Two Counterexamples in Semigroup Theory on Hilbert Space1

An iterative method for amenable semigroup and infinite family of non expansive mappings in Hilbert spaces

begin{abstract} In this paper, we introduce an iterative method for amenable semigroup of non expansive mappings and infinite family of non expansive mappings in the frame work of Hilbert spaces. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. The results present...

Stability of Infinite-dimensional Sampled-data Systems

Suppose that a static-state feedback stabilizes a continuous-time linear infinite-dimensional control system. We consider the following question: if we construct a sampled-data controller by applying an idealized sample-andhold process to a continuous-time stabilizing feedback, will this sampled-data controller stabilize the system for all sufficiently small sampling times? Here the state space...

Semigroups and Scattering Theory for a Scalar Field on Curved Space-time

A classical theory is developed for the time development and scattering of a minimally coupled scalar eld on closed spacetimes that evolve from initial, to nal static states. The time development is obtained by reformulating the eld equation as an abstract Cauchy problem on a Hilbert space. Constraints are imposed on the metric that enable the use of semigroup theory, and the eld solution is ob...

Dynamical Systems on Hilbert C*-Modules

What type of dynamics arise in E0-dilations of commuting quantum Markov processes?

Let H be a separable Hilbert space. Given two strongly commuting CP0-semigroups φ and θ on B(H), there is a Hilbert space K ⊇ H and two (strongly) commuting E0-semigroups α and β such that φs ◦ θt(PHAPH) = PHαs ◦ βt(A)PH for all s, t ≥ 0 and all A ∈ B(K). In this note we prove that if φ is not an automorphism semigroup then α is cocycle conjugate to the minimal ∗-endomorphic dilation of φ, and ...