Opuscula Mathematica
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.
Cite this article as:
Benedetto Silvestri, Fréchet differential of a power series in Banach algebras, Opuscula Math. 30, no. 2 (2010), 155-177, http://dx.doi.org/10.7494/OpMath.2010.30.2.155
 
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ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
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