A remark on operator-norm convergence of Trotter-Kato product formula (Q1583712)

It has been shown by Neidhardt and Zagrebnov that if the selfjoint operators \(A\) and \(B\) on some Hilbert space \(H\) with domains \(D(A)\) and \(D(B)\), respectively, satisfy, for some \(\alpha\in ]1/2,1[\) and \(a\in ]0,1[\), (i) \(A\geq I\), \(B\geq 0\), (ii) \(D(A^\alpha)\subset D(B^\alpha)\), \(\|B^\alpha x\|\leq a\|A^\alpha x\|\) for all \(x\in D(A)\), and (iii) \(D((A+ B)^\alpha)\subset D(A^\alpha)\), where \(A_B\) is the form-sum, then there exists a constant \(c>0\) such that \[ \|\exp\{- t(A+B)\}-(\exp\{- tA/n\}\exp\{- tB/n\})^n\|\leq c/n^{2\alpha- 1} \] uniformly in \(t\geq 0\). The author constructs an example showing that the bound in this estimate is the best possible as far as the power of \(n\) is concerned.