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

### Hagen Neidhardt

Karl-Weierstrass-Institut für Mathematik, Berlin, Germany### Artur Stephan

Humboldt-Universität zu Berlin, Germany### Valentin A. Zagrebnov

Université d'Aix-Marseille, Marseille, France

## 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 $L_{p}(I,X)$, $p∈[1,∞)$, 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 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.

## Cite this article

Hagen Neidhardt, Artur Stephan, Valentin A. Zagrebnov, Convergence Rate Estimates for Trotter Product Approximations of Solution Operators for Non-autonomous Cauchy Problems. Publ. Res. Inst. Math. Sci. 56 (2020), no. 1, pp. 83–135

DOI 10.4171/PRIMS/56-1-5