On error estimates for the Trotter-Kato product formula (Q1127614)

Let \(A\) and \(B\) be non-negative self-adjoint operators on the separable Hilbert space. \textit{T. Kato} [Adv. Math. Suppl. Stud. 3, 185-195 (1978; Zbl 0461.47018)] has shown that the Trotter product formula always converges in the strong operator sense to a semigroup which is generated by the form sum \(H := A \dot{+} B\), i.e. \[ s-\lim_{n\to\infty}(e^{-tA/n}e^{-tB/n})^n = e^{-tH}. \] Authors prove the following error estimate \[ \|(e^{-tA/n}e^{-tB/n})^n - e^{-tH} \|\leq C {\text{ln}(n) \over n} \] under additional assumption \[ \text{dom}(A) \subset \text{dom}(B), \;\|Bf \|\leq a\|Af\|, \forall f\in \text{dom}(A),\;\;a > 0. \] Similar results obtained for the generalization of Trotter formula à la Kato \[ s-\lim_{n\to\infty}(f(t A/n)g(t B/n))^n = e^{-tH}, \] \[ s-\lim_{n\to\infty}(\sqrt{f(t A/n)}g(t B/n)\sqrt{f(t A/n)})^n = e^{-tH}. \] These estimates improve recent results of \textit{D. L. Rogava} [Funkts. Anal. Prilozh. 27, No. 3, 84-86 (1993; Zbl 0814.47050)] and \textit{T. Ichinose} and \textit{H. Tamura} [Integral Equations Oper. Theory 27, No. 2, 195-207 (1997; Zbl 0906.47028)].