This is an announcement for the paper "Spectral and asymptotic properties of contractive semigroups on non-Hilbert spaces" by Jochen Gl"uck.

Abstract: We analyse $C_0$-semigroups of contractive operators on real-valued $L^p$-spaces for $p \not= 2$ and on other classes of non-Hilbert spaces. We show that, under some regularity assumptions on the semigroup, the geometry of the unit ball of those spaces forces the semigroup's generator to have only trivial (point) spectrum on the imaginary axis. This has interesting consequences for the asymptotic behaviour as $t \to \infty$. For example, we can show that a contractive and eventually norm continuous $C_0$-semigroup on a real-valued $L^p$-space automatically converges strongly if $p \not\in {1,2,\infty}$.

Archive classification: math.FA

Mathematics Subject Classification: 47D06

Remarks: 26 pages

Submitted from: jochen.glueck@uni-ulm.de

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1410.2502

or

http://arXiv.org/abs/1410.2502