Opuscula Math. 30, no. 2 (), 155-177
http://dx.doi.org/10.7494/OpMath.2010.30.2.155
Opuscula Mathematica

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.