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.
Keywords: Fréchet differentiation in Banach algebras, functional calculus.
Mathematics Subject Classification: 58C20, 46H, 47A60.