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 $P=\sum_{i=1}^\infty T^n_i$, where $T_i:E\to A$ ($1\leq i <\infty$) are bounded linear operators and $\sum_{i=1}^\infty \|T_i\|^n < \infty$). 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 $A$.