Weighted inversion of vector valued Dirichlet series

Abstract

Let $\hat{I}›\subset [0,\infty )$ be an additive semigroup with $0\in \hat{I}›$, $\ddot{I}‰$ be an admissible weight on $\hat{I}›$, $\mathcal{A}$ be a unital Banach algebra, and let $f(s)={\sum }_{\hat{I}»\in \hat{I}›}{f}_{\hat{I}}»e^{-\hat{I}»s}$ for $s\in \mathcal{H}=\{j+it\in \mathbb{C}:j\ge 0\}$ be a generalized Dirichlet series satisfying $\Vert f{\Vert }_{\ddot{I}}‰={\sum }_{\hat{I}»\in \hat{I}›}\Vert f_{\hat{I}}»\Vert \ddot{I}‰(\hat{I}»)<\infty ,$ where $f_{\hat{I}}»\in \mathcal{A}$ for all $\hat{I}»\in \hat{I}›$. We take $\mathcal{A}$ to be a commutative complex Banach algebra (with $\hat{I}›=\log \mathbb{N}$) and $M_d(\mathcal{X})$ - the Banach algebra of $d\times d$ matrices having entries from $\mathcal{X}$, where $\mathcal{X}$ is either the complex plane or the real algebra of bicomplex numbers or quaternions, and show that $f$ is invertible if and only if the closure of the image of $f$ is contained in the set of all invertible elements of $\mathcal{A}$.

11 pages