Fréchet differential of a power series in Banach algebras

Abstract

We present two new forms in which the Fréchet differential of a power series in a unitary Banach algebra can be expressed in terms of absolutely convergent series involving the commutant $C(T) : A \mapsto [A,T]$. Then we apply the results to study series of vector-valued functions on domains in Banach spaces and to the analytic functional calculus in a complex Banach space.