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Remarks on the Operator-Norm Convergence of the Trotter Product Formula

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

Abstract

We revise the operator-norm convergence of the Trotter product formula for a pair \(\{A,B\}\) of generators of semigroups on a Banach space. Operator-norm convergence holds true if the dominating operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded operator. Inspired by studies of evolution semigroups it is shown in the present paper that the operator-norm convergence generally fails even for bounded operators B if A is not a holomorphic generator. Moreover, it is shown that operator norm convergence of the Trotter product formula can be arbitrary slow.

Keywords

Semigroups Bounded perturbations Trotter product formula Darboux–Riemann sums Operator-norm convergence 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Hagen Neidhardt
    • 1
  • Artur Stephan
    • 2
  • Valentin A. Zagrebnov
    • 3
  1. 1.WIAS BerlinBerlinGermany
  2. 2.Institut für MathematikHumboldt Universität zu BerlinBerlinGermany
  3. 3.Institut de Mathématiques de Marseille (I2M-UMR7373)Université d’Aix-MarseilleMarseilleFrance

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