Convergence Rate Estimates for Trotter Product Approximations of Solution Operators for Non-autonomous Cauchy Problems

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.