Abstract
In this paper, we study the possibility of approximating some bi-continuous semigroup on a bi-admissible locally convex space \(X\) using Chernoff iterations \((\mathbf{F}_{t})_{t\geq 0}\) of some class of operator-valued functions \((\mathbf{F}_{t})_{t\geq 0}\) . We obtain results that do not require a direct verification of the complicated condition of the Chernoff theorem, which consists in the fact that for some \(\lambda>0\) the subspace \((\lambda\mathbf{I}-\mathbf{L})\mathcal{D}\) is bi-dense in the ambient space. However, we have to abandon the fact that the limit bi-continuous semigroup is defined on the \(X\) and restrict ourselves to some bi-closed subspace that is constructed from the system \((\mathbf{F}_{t})_{t\geq 0}\) based on vectors with certain properties connected with the behavior of \((\mathbf{F}_{t})_{t\geq 0}\) . The formulation of the problem of approximation of a semigroup for bi-continuous semigroups is motivated by the dynamic systems existing in quantum mechanics, the evolution of which is described in terms of bi-continuous semigroups, but not strongly continuous ones. Here we can, for example, include the example of the evolution of a quantum system under a continuous measurement process, described by the Schrödinger–Belavkin equation. The study of approximations of such dynamics can be carried out by the proposed method.
