  In the present paper we advocate the Howland-Evans approach to solution of
the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach
space X. The main idea is to reformulate this problem as an autonomous Cauchy
problem (ACP) in a new Banach space L^p(I,X), consisting of X-valued functions
on the time-interval I. The fundamental observation is a one-to-one
correspondence between solution operators (propagators) for a non-ACP and the
corresponding evolution semigroups for ACP in L^p(I,X). We show that the latter
also allows to apply a full power of the operator-theoretical methods to
scrutinise the non-ACP including the proof of the Trotter product approximation
formulae with operator-norm estimate of the rate of convergence. The paper
extends and improves some recent results in this direction in particular for
Hilbert spaces.
