Title: Weakly and almost weakly stable C₀-semigroups

Authors: Tanja Eisner, Balint Farkas, Rainer Nagel, Andras Sereny

Addresses: Mathematisches Institut, Universitat Tubingen, Auf der Morgenstelle 10, D-70176 Tubingen, Germany. ' AG Angewandte Analysis, Fachbereich Mathematik, Technische Universitat Darmstadt, Schlossgartenstrasse 7, D-64289 Darmstadt, Germany. ' Mathematisches Institut, Universitat Tubingen, Auf der Morgenstelle 10, D-70176 Tubingen, Germany. ' Department of Mathematics and its Applications, Central European University, Nador utca 9, H-1051 Budapest, Hungary

Abstract: In this paper we survey results concerning the asymptotic properties of C₀-semigroups on Banach spaces with respect to the weak operator topology. The property ||no eigenvalues of the generator on the imaginary axis|| is equivalent to weak stability for most time values; a phenomenon called |almost weak stability|. Further, sufficient conditions actually implying weak stability are also given. By several examples we explain weak and almost weak stability and illustrate the fundamental difference between them. Many historical and bibliographical remarks position the material in the literature. We conclude the paper with some open questions and comments.

Keywords: C₀ semigroups; weak operator topology; stability; mixing; asymptotic properties; Banach spaces.

DOI: 10.1504/IJDSDE.2007.013744

International Journal of Dynamical Systems and Differential Equations, 2007 Vol.1 No.1, pp.44 - 57

Published online: 23 May 2007 *

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