Math. Proc. Camb. Phil. Soc. (2003), 135, 493 c  2003 Cambridge Philosophical Society DOI: 10.1017/S0305004103006893 Printed in the United Kingdom 493 Stability of C 0 -semigroups and geometry of Banach spaces By RALPH CHILL† Abteilung Angewandte Analysis, Universit¨ at Ulm, 89069 Ulm, Germany. e-mail: chill@mathematik.uni-ulm.de and YURI TOMILOV‡ Faculty of Mathematics and Computer Science, Nicolas Copernicus University, ul. Chopina 12/18, 87-100 Torun, Poland. e-mail: tomilov@mat.uni.torun.pl (Received 14 February 2002; revised 26 July 2002) Abstract We obtain new stability conditions for C 0 -semigroups on Banach spaces hav- ing nontrivial Fourier type. On Hilbert spaces these conditions are sharp. For C 0 - semigroups on general Banach spaces, we prove a new individual stability criterion. We also show that stronger versions of the range stability condition from [4] are not necessary for stability. This answers an open problem from [4]. 1. Introduction In this paper we continue to study the stability of C 0 -semigroups on Banach spaces and their individual orbits. Given a C 0 -semigroup (T (t)) t0 on a Banach space X we call an orbit T (·)x (x ∈ X) stable if lim t→∞ ‖T (t)x‖ =0. (1·1) If (1·1) holds for every x ∈ X then we call (T (t)) t0 stable. Motivated by the applications to the study of asymptotic behaviour of solutions of abstract Cauchy problems and thus to various kinds of evolution problems including partial differential equations, our major task in this paper is to find properties of the generator of a C 0 -semigroup which are sufficient or even necessary for stability. These properties are strongly related to abstract operator theory, harmonic analysis and function theory (see, for instance, [1, 5, 6, 14]). Early stability criteria (e.g. in the famous Arendt–Batty–Lyubich–Vu (ABLV) theorem) were based mostly on spectral properties of the generator A of (T (t)) t0 . † This work was started during a Research in pair stay at the Mathematisches Forschungsinstitut Oberwolfach, Germany. Support by the Volkswagenstiftung is gratefully acknowledged. The first author is grateful for partial support by Deutscher Akademischer Austauschdienst (Sti- pendium im Rahmen des gemeinsamen Hochschulsonderprogramms III von Bund und L¨ andern ¨ uber den DAAD). ‡ The second author was partially supported by a KBN grant and an INTAS Young Scientists Fellowship. www.DownloadPaper.ir www.DownloadPaper.ir