Abstract.
We study the validity of the Trotter–Kato product formula in the setting of gradient flows on CAT(0) metric spaces. We follow the strategy of the proof of the Hilbert space version of this theorem given by Kato–Masuda, but instead of the linear structure and inner product we have only geodesics and the CAT(0) inequality available. Thus we construct a counterpart of the approximation semigroups and their resolvents in CAT(0) spaces. We show that the convergence of the approximated resolvents to the resolvents of the sum functional implies convergence of the approximation semigroups to the semigroup associated to the sum functional. We also show that this resolvents convergence implies the product Trotter–Kato formula in CAT(0) spaces. These approximation theorems are of independent interest. A major difficulty compared to the linear theory is the lack of an appropriate notion of weak convergence on such spaces. This difficulty is successfully overcome with the aid of an ultra-filters technique. The main results of our investigation are various versions of the Trotter–Kato product formula in CAT(0) spaces.
Comments (0)