Algebras of Polynomials Generated by Linear Operators

Abstract

Let $E$ be a Banach space and $A$ be a commutative Banach algebra with identity. Let $P(E,A)$ be the space of $A$-valued polynomials on $E$ generated by bounded linear operators (an $n$-homogenous polynomial in $P(E,A)$ is of the form $P={\sum }_{i=1}^{\infty }{T}_i^n$, where $T_i:E\to A$ ($1\le i<\infty$) are bounded linear operators and ${\sum }_{i=1}^{\infty }\Vert T_i{\Vert }^n<\infty$). For a compact set $K$ in $E$, we let $P(K,A)$ be the closure in $C(K,A)$ of the restrictions $P{|}_K$ of polynomials $P$ in $P(E,A)$. It is proved that $P(K,A)$ is an $A$-valued uniform algebra and that, under certain conditions, it is isometrically isomorphic to the injective tensor product ${\mathcal{P}}_N(K){\hat{\otimes }}_{\epsilon }A$, where ${\mathcal{P}}_N(K)$ is the uniform algebra on $K$ generated by nuclear scalar-valued polynomials. The character space of $P(K,A)$ is then identified with ${\hat{K}}_N\times \mathfrak{M}(A)$, where ${\hat{K}}_N$ is the nuclear polynomially convex hull of $K$ in $E$, and $\mathfrak{M}(A)$ is the character space of $A$.