Derivative of Map of Banach algebra

TL;DR

This paper develops the theory of Gateaux derivatives for maps on Banach algebras over a commutative ring, establishing conditions for differentiability and Taylor series expansion.

Contribution

It introduces a formal definition of higher-order Gateaux derivatives in Banach algebras and derives the Taylor series expansion for such maps.

Findings

Defined Gateaux derivatives of arbitrary order in Banach algebras.

Established the Taylor series expansion for differentiable maps.

Provided a framework for analyzing differentiability in Banach algebra contexts.

Abstract

Let $A$ be Banach algebra over commutative ring $D$. The map $f:A\rightarrow A\ $ is called differentiable in the Gateaux sense, if $$f(x+a)-f(x)=\partial f(x)\circ a+o(a)$$ where the Gateaux derivative $\partial f(x)$ of map $f$ is linear map of increment $a$ and $o$ is such continuous map that $$ \lim_{a\rightarrow 0}\frac{|o(a)|}{|a|}=0 $$ Assuming that we defined the Gateaux derivative $\partial^{n-1} f(x)$ of order $n-1$, we define $$ \partial^n f(x)\circ(a_1\otimes...\otimes a_n) =\partial(\partial^{n-1} f(x)\circ(a_1\otimes...\otimes a_{n-1}))\circ a_n $$ the Gateaux derivative of order $n$ of map $f$. Since the map $f(x)$ has all derivatives, then the map $f(x)$ has Taylor series expansion $$ f(x)=\sum_{n=0}^{\infty}(n!)^{-1}\partial^n f(x_0)\circ(x-x_0)^n $$