Trotter-Kato Product Formulae: Operator-Norm Topology and Error Bounds

In this chapter, we present two fundamental results about the operator-norm optimal error bound estimates for convergence rate of the Trotter-Kato product formul for semigroups with noncommuting self-adjoint generators. The ITTZ-Theorem (Sect. 7.1) shows that self-adjointness of the sum of a couple of generators, is a crucial condition for the operator-norm convergence of the Trotter-Kato product formul with the optimal error bound estimate for the rate of convergence. The INZ-Theorem (Sect. 7.4) concerns the case when it is the form-sum of a couple of generators is self-adjoint. In Sect. 7.4 we establish dependence of the optimal operator-norm error bound estimate for convergence rate of the Trotter-Kato product formul on relations between fractional powers of the self-adjoint generators.