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

  • Hagen Neidhardt
  • Artur Stephan
  • Valentin A. Zagrebnov


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.


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


  1. 1.
    Cachia, V., Zagrebnov, V.A.: Operator-norm convergence of the Trotter product formula for holomorphic semigroups. J. Oper. Theory 46(1), 199–213 (2001)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chernoff, P.R.: Product Formulas, Nonlinear Semigroups, and Addition of Unbounded Operators. Memoirs of the American Mathematical Society, No. 140. American Mathematical Society, Providence (1974)Google Scholar
  3. 3.
    Ichinose, T., Tamura, H., Tamura, H., Zagrebnov, V.A.: Note on the paper: the norm convergence of thea Trotter-Kato product formula with error bound by T. Ichinose and H. Tamura. Commun. Math. Phys. 221(3), 499–510 (2001)CrossRefzbMATHGoogle Scholar
  4. 4.
    Kato, T.: Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups. Topics in functional analysis, Essays dedic. M. G. Krein. Adv. Math. Suppl. Stud. 3, 185–195 (1978)Google Scholar
  5. 5.
    Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)CrossRefGoogle Scholar
  6. 6.
    Neidhardt, H., Stephan, A., Zagrebnov. V.A.: Convergence rate estimates for the Trotter product approximations of solution operators for non-autonomous Cauchy problems. arXiv:1612.06147v1 [math.FA] (2016 December)
  7. 7.
    Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Walsh, J.L., Sewell, W.E.: Note on degree of approximation to an integral by Riemann sums. Am. Math. Mon. 44(3), 155–160 (1937)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations