Typical properties of positive contractions and the invariant subspace problem

Abstract

In this paper, we first study some elementary properties of a typical positive contraction on $\ell_q$ for the Strong Operator Topology and the Strong* Operator Topology. Using these properties, we prove that a typical positive contraction on $\ell_1$ (resp. on $\ell_2$) has a non-trivial invariant subspace for the Strong Operator Topology (resp. for the Strong Operator Topology and the Strong* Operator Topology). We then focus on the case where $X$ is a Banach space with a basis. We prove that a typical positive contraction on a Banach space with an unconditional basis has no non-trivial closed invariant ideals for the Strong Operator Topology and the Strong* Operator Topology. In particular, this shows that when $X = \ell_q$ with $1 \leq q < \infty$, a typical positive contraction $T$ on $X$ for the Strong Operator Topology (resp. for the Strong* Operator Topology when $1 < q < \infty$) does not satisfy the Abramovich, Aliprantis and Burkinshaw criterion, that is, there is no non-zero positive operator in the commutant of $T$ which is quasinilpotent at a non-zero positive vector of $X$. Finally, we prove that, for the Strong* Operator Topology, a typical positive contraction on a reflexive Banach space with a monotone basis does not satisfy the Abramovich, Aliprantis and Burkinshaw criterion.