Algebras of polynomials generated by linear operators

Abstract

Let $E$ be a Banach space and $A$ be a commutative Banach algebra with identity. Let $\mathbb{P}(E, A)$ be the space of $A$-valued polynomials on $E$ generated by bounded linear operators (an $n$-homogenous polynomial in $\mathbb{P}(E,A)$ 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 $\mathbb{P}(K, A)$ be the closure in $\mathscr{C}(K,A)$ of the restrictions $P|_K$ of polynomials $P$ in $\mathbb{P}(E,A)$. It is proved that $\mathbb{P}(K, A)$ is an $A$-valued uniform algebra and that, under certain conditions, it is isometrically isomorphic to the injective tensor product $\mathcal{P}_N(K){\widehat\otimes}_\epsilon A$, where $\mathcal{P}_N(K)$ is the uniform algebra on $K$ generated by nuclear scalar-valued polynomials. The character space of $\mathbb{P}(K, A)$ 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 $A$.