A Class of Polynomials from Banach Spaces into Banach Algebras

Abstract

Let $E$ be a complex Banach space and $F$ be a complex Banach algebra. We will be interested in the subspace $P_f({}^{n}E,F)$ of $P({}^{n}E,F)$ generated by the collection of functions ${\phi }^n(n\in N,\phi \in L(E,F))$ where ${\phi }^n(x)=(\phi (x){)}^n$ for each $x\in E$.