Similar operator topologies on the space of positive contractions

Abstract

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

21 pages

References in corpus (1)

• Linear dynamical systems on Hilbert spaces: typical properties and explicit examples