Frechet differential of a power series in a Banach algebra

TL;DR

This paper introduces new formulas for the Frechet differential of power series in Banach algebras, expressed via absolutely convergent series involving the commutant, and applies these to the analytic functional calculus.

Contribution

It provides novel expressions for the Frechet differential in Banach algebras and extends these results to the analytic functional calculus in complex Banach spaces.

Findings

New formulas for Frechet differential in Banach algebras

Expressed via absolutely convergent series involving the commutant

Applications to analytic functional calculus in complex Banach spaces

Abstract

We present two new forms in which the Frechet differential of a power series in a 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 the analytic functional calculus in a complex Banach space.