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

Research paper by Hagen Neidhardt, Artur Stephan, Valentin A. Zagrebnov

Indexed on: 28 Mar '17Published on: 28 Mar '17Published in: arXiv - Mathematical Physics



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.