Similar operator topologies on the space of positive contractions

TL;DR

This paper investigates the similarity of various operator topologies on positive contractions in $ ext{ell}_p$ spaces, showing they share dense and comeager sets on $ ext{ell}_2$, and studies typical spectral properties under these topologies.

Contribution

It proves the topologies are similar on positive contractions in $ ext{ell}_2$, and analyzes the typical spectral properties of these operators in the Baire category sense.

Findings

Topologies are similar on $ ext{ell}_2$ positive contractions.

Typical positive contractions have no eigenvalues under certain topologies.

Contrast with general contractions where the spectrum is dense in the unit disk.

Abstract

In this article, we study the similarity of the Polish operator topologies $\texttt{WOT}$, $\texttt{SOT}$, $\texttt{SOT}\mbox{$_{*}$}$ 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 $T \in (\mathcal{P}_1(\ell_2), \texttt{SOT})$ has no eigenvalue. This stands in strong contrast to a result of Eisner and M\'atrai, stating that the point spectrum of a typical contraction $T \in (\mathcal{B}_1(\ell_2), \texttt{SOT})$ contains the whole unit disk. As a consequence of our results, we obtain that a typical positive contraction $T \in (\mathcal{P}_1(\ell_2), \texttt{WOT})$ (resp. $T \in (\mathcal{P}_1(\ell_2), \texttt{SOT}\mbox{$_{*}$})$) has no eigenvalue.