Convergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems

TL;DR

This paper analyzes the convergence rates of Trotter product approximations for solution operators of non-autonomous Cauchy problems by reformulating them as autonomous problems in a function space and applying operator-theoretical methods.

Contribution

It introduces a novel approach using evolution semigroups in L^p spaces to estimate convergence rates of Trotter product formulas for non-autonomous problems.

Findings

Established operator-norm convergence rate estimates for Trotter approximations.

Extended results to non-Hilbert space Banach spaces.

Improved existing convergence estimates for solution operators.

Abstract

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.