Spectral gaps for hyperbounded operators

TL;DR

This paper establishes spectral gap properties for positive, power-bounded operators on L^p spaces under certain embedding conditions, connecting spectral theory with Banach space geometry and extending previous results.

Contribution

It proves new spectral gap results for hyperbounded operators, generalizing prior work and introducing methods linking spectral theory with Banach space geometry.

Findings

Essential spectral radius of T is less than 1 under certain conditions.

The condition T L^p ⊆ L^q can be weakened to positive part of the unit ball mapping into a uniformly p-integrable set.

The results extend to non-finite measure spaces and include uniform bounds on spectral radius.

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 $L^q$.