Differential equations in Banach algebras

In a complex Banach algebra that is not assumed to be commutative, $n$th-order linear differential equations with constant coefficients are considered. The corresponding algebraic characteristic equation of the $n$th degree is assumed to have $n$ distinct roots for which the Vandermonde matrix is invertible. Analogues of Sylvester's and Vieta's theorems are proved, and a contour integral of Cauchy type is studied.