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