Aperiodicity of positive operators that increase the support of functions

Consider a positive operator $T$ on an $L^p$-space (or, more generally, a Banach lattice) which increases the support of functions in the sense that $supp(Tf) \supseteq supp{f}$ for every function $f \ge 0$. We show that this implies, under mild assumptions, that $T$ has no unimodular eigenvalues except for possibly the number $1$. This rules out periodic behaviour of any orbits of the powers of $T$, and thus enables us to prove convergence of those powers in many situations. For the proof we first perform a careful analysis of the action of lattice homomorphisms on the support of functions; then we split $T$ into an invertible and a weakly stable part, and apply the aforementioned analysis to the invertible part. An appropriate adaptation of this argument allows us to prove another version of our main result which is useful for the study of so-called irreducible operators.