# 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

Contents
Full-Text PDF