Typical properties of positive contractions and the invariant subspace problem

TL;DR

This paper investigates the typical properties of positive contractions on various Banach spaces, revealing their invariant subspace structures and their relation to the Abramovich, Aliprantis, and Burkinshaw criterion.

Contribution

It provides new insights into the invariant subspace problem for positive contractions on Banach spaces with bases, especially regarding typical behaviors under different topologies.

Findings

Typical positive contractions on ll_1 and ll_2 have non-trivial invariant subspaces.

On Banach spaces with an unconditional basis, typical positive contractions lack non-trivial invariant ideals.

Most positive contractions do not satisfy the Abramovich, Aliprantis, and Burkinshaw criterion under the studied topologies.

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.