Similar operator topologies on the space of positive contractions

In this article, we study the similarity of the Polish operator topologies $\mathtt{\text{WOT}}$, $\mathtt{\text{SOT}}$, \texttt{SOT}\mbox{{}} and \texttt{SOT}\mbox{^{}} on the set of the positive contractions on ${\mathrm{\ell }}_p$ with $p>1$. Using the notion of norming vector for a positive operator, we prove that these topologies are similar on ${\mathcal{P}}_1({\mathrm{\ell }}_2)$, that is, they have the same dense sets in ${\mathcal{P}}_1({\mathrm{\ell }}_2)$. In particular, these topologies will share the same comeager sets in ${\mathcal{P}}_1({\mathrm{\ell }}_2)$. We then apply these results to the study of typical properties of positive contractions on ${\mathrm{\ell }}_p$-spaces in the Baire category sense. In particular, we prove that a typical positive contraction $T\in ({\mathcal{P}}_1({\mathrm{\ell }}_2),\mathtt{\text{SOT}})$ has no eigenvalue. This stands in strong contrast to a result of Eisner and Mátrai, stating that the point spectrum of a typical contraction $T\in ({\mathcal{B}}_1({\mathrm{\ell }}_2),\mathtt{\text{SOT}})$ contains the whole unit disk. As a consequence of our results, we obtain that a typical positive contraction $T\in ({\mathcal{P}}_1({\mathrm{\ell }}_2),\mathtt{\text{WOT}})$ (resp. T \in (\mathcal{P}_1(\ell_2), \texttt{SOT}\mbox{{*}})) has no eigenvalue.