Differentiability in Banach Algebras

Abstract

In this note a definition from the finite dimensional setting [see 3] is generalized to the infinite dimensional setting and a class of functions is exhibited which have this property. Although these results appear in [2], the use of the right and left multiplication operator and the theory of several commuting operators is avoided and only elementary techniques are employed. Let W be a Banach algebra with an identity 1, and let F be defined on a domain 9, an open connected set, of W, with range in W. Fundamental in the theory of differentiation on W is the Frechet derivative. The function F is said to be Frechet differentiable at a point x E 9 if there exists a continuous linear function of t, say dF(x, ), such that