# Journal of Operator Theory

Volume 76, Issue 1, Summer 2016  pp. 3-31.

Spectral and asymptotic properties of contractive semigroups on non-Hilbert spaces

Authors:  Jochen Glueck
Author institution: Institute of Applied Analysis, Ulm University, 89069 Ulm, Germany

Summary:  We analyse $C_0$-semigroups of contractive operators on $\mathbb{R}$-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 $p \not\in \{1,2,\infty\}$.

DOI: http://dx.doi.org/10.7900/jot.2014oct09.2087
Keywords: asymtotics of contractive semigroups, geometry of unit balls, triviality of the peripheral spectrum