  Due to the seminal works of Hochbruck and Ostermann exponential splittings
are well established numerical methods utilizing operator semigroup theory for
the treatment of semilinear evolution equations whose principal linear part
involves a sectorial operator with angle greater than ${\pi}/2$ (meaning
essentially the holomorphy of the underlying semigroup). The present paper
contributes to this subject by relaxing on the sectoriality condition, but in
turn requiring that the semigroup operators act consistently on an
interpolation couple (or on a scale of Banach spaces). Our conditions (on the
semigroup and on the semilinearity) are inspired by the approach of T. Kato to
the local solvability of the Navier-Stokes equation, where the $L^p-L^r$
smoothing of the Stokes semigroup was fundamental. The present abstract
operator theoretic result is applicable for this latter problem (as was already
the result of Ostermann and Hochbruck), or more generally in the setting of
holomorphic semigroups, but also allows the consideration of examples, such as
non-analytic Ornstein-Uhlenbeck semigroups or the Navier-Stokes flow around
rotating bodies.
