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Frechet differential of a power series in Banach algebras

Silvestri B.

Opuscula Mathematica

2010

Vol. 30, no. 2

155-177

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 → [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.