Similar operator topologies on the space of positive contractions

TL;DR

The paper investigates the similarity of Polish operator topologies on the set of positive contractions of $l_p$, with a focus on $p>1$ and the critical case $p=2$. Using norming vectors and the study of continuity points of the identity map, it proves that $WOT$, $SOT$, $SOT^*$ and $SOT_*$ are similar on ${ m P}_1(l_2)$, and that similar relations hold for $p$ away from 2 in the expected pairs; any Polish topology between $WOT$ and $SOT^*$ is then shown to be similar to these on ${ m P}_1(l_2)$. These similarity results enable transfer of comeager sets and typical properties across topologies, yielding, in particular, that a typical positive contraction on $l_2$ has no eigenvalues under several topologies and possesses nontrivial invariant subspaces, contrasting with known behaviors for general contractions. The authors also construct large classes of continuity points (notably $M$ and $M'$) and describe how these influence the structure of continuity sets, while highlighting open problems for complete descriptions in the positive setting and for $p eq 2$. Overall, the work connects topology similarity to concrete spectral and invariant-subspace conclusions for positive contractions in $oldsymbol{l_p}$ spaces, with potential implications for broader operator-theoretic normality and ergodic-type questions.

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 Mátrai, 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.

Theorems & Definitions (56)

• Theorem 1.1: GMM2
• Proposition 1.2
• Proposition 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Proposition 2.1