Some Remarks on Operator-Norm Convergence of the Trotter Product Formula on Banach Spaces

We extend the Trotter–Kato–Chernoff theory of strong approximation of C0 semigroups on Banach spaces to operator-norm approximation of analytic semigroups with error estimate. As application we obtain a criterion for the operator-norm convergence of the Trotter product formula on Banach spaces with error estimate n−1 log n, provided one of the generators has a bounded H∞ functional calculus. For both results, we present versions for approximation in operator-ideal-norms such as the trace norm or the Hilbert–Schmidt norm. Finally, we give some remarks on the operator-norm convergence of the Trotter product formula for semigroups acting on a scale of Lp-spaces.