Title: Remarks on the operator-norm convergence of the Trotter product formula,
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.