Semigroups in Möbius and Lorentzian Geometry

The Möbius semigroup studied in this paper arises very naturally geometrically as the (compression) subsemigroup of the group of Möbius transformations which carry some fixed open Möbius ball into itself. It is shown, using geometric arguments, that this semigroup is a maximal subsemigroup. A detailed analysis of the semigroup is carried out via the Lorentz representation, in which the semigroup resurfaces as the semigroup carrying a fixed half of a Lorentzian cone into itself. Close ties with the Lie theory of semigroups are established by showing that the semigroup in question admits the structure of an Ol'shanskii semigroup, the most widely studied class of Lie semigroups.     

An Index Theory For Quantum Dynamical Semigroups

W. Arveson showed a way of associating continuous tensor product systems of Hilbert spaces with endomorphism semigroups of type I factors. We do the same for general quantum dynamical semigroups through a dilation procedure. The product system so obtained is the index and its dimension is a numerical invariant for the original semigroup.

     

Randomly flashing diffusion: Asymptotic properties

The theory of abstract Markov operators and semigroups is applied for studying asymptotics of a randomly flashing diffusion process. The probability distribution of the process is determined by a set of two partial differential equations and sufficient conditions for the existence of a stationary solution of the equations are formulated, and convergence of solutions to the stationary solution is proved.     

On a relationship between Uspensky's theorem and poisson approximations

In this paper we show that Uspensky's expansion theorem for the Poisson approximation of the distribution of sums of independent Bernoulli random variables can be rewritten in terms of the Poisson convolution semigroup. This gives rise to exact evaluations and simple remainder term estimations for the deviations of the distributions in study with respect to various probability metrics, generalizing results of Shorgin (1977, Theory Probab. Appl., 22, 846–850). Finally, we compare the sharpness of Poisson versus normal approximations.     

Uniformly bounded limit of fractional homomorphisms

We show that a bounded homomorphism is equivalent to a uniformly bounded family of fractional homomorphisms for any 0$">. We add this characterization to the Widder-Arendt-Kisynski theorem and relate it to -times integrated semigroups.

     

Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semigroups without Bochner integrals

In this paper, we prove Krasnoselskii and Mann's type convergence theorems for nonexpansive semigroups without using Bochner integral and without assuming the strict convexity of Banach spaces. One of our main results is the following: let C be a compact convex subset of a Banach space E and let be a one-parameter strongly continuous semigroup of nonexpansive mappings on C. Let {tn} be a sequence in [0,∞) satisfying