A faithful Hille-Yosida theorem for finite-dimensional evolutions

As a natural generalization of the classical Hille-Yosida theorem to evolution operators, necessary and sufficient conditions are found for an evolution $U$ acting in ${R^N}$ so that for each $s \geqslant t$, $U(s,t)$ can be uniquely represented as a product integral $\prod _t^s{[I + V]^{ - 1}}$ for some additive, accretive generator $V$. Under these conditions, we further show that $U(\xi ,\zeta )$ is differentiable a.e.