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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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