  The time-ordered exponential of a time-dependent matrix $\mathsf{A}(t)$ is
defined as the function of $\mathsf{A}(t)$ that solves the first-order system
of coupled linear differential equations with non-constant coefficients encoded
in $\mathsf{A}(t)$. The authors recently proposed the first Lanczos-like
algorithm capable of evaluating this function. This algorithm relies on
inverses of time-dependent functions with respect to a non-commutative
convolution-like product, denoted $\ast$. Yet, the existence of such inverses,
crucial to avoid algorithmic breakdowns, still needed to be proved. Here we
constructively prove that $\ast$-inverses exist for all non-identically null,
smooth, separable functions of two variables. As a corollary, we partially
solve the Green's function inverse problem which, given a distribution $G$,
asks for the differential operator whose fundamental solution is $G$. Our
results are abundantly illustrated by examples.
