Minimal vectors of positive operators Razvan AniscaVladimir Troitsky 47A1546B4247B65invariant subspaceminimal vectorBanach latticepositive operator 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. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2544 10.1512/iumj.2005.54.2544 en Indiana Univ. Math. J. 54 (2005) 861 - 872 state-of-the-art mathematics http://iumj.org/access/