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/