On a Banach algebra of entire functions with a weighted Hadamard multiplication

Abstract

New algebraic-analytic properties of a previously studied Banach algebra $\mathcal{A}({\bf{p}})$ of entire functions are established. For a given fixed sequence $(\bf{p}(n))_{n\geq 0}$ of positive real numbers, such that $\lim_{n\rightarrow \infty} {\bf{p}}(n)^{\frac{1}{n}}=\infty$, the Banach algebra $\mathcal{A}({\bf{p}})$ is the set of all entire functions $f$ such that $f(z)=\sum_{n=0}^\infty \hat{f}(n) z^n $ ($z\in \mathbb{C}$), where the sequence $(\hat{f}(n))_{n\geq 0}$ of Taylor coefficients of $f$ satisfies $\hat{f}(n)=O({\bf{p}}(n)^{-1})$ for $n\rightarrow \infty$, with pointwise addition and scalar multiplication, a weighted Hadamard multiplication $\ast$ with weight given by ${\bf{p}}$ (i.e., $(f\ast g)(z)=\sum_{n=0}^\infty {\bf{p}}(n) \hat{f}(n)\hat{g}(n)z^n$ for all $z\in \mathbb{C}$), and the norm $\|f\|=\sup_{n\geq 0} {\bf{p}}(n)|\hat{f}(n)|$. The following results are shown: The Bass and the topological stable ranks of $\mathcal{A}({\bf{p}})$ are both $1$. $\mathcal{A}({\bf{p}})$ is a Hermite ring, but not a projective-free ring. Idempotents and exponentials in $\mathcal{A}({\bf{p}})$ are described, and it is shown that every invertible element of $\mathcal{A}({\bf{p}})$ has a logarithm. A generalised necessary and sufficient `corona-type condition' on the matricial data $(A,b)$ with entries from $\mathcal{A}({\bf{p}})$ is given for the solvability of $Ax = b$ with $x$ also having entries from $\mathcal{A}({\bf{p}})$. The Krull dimension of $\mathcal{A}({\bf{p}})$ is infinite. $\mathcal{A}({\bf{p}})$ is neither Artinian nor Noetherian, but is coherent. The special linear group over $\mathcal{A}({\bf{p}})$ is generated by elementary matrices.