Opuscula Math. 30, no. 2 (2010), 155-177
http://dx.doi.org/10.7494/OpMath.2010.30.2.155

Opuscula Mathematica

# Fréchet differential of a power series in Banach algebras

Benedetto Silvestri

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.

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• Benedetto Silvestri
• Dipartimento di Matematica, Pura ed Applicata via Trieste, 63, 35121 Padova, Italy
• Revised: 2010-01-08.
• Accepted: 2010-01-10.