article

Title: Minimal vectors of positive operators

Authors: Razvan Anisca and Vladimir G. Troitsky

Issue: Volume 54 (2005), Issue 3, 861-872

Abstract:

$We use the method of minimal vectors to prove that certain classes of positive quasinilpotent operators on Banach lattices have invariant subspaces. We say that a collection of operators $\mathcal{F}$ on a Banach lattice $X$ satisfies condition ($\boldsymbol{*}$) if there exists a closed ball $B(x_0,r)$ in $X$ such that $x_0\geq0$ and $\norm{x_0}>r$, and for every sequence $(x_n)$ in $B(x_0,r)\cap[0,x_0]$ there exists a subsequence $(x_{n_i})$ and a sequence $K_i\in\mathcal{F}$ such that $K_ix_{n_i}$ converges to a non-zero vector. Let $Q$ be a positive quasinilpotent operator on $X$, one-to-one, with dense range. Denote $\Q=\{T\geq0\mid TQ\leq QT\}$. If either the set of all operators dominated by $Q$ or the set of all contractions in $\Q$ satisfies ($\boldsymbol{*}$), then $\Q$ has a common invariant subspace. We also show that if $Q$ is a one-to-one quasinilpotent interval preserving operator on $C_0(\Omega)$, then $\Q$ has a common invariant subspace.$