What do we actually know about the operator-norm convergent Trotter-Kato product formula?

Since 1875 due to Sophus Lie it is known that for any pair of (noncommutative) finite square matrices $A$ and $B$ as generators one has the norm estimate $O(1/n)$ for convergence rate of the exponential product formula. In 1959 H.Trotter proved this formula in the strong operator topology on the Banach space for strongly continuous semigroups and unbounded generators $A$ and B. Further, in 1978 T. Kato extended this result (still in the strong operator topology) to the non-exponential product formulae. A breakthrough result in this direction was presented in the Dzh. L. Rogava theorem (1993). It says that on a separable Hilbert space the exponential Trotter product formula may converge in the operator-norm topology with convergence rate of the order $O(\ln{n}/\sqrt{n})$. This discovery initiated a number of papers addressed to the study of conditions on generators A and B aiming to optimise the rate of convergence in Rogava’s assertion. Motivated by this discovery the optimal rate of convergence $O(1/n)$ in the operator-norm topology under conditions of the Rogava theorem was proved only in 2001 (the Ichinose-Tamura-Tamura-Zagrebnov theorem) for both the Trotter and the Trotter-Kato product formulae. Under new fractional conditions on generators A and B the optimal rate of the Trotter-Kato product formulae convergence in the operator-norm topology on a Hilbert space was established in the Ichinose-Neidhardt-Zagrebnov (INZ)-theorem (2004).
I shall present these and some other recent results about the Lie-Trotter-Kato product formulae on Hilbert and Banach spaces, which are collected in the book: V. A. Zagrebnov, H. Neidhardt, T. Ichinose, Trotter-Kato Product Formulae, 2024.