  We consider a positive and power-bounded linear operator $T$ on $L^p$ over a
finite measure space and prove that, if $TL^p \subseteq L^q$ for some $q > p$,
then the essential spectral radius of $T$ is strictly smaller than $1$. As a
special case, we obtain a recent result of Miclo who proved this assertion for
self-adjoint ergodic Markov operators in the case $p=2$ and thereby solved a
long-open problem of Simon and H{\o}egh-Krohn.
  Our methods draw a connection between spectral theory and the geometry of
Banach spaces: they rely on a result going back to Groh that encodes spectral
gap properties via ultrapowers, and on the fact that an infinite dimensional
$L^p$-space cannot by isomorphic to an $L^q$-space for $q \not= p$.
  We also prove a number of variations of our main result: (i) it follows from
theorems of Lotz and Mart\'{i}nez that the condition $TL^p \subseteq L^q$ can
be replaced with the weaker assumption that $T$ maps the positive part of the
$L^p$-unit ball into a uniformly $p$-integrable set; (ii) while it is known
that the positivity assumption on $T$ cannot in general be omitted, we show
that we can replace it with the assumption that $T$ is contractive both on
$L^p$ and on $L^q$; (iii) we prove a version of the theorem which allows us,
under appropriate circumstances, to also consider non-finite measures spaces;
(iv) our result also has a uniform version: there exists an upper bound $c \in
[0,1)$ for the essential spectral radius of $T$, where $c$ depends on certain
quantitative properties of $T$, $L^p$ and $L^q$.
