Weighted inversion of vector valued Dirichlet series

Abstract

Let $\Lambda\subset[0,\infty)$ be an additive semigroup with $0\in\Lambda$, $\omega$ be an admissible weight on $\Lambda$, $\mathcal A$ be a unital Banach algebra, and let $f(s)=\sum_{\lambda\in\Lambda} f_\lambda e^{-\lambda s}$ for $s\in\mathcal{H}=\{j+it\in\mathbb{C}:j\geq0\}$ be a generalized Dirichlet series satisfying $\|f\|_\omega=\sum_{\lambda\in\Lambda}\|f_\lambda\|\omega(\lambda)<\infty,$ where $f_\lambda\in\mathcal{A}$ for all $\lambda\in\Lambda$. We take $\mathcal{A}$ to be a commutative complex Banach algebra (with $\Lambda=\log\mathbb{N}$) 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 $\mathcal{A}$.