In this paper, we first study some elementary properties of a typical positive contraction on \(\ell _q\) for the SOT and the SOT \(^{*}\) topologies. 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 SOT topology (resp. the SOT and the SOT \(^{*}\) topologies). 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 SOT and the SOT \(^{*}\) topologies. In particular, this shows that when \(X = \ell _q\) with \(1 \le q < \infty \) , a typical positive contraction \(T \in ({\mathcal {P}}_{1}(X),\texttt {SOT} )\) (resp. \(T \in ({\mathcal {P}}_{1}(X), \texttt {SOT} ^{*})\) 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 SOT \(^{*}\) topology, a typical positive contraction on a reflexive Banach space with a monotone basis does not satisfy the Abramovich, Aliprantis and Burkinshaw criterion.
