Weighted inversion of vector valued Dirichlet series

Let Λ⊂[0,∞) be an additive semigroup with 0∈Λ, $ω$ be an admissible weight on $Λ$, A be a unital Banach algebra, and let f(s)=∑λ∈Λ f_λe-λs for s=+it:j≥0\ be a generalized Dirichlet series satisfying \|f\|_ω=∑λ∈Λ\|f_λ\|ω(λ)<∞, where f_λ for all λ∈Λ. We take A to be a commutative complex Banach algebra (with Λ=) and Md(X) - the Banach algebra of d × d matrices having entries from X, where 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 A.