WIAS Preprint No. 2356, (2016)

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



Authors

  • Neidhardt, Hagen
  • Stephan, Artur
  • Zagrebnov, Valentin A.

2010 Mathematics Subject Classification

  • 47D06 34G10 34K30 47A55

Keywords

  • Trotter product formula, convergence rate, approximation, evolution equations, solution operator, extension theory, perturbation theory, operator splitting

DOI

10.20347/WIAS.PREPRINT.2356

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 Lp(J,X), consisting of X-valued functions on the time-interval J. 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 Lp(J,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.

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