Weighted inversion of vector valued Dirichlet series

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