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

## TL;DR

This paper investigates the conditions under which the Trotter product formula converges in operator norm, revealing that convergence can fail or be arbitrarily slow if certain holomorphic properties are not met.

## Contribution

It clarifies the precise conditions for operator-norm convergence of the Trotter product formula, especially highlighting the importance of A being a holomorphic generator.

## Key findings

- Operator-norm convergence holds if A generates a holomorphic contraction semigroup and B is A-infinitesimally small.
- Convergence generally fails for bounded B if A is not a holomorphic generator.
- Operator-norm convergence can be arbitrarily slow.

## 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.