Spectral gaps for hyperbounded operators

Abstract

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 $L^q$.