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We revise the operator-norm convergence of the Trotter product formula for apair {A,B} of generators of semigroups on a Banach space. Operator-normconvergence holds true if the dominating operator A generates a holomorphiccontraction semigroup and B is a A-infinitesimally small generator of acontraction semigroup, in particular, if B is a bounded operator. Inspired bystudies of evolution semigroups it is shown in the present paper that theoperator-norm convergence generally fails even for bounded operators B if A isnot a holomorphic generator. Moreover, it is shown that operator normconvergence of the Trotter product formula can be arbitrary slow.

Remarks on the operator-norm convergence of the Trotter product formula