Ulrich Groh

One-parameter semigroups of positive operators

Basic results on semigroups on banach spaces.- Characterization of semigroups on banach spaces.- Spectral theory.- Asymptotics of semigroups on banach spaces.- Basic results on spaces Co(X).- Characterization of positive semigroups on Co(X).- Spectral theory of positive semigroups on Co(X).- Asymptotics of positive semigroups on Co(X).- Basic results on banach lattices and positive operators.- Characterization of positive semigroups on banach lattices.- Spectral theory of positive semigroups on banach lattices.- Asymptotics of positive semigroups on banach lattices.- Basic results on semigroups and operator algebras.- Characterization of positive semigroups on w*-algebras.- Spectral theory of positive semigroups on w*-algebras and their preduals.- Asymptotics of positive semigroups on c*-and w*-algebras.

Some observations on the spectra of positive operators on finite-dimensional C∗-algebras

Abstract The classical theorems of O. Perron and G. Frobenius about spectral properties of matrices with positive entries have been studied and generalized by various authors. In the book of A. Berman and R.J. Plemmons [3] the finite-dimensional aspects of this theory are described, whereas in the two monographs of H.H. Schaefer [20, 21] the infinite-dimensional theory is developed. Our purpose is to continue these extensions to finite-dimensional C ∗ -algebras, obtaining a complete description of the spectrum of positive irreducible operators on such ordered vector spaces.

Decomposition of operator semigroups on W*-algebras

We consider semigroups of operators on a W∗-algebra and prove, under appropriate assumptions, the existence of a Jacobs-DeLeeuw-Glicksberg type decomposition. This decomposition splits the algebra into a “stable” and “reversible” part with respect to the semigroup and yields, among others, a structural approach to the Perron-Frobenius spectral theory for completely positive operators on W∗-algebras.

Uniformly ergodic maps onC*-algebras 027

LetT be an identity preserving Schwarz map on aC*-algebra. The following conditions are proved to be equivalent: (a)T is uniformly ergodic with finite-dimensional fixed space. (b)T is quasi-compact.