the Banach algebra (C(K), * _n), with the Jⁿ-multiplication * _n. THEOREM 1. The element a is of class Cⁿ if and only if there exists a continuous representation U of the Banach algebra (C(K), * _n) on A such that Ul = aⁿ/n!; U is uniquely determined and precisely the “weak representation of a on C(K).” A new perspective to the universal model provided by that representation is obtained as a consequence of a “non-commutative” Taylor formula for the analytical operational calculus. This formula is also used to obtain a perturbation result for semigroups of operators (Theorem 4).