Typical properties of positive contractions and the invariant subspace problem

TL;DR

The paper investigates typical properties of positive contractions on Banach spaces under the Strong Operator Topology and the Strong* topology, with a focus on invariant subspaces. Using a framework of Polish spaces for positive contractions, a density-approximation lemma, and a topological 0-1 law, it shows that on $\ell_1$ (and $\ell_2$) a typical positive contraction has a non-trivial invariant subspace, while on spaces with an unconditional basis a typical positive contraction has no non-trivial closed invariant ideals and typically fails the Abramovich–Aliprantis–Burkinshaw (AAB) criterion. The results are extended to Banach spaces with a basis, establishing that typical positive contractions neither commute with non-zero compact operators nor satisfy AAB in many reflexive or monotone-basis settings. Collectively, these findings illuminate how invariant-subspace behavior for typical positive contractions can starkly depend on the underlying space structure, contributing to the broader understanding of the invariant subspace problem in restricted operator classes.

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.

Theorems & Definitions (58)

• Theorem 1.1
• Corollary 1.2
• Proposition 1.3
• Proposition 1.4
• Proposition 1.5
• Proposition 1.6
• Corollary 1.7
• Theorem 1.8
• Corollary 1.9
• Proposition 2.1