Abstract
In the present paper we advocate the Howland–Evans approach to the 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(\mathcal I,X) , p \in [1,\infty) , consisting of X -valued functions on the time interval \mathcal 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(\mathcal I,X) . We show that the latter also allows us to apply the full power of 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.
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