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

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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.

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References

  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)

    MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  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. Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)

    Book  Google Scholar 

  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. Trotter, H.F.: On the product of semi-groups of operators. Proc. Am. Math. Soc. 10, 545–551 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Hagen Neidhardt.

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Neidhardt, H., Stephan, A. & Zagrebnov, V.A. Remarks on the Operator-Norm Convergence of the Trotter Product Formula. Integr. Equ. Oper. Theory 90, 15 (2018). https://doi.org/10.1007/s00020-018-2424-z

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  • DOI: https://doi.org/10.1007/s00020-018-2424-z

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